1 #define ANSI_DECLARATORS
2 /*****************************************************************************/
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6 /* 888 888 888 88b 888 888 888 888 888 d888 88b */
7 /* 888 888 888 o88^o888 888 888 "88888" 888 8888oo888 */
8 /* 888 888 888 C888 888 888 888 / 888 q888 */
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12 /* A Two-Dimensional Quality Mesh Generator and Delaunay Triangulator. */
19 /* Jonathan Richard Shewchuk */
20 /* School of Computer Science */
21 /* Carnegie Mellon University */
22 /* 5000 Forbes Avenue */
23 /* Pittsburgh, Pennsylvania 15213-3891 */
26 /* This program may be freely redistributed under the condition that the */
27 /* copyright notices (including this entire header and the copyright */
28 /* notice printed when the `-h' switch is selected) are not removed, and */
29 /* no compensation is received. Private, research, and institutional */
30 /* use is free. You may distribute modified versions of this code UNDER */
31 /* THE CONDITION THAT THIS CODE AND ANY MODIFICATIONS MADE TO IT IN THE */
32 /* SAME FILE REMAIN UNDER COPYRIGHT OF THE ORIGINAL AUTHOR, BOTH SOURCE */
33 /* AND OBJECT CODE ARE MADE FREELY AVAILABLE WITHOUT CHARGE, AND CLEAR */
34 /* NOTICE IS GIVEN OF THE MODIFICATIONS. Distribution of this code as */
35 /* part of a commercial system is permissible ONLY BY DIRECT ARRANGEMENT */
36 /* WITH THE AUTHOR. (If you are not directly supplying this code to a */
37 /* customer, and you are instead telling them how they can obtain it for */
38 /* free, then you are not required to make any arrangement with me.) */
40 /* Hypertext instructions for Triangle are available on the Web at */
42 /* http://www.cs.cmu.edu/~quake/triangle.html */
44 /* Some of the references listed below are marked [*]. These are available */
45 /* for downloading from the Web page */
47 /* http://www.cs.cmu.edu/~quake/triangle.research.html */
49 /* A paper discussing some aspects of Triangle is available. See Jonathan */
50 /* Richard Shewchuk, "Triangle: Engineering a 2D Quality Mesh Generator */
51 /* and Delaunay Triangulator," First Workshop on Applied Computational */
52 /* Geometry, ACM, May 1996. [*] */
54 /* Triangle was created as part of the Archimedes project in the School of */
55 /* Computer Science at Carnegie Mellon University. Archimedes is a */
56 /* system for compiling parallel finite element solvers. For further */
57 /* information, see Anja Feldmann, Omar Ghattas, John R. Gilbert, Gary L. */
58 /* Miller, David R. O'Hallaron, Eric J. Schwabe, Jonathan R. Shewchuk, */
59 /* and Shang-Hua Teng, "Automated Parallel Solution of Unstructured PDE */
60 /* Problems." To appear in Communications of the ACM, we hope. */
62 /* The quality mesh generation algorithm is due to Jim Ruppert, "A */
63 /* Delaunay Refinement Algorithm for Quality 2-Dimensional Mesh */
64 /* Generation," Journal of Algorithms 18(3):548-585, May 1995. [*] */
66 /* My implementation of the divide-and-conquer and incremental Delaunay */
67 /* triangulation algorithms follows closely the presentation of Guibas */
68 /* and Stolfi, even though I use a triangle-based data structure instead */
69 /* of their quad-edge data structure. (In fact, I originally implemented */
70 /* Triangle using the quad-edge data structure, but switching to a */
71 /* triangle-based data structure sped Triangle by a factor of two.) The */
72 /* mesh manipulation primitives and the two aforementioned Delaunay */
73 /* triangulation algorithms are described by Leonidas J. Guibas and Jorge */
74 /* Stolfi, "Primitives for the Manipulation of General Subdivisions and */
75 /* the Computation of Voronoi Diagrams," ACM Transactions on Graphics */
76 /* 4(2):74-123, April 1985. */
78 /* Their O(n log n) divide-and-conquer algorithm is adapted from Der-Tsai */
79 /* Lee and Bruce J. Schachter, "Two Algorithms for Constructing the */
80 /* Delaunay Triangulation," International Journal of Computer and */
81 /* Information Science 9(3):219-242, 1980. The idea to improve the */
82 /* divide-and-conquer algorithm by alternating between vertical and */
83 /* horizontal cuts was introduced by Rex A. Dwyer, "A Faster Divide-and- */
84 /* Conquer Algorithm for Constructing Delaunay Triangulations," */
85 /* Algorithmica 2(2):137-151, 1987. */
87 /* The incremental insertion algorithm was first proposed by C. L. Lawson, */
88 /* "Software for C1 Surface Interpolation," in Mathematical Software III, */
89 /* John R. Rice, editor, Academic Press, New York, pp. 161-194, 1977. */
90 /* For point location, I use the algorithm of Ernst P. Mucke, Isaac */
91 /* Saias, and Binhai Zhu, "Fast Randomized Point Location Without */
92 /* Preprocessing in Two- and Three-dimensional Delaunay Triangulations," */
93 /* Proceedings of the Twelfth Annual Symposium on Computational Geometry, */
94 /* ACM, May 1996. [*] If I were to randomize the order of point */
95 /* insertion (I currently don't bother), their result combined with the */
96 /* result of Leonidas J. Guibas, Donald E. Knuth, and Micha Sharir, */
97 /* "Randomized Incremental Construction of Delaunay and Voronoi */
98 /* Diagrams," Algorithmica 7(4):381-413, 1992, would yield an expected */
99 /* O(n^{4/3}) bound on running time. */
101 /* The O(n log n) sweepline Delaunay triangulation algorithm is taken from */
102 /* Steven Fortune, "A Sweepline Algorithm for Voronoi Diagrams", */
103 /* Algorithmica 2(2):153-174, 1987. A random sample of edges on the */
104 /* boundary of the triangulation are maintained in a splay tree for the */
105 /* purpose of point location. Splay trees are described by Daniel */
106 /* Dominic Sleator and Robert Endre Tarjan, "Self-Adjusting Binary Search */
107 /* Trees," Journal of the ACM 32(3):652-686, July 1985. */
109 /* The algorithms for exact computation of the signs of determinants are */
110 /* described in Jonathan Richard Shewchuk, "Adaptive Precision Floating- */
111 /* Point Arithmetic and Fast Robust Geometric Predicates," Technical */
112 /* Report CMU-CS-96-140, School of Computer Science, Carnegie Mellon */
113 /* University, Pittsburgh, Pennsylvania, May 1996. [*] (Submitted to */
114 /* Discrete & Computational Geometry.) An abbreviated version appears as */
115 /* Jonathan Richard Shewchuk, "Robust Adaptive Floating-Point Geometric */
116 /* Predicates," Proceedings of the Twelfth Annual Symposium on Computa- */
117 /* tional Geometry, ACM, May 1996. [*] Many of the ideas for my exact */
118 /* arithmetic routines originate with Douglas M. Priest, "Algorithms for */
119 /* Arbitrary Precision Floating Point Arithmetic," Tenth Symposium on */
120 /* Computer Arithmetic, 132-143, IEEE Computer Society Press, 1991. [*] */
121 /* Many of the ideas for the correct evaluation of the signs of */
122 /* determinants are taken from Steven Fortune and Christopher J. Van Wyk, */
123 /* "Efficient Exact Arithmetic for Computational Geometry," Proceedings */
124 /* of the Ninth Annual Symposium on Computational Geometry, ACM, */
125 /* pp. 163-172, May 1993, and from Steven Fortune, "Numerical Stability */
126 /* of Algorithms for 2D Delaunay Triangulations," International Journal */
127 /* of Computational Geometry & Applications 5(1-2):193-213, March-June */
130 /* For definitions of and results involving Delaunay triangulations, */
131 /* constrained and conforming versions thereof, and other aspects of */
132 /* triangular mesh generation, see the excellent survey by Marshall Bern */
133 /* and David Eppstein, "Mesh Generation and Optimal Triangulation," in */
134 /* Computing and Euclidean Geometry, Ding-Zhu Du and Frank Hwang, */
135 /* editors, World Scientific, Singapore, pp. 23-90, 1992. */
137 /* The time for incrementally adding PSLG (planar straight line graph) */
138 /* segments to create a constrained Delaunay triangulation is probably */
139 /* O(n^2) per segment in the worst case and O(n) per edge in the common */
140 /* case, where n is the number of triangles that intersect the segment */
141 /* before it is inserted. This doesn't count point location, which can */
142 /* be much more expensive. (This note does not apply to conforming */
143 /* Delaunay triangulations, for which a different method is used to */
144 /* insert segments.) */
146 /* The time for adding segments to a conforming Delaunay triangulation is */
147 /* not clear, but does not depend upon n alone. In some cases, very */
148 /* small features (like a point lying next to a segment) can cause a */
149 /* single segment to be split an arbitrary number of times. Of course, */
150 /* floating-point precision is a practical barrier to how much this can */
153 /* The time for deleting a point from a Delaunay triangulation is O(n^2) in */
154 /* the worst case and O(n) in the common case, where n is the degree of */
155 /* the point being deleted. I could improve this to expected O(n) time */
156 /* by "inserting" the neighboring vertices in random order, but n is */
157 /* usually quite small, so it's not worth the bother. (The O(n) time */
158 /* for random insertion follows from L. Paul Chew, "Building Voronoi */
159 /* Diagrams for Convex Polygons in Linear Expected Time," Technical */
160 /* Report PCS-TR90-147, Department of Mathematics and Computer Science, */
161 /* Dartmouth College, 1990. */
163 /* Ruppert's Delaunay refinement algorithm typically generates triangles */
164 /* at a linear rate (constant time per triangle) after the initial */
165 /* triangulation is formed. There may be pathological cases where more */
166 /* time is required, but these never arise in practice. */
168 /* The segment intersection formulae are straightforward. If you want to */
169 /* see them derived, see Franklin Antonio. "Faster Line Segment */
170 /* Intersection." In Graphics Gems III (David Kirk, editor), pp. 199- */
171 /* 202. Academic Press, Boston, 1992. */
173 /* If you make any improvements to this code, please please please let me */
174 /* know, so that I may obtain the improvements. Even if you don't change */
175 /* the code, I'd still love to hear what it's being used for. */
177 /* Disclaimer: Neither I nor Carnegie Mellon warrant this code in any way */
178 /* whatsoever. This code is provided "as-is". Use at your own risk. */
180 /*****************************************************************************/
182 /* For single precision (which will save some memory and reduce paging), */
183 /* define the symbol SINGLE by using the -DSINGLE compiler switch or by */
184 /* writing "#define SINGLE" below. */
186 /* For double precision (which will allow you to refine meshes to a smaller */
187 /* edge length), leave SINGLE undefined. */
189 /* Double precision uses more memory, but improves the resolution of the */
190 /* meshes you can generate with Triangle. It also reduces the likelihood */
191 /* of a floating exception due to overflow. Finally, it is much faster */
192 /* than single precision on 64-bit architectures like the DEC Alpha. I */
193 /* recommend double precision unless you want to generate a mesh for which */
194 /* you do not have enough memory. */
200 #else /* not SINGLE */
202 #endif /* not SINGLE */
204 /* If yours is not a Unix system, define the NO_TIMER compiler switch to */
205 /* remove the Unix-specific timing code. */
209 /* To insert lots of self-checks for internal errors, define the SELF_CHECK */
210 /* symbol. This will slow down the program significantly. It is best to */
211 /* define the symbol using the -DSELF_CHECK compiler switch, but you could */
212 /* write "#define SELF_CHECK" below. If you are modifying this code, I */
213 /* recommend you turn self-checks on. */
215 /* #define SELF_CHECK */
217 /* To compile Triangle as a callable object library (triangle.o), define the */
218 /* TRILIBRARY symbol. Read the file triangle.h for details on how to call */
219 /* the procedure triangulate() that results. */
223 /* It is possible to generate a smaller version of Triangle using one or */
224 /* both of the following symbols. Define the REDUCED symbol to eliminate */
225 /* all features that are primarily of research interest; specifically, the */
226 /* -i, -F, -s, and -C switches. Define the CDT_ONLY symbol to eliminate */
227 /* all meshing algorithms above and beyond constrained Delaunay */
228 /* triangulation; specifically, the -r, -q, -a, -S, and -s switches. */
229 /* These reductions are most likely to be useful when generating an object */
230 /* library (triangle.o) by defining the TRILIBRARY symbol. */
235 /* On some machines, the exact arithmetic routines might be defeated by the */
236 /* use of internal extended precision floating-point registers. Sometimes */
237 /* this problem can be fixed by defining certain values to be volatile, */
238 /* thus forcing them to be stored to memory and rounded off. This isn't */
239 /* a great solution, though, as it slows Triangle down. */
241 /* To try this out, write "#define INEXACT volatile" below. Normally, */
242 /* however, INEXACT should be defined to be nothing. ("#define INEXACT".) */
244 #define INEXACT /* Nothing */
245 /* #define INEXACT volatile */
247 /* Maximum number of characters in a file name (including the null). */
249 #define FILENAMESIZE 512
251 /* Maximum number of characters in a line read from a file (including the */
254 #define INPUTLINESIZE 512
256 /* For efficiency, a variety of data structures are allocated in bulk. The */
257 /* following constants determine how many of each structure is allocated */
260 #define TRIPERBLOCK 4092 /* Number of triangles allocated at once. */
261 #define SHELLEPERBLOCK 508 /* Number of shell edges allocated at once. */
262 #define POINTPERBLOCK 4092 /* Number of points allocated at once. */
263 #define VIRUSPERBLOCK 1020 /* Number of virus triangles allocated at once. */
264 /* Number of encroached segments allocated at once. */
265 #define BADSEGMENTPERBLOCK 252
266 /* Number of skinny triangles allocated at once. */
267 #define BADTRIPERBLOCK 4092
268 /* Number of splay tree nodes allocated at once. */
269 #define SPLAYNODEPERBLOCK 508
271 /* The point marker DEADPOINT is an arbitrary number chosen large enough to */
272 /* (hopefully) not conflict with user boundary markers. Make sure that it */
273 /* is small enough to fit into your machine's integer size. */
275 #define DEADPOINT -1073741824
277 /* The next line is used to outsmart some very stupid compilers. If your */
278 /* compiler is smarter, feel free to replace the "int" with "void". */
279 /* Not that it matters. */
283 /* Two constants for algorithms based on random sampling. Both constants */
284 /* have been chosen empirically to optimize their respective algorithms. */
286 /* Used for the point location scheme of Mucke, Saias, and Zhu, to decide */
287 /* how large a random sample of triangles to inspect. */
288 #define SAMPLEFACTOR 11
289 /* Used in Fortune's sweepline Delaunay algorithm to determine what fraction */
290 /* of boundary edges should be maintained in the splay tree for point */
291 /* location on the front. */
292 #define SAMPLERATE 10
294 /* A number that speaks for itself, every kissable digit. */
296 #define PI 3.141592653589793238462643383279502884197169399375105820974944592308
300 #define SQUAREROOTTWO 1.4142135623730950488016887242096980785696718753769480732
302 /* And here's one for those of you who are intimidated by math. */
304 #define ONETHIRD 0.333333333333333333333333333333333333333333333333333333333333
310 #include <sys/time.h>
311 #endif /* NO_TIMER */
313 #include "triangle.h"
314 #endif /* TRILIBRARY */
316 /* The following obscenity seems to be necessary to ensure that this program */
317 /* will port to Dec Alphas running OSF/1, because their stdio.h file commits */
318 /* the unpardonable sin of including stdlib.h. Hence, malloc(), free(), and */
319 /* exit() may or may not already be defined at this point. I declare these */
320 /* functions explicitly because some non-ANSI C compilers lack stdlib.h. */
323 extern void *malloc();
326 extern double strtod();
327 extern long strtol();
328 #endif /* _STDLIB_H_ */
330 /* A few forward declarations. */
336 #endif /* not TRILIBRARY */
338 /* Labels that signify whether a record consists primarily of pointers or of */
339 /* floating-point words. Used to make decisions about data alignment. */
341 enum wordtype {POINTER, FLOATINGPOINT};
343 /* Labels that signify the result of point location. The result of a */
344 /* search indicates that the point falls in the interior of a triangle, on */
345 /* an edge, on a vertex, or outside the mesh. */
347 enum locateresult {INTRIANGLE, ONEDGE, ONVERTEX, OUTSIDE};
349 /* Labels that signify the result of site insertion. The result indicates */
350 /* that the point was inserted with complete success, was inserted but */
351 /* encroaches on a segment, was not inserted because it lies on a segment, */
352 /* or was not inserted because another point occupies the same location. */
354 enum insertsiteresult {SUCCESSFULPOINT, ENCROACHINGPOINT, VIOLATINGPOINT,
357 /* Labels that signify the result of direction finding. The result */
358 /* indicates that a segment connecting the two query points falls within */
359 /* the direction triangle, along the left edge of the direction triangle, */
360 /* or along the right edge of the direction triangle. */
362 enum finddirectionresult {WITHIN, LEFTCOLLINEAR, RIGHTCOLLINEAR};
364 /* Labels that signify the result of the circumcenter computation routine. */
365 /* The return value indicates which edge of the triangle is shortest. */
367 enum circumcenterresult {OPPOSITEORG, OPPOSITEDEST, OPPOSITEAPEX};
369 /*****************************************************************************/
371 /* The basic mesh data structures */
373 /* There are three: points, triangles, and shell edges (abbreviated */
374 /* `shelle'). These three data structures, linked by pointers, comprise */
375 /* the mesh. A point simply represents a point in space and its properties.*/
376 /* A triangle is a triangle. A shell edge is a special data structure used */
377 /* to represent impenetrable segments in the mesh (including the outer */
378 /* boundary, boundaries of holes, and internal boundaries separating two */
379 /* triangulated regions). Shell edges represent boundaries defined by the */
380 /* user that triangles may not lie across. */
382 /* A triangle consists of a list of three vertices, a list of three */
383 /* adjoining triangles, a list of three adjoining shell edges (when shell */
384 /* edges are used), an arbitrary number of optional user-defined floating- */
385 /* point attributes, and an optional area constraint. The latter is an */
386 /* upper bound on the permissible area of each triangle in a region, used */
387 /* for mesh refinement. */
389 /* For a triangle on a boundary of the mesh, some or all of the neighboring */
390 /* triangles may not be present. For a triangle in the interior of the */
391 /* mesh, often no neighboring shell edges are present. Such absent */
392 /* triangles and shell edges are never represented by NULL pointers; they */
393 /* are represented by two special records: `dummytri', the triangle that */
394 /* fills "outer space", and `dummysh', the omnipresent shell edge. */
395 /* `dummytri' and `dummysh' are used for several reasons; for instance, */
396 /* they can be dereferenced and their contents examined without causing the */
397 /* memory protection exception that would occur if NULL were dereferenced. */
399 /* However, it is important to understand that a triangle includes other */
400 /* information as well. The pointers to adjoining vertices, triangles, and */
401 /* shell edges are ordered in a way that indicates their geometric relation */
402 /* to each other. Furthermore, each of these pointers contains orientation */
403 /* information. Each pointer to an adjoining triangle indicates which face */
404 /* of that triangle is contacted. Similarly, each pointer to an adjoining */
405 /* shell edge indicates which side of that shell edge is contacted, and how */
406 /* the shell edge is oriented relative to the triangle. */
408 /* Shell edges are found abutting edges of triangles; either sandwiched */
409 /* between two triangles, or resting against one triangle on an exterior */
410 /* boundary or hole boundary. */
412 /* A shell edge consists of a list of two vertices, a list of two */
413 /* adjoining shell edges, and a list of two adjoining triangles. One of */
414 /* the two adjoining triangles may not be present (though there should */
415 /* always be one), and neighboring shell edges might not be present. */
416 /* Shell edges also store a user-defined integer "boundary marker". */
417 /* Typically, this integer is used to indicate what sort of boundary */
418 /* conditions are to be applied at that location in a finite element */
421 /* Like triangles, shell edges maintain information about the relative */
422 /* orientation of neighboring objects. */
424 /* Points are relatively simple. A point is a list of floating point */
425 /* numbers, starting with the x, and y coordinates, followed by an */
426 /* arbitrary number of optional user-defined floating-point attributes, */
427 /* followed by an integer boundary marker. During the segment insertion */
428 /* phase, there is also a pointer from each point to a triangle that may */
429 /* contain it. Each pointer is not always correct, but when one is, it */
430 /* speeds up segment insertion. These pointers are assigned values once */
431 /* at the beginning of the segment insertion phase, and are not used or */
432 /* updated at any other time. Edge swapping during segment insertion will */
433 /* render some of them incorrect. Hence, don't rely upon them for */
434 /* anything. For the most part, points do not have any information about */
435 /* what triangles or shell edges they are linked to. */
437 /*****************************************************************************/
439 /*****************************************************************************/
443 /* The oriented triangle (`triedge') and oriented shell edge (`edge') data */
444 /* structures defined below do not themselves store any part of the mesh. */
445 /* The mesh itself is made of `triangle's, `shelle's, and `point's. */
447 /* Oriented triangles and oriented shell edges will usually be referred to */
448 /* as "handles". A handle is essentially a pointer into the mesh; it */
449 /* allows you to "hold" one particular part of the mesh. Handles are used */
450 /* to specify the regions in which one is traversing and modifying the mesh.*/
451 /* A single `triangle' may be held by many handles, or none at all. (The */
452 /* latter case is not a memory leak, because the triangle is still */
453 /* connected to other triangles in the mesh.) */
455 /* A `triedge' is a handle that holds a triangle. It holds a specific side */
456 /* of the triangle. An `edge' is a handle that holds a shell edge. It */
457 /* holds either the left or right side of the edge. */
459 /* Navigation about the mesh is accomplished through a set of mesh */
460 /* manipulation primitives, further below. Many of these primitives take */
461 /* a handle and produce a new handle that holds the mesh near the first */
462 /* handle. Other primitives take two handles and glue the corresponding */
463 /* parts of the mesh together. The exact position of the handles is */
464 /* important. For instance, when two triangles are glued together by the */
465 /* bond() primitive, they are glued by the sides on which the handles lie. */
467 /* Because points have no information about which triangles they are */
468 /* attached to, I commonly represent a point by use of a handle whose */
469 /* origin is the point. A single handle can simultaneously represent a */
470 /* triangle, an edge, and a point. */
472 /*****************************************************************************/
474 /* The triangle data structure. Each triangle contains three pointers to */
475 /* adjoining triangles, plus three pointers to vertex points, plus three */
476 /* pointers to shell edges (defined below; these pointers are usually */
477 /* `dummysh'). It may or may not also contain user-defined attributes */
478 /* and/or a floating-point "area constraint". It may also contain extra */
479 /* pointers for nodes, when the user asks for high-order elements. */
480 /* Because the size and structure of a `triangle' is not decided until */
481 /* runtime, I haven't simply defined the type `triangle' to be a struct. */
483 typedef REAL **triangle; /* Really: typedef triangle *triangle */
485 /* An oriented triangle: includes a pointer to a triangle and orientation. */
486 /* The orientation denotes an edge of the triangle. Hence, there are */
487 /* three possible orientations. By convention, each edge is always */
488 /* directed to point counterclockwise about the corresponding triangle. */
492 int orient; /* Ranges from 0 to 2. */
495 /* The shell data structure. Each shell edge contains two pointers to */
496 /* adjoining shell edges, plus two pointers to vertex points, plus two */
497 /* pointers to adjoining triangles, plus one shell marker. */
499 typedef REAL **shelle; /* Really: typedef shelle *shelle */
501 /* An oriented shell edge: includes a pointer to a shell edge and an */
502 /* orientation. The orientation denotes a side of the edge. Hence, there */
503 /* are two possible orientations. By convention, the edge is always */
504 /* directed so that the "side" denoted is the right side of the edge. */
508 int shorient; /* Ranges from 0 to 1. */
511 /* The point data structure. Each point is actually an array of REALs. */
512 /* The number of REALs is unknown until runtime. An integer boundary */
513 /* marker, and sometimes a pointer to a triangle, is appended after the */
518 /* A queue used to store encroached segments. Each segment's vertices are */
519 /* stored so that one can check whether a segment is still the same. */
522 struct edge encsegment; /* An encroached segment. */
523 point segorg, segdest; /* The two vertices. */
524 struct badsegment *nextsegment; /* Pointer to next encroached segment. */
527 /* A queue used to store bad triangles. The key is the square of the cosine */
528 /* of the smallest angle of the triangle. Each triangle's vertices are */
529 /* stored so that one can check whether a triangle is still the same. */
532 struct triedge badfacetri; /* A bad triangle. */
533 REAL key; /* cos^2 of smallest (apical) angle. */
534 point faceorg, facedest, faceapex; /* The three vertices. */
535 struct badface *nextface; /* Pointer to next bad triangle. */
538 /* A node in a heap used to store events for the sweepline Delaunay */
539 /* algorithm. Nodes do not point directly to their parents or children in */
540 /* the heap. Instead, each node knows its position in the heap, and can */
541 /* look up its parent and children in a separate array. The `eventptr' */
542 /* points either to a `point' or to a triangle (in encoded format, so that */
543 /* an orientation is included). In the latter case, the origin of the */
544 /* oriented triangle is the apex of a "circle event" of the sweepline */
545 /* algorithm. To distinguish site events from circle events, all circle */
546 /* events are given an invalid (smaller than `xmin') x-coordinate `xkey'. */
549 REAL xkey, ykey; /* Coordinates of the event. */
550 VOID *eventptr; /* Can be a point or the location of a circle event. */
551 int heapposition; /* Marks this event's position in the heap. */
554 /* A node in the splay tree. Each node holds an oriented ghost triangle */
555 /* that represents a boundary edge of the growing triangulation. When a */
556 /* circle event covers two boundary edges with a triangle, so that they */
557 /* are no longer boundary edges, those edges are not immediately deleted */
558 /* from the tree; rather, they are lazily deleted when they are next */
559 /* encountered. (Since only a random sample of boundary edges are kept */
560 /* in the tree, lazy deletion is faster.) `keydest' is used to verify */
561 /* that a triangle is still the same as when it entered the splay tree; if */
562 /* it has been rotated (due to a circle event), it no longer represents a */
563 /* boundary edge and should be deleted. */
566 struct triedge keyedge; /* Lprev of an edge on the front. */
567 point keydest; /* Used to verify that splay node is still live. */
568 struct splaynode *lchild, *rchild; /* Children in splay tree. */
571 /* A type used to allocate memory. firstblock is the first block of items. */
572 /* nowblock is the block from which items are currently being allocated. */
573 /* nextitem points to the next slab of free memory for an item. */
574 /* deaditemstack is the head of a linked list (stack) of deallocated items */
575 /* that can be recycled. unallocateditems is the number of items that */
576 /* remain to be allocated from nowblock. */
578 /* Traversal is the process of walking through the entire list of items, and */
579 /* is separate from allocation. Note that a traversal will visit items on */
580 /* the "deaditemstack" stack as well as live items. pathblock points to */
581 /* the block currently being traversed. pathitem points to the next item */
582 /* to be traversed. pathitemsleft is the number of items that remain to */
583 /* be traversed in pathblock. */
585 /* itemwordtype is set to POINTER or FLOATINGPOINT, and is used to suggest */
586 /* what sort of word the record is primarily made up of. alignbytes */
587 /* determines how new records should be aligned in memory. itembytes and */
588 /* itemwords are the length of a record in bytes (after rounding up) and */
589 /* words. itemsperblock is the number of items allocated at once in a */
590 /* single block. items is the number of currently allocated items. */
591 /* maxitems is the maximum number of items that have been allocated at */
592 /* once; it is the current number of items plus the number of records kept */
593 /* on deaditemstack. */
596 VOID **firstblock, **nowblock;
601 enum wordtype itemwordtype;
603 int itembytes, itemwords;
605 long items, maxitems;
606 int unallocateditems;
610 /* Variables used to allocate memory for triangles, shell edges, points, */
611 /* viri (triangles being eaten), bad (encroached) segments, bad (skinny */
612 /* or too large) triangles, and splay tree nodes. */
614 static struct memorypool triangles;
615 static struct memorypool shelles;
616 static struct memorypool points;
617 static struct memorypool viri;
618 static struct memorypool badsegments;
619 static struct memorypool badtriangles;
620 static struct memorypool splaynodes;
622 /* Variables that maintain the bad triangle queues. The tails are pointers */
623 /* to the pointers that have to be filled in to enqueue an item. */
625 static struct badface *queuefront[64];
626 static struct badface **queuetail[64];
628 static REAL xmin, xmax, ymin, ymax; /* x and y bounds. */
629 static REAL xminextreme; /* Nonexistent x value used as a flag in sweepline. */
630 static int inpoints; /* Number of input points. */
631 static int inelements; /* Number of input triangles. */
632 static int insegments; /* Number of input segments. */
633 static int holes; /* Number of input holes. */
634 static int regions; /* Number of input regions. */
635 static long edges; /* Number of output edges. */
636 static int mesh_dim; /* Dimension (ought to be 2). */
637 static int nextras; /* Number of attributes per point. */
638 static int eextras; /* Number of attributes per triangle. */
639 static long hullsize; /* Number of edges of convex hull. */
640 static int triwords; /* Total words per triangle. */
641 static int shwords; /* Total words per shell edge. */
642 static int pointmarkindex; /* Index to find boundary marker of a point. */
643 static int point2triindex; /* Index to find a triangle adjacent to a point. */
644 static int highorderindex; /* Index to find extra nodes for high-order elements. */
645 static int elemattribindex; /* Index to find attributes of a triangle. */
646 static int areaboundindex; /* Index to find area bound of a triangle. */
647 static int checksegments; /* Are there segments in the triangulation yet? */
648 static int readnodefile; /* Has a .node file been read? */
649 static long samples; /* Number of random samples for point location. */
650 static unsigned long randomseed; /* Current random number seed. */
652 static REAL splitter; /* Used to split REAL factors for exact multiplication. */
653 static REAL epsilon; /* Floating-point machine epsilon. */
654 static REAL resulterrbound;
655 static REAL ccwerrboundA, ccwerrboundB, ccwerrboundC;
656 static REAL iccerrboundA, iccerrboundB, iccerrboundC;
658 static long incirclecount; /* Number of incircle tests performed. */
659 static long counterclockcount; /* Number of counterclockwise tests performed. */
660 static long hyperbolacount; /* Number of right-of-hyperbola tests performed. */
661 static long circumcentercount; /* Number of circumcenter calculations performed. */
662 static long circletopcount; /* Number of circle top calculations performed. */
664 /* Switches for the triangulator. */
665 /* poly: -p switch. refine: -r switch. */
666 /* quality: -q switch. */
667 /* minangle: minimum angle bound, specified after -q switch. */
668 /* goodangle: cosine squared of minangle. */
669 /* vararea: -a switch without number. */
670 /* fixedarea: -a switch with number. */
671 /* maxarea: maximum area bound, specified after -a switch. */
672 /* regionattrib: -A switch. convex: -c switch. */
673 /* firstnumber: inverse of -z switch. All items are numbered starting */
674 /* from firstnumber. */
675 /* edgesout: -e switch. voronoi: -v switch. */
676 /* neighbors: -n switch. geomview: -g switch. */
677 /* nobound: -B switch. nopolywritten: -P switch. */
678 /* nonodewritten: -N switch. noelewritten: -E switch. */
679 /* noiterationnum: -I switch. noholes: -O switch. */
680 /* noexact: -X switch. */
681 /* order: element order, specified after -o switch. */
682 /* nobisect: count of how often -Y switch is selected. */
683 /* steiner: maximum number of Steiner points, specified after -S switch. */
684 /* steinerleft: number of Steiner points not yet used. */
685 /* incremental: -i switch. sweepline: -F switch. */
686 /* dwyer: inverse of -l switch. */
687 /* splitseg: -s switch. */
688 /* docheck: -C switch. */
689 /* quiet: -Q switch. verbose: count of how often -V switch is selected. */
690 /* useshelles: -p, -r, -q, or -c switch; determines whether shell edges */
691 /* are used at all. */
693 /* Read the instructions to find out the meaning of these switches. */
695 static int poly, refine, quality, vararea, fixedarea, regionattrib, convex;
696 static int firstnumber;
697 static int edgesout, voronoi, neighbors, geomview;
698 static int nobound, nopolywritten, nonodewritten, noelewritten, noiterationnum;
699 static int noholes, noexact;
700 static int incremental, sweepline, dwyer;
703 static int quiet, verbose;
704 static int useshelles;
707 static int steiner, steinerleft;
708 static REAL minangle, goodangle;
711 /* Variables for file names. */
714 char innodefilename[FILENAMESIZE];
715 char inelefilename[FILENAMESIZE];
716 char inpolyfilename[FILENAMESIZE];
717 char areafilename[FILENAMESIZE];
718 char outnodefilename[FILENAMESIZE];
719 char outelefilename[FILENAMESIZE];
720 char outpolyfilename[FILENAMESIZE];
721 char edgefilename[FILENAMESIZE];
722 char vnodefilename[FILENAMESIZE];
723 char vedgefilename[FILENAMESIZE];
724 char neighborfilename[FILENAMESIZE];
725 char offfilename[FILENAMESIZE];
726 #endif /* not TRILIBRARY */
728 /* Triangular bounding box points. */
730 static point infpoint1, infpoint2, infpoint3;
732 /* Pointer to the `triangle' that occupies all of "outer space". */
734 static triangle *dummytri;
735 static triangle *dummytribase; /* Keep base address so we can free() it later. */
737 /* Pointer to the omnipresent shell edge. Referenced by any triangle or */
738 /* shell edge that isn't really connected to a shell edge at that */
741 static shelle *dummysh;
742 static shelle *dummyshbase; /* Keep base address so we can free() it later. */
744 /* Pointer to a recently visited triangle. Improves point location if */
745 /* proximate points are inserted sequentially. */
747 static struct triedge recenttri;
749 /*****************************************************************************/
751 /* Mesh manipulation primitives. Each triangle contains three pointers to */
752 /* other triangles, with orientations. Each pointer points not to the */
753 /* first byte of a triangle, but to one of the first three bytes of a */
754 /* triangle. It is necessary to extract both the triangle itself and the */
755 /* orientation. To save memory, I keep both pieces of information in one */
756 /* pointer. To make this possible, I assume that all triangles are aligned */
757 /* to four-byte boundaries. The `decode' routine below decodes a pointer, */
758 /* extracting an orientation (in the range 0 to 2) and a pointer to the */
759 /* beginning of a triangle. The `encode' routine compresses a pointer to a */
760 /* triangle and an orientation into a single pointer. My assumptions that */
761 /* triangles are four-byte-aligned and that the `unsigned long' type is */
762 /* long enough to hold a pointer are two of the few kludges in this program.*/
764 /* Shell edges are manipulated similarly. A pointer to a shell edge */
765 /* carries both an address and an orientation in the range 0 to 1. */
767 /* The other primitives take an oriented triangle or oriented shell edge, */
768 /* and return an oriented triangle or oriented shell edge or point; or they */
769 /* change the connections in the data structure. */
771 /*****************************************************************************/
773 /********* Mesh manipulation primitives begin here *********/
777 /* Fast lookup arrays to speed some of the mesh manipulation primitives. */
779 int plus1mod3[3] = {1, 2, 0};
780 int minus1mod3[3] = {2, 0, 1};
782 /********* Primitives for triangles *********/
786 /* decode() converts a pointer to an oriented triangle. The orientation is */
787 /* extracted from the two least significant bits of the pointer. */
789 #define decode( ptr, triedge ) \
790 ( triedge ).orient = (int) ( (unsigned long) ( ptr ) & (unsigned long) 3l ); \
791 ( triedge ).tri = (triangle *) \
792 ( (unsigned long) ( ptr ) ^ (unsigned long) ( triedge ).orient )
794 /* encode() compresses an oriented triangle into a single pointer. It */
795 /* relies on the assumption that all triangles are aligned to four-byte */
796 /* boundaries, so the two least significant bits of (triedge).tri are zero.*/
798 #define encode( triedge ) \
799 (triangle) ( (unsigned long) ( triedge ).tri | (unsigned long) ( triedge ).orient )
801 /* The following edge manipulation primitives are all described by Guibas */
802 /* and Stolfi. However, they use an edge-based data structure, whereas I */
803 /* am using a triangle-based data structure. */
805 /* sym() finds the abutting triangle, on the same edge. Note that the */
806 /* edge direction is necessarily reversed, because triangle/edge handles */
807 /* are always directed counterclockwise around the triangle. */
809 #define sym( triedge1, triedge2 ) \
810 ptr = ( triedge1 ).tri[( triedge1 ).orient]; \
811 decode( ptr, triedge2 );
813 #define symself( triedge ) \
814 ptr = ( triedge ).tri[( triedge ).orient]; \
815 decode( ptr, triedge );
817 /* lnext() finds the next edge (counterclockwise) of a triangle. */
819 #define lnext( triedge1, triedge2 ) \
820 ( triedge2 ).tri = ( triedge1 ).tri; \
821 ( triedge2 ).orient = plus1mod3[( triedge1 ).orient]
823 #define lnextself( triedge ) \
824 ( triedge ).orient = plus1mod3[( triedge ).orient]
826 /* lprev() finds the previous edge (clockwise) of a triangle. */
828 #define lprev( triedge1, triedge2 ) \
829 ( triedge2 ).tri = ( triedge1 ).tri; \
830 ( triedge2 ).orient = minus1mod3[( triedge1 ).orient]
832 #define lprevself( triedge ) \
833 ( triedge ).orient = minus1mod3[( triedge ).orient]
835 /* onext() spins counterclockwise around a point; that is, it finds the next */
836 /* edge with the same origin in the counterclockwise direction. This edge */
837 /* will be part of a different triangle. */
839 #define onext( triedge1, triedge2 ) \
840 lprev( triedge1, triedge2 ); \
843 #define onextself( triedge ) \
844 lprevself( triedge ); \
847 /* oprev() spins clockwise around a point; that is, it finds the next edge */
848 /* with the same origin in the clockwise direction. This edge will be */
849 /* part of a different triangle. */
851 #define oprev( triedge1, triedge2 ) \
852 sym( triedge1, triedge2 ); \
853 lnextself( triedge2 );
855 #define oprevself( triedge ) \
856 symself( triedge ); \
857 lnextself( triedge );
859 /* dnext() spins counterclockwise around a point; that is, it finds the next */
860 /* edge with the same destination in the counterclockwise direction. This */
861 /* edge will be part of a different triangle. */
863 #define dnext( triedge1, triedge2 ) \
864 sym( triedge1, triedge2 ); \
865 lprevself( triedge2 );
867 #define dnextself( triedge ) \
868 symself( triedge ); \
869 lprevself( triedge );
871 /* dprev() spins clockwise around a point; that is, it finds the next edge */
872 /* with the same destination in the clockwise direction. This edge will */
873 /* be part of a different triangle. */
875 #define dprev( triedge1, triedge2 ) \
876 lnext( triedge1, triedge2 ); \
879 #define dprevself( triedge ) \
880 lnextself( triedge ); \
883 /* rnext() moves one edge counterclockwise about the adjacent triangle. */
884 /* (It's best understood by reading Guibas and Stolfi. It involves */
885 /* changing triangles twice.) */
887 #define rnext( triedge1, triedge2 ) \
888 sym( triedge1, triedge2 ); \
889 lnextself( triedge2 ); \
892 #define rnextself( triedge ) \
893 symself( triedge ); \
894 lnextself( triedge ); \
897 /* rnext() moves one edge clockwise about the adjacent triangle. */
898 /* (It's best understood by reading Guibas and Stolfi. It involves */
899 /* changing triangles twice.) */
901 #define rprev( triedge1, triedge2 ) \
902 sym( triedge1, triedge2 ); \
903 lprevself( triedge2 ); \
906 #define rprevself( triedge ) \
907 symself( triedge ); \
908 lprevself( triedge ); \
911 /* These primitives determine or set the origin, destination, or apex of a */
914 #define org( triedge, pointptr ) \
915 pointptr = (point) ( triedge ).tri[plus1mod3[( triedge ).orient] + 3]
917 #define dest( triedge, pointptr ) \
918 pointptr = (point) ( triedge ).tri[minus1mod3[( triedge ).orient] + 3]
920 #define apex( triedge, pointptr ) \
921 pointptr = (point) ( triedge ).tri[( triedge ).orient + 3]
923 #define setorg( triedge, pointptr ) \
924 ( triedge ).tri[plus1mod3[( triedge ).orient] + 3] = (triangle) pointptr
926 #define setdest( triedge, pointptr ) \
927 ( triedge ).tri[minus1mod3[( triedge ).orient] + 3] = (triangle) pointptr
929 #define setapex( triedge, pointptr ) \
930 ( triedge ).tri[( triedge ).orient + 3] = (triangle) pointptr
932 #define setvertices2null( triedge ) \
933 ( triedge ).tri[3] = (triangle) NULL; \
934 ( triedge ).tri[4] = (triangle) NULL; \
935 ( triedge ).tri[5] = (triangle) NULL;
937 /* Bond two triangles together. */
939 #define bond( triedge1, triedge2 ) \
940 ( triedge1 ).tri[( triedge1 ).orient] = encode( triedge2 ); \
941 ( triedge2 ).tri[( triedge2 ).orient] = encode( triedge1 )
943 /* Dissolve a bond (from one side). Note that the other triangle will still */
944 /* think it's connected to this triangle. Usually, however, the other */
945 /* triangle is being deleted entirely, or bonded to another triangle, so */
946 /* it doesn't matter. */
948 #define dissolve( triedge ) \
949 ( triedge ).tri[( triedge ).orient] = (triangle) dummytri
951 /* Copy a triangle/edge handle. */
953 #define triedgecopy( triedge1, triedge2 ) \
954 ( triedge2 ).tri = ( triedge1 ).tri; \
955 ( triedge2 ).orient = ( triedge1 ).orient
957 /* Test for equality of triangle/edge handles. */
959 #define triedgeequal( triedge1, triedge2 ) \
960 ( ( ( triedge1 ).tri == ( triedge2 ).tri ) && \
961 ( ( triedge1 ).orient == ( triedge2 ).orient ) )
963 /* Primitives to infect or cure a triangle with the virus. These rely on */
964 /* the assumption that all shell edges are aligned to four-byte boundaries.*/
966 #define infect( triedge ) \
967 ( triedge ).tri[6] = (triangle) \
968 ( (unsigned long) ( triedge ).tri[6] | (unsigned long) 2l )
970 #define uninfect( triedge ) \
971 ( triedge ).tri[6] = (triangle) \
972 ( (unsigned long) ( triedge ).tri[6] & ~(unsigned long) 2l )
974 /* Test a triangle for viral infection. */
976 #define infected( triedge ) \
977 ( ( (unsigned long) ( triedge ).tri[6] & (unsigned long) 2l ) != 0 )
979 /* Check or set a triangle's attributes. */
981 #define elemattribute( triedge, attnum ) \
982 ( (REAL *) ( triedge ).tri )[elemattribindex + ( attnum )]
984 #define setelemattribute( triedge, attnum, value ) \
985 ( (REAL *) ( triedge ).tri )[elemattribindex + ( attnum )] = (REAL)value
987 /* Check or set a triangle's maximum area bound. */
989 #define areabound( triedge ) ( (REAL *) ( triedge ).tri )[areaboundindex]
991 #define setareabound( triedge, value ) \
992 ( (REAL *) ( triedge ).tri )[areaboundindex] = (REAL)value
994 /********* Primitives for shell edges *********/
998 /* sdecode() converts a pointer to an oriented shell edge. The orientation */
999 /* is extracted from the least significant bit of the pointer. The two */
1000 /* least significant bits (one for orientation, one for viral infection) */
1001 /* are masked out to produce the real pointer. */
1003 #define sdecode( sptr, edge ) \
1004 ( edge ).shorient = (int) ( (unsigned long) ( sptr ) & (unsigned long) 1l ); \
1005 ( edge ).sh = (shelle *) \
1006 ( (unsigned long) ( sptr ) & ~(unsigned long) 3l )
1008 /* sencode() compresses an oriented shell edge into a single pointer. It */
1009 /* relies on the assumption that all shell edges are aligned to two-byte */
1010 /* boundaries, so the least significant bit of (edge).sh is zero. */
1012 #define sencode( edge ) \
1013 (shelle) ( (unsigned long) ( edge ).sh | (unsigned long) ( edge ).shorient )
1015 /* ssym() toggles the orientation of a shell edge. */
1017 #define ssym( edge1, edge2 ) \
1018 ( edge2 ).sh = ( edge1 ).sh; \
1019 ( edge2 ).shorient = 1 - ( edge1 ).shorient
1021 #define ssymself( edge ) \
1022 ( edge ).shorient = 1 - ( edge ).shorient
1024 /* spivot() finds the other shell edge (from the same segment) that shares */
1025 /* the same origin. */
1027 #define spivot( edge1, edge2 ) \
1028 sptr = ( edge1 ).sh[( edge1 ).shorient]; \
1029 sdecode( sptr, edge2 )
1031 #define spivotself( edge ) \
1032 sptr = ( edge ).sh[( edge ).shorient]; \
1033 sdecode( sptr, edge )
1035 /* snext() finds the next shell edge (from the same segment) in sequence; */
1036 /* one whose origin is the input shell edge's destination. */
1038 #define snext( edge1, edge2 ) \
1039 sptr = ( edge1 ).sh[1 - ( edge1 ).shorient]; \
1040 sdecode( sptr, edge2 )
1042 #define snextself( edge ) \
1043 sptr = ( edge ).sh[1 - ( edge ).shorient]; \
1044 sdecode( sptr, edge )
1046 /* These primitives determine or set the origin or destination of a shell */
1049 #define sorg( edge, pointptr ) \
1050 pointptr = (point) ( edge ).sh[2 + ( edge ).shorient]
1052 #define sdest( edge, pointptr ) \
1053 pointptr = (point) ( edge ).sh[3 - ( edge ).shorient]
1055 #define setsorg( edge, pointptr ) \
1056 ( edge ).sh[2 + ( edge ).shorient] = (shelle) pointptr
1058 #define setsdest( edge, pointptr ) \
1059 ( edge ).sh[3 - ( edge ).shorient] = (shelle) pointptr
1061 /* These primitives read or set a shell marker. Shell markers are used to */
1062 /* hold user boundary information. */
1064 #define mark( edge ) ( *(int *) ( ( edge ).sh + 6 ) )
1066 #define setmark( edge, value ) \
1067 *(int *) ( ( edge ).sh + 6 ) = value
1069 /* Bond two shell edges together. */
1071 #define sbond( edge1, edge2 ) \
1072 ( edge1 ).sh[( edge1 ).shorient] = sencode( edge2 ); \
1073 ( edge2 ).sh[( edge2 ).shorient] = sencode( edge1 )
1075 /* Dissolve a shell edge bond (from one side). Note that the other shell */
1076 /* edge will still think it's connected to this shell edge. */
1078 #define sdissolve( edge ) \
1079 ( edge ).sh[( edge ).shorient] = (shelle) dummysh
1081 /* Copy a shell edge. */
1083 #define shellecopy( edge1, edge2 ) \
1084 ( edge2 ).sh = ( edge1 ).sh; \
1085 ( edge2 ).shorient = ( edge1 ).shorient
1087 /* Test for equality of shell edges. */
1089 #define shelleequal( edge1, edge2 ) \
1090 ( ( ( edge1 ).sh == ( edge2 ).sh ) && \
1091 ( ( edge1 ).shorient == ( edge2 ).shorient ) )
1093 /********* Primitives for interacting triangles and shell edges *********/
1097 /* tspivot() finds a shell edge abutting a triangle. */
1099 #define tspivot( triedge, edge ) \
1100 sptr = (shelle) ( triedge ).tri[6 + ( triedge ).orient]; \
1101 sdecode( sptr, edge )
1103 /* stpivot() finds a triangle abutting a shell edge. It requires that the */
1104 /* variable `ptr' of type `triangle' be defined. */
1106 #define stpivot( edge, triedge ) \
1107 ptr = (triangle) ( edge ).sh[4 + ( edge ).shorient]; \
1108 decode( ptr, triedge )
1110 /* Bond a triangle to a shell edge. */
1112 #define tsbond( triedge, edge ) \
1113 ( triedge ).tri[6 + ( triedge ).orient] = (triangle) sencode( edge ); \
1114 ( edge ).sh[4 + ( edge ).shorient] = (shelle) encode( triedge )
1116 /* Dissolve a bond (from the triangle side). */
1118 #define tsdissolve( triedge ) \
1119 ( triedge ).tri[6 + ( triedge ).orient] = (triangle) dummysh
1121 /* Dissolve a bond (from the shell edge side). */
1123 #define stdissolve( edge ) \
1124 ( edge ).sh[4 + ( edge ).shorient] = (shelle) dummytri
1126 /********* Primitives for points *********/
1130 #define pointmark( pt ) ( (int *) ( pt ) )[pointmarkindex]
1132 #define setpointmark( pt, value ) \
1133 ( (int *) ( pt ) )[pointmarkindex] = value
1135 #define point2tri( pt ) ( (triangle *) ( pt ) )[point2triindex]
1137 #define setpoint2tri( pt, value ) \
1138 ( (triangle *) ( pt ) )[point2triindex] = value
1142 /********* Mesh manipulation primitives end here *********/
1144 /********* User interaction routines begin here *********/
1148 /*****************************************************************************/
1150 /* syntax() Print list of command line switches. */
1152 /*****************************************************************************/
1159 printf( "triangle [-pAcevngBPNEIOXzo_lQVh] input_file\n" );
1160 #else /* not REDUCED */
1161 printf( "triangle [-pAcevngBPNEIOXzo_iFlCQVh] input_file\n" );
1162 #endif /* not REDUCED */
1163 #else /* not CDT_ONLY */
1165 printf( "triangle [-prq__a__AcevngBPNEIOXzo_YS__lQVh] input_file\n" );
1166 #else /* not REDUCED */
1167 printf( "triangle [-prq__a__AcevngBPNEIOXzo_YS__iFlsCQVh] input_file\n" );
1168 #endif /* not REDUCED */
1169 #endif /* not CDT_ONLY */
1171 printf( " -p Triangulates a Planar Straight Line Graph (.poly file).\n" );
1173 printf( " -r Refines a previously generated mesh.\n" );
1175 " -q Quality mesh generation. A minimum angle may be specified.\n" );
1176 printf( " -a Applies a maximum triangle area constraint.\n" );
1177 #endif /* not CDT_ONLY */
1179 " -A Applies attributes to identify elements in certain regions.\n" );
1180 printf( " -c Encloses the convex hull with segments.\n" );
1181 printf( " -e Generates an edge list.\n" );
1182 printf( " -v Generates a Voronoi diagram.\n" );
1183 printf( " -n Generates a list of triangle neighbors.\n" );
1184 printf( " -g Generates an .off file for Geomview.\n" );
1185 printf( " -B Suppresses output of boundary information.\n" );
1186 printf( " -P Suppresses output of .poly file.\n" );
1187 printf( " -N Suppresses output of .node file.\n" );
1188 printf( " -E Suppresses output of .ele file.\n" );
1189 printf( " -I Suppresses mesh iteration numbers.\n" );
1190 printf( " -O Ignores holes in .poly file.\n" );
1191 printf( " -X Suppresses use of exact arithmetic.\n" );
1192 printf( " -z Numbers all items starting from zero (rather than one).\n" );
1193 printf( " -o2 Generates second-order subparametric elements.\n" );
1195 printf( " -Y Suppresses boundary segment splitting.\n" );
1196 printf( " -S Specifies maximum number of added Steiner points.\n" );
1197 #endif /* not CDT_ONLY */
1199 printf( " -i Uses incremental method, rather than divide-and-conquer.\n" );
1200 printf( " -F Uses Fortune's sweepline algorithm, rather than d-and-c.\n" );
1201 #endif /* not REDUCED */
1202 printf( " -l Uses vertical cuts only, rather than alternating cuts.\n" );
1206 " -s Force segments into mesh by splitting (instead of using CDT).\n" );
1207 #endif /* not CDT_ONLY */
1208 printf( " -C Check consistency of final mesh.\n" );
1209 #endif /* not REDUCED */
1210 printf( " -Q Quiet: No terminal output except errors.\n" );
1211 printf( " -V Verbose: Detailed information on what I'm doing.\n" );
1212 printf( " -h Help: Detailed instructions for Triangle.\n" );
1216 #endif /* not TRILIBRARY */
1218 /*****************************************************************************/
1220 /* info() Print out complete instructions. */
1222 /*****************************************************************************/
1227 printf( "Triangle\n" );
1229 "A Two-Dimensional Quality Mesh Generator and Delaunay Triangulator.\n" );
1230 printf( "Version 1.3\n\n" );
1232 "Copyright 1996 Jonathan Richard Shewchuk (bugs/comments to jrs@cs.cmu.edu)\n"
1234 printf( "School of Computer Science / Carnegie Mellon University\n" );
1235 printf( "5000 Forbes Avenue / Pittsburgh, Pennsylvania 15213-3891\n" );
1237 "Created as part of the Archimedes project (tools for parallel FEM).\n" );
1239 "Supported in part by NSF Grant CMS-9318163 and an NSERC 1967 Scholarship.\n" );
1240 printf( "There is no warranty whatsoever. Use at your own risk.\n" );
1242 printf( "This executable is compiled for single precision arithmetic.\n\n\n" );
1243 #else /* not SINGLE */
1244 printf( "This executable is compiled for double precision arithmetic.\n\n\n" );
1245 #endif /* not SINGLE */
1247 "Triangle generates exact Delaunay triangulations, constrained Delaunay\n" );
1249 "triangulations, and quality conforming Delaunay triangulations. The latter\n"
1252 "can be generated with no small angles, and are thus suitable for finite\n" );
1254 "element analysis. If no command line switches are specified, your .node\n" );
1256 "input file will be read, and the Delaunay triangulation will be returned in\n"
1258 printf( ".node and .ele output files. The command syntax is:\n\n" );
1261 printf( "triangle [-pAcevngBPNEIOXzo_lQVh] input_file\n\n" );
1262 #else /* not REDUCED */
1263 printf( "triangle [-pAcevngBPNEIOXzo_iFlCQVh] input_file\n\n" );
1264 #endif /* not REDUCED */
1265 #else /* not CDT_ONLY */
1267 printf( "triangle [-prq__a__AcevngBPNEIOXzo_YS__lQVh] input_file\n\n" );
1268 #else /* not REDUCED */
1269 printf( "triangle [-prq__a__AcevngBPNEIOXzo_YS__iFlsCQVh] input_file\n\n" );
1270 #endif /* not REDUCED */
1271 #endif /* not CDT_ONLY */
1273 "Underscores indicate that numbers may optionally follow certain switches;\n" );
1275 "do not leave any space between a switch and its numeric parameter.\n" );
1277 "input_file must be a file with extension .node, or extension .poly if the\n" );
1279 "-p switch is used. If -r is used, you must supply .node and .ele files,\n" );
1281 "and possibly a .poly file and .area file as well. The formats of these\n" );
1282 printf( "files are described below.\n\n" );
1283 printf( "Command Line Switches:\n\n" );
1285 " -p Reads a Planar Straight Line Graph (.poly file), which can specify\n"
1288 " points, segments, holes, and regional attributes and area\n" );
1290 " constraints. Will generate a constrained Delaunay triangulation\n" );
1292 " fitting the input; or, if -s, -q, or -a is used, a conforming\n" );
1294 " Delaunay triangulation. If -p is not used, Triangle reads a .node\n"
1296 printf( " file by default.\n" );
1298 " -r Refines a previously generated mesh. The mesh is read from a .node\n"
1301 " file and an .ele file. If -p is also used, a .poly file is read\n" );
1303 " and used to constrain edges in the mesh. Further details on\n" );
1304 printf( " refinement are given below.\n" );
1306 " -q Quality mesh generation by Jim Ruppert's Delaunay refinement\n" );
1308 " algorithm. Adds points to the mesh to ensure that no angles\n" );
1310 " smaller than 20 degrees occur. An alternative minimum angle may be\n"
1313 " specified after the `q'. If the minimum angle is 20.7 degrees or\n" );
1315 " smaller, the triangulation algorithm is theoretically guaranteed to\n"
1318 " terminate (assuming infinite precision arithmetic - Triangle may\n" );
1320 " fail to terminate if you run out of precision). In practice, the\n" );
1322 " algorithm often succeeds for minimum angles up to 33.8 degrees.\n" );
1324 " For highly refined meshes, however, it may be necessary to reduce\n" );
1326 " the minimum angle to well below 20 to avoid problems associated\n" );
1328 " with insufficient floating-point precision. The specified angle\n" );
1329 printf( " may include a decimal point.\n" );
1331 " -a Imposes a maximum triangle area. If a number follows the `a', no\n" );
1333 " triangle will be generated whose area is larger than that number.\n" );
1335 " If no number is specified, an .area file (if -r is used) or .poly\n" );
1337 " file (if -r is not used) specifies a number of maximum area\n" );
1339 " constraints. An .area file contains a separate area constraint for\n"
1342 " each triangle, and is useful for refining a finite element mesh\n" );
1344 " based on a posteriori error estimates. A .poly file can optionally\n"
1347 " contain an area constraint for each segment-bounded region, thereby\n"
1350 " enforcing triangle densities in a first triangulation. You can\n" );
1352 " impose both a fixed area constraint and a varying area constraint\n" );
1354 " by invoking the -a switch twice, once with and once without a\n" );
1356 " number following. Each area specified may include a decimal point.\n"
1359 " -A Assigns an additional attribute to each triangle that identifies\n" );
1361 " what segment-bounded region each triangle belongs to. Attributes\n" );
1363 " are assigned to regions by the .poly file. If a region is not\n" );
1365 " explicitly marked by the .poly file, triangles in that region are\n" );
1367 " assigned an attribute of zero. The -A switch has an effect only\n" );
1368 printf( " when the -p switch is used and the -r switch is not.\n" );
1370 " -c Creates segments on the convex hull of the triangulation. If you\n" );
1372 " are triangulating a point set, this switch causes a .poly file to\n" );
1374 " be written, containing all edges in the convex hull. (By default,\n"
1377 " a .poly file is written only if a .poly file is read.) If you are\n"
1380 " triangulating a PSLG, this switch specifies that the interior of\n" );
1382 " the convex hull of the PSLG should be triangulated. If you do not\n"
1385 " use this switch when triangulating a PSLG, it is assumed that you\n" );
1387 " have identified the region to be triangulated by surrounding it\n" );
1389 " with segments of the input PSLG. Beware: if you are not careful,\n"
1392 " this switch can cause the introduction of an extremely thin angle\n" );
1394 " between a PSLG segment and a convex hull segment, which can cause\n" );
1396 " overrefinement or failure if Triangle runs out of precision. If\n" );
1398 " you are refining a mesh, the -c switch works differently; it\n" );
1400 " generates the set of boundary edges of the mesh, rather than the\n" );
1401 printf( " convex hull.\n" );
1403 " -e Outputs (to an .edge file) a list of edges of the triangulation.\n" );
1405 " -v Outputs the Voronoi diagram associated with the triangulation.\n" );
1406 printf( " Does not attempt to detect degeneracies.\n" );
1408 " -n Outputs (to a .neigh file) a list of triangles neighboring each\n" );
1409 printf( " triangle.\n" );
1411 " -g Outputs the mesh to an Object File Format (.off) file, suitable for\n"
1413 printf( " viewing with the Geometry Center's Geomview package.\n" );
1415 " -B No boundary markers in the output .node, .poly, and .edge output\n" );
1417 " files. See the detailed discussion of boundary markers below.\n" );
1419 " -P No output .poly file. Saves disk space, but you lose the ability\n" );
1421 " to impose segment constraints on later refinements of the mesh.\n" );
1422 printf( " -N No output .node file.\n" );
1423 printf( " -E No output .ele file.\n" );
1425 " -I No iteration numbers. Suppresses the output of .node and .poly\n" );
1427 " files, so your input files won't be overwritten. (If your input is\n"
1430 " a .poly file only, a .node file will be written.) Cannot be used\n" );
1432 " with the -r switch, because that would overwrite your input .ele\n" );
1434 " file. Shouldn't be used with the -s, -q, or -a switch if you are\n" );
1436 " using a .node file for input, because no .node file will be\n" );
1437 printf( " written, so there will be no record of any added points.\n" );
1438 printf( " -O No holes. Ignores the holes in the .poly file.\n" );
1440 " -X No exact arithmetic. Normally, Triangle uses exact floating-point\n"
1443 " arithmetic for certain tests if it thinks the inexact tests are not\n"
1446 " accurate enough. Exact arithmetic ensures the robustness of the\n" );
1448 " triangulation algorithms, despite floating-point roundoff error.\n" );
1450 " Disabling exact arithmetic with the -X switch will cause a small\n" );
1452 " improvement in speed and create the possibility (albeit small) that\n"
1455 " Triangle will fail to produce a valid mesh. Not recommended.\n" );
1457 " -z Numbers all items starting from zero (rather than one). Note that\n"
1460 " this switch is normally overrided by the value used to number the\n" );
1462 " first point of the input .node or .poly file. However, this switch\n"
1464 printf( " is useful when calling Triangle from another program.\n" );
1466 " -o2 Generates second-order subparametric elements with six nodes each.\n"
1469 " -Y No new points on the boundary. This switch is useful when the mesh\n"
1472 " boundary must be preserved so that it conforms to some adjacent\n" );
1474 " mesh. Be forewarned that you will probably sacrifice some of the\n" );
1476 " quality of the mesh; Triangle will try, but the resulting mesh may\n"
1479 " contain triangles of poor aspect ratio. Works well if all the\n" );
1481 " boundary points are closely spaced. Specify this switch twice\n" );
1483 " (`-YY') to prevent all segment splitting, including internal\n" );
1484 printf( " boundaries.\n" );
1486 " -S Specifies the maximum number of Steiner points (points that are not\n"
1489 " in the input, but are added to meet the constraints of minimum\n" );
1491 " angle and maximum area). The default is to allow an unlimited\n" );
1493 " number. If you specify this switch with no number after it,\n" );
1495 " the limit is set to zero. Triangle always adds points at segment\n" );
1497 " intersections, even if it needs to use more points than the limit\n" );
1499 " you set. When Triangle inserts segments by splitting (-s), it\n" );
1501 " always adds enough points to ensure that all the segments appear in\n"
1504 " the triangulation, again ignoring the limit. Be forewarned that\n" );
1506 " the -S switch may result in a conforming triangulation that is not\n"
1509 " truly Delaunay, because Triangle may be forced to stop adding\n" );
1511 " points when the mesh is in a state where a segment is non-Delaunay\n"
1514 " and needs to be split. If so, Triangle will print a warning.\n" );
1516 " -i Uses an incremental rather than divide-and-conquer algorithm to\n" );
1518 " form a Delaunay triangulation. Try it if the divide-and-conquer\n" );
1519 printf( " algorithm fails.\n" );
1521 " -F Uses Steven Fortune's sweepline algorithm to form a Delaunay\n" );
1523 " triangulation. Warning: does not use exact arithmetic for all\n" );
1524 printf( " calculations. An exact result is not guaranteed.\n" );
1526 " -l Uses only vertical cuts in the divide-and-conquer algorithm. By\n" );
1528 " default, Triangle uses alternating vertical and horizontal cuts,\n" );
1530 " which usually improve the speed except with point sets that are\n" );
1532 " small or short and wide. This switch is primarily of theoretical\n" );
1533 printf( " interest.\n" );
1535 " -s Specifies that segments should be forced into the triangulation by\n"
1538 " recursively splitting them at their midpoints, rather than by\n" );
1540 " generating a constrained Delaunay triangulation. Segment splitting\n"
1543 " is true to Ruppert's original algorithm, but can create needlessly\n"
1545 printf( " small triangles near external small features.\n" );
1547 " -C Check the consistency of the final mesh. Uses exact arithmetic for\n"
1550 " checking, even if the -X switch is used. Useful if you suspect\n" );
1551 printf( " Triangle is buggy.\n" );
1553 " -Q Quiet: Suppresses all explanation of what Triangle is doing, unless\n"
1555 printf( " an error occurs.\n" );
1557 " -V Verbose: Gives detailed information about what Triangle is doing.\n" );
1559 " Add more `V's for increasing amount of detail. `-V' gives\n" );
1561 " information on algorithmic progress and more detailed statistics.\n" );
1563 " `-VV' gives point-by-point details, and will print so much that\n" );
1565 " Triangle will run much more slowly. `-VVV' gives information only\n"
1567 printf( " a debugger could love.\n" );
1568 printf( " -h Help: Displays these instructions.\n" );
1570 printf( "Definitions:\n" );
1573 " A Delaunay triangulation of a point set is a triangulation whose vertices\n"
1576 " are the point set, having the property that no point in the point set\n" );
1578 " falls in the interior of the circumcircle (circle that passes through all\n"
1580 printf( " three vertices) of any triangle in the triangulation.\n\n" );
1582 " A Voronoi diagram of a point set is a subdivision of the plane into\n" );
1584 " polygonal regions (some of which may be infinite), where each region is\n" );
1586 " the set of points in the plane that are closer to some input point than\n" );
1588 " to any other input point. (The Voronoi diagram is the geometric dual of\n"
1590 printf( " the Delaunay triangulation.)\n\n" );
1592 " A Planar Straight Line Graph (PSLG) is a collection of points and\n" );
1594 " segments. Segments are simply edges, whose endpoints are points in the\n" );
1596 " PSLG. The file format for PSLGs (.poly files) is described below.\n" );
1599 " A constrained Delaunay triangulation of a PSLG is similar to a Delaunay\n" );
1601 " triangulation, but each PSLG segment is present as a single edge in the\n" );
1603 " triangulation. (A constrained Delaunay triangulation is not truly a\n" );
1604 printf( " Delaunay triangulation.)\n\n" );
1606 " A conforming Delaunay triangulation of a PSLG is a true Delaunay\n" );
1608 " triangulation in which each PSLG segment may have been subdivided into\n" );
1610 " several edges by the insertion of additional points. These inserted\n" );
1612 " points are necessary to allow the segments to exist in the mesh while\n" );
1613 printf( " maintaining the Delaunay property.\n\n" );
1614 printf( "File Formats:\n\n" );
1616 " All files may contain comments prefixed by the character '#'. Points,\n" );
1618 " triangles, edges, holes, and maximum area constraints must be numbered\n" );
1620 " consecutively, starting from either 1 or 0. Whichever you choose, all\n" );
1622 " input files must be consistent; if the nodes are numbered from 1, so must\n"
1625 " be all other objects. Triangle automatically detects your choice while\n" );
1627 " reading the .node (or .poly) file. (When calling Triangle from another\n" );
1629 " program, use the -z switch if you wish to number objects from zero.)\n" );
1630 printf( " Examples of these file formats are given below.\n\n" );
1631 printf( " .node files:\n" );
1633 " First line: <# of points> <dimension (must be 2)> <# of attributes>\n" );
1635 " <# of boundary markers (0 or 1)>\n"
1638 " Remaining lines: <point #> <x> <y> [attributes] [boundary marker]\n" );
1641 " The attributes, which are typically floating-point values of physical\n" );
1643 " quantities (such as mass or conductivity) associated with the nodes of\n"
1646 " a finite element mesh, are copied unchanged to the output mesh. If -s,\n"
1649 " -q, or -a is selected, each new Steiner point added to the mesh will\n" );
1650 printf( " have attributes assigned to it by linear interpolation.\n\n" );
1652 " If the fourth entry of the first line is `1', the last column of the\n" );
1654 " remainder of the file is assumed to contain boundary markers. Boundary\n"
1657 " markers are used to identify boundary points and points resting on PSLG\n"
1660 " segments; a complete description appears in a section below. The .node\n"
1663 " file produced by Triangle will contain boundary markers in the last\n" );
1664 printf( " column unless they are suppressed by the -B switch.\n\n" );
1665 printf( " .ele files:\n" );
1667 " First line: <# of triangles> <points per triangle> <# of attributes>\n" );
1669 " Remaining lines: <triangle #> <point> <point> <point> ... [attributes]\n"
1673 " Points are indices into the corresponding .node file. The first three\n"
1676 " points are the corners, and are listed in counterclockwise order around\n"
1679 " each triangle. (The remaining points, if any, depend on the type of\n" );
1681 " finite element used.) The attributes are just like those of .node\n" );
1683 " files. Because there is no simple mapping from input to output\n" );
1685 " triangles, an attempt is made to interpolate attributes, which may\n" );
1687 " result in a good deal of diffusion of attributes among nearby triangles\n"
1690 " as the triangulation is refined. Diffusion does not occur across\n" );
1692 " segments, so attributes used to identify segment-bounded regions remain\n"
1695 " intact. In output .ele files, all triangles have three points each\n" );
1697 " unless the -o2 switch is used, in which case they have six, and the\n" );
1699 " fourth, fifth, and sixth points lie on the midpoints of the edges\n" );
1700 printf( " opposite the first, second, and third corners.\n\n" );
1701 printf( " .poly files:\n" );
1703 " First line: <# of points> <dimension (must be 2)> <# of attributes>\n" );
1705 " <# of boundary markers (0 or 1)>\n"
1708 " Following lines: <point #> <x> <y> [attributes] [boundary marker]\n" );
1709 printf( " One line: <# of segments> <# of boundary markers (0 or 1)>\n" );
1711 " Following lines: <segment #> <endpoint> <endpoint> [boundary marker]\n" );
1712 printf( " One line: <# of holes>\n" );
1713 printf( " Following lines: <hole #> <x> <y>\n" );
1715 " Optional line: <# of regional attributes and/or area constraints>\n" );
1717 " Optional following lines: <constraint #> <x> <y> <attrib> <max area>\n" );
1720 " A .poly file represents a PSLG, as well as some additional information.\n"
1723 " The first section lists all the points, and is identical to the format\n"
1726 " of .node files. <# of points> may be set to zero to indicate that the\n"
1729 " points are listed in a separate .node file; .poly files produced by\n" );
1731 " Triangle always have this format. This has the advantage that a point\n"
1734 " set may easily be triangulated with or without segments. (The same\n" );
1736 " effect can be achieved, albeit using more disk space, by making a copy\n"
1739 " of the .poly file with the extension .node; all sections of the file\n" );
1740 printf( " but the first are ignored.)\n\n" );
1742 " The second section lists the segments. Segments are edges whose\n" );
1744 " presence in the triangulation is enforced. Each segment is specified\n" );
1746 " by listing the indices of its two endpoints. This means that you must\n"
1749 " include its endpoints in the point list. If -s, -q, and -a are not\n" );
1751 " selected, Triangle will produce a constrained Delaunay triangulation,\n" );
1753 " in which each segment appears as a single edge in the triangulation.\n" );
1755 " If -q or -a is selected, Triangle will produce a conforming Delaunay\n" );
1757 " triangulation, in which segments may be subdivided into smaller edges.\n"
1759 printf( " Each segment, like each point, may have a boundary marker.\n\n" );
1761 " The third section lists holes (and concavities, if -c is selected) in\n" );
1763 " the triangulation. Holes are specified by identifying a point inside\n" );
1765 " each hole. After the triangulation is formed, Triangle creates holes\n" );
1767 " by eating triangles, spreading out from each hole point until its\n" );
1769 " progress is blocked by PSLG segments; you must be careful to enclose\n" );
1771 " each hole in segments, or your whole triangulation may be eaten away.\n" );
1773 " If the two triangles abutting a segment are eaten, the segment itself\n" );
1775 " is also eaten. Do not place a hole directly on a segment; if you do,\n" );
1776 printf( " Triangle will choose one side of the segment arbitrarily.\n\n" );
1778 " The optional fourth section lists regional attributes (to be assigned\n" );
1780 " to all triangles in a region) and regional constraints on the maximum\n" );
1782 " triangle area. Triangle will read this section only if the -A switch\n" );
1784 " is used or the -a switch is used without a number following it, and the\n"
1787 " -r switch is not used. Regional attributes and area constraints are\n" );
1789 " propagated in the same manner as holes; you specify a point for each\n" );
1791 " attribute and/or constraint, and the attribute and/or constraint will\n" );
1793 " affect the whole region (bounded by segments) containing the point. If\n"
1796 " two values are written on a line after the x and y coordinate, the\n" );
1798 " former is assumed to be a regional attribute (but will only be applied\n"
1801 " if the -A switch is selected), and the latter is assumed to be a\n" );
1803 " regional area constraint (but will only be applied if the -a switch is\n"
1806 " selected). You may also specify just one value after the coordinates,\n"
1809 " which can serve as both an attribute and an area constraint, depending\n"
1812 " on the choice of switches. If you are using the -A and -a switches\n" );
1814 " simultaneously and wish to assign an attribute to some region without\n" );
1815 printf( " imposing an area constraint, use a negative maximum area.\n\n" );
1817 " When a triangulation is created from a .poly file, you must either\n" );
1819 " enclose the entire region to be triangulated in PSLG segments, or\n" );
1821 " use the -c switch, which encloses the convex hull of the input point\n" );
1823 " set. If you do not use the -c switch, Triangle will eat all triangles\n"
1826 " on the outer boundary that are not protected by segments; if you are\n" );
1828 " not careful, your whole triangulation may be eaten away. If you do\n" );
1830 " use the -c switch, you can still produce concavities by appropriate\n" );
1831 printf( " placement of holes just inside the convex hull.\n\n" );
1833 " An ideal PSLG has no intersecting segments, nor any points that lie\n" );
1835 " upon segments (except, of course, the endpoints of each segment.) You\n"
1838 " aren't required to make your .poly files ideal, but you should be aware\n"
1841 " of what can go wrong. Segment intersections are relatively safe -\n" );
1843 " Triangle will calculate the intersection points for you and add them to\n"
1846 " the triangulation - as long as your machine's floating-point precision\n"
1849 " doesn't become a problem. You are tempting the fates if you have three\n"
1852 " segments that cross at the same location, and expect Triangle to figure\n"
1855 " out where the intersection point is. Thanks to floating-point roundoff\n"
1858 " error, Triangle will probably decide that the three segments intersect\n"
1861 " at three different points, and you will find a minuscule triangle in\n" );
1863 " your output - unless Triangle tries to refine the tiny triangle, uses\n" );
1865 " up the last bit of machine precision, and fails to terminate at all.\n" );
1867 " You're better off putting the intersection point in the input files,\n" );
1869 " and manually breaking up each segment into two. Similarly, if you\n" );
1871 " place a point at the middle of a segment, and hope that Triangle will\n" );
1873 " break up the segment at that point, you might get lucky. On the other\n"
1876 " hand, Triangle might decide that the point doesn't lie precisely on the\n"
1879 " line, and you'll have a needle-sharp triangle in your output - or a lot\n"
1881 printf( " of tiny triangles if you're generating a quality mesh.\n\n" );
1883 " When Triangle reads a .poly file, it also writes a .poly file, which\n" );
1885 " includes all edges that are part of input segments. If the -c switch\n" );
1887 " is used, the output .poly file will also include all of the edges on\n" );
1889 " the convex hull. Hence, the output .poly file is useful for finding\n" );
1891 " edges associated with input segments and setting boundary conditions in\n"
1894 " finite element simulations. More importantly, you will need it if you\n"
1897 " plan to refine the output mesh, and don't want segments to be missing\n" );
1898 printf( " in later triangulations.\n\n" );
1899 printf( " .area files:\n" );
1900 printf( " First line: <# of triangles>\n" );
1901 printf( " Following lines: <triangle #> <maximum area>\n\n" );
1903 " An .area file associates with each triangle a maximum area that is used\n"
1906 " for mesh refinement. As with other file formats, every triangle must\n" );
1908 " be represented, and they must be numbered consecutively. A triangle\n" );
1910 " may be left unconstrained by assigning it a negative maximum area.\n" );
1912 printf( " .edge files:\n" );
1913 printf( " First line: <# of edges> <# of boundary markers (0 or 1)>\n" );
1915 " Following lines: <edge #> <endpoint> <endpoint> [boundary marker]\n" );
1918 " Endpoints are indices into the corresponding .node file. Triangle can\n"
1921 " produce .edge files (use the -e switch), but cannot read them. The\n" );
1923 " optional column of boundary markers is suppressed by the -B switch.\n" );
1926 " In Voronoi diagrams, one also finds a special kind of edge that is an\n" );
1928 " infinite ray with only one endpoint. For these edges, a different\n" );
1929 printf( " format is used:\n\n" );
1930 printf( " <edge #> <endpoint> -1 <direction x> <direction y>\n\n" );
1932 " The `direction' is a floating-point vector that indicates the direction\n"
1934 printf( " of the infinite ray.\n\n" );
1935 printf( " .neigh files:\n" );
1937 " First line: <# of triangles> <# of neighbors per triangle (always 3)>\n"
1940 " Following lines: <triangle #> <neighbor> <neighbor> <neighbor>\n" );
1943 " Neighbors are indices into the corresponding .ele file. An index of -1\n"
1946 " indicates a mesh boundary, and therefore no neighbor. Triangle can\n" );
1948 " produce .neigh files (use the -n switch), but cannot read them.\n" );
1951 " The first neighbor of triangle i is opposite the first corner of\n" );
1952 printf( " triangle i, and so on.\n\n" );
1953 printf( "Boundary Markers:\n\n" );
1955 " Boundary markers are tags used mainly to identify which output points and\n"
1958 " edges are associated with which PSLG segment, and to identify which\n" );
1960 " points and edges occur on a boundary of the triangulation. A common use\n"
1963 " is to determine where boundary conditions should be applied to a finite\n" );
1965 " element mesh. You can prevent boundary markers from being written into\n" );
1966 printf( " files produced by Triangle by using the -B switch.\n\n" );
1968 " The boundary marker associated with each segment in an output .poly file\n"
1970 printf( " or edge in an output .edge file is chosen as follows:\n" );
1972 " - If an output edge is part or all of a PSLG segment with a nonzero\n" );
1974 " boundary marker, then the edge is assigned the same marker.\n" );
1976 " - Otherwise, if the edge occurs on a boundary of the triangulation\n" );
1978 " (including boundaries of holes), then the edge is assigned the marker\n"
1980 printf( " one (1).\n" );
1981 printf( " - Otherwise, the edge is assigned the marker zero (0).\n" );
1983 " The boundary marker associated with each point in an output .node file is\n"
1985 printf( " chosen as follows:\n" );
1987 " - If a point is assigned a nonzero boundary marker in the input file,\n" );
1989 " then it is assigned the same marker in the output .node file.\n" );
1991 " - Otherwise, if the point lies on a PSLG segment (including the\n" );
1993 " segment's endpoints) with a nonzero boundary marker, then the point\n" );
1995 " is assigned the same marker. If the point lies on several such\n" );
1996 printf( " segments, one of the markers is chosen arbitrarily.\n" );
1998 " - Otherwise, if the point occurs on a boundary of the triangulation,\n" );
1999 printf( " then the point is assigned the marker one (1).\n" );
2000 printf( " - Otherwise, the point is assigned the marker zero (0).\n" );
2003 " If you want Triangle to determine for you which points and edges are on\n" );
2005 " the boundary, assign them the boundary marker zero (or use no markers at\n"
2008 " all) in your input files. Alternatively, you can mark some of them and\n" );
2009 printf( " leave others marked zero, allowing Triangle to label them.\n\n" );
2010 printf( "Triangulation Iteration Numbers:\n\n" );
2012 " Because Triangle can read and refine its own triangulations, input\n" );
2014 " and output files have iteration numbers. For instance, Triangle might\n" );
2016 " read the files mesh.3.node, mesh.3.ele, and mesh.3.poly, refine the\n" );
2018 " triangulation, and output the files mesh.4.node, mesh.4.ele, and\n" );
2019 printf( " mesh.4.poly. Files with no iteration number are treated as if\n" );
2021 " their iteration number is zero; hence, Triangle might read the file\n" );
2023 " points.node, triangulate it, and produce the files points.1.node and\n" );
2024 printf( " points.1.ele.\n\n" );
2026 " Iteration numbers allow you to create a sequence of successively finer\n" );
2028 " meshes suitable for multigrid methods. They also allow you to produce a\n"
2031 " sequence of meshes using error estimate-driven mesh refinement.\n" );
2034 " If you're not using refinement or quality meshing, and you don't like\n" );
2036 " iteration numbers, use the -I switch to disable them. This switch will\n" );
2038 " also disable output of .node and .poly files to prevent your input files\n"
2041 " from being overwritten. (If the input is a .poly file that contains its\n"
2043 printf( " own points, a .node file will be written.)\n\n" );
2044 printf( "Examples of How to Use Triangle:\n\n" );
2046 " `triangle dots' will read points from dots.node, and write their Delaunay\n"
2049 " triangulation to dots.1.node and dots.1.ele. (dots.1.node will be\n" );
2051 " identical to dots.node.) `triangle -I dots' writes the triangulation to\n"
2054 " dots.ele instead. (No additional .node file is needed, so none is\n" );
2055 printf( " written.)\n\n" );
2057 " `triangle -pe object.1' will read a PSLG from object.1.poly (and possibly\n"
2060 " object.1.node, if the points are omitted from object.1.poly) and write\n" );
2061 printf( " their constrained Delaunay triangulation to object.2.node and\n" );
2063 " object.2.ele. The segments will be copied to object.2.poly, and all\n" );
2064 printf( " edges will be written to object.2.edge.\n\n" );
2066 " `triangle -pq31.5a.1 object' will read a PSLG from object.poly (and\n" );
2068 " possibly object.node), generate a mesh whose angles are all greater than\n"
2071 " 31.5 degrees and whose triangles all have area smaller than 0.1, and\n" );
2073 " write the mesh to object.1.node and object.1.ele. Each segment may have\n"
2076 " been broken up into multiple edges; the resulting constrained edges are\n" );
2077 printf( " written to object.1.poly.\n\n" );
2079 " Here is a sample file `box.poly' describing a square with a square hole:\n"
2083 " # A box with eight points in 2D, no attributes, one boundary marker.\n" );
2084 printf( " 8 2 0 1\n" );
2085 printf( " # Outer box has these vertices:\n" );
2086 printf( " 1 0 0 0\n" );
2087 printf( " 2 0 3 0\n" );
2088 printf( " 3 3 0 0\n" );
2089 printf( " 4 3 3 33 # A special marker for this point.\n" );
2090 printf( " # Inner square has these vertices:\n" );
2091 printf( " 5 1 1 0\n" );
2092 printf( " 6 1 2 0\n" );
2093 printf( " 7 2 1 0\n" );
2094 printf( " 8 2 2 0\n" );
2095 printf( " # Five segments with boundary markers.\n" );
2097 printf( " 1 1 2 5 # Left side of outer box.\n" );
2098 printf( " 2 5 7 0 # Segments 2 through 5 enclose the hole.\n" );
2099 printf( " 3 7 8 0\n" );
2100 printf( " 4 8 6 10\n" );
2101 printf( " 5 6 5 0\n" );
2102 printf( " # One hole in the middle of the inner square.\n" );
2104 printf( " 1 1.5 1.5\n\n" );
2106 " Note that some segments are missing from the outer square, so one must\n" );
2108 " use the `-c' switch. After `triangle -pqc box.poly', here is the output\n"
2111 " file `box.1.node', with twelve points. The last four points were added\n" );
2113 " to meet the angle constraint. Points 1, 2, and 9 have markers from\n" );
2115 " segment 1. Points 6 and 8 have markers from segment 4. All the other\n" );
2117 " points but 4 have been marked to indicate that they lie on a boundary.\n" );
2119 printf( " 12 2 0 1\n" );
2120 printf( " 1 0 0 5\n" );
2121 printf( " 2 0 3 5\n" );
2122 printf( " 3 3 0 1\n" );
2123 printf( " 4 3 3 33\n" );
2124 printf( " 5 1 1 1\n" );
2125 printf( " 6 1 2 10\n" );
2126 printf( " 7 2 1 1\n" );
2127 printf( " 8 2 2 10\n" );
2128 printf( " 9 0 1.5 5\n" );
2129 printf( " 10 1.5 0 1\n" );
2130 printf( " 11 3 1.5 1\n" );
2131 printf( " 12 1.5 3 1\n" );
2132 printf( " # Generated by triangle -pqc box.poly\n\n" );
2133 printf( " Here is the output file `box.1.ele', with twelve triangles.\n\n" );
2134 printf( " 12 3 0\n" );
2135 printf( " 1 5 6 9\n" );
2136 printf( " 2 10 3 7\n" );
2137 printf( " 3 6 8 12\n" );
2138 printf( " 4 9 1 5\n" );
2139 printf( " 5 6 2 9\n" );
2140 printf( " 6 7 3 11\n" );
2141 printf( " 7 11 4 8\n" );
2142 printf( " 8 7 5 10\n" );
2143 printf( " 9 12 2 6\n" );
2144 printf( " 10 8 7 11\n" );
2145 printf( " 11 5 1 10\n" );
2146 printf( " 12 8 4 12\n" );
2147 printf( " # Generated by triangle -pqc box.poly\n\n" );
2149 " Here is the output file `box.1.poly'. Note that segments have been added\n"
2152 " to represent the convex hull, and some segments have been split by newly\n"
2155 " added points. Note also that <# of points> is set to zero to indicate\n" );
2156 printf( " that the points should be read from the .node file.\n\n" );
2157 printf( " 0 2 0 1\n" );
2158 printf( " 12 1\n" );
2159 printf( " 1 1 9 5\n" );
2160 printf( " 2 5 7 1\n" );
2161 printf( " 3 8 7 1\n" );
2162 printf( " 4 6 8 10\n" );
2163 printf( " 5 5 6 1\n" );
2164 printf( " 6 3 10 1\n" );
2165 printf( " 7 4 11 1\n" );
2166 printf( " 8 2 12 1\n" );
2167 printf( " 9 9 2 5\n" );
2168 printf( " 10 10 1 1\n" );
2169 printf( " 11 11 3 1\n" );
2170 printf( " 12 12 4 1\n" );
2172 printf( " 1 1.5 1.5\n" );
2173 printf( " # Generated by triangle -pqc box.poly\n\n" );
2174 printf( "Refinement and Area Constraints:\n\n" );
2176 " The -r switch causes a mesh (.node and .ele files) to be read and\n" );
2178 " refined. If the -p switch is also used, a .poly file is read and used to\n"
2181 " specify edges that are constrained and cannot be eliminated (although\n" );
2183 " they can be divided into smaller edges) by the refinement process.\n" );
2186 " When you refine a mesh, you generally want to impose tighter quality\n" );
2188 " constraints. One way to accomplish this is to use -q with a larger\n" );
2190 " angle, or -a followed by a smaller area than you used to generate the\n" );
2192 " mesh you are refining. Another way to do this is to create an .area\n" );
2194 " file, which specifies a maximum area for each triangle, and use the -a\n" );
2196 " switch (without a number following). Each triangle's area constraint is\n"
2199 " applied to that triangle. Area constraints tend to diffuse as the mesh\n" );
2201 " is refined, so if there are large variations in area constraint between\n" );
2202 printf( " adjacent triangles, you may not get the results you want.\n\n" );
2204 " If you are refining a mesh composed of linear (three-node) elements, the\n"
2207 " output mesh will contain all the nodes present in the input mesh, in the\n"
2210 " same order, with new nodes added at the end of the .node file. However,\n"
2213 " there is no guarantee that each output element is contained in a single\n" );
2215 " input element. Often, output elements will overlap two input elements,\n" );
2217 " and input edges are not present in the output mesh. Hence, a sequence of\n"
2220 " refined meshes will form a hierarchy of nodes, but not a hierarchy of\n" );
2222 " elements. If you a refining a mesh of higher-order elements, the\n" );
2224 " hierarchical property applies only to the nodes at the corners of an\n" );
2225 printf( " element; other nodes may not be present in the refined mesh.\n\n" );
2227 " It is important to understand that maximum area constraints in .poly\n" );
2229 " files are handled differently from those in .area files. A maximum area\n"
2232 " in a .poly file applies to the whole (segment-bounded) region in which a\n"
2235 " point falls, whereas a maximum area in an .area file applies to only one\n"
2238 " triangle. Area constraints in .poly files are used only when a mesh is\n" );
2240 " first generated, whereas area constraints in .area files are used only to\n"
2243 " refine an existing mesh, and are typically based on a posteriori error\n" );
2245 " estimates resulting from a finite element simulation on that mesh.\n" );
2248 " `triangle -rq25 object.1' will read object.1.node and object.1.ele, then\n"
2251 " refine the triangulation to enforce a 25 degree minimum angle, and then\n" );
2253 " write the refined triangulation to object.2.node and object.2.ele.\n" );
2256 " `triangle -rpaa6.2 z.3' will read z.3.node, z.3.ele, z.3.poly, and\n" );
2258 " z.3.area. After reconstructing the mesh and its segments, Triangle will\n"
2261 " refine the mesh so that no triangle has area greater than 6.2, and\n" );
2263 " furthermore the triangles satisfy the maximum area constraints in\n" );
2265 " z.3.area. The output is written to z.4.node, z.4.ele, and z.4.poly.\n" );
2268 " The sequence `triangle -qa1 x', `triangle -rqa.3 x.1', `triangle -rqa.1\n" );
2270 " x.2' creates a sequence of successively finer meshes x.1, x.2, and x.3,\n" );
2271 printf( " suitable for multigrid.\n\n" );
2272 printf( "Convex Hulls and Mesh Boundaries:\n\n" );
2274 " If the input is a point set (rather than a PSLG), Triangle produces its\n" );
2276 " convex hull as a by-product in the output .poly file if you use the -c\n" );
2278 " switch. There are faster algorithms for finding a two-dimensional convex\n"
2281 " hull than triangulation, of course, but this one comes for free. If the\n"
2284 " input is an unconstrained mesh (you are using the -r switch but not the\n" );
2286 " -p switch), Triangle produces a list of its boundary edges (including\n" );
2287 printf( " hole boundaries) as a by-product if you use the -c switch.\n\n" );
2288 printf( "Voronoi Diagrams:\n\n" );
2290 " The -v switch produces a Voronoi diagram, in files suffixed .v.node and\n" );
2292 " .v.edge. For example, `triangle -v points' will read points.node,\n" );
2294 " produce its Delaunay triangulation in points.1.node and points.1.ele,\n" );
2296 " and produce its Voronoi diagram in points.1.v.node and points.1.v.edge.\n" );
2298 " The .v.node file contains a list of all Voronoi vertices, and the .v.edge\n"
2301 " file contains a list of all Voronoi edges, some of which may be infinite\n"
2304 " rays. (The choice of filenames makes it easy to run the set of Voronoi\n" );
2305 printf( " vertices through Triangle, if so desired.)\n\n" );
2307 " This implementation does not use exact arithmetic to compute the Voronoi\n"
2310 " vertices, and does not check whether neighboring vertices are identical.\n"
2313 " Be forewarned that if the Delaunay triangulation is degenerate or\n" );
2315 " near-degenerate, the Voronoi diagram may have duplicate points, crossing\n"
2318 " edges, or infinite rays whose direction vector is zero. Also, if you\n" );
2320 " generate a constrained (as opposed to conforming) Delaunay triangulation,\n"
2323 " or if the triangulation has holes, the corresponding Voronoi diagram is\n" );
2324 printf( " likely to have crossing edges and unlikely to make sense.\n\n" );
2325 printf( "Mesh Topology:\n\n" );
2327 " You may wish to know which triangles are adjacent to a certain Delaunay\n" );
2329 " edge in an .edge file, which Voronoi regions are adjacent to a certain\n" );
2331 " Voronoi edge in a .v.edge file, or which Voronoi regions are adjacent to\n"
2334 " each other. All of this information can be found by cross-referencing\n" );
2336 " output files with the recollection that the Delaunay triangulation and\n" );
2337 printf( " the Voronoi diagrams are planar duals.\n\n" );
2339 " Specifically, edge i of an .edge file is the dual of Voronoi edge i of\n" );
2341 " the corresponding .v.edge file, and is rotated 90 degrees counterclock-\n" );
2343 " wise from the Voronoi edge. Triangle j of an .ele file is the dual of\n" );
2345 " vertex j of the corresponding .v.node file; and Voronoi region k is the\n" );
2346 printf( " dual of point k of the corresponding .node file.\n\n" );
2348 " Hence, to find the triangles adjacent to a Delaunay edge, look at the\n" );
2350 " vertices of the corresponding Voronoi edge; their dual triangles are on\n" );
2352 " the left and right of the Delaunay edge, respectively. To find the\n" );
2354 " Voronoi regions adjacent to a Voronoi edge, look at the endpoints of the\n"
2357 " corresponding Delaunay edge; their dual regions are on the right and left\n"
2360 " of the Voronoi edge, respectively. To find which Voronoi regions are\n" );
2361 printf( " adjacent to each other, just read the list of Delaunay edges.\n" );
2363 printf( "Statistics:\n" );
2366 " After generating a mesh, Triangle prints a count of the number of points,\n"
2369 " triangles, edges, boundary edges, and segments in the output mesh. If\n" );
2371 " you've forgotten the statistics for an existing mesh, the -rNEP switches\n"
2374 " (or -rpNEP if you've got a .poly file for the existing mesh) will\n" );
2375 printf( " regenerate these statistics without writing any output.\n\n" );
2377 " The -V switch produces extended statistics, including a rough estimate\n" );
2379 " of memory use and a histogram of triangle aspect ratios and angles in the\n"
2381 printf( " mesh.\n\n" );
2382 printf( "Exact Arithmetic:\n\n" );
2384 " Triangle uses adaptive exact arithmetic to perform what computational\n" );
2386 " geometers call the `orientation' and `incircle' tests. If the floating-\n"
2389 " point arithmetic of your machine conforms to the IEEE 754 standard (as\n" );
2391 " most workstations do), and does not use extended precision internal\n" );
2393 " registers, then your output is guaranteed to be an absolutely true\n" );
2394 printf( " Delaunay or conforming Delaunay triangulation, roundoff error\n" );
2396 " notwithstanding. The word `adaptive' implies that these arithmetic\n" );
2398 " routines compute the result only to the precision necessary to guarantee\n"
2401 " correctness, so they are usually nearly as fast as their approximate\n" );
2403 " counterparts. The exact tests can be disabled with the -X switch. On\n" );
2405 " most inputs, this switch will reduce the computation time by about eight\n"
2408 " percent - it's not worth the risk. There are rare difficult inputs\n" );
2410 " (having many collinear and cocircular points), however, for which the\n" );
2412 " difference could be a factor of two. These are precisely the inputs most\n"
2414 printf( " likely to cause errors if you use the -X switch.\n\n" );
2416 " Unfortunately, these routines don't solve every numerical problem. Exact\n"
2419 " arithmetic is not used to compute the positions of points, because the\n" );
2421 " bit complexity of point coordinates would grow without bound. Hence,\n" );
2423 " segment intersections aren't computed exactly; in very unusual cases,\n" );
2425 " roundoff error in computing an intersection point might actually lead to\n"
2428 " an inverted triangle and an invalid triangulation. (This is one reason\n" );
2430 " to compute your own intersection points in your .poly files.) Similarly,\n"
2433 " exact arithmetic is not used to compute the vertices of the Voronoi\n" );
2434 printf( " diagram.\n\n" );
2436 " Underflow and overflow can also cause difficulties; the exact arithmetic\n"
2439 " routines do not ameliorate out-of-bounds exponents, which can arise\n" );
2441 " during the orientation and incircle tests. As a rule of thumb, you\n" );
2443 " should ensure that your input values are within a range such that their\n" );
2445 " third powers can be taken without underflow or overflow. Underflow can\n" );
2447 " silently prevent the tests from being performed exactly, while overflow\n" );
2448 printf( " will typically cause a floating exception.\n\n" );
2449 printf( "Calling Triangle from Another Program:\n\n" );
2450 printf( " Read the file triangle.h for details.\n\n" );
2451 printf( "Troubleshooting:\n\n" );
2452 printf( " Please read this section before mailing me bugs.\n\n" );
2453 printf( " `My output mesh has no triangles!'\n\n" );
2455 " If you're using a PSLG, you've probably failed to specify a proper set\n"
2458 " of bounding segments, or forgotten to use the -c switch. Or you may\n" );
2460 " have placed a hole badly. To test these possibilities, try again with\n"
2463 " the -c and -O switches. Alternatively, all your input points may be\n" );
2465 " collinear, in which case you can hardly expect to triangulate them.\n" );
2467 printf( " `Triangle doesn't terminate, or just crashes.'\n" );
2470 " Bad things can happen when triangles get so small that the distance\n" );
2472 " between their vertices isn't much larger than the precision of your\n" );
2474 " machine's arithmetic. If you've compiled Triangle for single-precision\n"
2477 " arithmetic, you might do better by recompiling it for double-precision.\n"
2480 " Then again, you might just have to settle for more lenient constraints\n"
2483 " on the minimum angle and the maximum area than you had planned.\n" );
2486 " You can minimize precision problems by ensuring that the origin lies\n" );
2488 " inside your point set, or even inside the densest part of your\n" );
2490 " mesh. On the other hand, if you're triangulating an object whose x\n" );
2492 " coordinates all fall between 6247133 and 6247134, you're not leaving\n" );
2493 printf( " much floating-point precision for Triangle to work with.\n\n" );
2495 " Precision problems can occur covertly if the input PSLG contains two\n" );
2497 " segments that meet (or intersect) at a very small angle, or if such an\n"
2500 " angle is introduced by the -c switch, which may occur if a point lies\n" );
2502 " ever-so-slightly inside the convex hull, and is connected by a PSLG\n" );
2504 " segment to a point on the convex hull. If you don't realize that a\n" );
2506 " small angle is being formed, you might never discover why Triangle is\n" );
2508 " crashing. To check for this possibility, use the -S switch (with an\n" );
2510 " appropriate limit on the number of Steiner points, found by trial-and-\n"
2513 " error) to stop Triangle early, and view the output .poly file with\n" );
2515 " Show Me (described below). Look carefully for small angles between\n" );
2517 " segments; zoom in closely, as such segments might look like a single\n" );
2518 printf( " segment from a distance.\n\n" );
2520 " If some of the input values are too large, Triangle may suffer a\n" );
2522 " floating exception due to overflow when attempting to perform an\n" );
2524 " orientation or incircle test. (Read the section on exact arithmetic\n" );
2526 " above.) Again, I recommend compiling Triangle for double (rather\n" );
2527 printf( " than single) precision arithmetic.\n\n" );
2529 " `The numbering of the output points doesn't match the input points.'\n" );
2532 " You may have eaten some of your input points with a hole, or by placing\n"
2534 printf( " them outside the area enclosed by segments.\n\n" );
2536 " `Triangle executes without incident, but when I look at the resulting\n" );
2538 " mesh, it has overlapping triangles or other geometric inconsistencies.'\n" );
2541 " If you select the -X switch, Triangle's divide-and-conquer Delaunay\n" );
2543 " triangulation algorithm occasionally makes mistakes due to floating-\n" );
2545 " point roundoff error. Although these errors are rare, don't use the -X\n"
2547 printf( " switch. If you still have problems, please report the bug.\n" );
2550 " Strange things can happen if you've taken liberties with your PSLG. Do\n" );
2552 " you have a point lying in the middle of a segment? Triangle sometimes\n" );
2554 " copes poorly with that sort of thing. Do you want to lay out a collinear\n"
2557 " row of evenly spaced, segment-connected points? Have you simply defined\n"
2560 " one long segment connecting the leftmost point to the rightmost point,\n" );
2562 " and a bunch of points lying along it? This method occasionally works,\n" );
2564 " especially with horizontal and vertical lines, but often it doesn't, and\n"
2567 " you'll have to connect each adjacent pair of points with a separate\n" );
2568 printf( " segment. If you don't like it, tough.\n\n" );
2570 " Furthermore, if you have segments that intersect other than at their\n" );
2572 " endpoints, try not to let the intersections fall extremely close to PSLG\n"
2574 printf( " points or each other.\n\n" );
2576 " If you have problems refining a triangulation not produced by Triangle:\n" );
2578 " Are you sure the triangulation is geometrically valid? Is it formatted\n" );
2580 " correctly for Triangle? Are the triangles all listed so the first three\n"
2582 printf( " points are their corners in counterclockwise order?\n\n" );
2583 printf( "Show Me:\n\n" );
2585 " Triangle comes with a separate program named `Show Me', whose primary\n" );
2587 " purpose is to draw meshes on your screen or in PostScript. Its secondary\n"
2590 " purpose is to check the validity of your input files, and do so more\n" );
2592 " thoroughly than Triangle does. Show Me requires that you have the X\n" );
2594 " Windows system. If you didn't receive Show Me with Triangle, complain to\n"
2596 printf( " whomever you obtained Triangle from, then send me mail.\n\n" );
2597 printf( "Triangle on the Web:\n\n" );
2599 " To see an illustrated, updated version of these instructions, check out\n" );
2601 printf( " http://www.cs.cmu.edu/~quake/triangle.html\n" );
2603 printf( "A Brief Plea:\n" );
2606 " If you use Triangle, and especially if you use it to accomplish real\n" );
2608 " work, I would like very much to hear from you. A short letter or email\n" );
2610 " (to jrs@cs.cmu.edu) describing how you use Triangle will mean a lot to\n" );
2612 " me. The more people I know are using this program, the more easily I can\n"
2615 " justify spending time on improvements and on the three-dimensional\n" );
2617 " successor to Triangle, which in turn will benefit you. Also, I can put\n" );
2619 " you on a list to receive email whenever a new version of Triangle is\n" );
2620 printf( " available.\n\n" );
2622 " If you use a mesh generated by Triangle in a publication, please include\n"
2624 printf( " an acknowledgment as well.\n\n" );
2625 printf( "Research credit:\n\n" );
2627 " Of course, I can take credit for only a fraction of the ideas that made\n" );
2629 " this mesh generator possible. Triangle owes its existence to the efforts\n"
2632 " of many fine computational geometers and other researchers, including\n" );
2634 " Marshall Bern, L. Paul Chew, Boris Delaunay, Rex A. Dwyer, David\n" );
2636 " Eppstein, Steven Fortune, Leonidas J. Guibas, Donald E. Knuth, C. L.\n" );
2638 " Lawson, Der-Tsai Lee, Ernst P. Mucke, Douglas M. Priest, Jim Ruppert,\n" );
2640 " Isaac Saias, Bruce J. Schachter, Micha Sharir, Jorge Stolfi, Christopher\n"
2643 " J. Van Wyk, David F. Watson, and Binhai Zhu. See the comments at the\n" );
2644 printf( " beginning of the source code for references.\n\n" );
2648 #endif /* not TRILIBRARY */
2650 /*****************************************************************************/
2652 /* internalerror() Ask the user to send me the defective product. Exit. */
2654 /*****************************************************************************/
2656 void internalerror(){
2657 printf( " Please report this bug to jrs@cs.cmu.edu\n" );
2658 printf( " Include the message above, your input data set, and the exact\n" );
2659 printf( " command line you used to run Triangle.\n" );
2663 /*****************************************************************************/
2665 /* parsecommandline() Read the command line, identify switches, and set */
2666 /* up options and file names. */
2668 /* The effects of this routine are felt entirely through global variables. */
2670 /*****************************************************************************/
2672 void parsecommandline( argc, argv )
2677 #define STARTINDEX 0
2678 #else /* not TRILIBRARY */
2679 #define STARTINDEX 1
2682 #endif /* not TRILIBRARY */
2686 char workstring[FILENAMESIZE];
2689 poly = refine = quality = vararea = fixedarea = regionattrib = convex = 0;
2691 edgesout = voronoi = neighbors = geomview = 0;
2692 nobound = nopolywritten = nonodewritten = noelewritten = noiterationnum = 0;
2693 noholes = noexact = 0;
2694 incremental = sweepline = 0;
2703 quiet = verbose = 0;
2705 innodefilename[0] = '\0';
2706 #endif /* not TRILIBRARY */
2708 for ( i = STARTINDEX; i < argc; i++ ) {
2710 if ( argv[i][0] == '-' ) {
2711 #endif /* not TRILIBRARY */
2712 for ( j = STARTINDEX; argv[i][j] != '\0'; j++ ) {
2713 if ( argv[i][j] == 'p' ) {
2717 if ( argv[i][j] == 'r' ) {
2720 if ( argv[i][j] == 'q' ) {
2722 if ( ( ( argv[i][j + 1] >= '0' ) && ( argv[i][j + 1] <= '9' ) ) ||
2723 ( argv[i][j + 1] == '.' ) ) {
2725 while ( ( ( argv[i][j + 1] >= '0' ) && ( argv[i][j + 1] <= '9' ) ) ||
2726 ( argv[i][j + 1] == '.' ) ) {
2728 workstring[k] = argv[i][j];
2731 workstring[k] = '\0';
2732 minangle = (REAL) strtod( workstring, (char **) NULL );
2738 if ( argv[i][j] == 'a' ) {
2740 if ( ( ( argv[i][j + 1] >= '0' ) && ( argv[i][j + 1] <= '9' ) ) ||
2741 ( argv[i][j + 1] == '.' ) ) {
2744 while ( ( ( argv[i][j + 1] >= '0' ) && ( argv[i][j + 1] <= '9' ) ) ||
2745 ( argv[i][j + 1] == '.' ) ) {
2747 workstring[k] = argv[i][j];
2750 workstring[k] = '\0';
2751 maxarea = (REAL) strtod( workstring, (char **) NULL );
2752 if ( maxarea <= 0.0 ) {
2753 printf( "Error: Maximum area must be greater than zero.\n" );
2761 #endif /* not CDT_ONLY */
2762 if ( argv[i][j] == 'A' ) {
2765 if ( argv[i][j] == 'c' ) {
2768 if ( argv[i][j] == 'z' ) {
2771 if ( argv[i][j] == 'e' ) {
2774 if ( argv[i][j] == 'v' ) {
2777 if ( argv[i][j] == 'n' ) {
2780 if ( argv[i][j] == 'g' ) {
2783 if ( argv[i][j] == 'B' ) {
2786 if ( argv[i][j] == 'P' ) {
2789 if ( argv[i][j] == 'N' ) {
2792 if ( argv[i][j] == 'E' ) {
2796 if ( argv[i][j] == 'I' ) {
2799 #endif /* not TRILIBRARY */
2800 if ( argv[i][j] == 'O' ) {
2803 if ( argv[i][j] == 'X' ) {
2806 if ( argv[i][j] == 'o' ) {
2807 if ( argv[i][j + 1] == '2' ) {
2813 if ( argv[i][j] == 'Y' ) {
2816 if ( argv[i][j] == 'S' ) {
2818 while ( ( argv[i][j + 1] >= '0' ) && ( argv[i][j + 1] <= '9' ) ) {
2820 steiner = steiner * 10 + (int) ( argv[i][j] - '0' );
2823 #endif /* not CDT_ONLY */
2825 if ( argv[i][j] == 'i' ) {
2828 if ( argv[i][j] == 'F' ) {
2831 #endif /* not REDUCED */
2832 if ( argv[i][j] == 'l' ) {
2837 if ( argv[i][j] == 's' ) {
2840 #endif /* not CDT_ONLY */
2841 if ( argv[i][j] == 'C' ) {
2844 #endif /* not REDUCED */
2845 if ( argv[i][j] == 'Q' ) {
2848 if ( argv[i][j] == 'V' ) {
2852 if ( ( argv[i][j] == 'h' ) || ( argv[i][j] == 'H' ) ||
2853 ( argv[i][j] == '?' ) ) {
2856 #endif /* not TRILIBRARY */
2860 strncpy( innodefilename, argv[i], FILENAMESIZE - 1 );
2861 innodefilename[FILENAMESIZE - 1] = '\0';
2863 #endif /* not TRILIBRARY */
2866 if ( innodefilename[0] == '\0' ) {
2869 if ( !strcmp( &innodefilename[strlen( innodefilename ) - 5], ".node" ) ) {
2870 innodefilename[strlen( innodefilename ) - 5] = '\0';
2872 if ( !strcmp( &innodefilename[strlen( innodefilename ) - 5], ".poly" ) ) {
2873 innodefilename[strlen( innodefilename ) - 5] = '\0';
2877 if ( !strcmp( &innodefilename[strlen( innodefilename ) - 4], ".ele" ) ) {
2878 innodefilename[strlen( innodefilename ) - 4] = '\0';
2881 if ( !strcmp( &innodefilename[strlen( innodefilename ) - 5], ".area" ) ) {
2882 innodefilename[strlen( innodefilename ) - 5] = '\0';
2887 #endif /* not CDT_ONLY */
2888 #endif /* not TRILIBRARY */
2889 steinerleft = steiner;
2890 useshelles = poly || refine || quality || convex;
2891 goodangle = (REAL)cos( minangle * PI / 180.0 );
2892 goodangle *= goodangle;
2893 if ( refine && noiterationnum ) {
2895 "Error: You cannot use the -I switch when refining a triangulation.\n" );
2898 /* Be careful not to allocate space for element area constraints that */
2899 /* will never be assigned any value (other than the default -1.0). */
2900 if ( !refine && !poly ) {
2903 /* Be careful not to add an extra attribute to each element unless the */
2904 /* input supports it (PSLG in, but not refining a preexisting mesh). */
2905 if ( refine || !poly ) {
2910 strcpy( inpolyfilename, innodefilename );
2911 strcpy( inelefilename, innodefilename );
2912 strcpy( areafilename, innodefilename );
2914 strcpy( workstring, innodefilename );
2916 while ( workstring[j] != '\0' ) {
2917 if ( ( workstring[j] == '.' ) && ( workstring[j + 1] != '\0' ) ) {
2923 if ( increment > 0 ) {
2926 if ( ( workstring[j] >= '0' ) && ( workstring[j] <= '9' ) ) {
2927 meshnumber = meshnumber * 10 + (int) ( workstring[j] - '0' );
2933 } while ( workstring[j] != '\0' );
2935 if ( noiterationnum ) {
2936 strcpy( outnodefilename, innodefilename );
2937 strcpy( outelefilename, innodefilename );
2938 strcpy( edgefilename, innodefilename );
2939 strcpy( vnodefilename, innodefilename );
2940 strcpy( vedgefilename, innodefilename );
2941 strcpy( neighborfilename, innodefilename );
2942 strcpy( offfilename, innodefilename );
2943 strcat( outnodefilename, ".node" );
2944 strcat( outelefilename, ".ele" );
2945 strcat( edgefilename, ".edge" );
2946 strcat( vnodefilename, ".v.node" );
2947 strcat( vedgefilename, ".v.edge" );
2948 strcat( neighborfilename, ".neigh" );
2949 strcat( offfilename, ".off" );
2951 else if ( increment == 0 ) {
2952 strcpy( outnodefilename, innodefilename );
2953 strcpy( outpolyfilename, innodefilename );
2954 strcpy( outelefilename, innodefilename );
2955 strcpy( edgefilename, innodefilename );
2956 strcpy( vnodefilename, innodefilename );
2957 strcpy( vedgefilename, innodefilename );
2958 strcpy( neighborfilename, innodefilename );
2959 strcpy( offfilename, innodefilename );
2960 strcat( outnodefilename, ".1.node" );
2961 strcat( outpolyfilename, ".1.poly" );
2962 strcat( outelefilename, ".1.ele" );
2963 strcat( edgefilename, ".1.edge" );
2964 strcat( vnodefilename, ".1.v.node" );
2965 strcat( vedgefilename, ".1.v.edge" );
2966 strcat( neighborfilename, ".1.neigh" );
2967 strcat( offfilename, ".1.off" );
2970 workstring[increment] = '%';
2971 workstring[increment + 1] = 'd';
2972 workstring[increment + 2] = '\0';
2973 sprintf( outnodefilename, workstring, meshnumber + 1 );
2974 strcpy( outpolyfilename, outnodefilename );
2975 strcpy( outelefilename, outnodefilename );
2976 strcpy( edgefilename, outnodefilename );
2977 strcpy( vnodefilename, outnodefilename );
2978 strcpy( vedgefilename, outnodefilename );
2979 strcpy( neighborfilename, outnodefilename );
2980 strcpy( offfilename, outnodefilename );
2981 strcat( outnodefilename, ".node" );
2982 strcat( outpolyfilename, ".poly" );
2983 strcat( outelefilename, ".ele" );
2984 strcat( edgefilename, ".edge" );
2985 strcat( vnodefilename, ".v.node" );
2986 strcat( vedgefilename, ".v.edge" );
2987 strcat( neighborfilename, ".neigh" );
2988 strcat( offfilename, ".off" );
2990 strcat( innodefilename, ".node" );
2991 strcat( inpolyfilename, ".poly" );
2992 strcat( inelefilename, ".ele" );
2993 strcat( areafilename, ".area" );
2994 #endif /* not TRILIBRARY */
2999 /********* User interaction routines begin here *********/
3001 /********* Debugging routines begin here *********/
3005 /*****************************************************************************/
3007 /* printtriangle() Print out the details of a triangle/edge handle. */
3009 /* I originally wrote this procedure to simplify debugging; it can be */
3010 /* called directly from the debugger, and presents information about a */
3011 /* triangle/edge handle in digestible form. It's also used when the */
3012 /* highest level of verbosity (`-VVV') is specified. */
3014 /*****************************************************************************/
3016 void printtriangle( t )
3019 struct triedge printtri;
3020 struct edge printsh;
3023 printf( "triangle x%lx with orientation %d:\n", (unsigned long) t->tri,
3025 decode( t->tri[0], printtri );
3026 if ( printtri.tri == dummytri ) {
3027 printf( " [0] = Outer space\n" );
3030 printf( " [0] = x%lx %d\n", (unsigned long) printtri.tri,
3033 decode( t->tri[1], printtri );
3034 if ( printtri.tri == dummytri ) {
3035 printf( " [1] = Outer space\n" );
3038 printf( " [1] = x%lx %d\n", (unsigned long) printtri.tri,
3041 decode( t->tri[2], printtri );
3042 if ( printtri.tri == dummytri ) {
3043 printf( " [2] = Outer space\n" );
3046 printf( " [2] = x%lx %d\n", (unsigned long) printtri.tri,
3049 org( *t, printpoint );
3050 if ( printpoint == (point) NULL ) {
3051 printf( " Origin[%d] = NULL\n", ( t->orient + 1 ) % 3 + 3 );
3054 printf( " Origin[%d] = x%lx (%.12g, %.12g)\n",
3055 ( t->orient + 1 ) % 3 + 3, (unsigned long) printpoint,
3056 printpoint[0], printpoint[1] );
3058 dest( *t, printpoint );
3059 if ( printpoint == (point) NULL ) {
3060 printf( " Dest [%d] = NULL\n", ( t->orient + 2 ) % 3 + 3 );
3063 printf( " Dest [%d] = x%lx (%.12g, %.12g)\n",
3064 ( t->orient + 2 ) % 3 + 3, (unsigned long) printpoint,
3065 printpoint[0], printpoint[1] );
3067 apex( *t, printpoint );
3068 if ( printpoint == (point) NULL ) {
3069 printf( " Apex [%d] = NULL\n", t->orient + 3 );
3072 printf( " Apex [%d] = x%lx (%.12g, %.12g)\n",
3073 t->orient + 3, (unsigned long) printpoint,
3074 printpoint[0], printpoint[1] );
3077 sdecode( t->tri[6], printsh );
3078 if ( printsh.sh != dummysh ) {
3079 printf( " [6] = x%lx %d\n", (unsigned long) printsh.sh,
3082 sdecode( t->tri[7], printsh );
3083 if ( printsh.sh != dummysh ) {
3084 printf( " [7] = x%lx %d\n", (unsigned long) printsh.sh,
3087 sdecode( t->tri[8], printsh );
3088 if ( printsh.sh != dummysh ) {
3089 printf( " [8] = x%lx %d\n", (unsigned long) printsh.sh,
3094 printf( " Area constraint: %.4g\n", areabound( *t ) );
3098 /*****************************************************************************/
3100 /* printshelle() Print out the details of a shell edge handle. */
3102 /* I originally wrote this procedure to simplify debugging; it can be */
3103 /* called directly from the debugger, and presents information about a */
3104 /* shell edge handle in digestible form. It's also used when the highest */
3105 /* level of verbosity (`-VVV') is specified. */
3107 /*****************************************************************************/
3109 void printshelle( s )
3112 struct edge printsh;
3113 struct triedge printtri;
3116 printf( "shell edge x%lx with orientation %d and mark %d:\n",
3117 (unsigned long) s->sh, s->shorient, mark( *s ) );
3118 sdecode( s->sh[0], printsh );
3119 if ( printsh.sh == dummysh ) {
3120 printf( " [0] = No shell\n" );
3123 printf( " [0] = x%lx %d\n", (unsigned long) printsh.sh,
3126 sdecode( s->sh[1], printsh );
3127 if ( printsh.sh == dummysh ) {
3128 printf( " [1] = No shell\n" );
3131 printf( " [1] = x%lx %d\n", (unsigned long) printsh.sh,
3134 sorg( *s, printpoint );
3135 if ( printpoint == (point) NULL ) {
3136 printf( " Origin[%d] = NULL\n", 2 + s->shorient );
3139 printf( " Origin[%d] = x%lx (%.12g, %.12g)\n",
3140 2 + s->shorient, (unsigned long) printpoint,
3141 printpoint[0], printpoint[1] );
3143 sdest( *s, printpoint );
3144 if ( printpoint == (point) NULL ) {
3145 printf( " Dest [%d] = NULL\n", 3 - s->shorient );
3148 printf( " Dest [%d] = x%lx (%.12g, %.12g)\n",
3149 3 - s->shorient, (unsigned long) printpoint,
3150 printpoint[0], printpoint[1] );
3152 decode( s->sh[4], printtri );
3153 if ( printtri.tri == dummytri ) {
3154 printf( " [4] = Outer space\n" );
3157 printf( " [4] = x%lx %d\n", (unsigned long) printtri.tri,
3160 decode( s->sh[5], printtri );
3161 if ( printtri.tri == dummytri ) {
3162 printf( " [5] = Outer space\n" );
3165 printf( " [5] = x%lx %d\n", (unsigned long) printtri.tri,
3172 /********* Debugging routines end here *********/
3174 /********* Memory management routines begin here *********/
3178 /*****************************************************************************/
3180 /* poolinit() Initialize a pool of memory for allocation of items. */
3182 /* This routine initializes the machinery for allocating items. A `pool' */
3183 /* is created whose records have size at least `bytecount'. Items will be */
3184 /* allocated in `itemcount'-item blocks. Each item is assumed to be a */
3185 /* collection of words, and either pointers or floating-point values are */
3186 /* assumed to be the "primary" word type. (The "primary" word type is used */
3187 /* to determine alignment of items.) If `alignment' isn't zero, all items */
3188 /* will be `alignment'-byte aligned in memory. `alignment' must be either */
3189 /* a multiple or a factor of the primary word size; powers of two are safe. */
3190 /* `alignment' is normally used to create a few unused bits at the bottom */
3191 /* of each item's pointer, in which information may be stored. */
3193 /* Don't change this routine unless you understand it. */
3195 /*****************************************************************************/
3197 void poolinit( pool, bytecount, itemcount, wtype, alignment )
3198 struct memorypool *pool;
3201 enum wordtype wtype;
3206 /* Initialize values in the pool. */
3207 pool->itemwordtype = wtype;
3208 wordsize = ( pool->itemwordtype == POINTER ) ? sizeof( VOID * ) : sizeof( REAL );
3209 /* Find the proper alignment, which must be at least as large as: */
3210 /* - The parameter `alignment'. */
3211 /* - The primary word type, to avoid unaligned accesses. */
3212 /* - sizeof(VOID *), so the stack of dead items can be maintained */
3213 /* without unaligned accesses. */
3214 if ( alignment > wordsize ) {
3215 pool->alignbytes = alignment;
3218 pool->alignbytes = wordsize;
3220 if ( sizeof( VOID * ) > pool->alignbytes ) {
3221 pool->alignbytes = sizeof( VOID * );
3223 pool->itemwords = ( ( bytecount + pool->alignbytes - 1 ) / pool->alignbytes )
3224 * ( pool->alignbytes / wordsize );
3225 pool->itembytes = pool->itemwords * wordsize;
3226 pool->itemsperblock = itemcount;
3228 /* Allocate a block of items. Space for `itemsperblock' items and one */
3229 /* pointer (to point to the next block) are allocated, as well as space */
3230 /* to ensure alignment of the items. */
3231 pool->firstblock = (VOID **) malloc( pool->itemsperblock * pool->itembytes
3232 + sizeof( VOID * ) + pool->alignbytes );
3233 if ( pool->firstblock == (VOID **) NULL ) {
3234 printf( "Error: Out of memory.\n" );
3237 /* Set the next block pointer to NULL. */
3238 *( pool->firstblock ) = (VOID *) NULL;
3239 poolrestart( pool );
3242 /*****************************************************************************/
3244 /* poolrestart() Deallocate all items in a pool. */
3246 /* The pool is returned to its starting state, except that no memory is */
3247 /* freed to the operating system. Rather, the previously allocated blocks */
3248 /* are ready to be reused. */
3250 /*****************************************************************************/
3252 void poolrestart( pool )
3253 struct memorypool *pool;
3255 unsigned long alignptr;
3260 /* Set the currently active block. */
3261 pool->nowblock = pool->firstblock;
3262 /* Find the first item in the pool. Increment by the size of (VOID *). */
3263 alignptr = (unsigned long) ( pool->nowblock + 1 );
3264 /* Align the item on an `alignbytes'-byte boundary. */
3265 pool->nextitem = (VOID *)
3266 ( alignptr + (unsigned long) pool->alignbytes
3267 - ( alignptr % (unsigned long) pool->alignbytes ) );
3268 /* There are lots of unallocated items left in this block. */
3269 pool->unallocateditems = pool->itemsperblock;
3270 /* The stack of deallocated items is empty. */
3271 pool->deaditemstack = (VOID *) NULL;
3274 /*****************************************************************************/
3276 /* pooldeinit() Free to the operating system all memory taken by a pool. */
3278 /*****************************************************************************/
3280 void pooldeinit( pool )
3281 struct memorypool *pool;
3283 while ( pool->firstblock != (VOID **) NULL ) {
3284 pool->nowblock = (VOID **) *( pool->firstblock );
3285 free( pool->firstblock );
3286 pool->firstblock = pool->nowblock;
3290 /*****************************************************************************/
3292 /* poolalloc() Allocate space for an item. */
3294 /*****************************************************************************/
3296 VOID *poolalloc( pool )
3297 struct memorypool *pool;
3301 unsigned long alignptr;
3303 /* First check the linked list of dead items. If the list is not */
3304 /* empty, allocate an item from the list rather than a fresh one. */
3305 if ( pool->deaditemstack != (VOID *) NULL ) {
3306 newitem = pool->deaditemstack; /* Take first item in list. */
3307 pool->deaditemstack = *(VOID **) pool->deaditemstack;
3310 /* Check if there are any free items left in the current block. */
3311 if ( pool->unallocateditems == 0 ) {
3312 /* Check if another block must be allocated. */
3313 if ( *( pool->nowblock ) == (VOID *) NULL ) {
3314 /* Allocate a new block of items, pointed to by the previous block. */
3315 newblock = (VOID **) malloc( pool->itemsperblock * pool->itembytes
3316 + sizeof( VOID * ) + pool->alignbytes );
3317 if ( newblock == (VOID **) NULL ) {
3318 printf( "Error: Out of memory.\n" );
3321 *( pool->nowblock ) = (VOID *) newblock;
3322 /* The next block pointer is NULL. */
3323 *newblock = (VOID *) NULL;
3325 /* Move to the new block. */
3326 pool->nowblock = (VOID **) *( pool->nowblock );
3327 /* Find the first item in the block. */
3328 /* Increment by the size of (VOID *). */
3329 alignptr = (unsigned long) ( pool->nowblock + 1 );
3330 /* Align the item on an `alignbytes'-byte boundary. */
3331 pool->nextitem = (VOID *)
3332 ( alignptr + (unsigned long) pool->alignbytes
3333 - ( alignptr % (unsigned long) pool->alignbytes ) );
3334 /* There are lots of unallocated items left in this block. */
3335 pool->unallocateditems = pool->itemsperblock;
3337 /* Allocate a new item. */
3338 newitem = pool->nextitem;
3339 /* Advance `nextitem' pointer to next free item in block. */
3340 if ( pool->itemwordtype == POINTER ) {
3341 pool->nextitem = (VOID *) ( (VOID **) pool->nextitem + pool->itemwords );
3344 pool->nextitem = (VOID *) ( (REAL *) pool->nextitem + pool->itemwords );
3346 pool->unallocateditems--;
3353 /*****************************************************************************/
3355 /* pooldealloc() Deallocate space for an item. */
3357 /* The deallocated space is stored in a queue for later reuse. */
3359 /*****************************************************************************/
3361 void pooldealloc( pool, dyingitem )
3362 struct memorypool *pool;
3365 /* Push freshly killed item onto stack. */
3366 *( (VOID **) dyingitem ) = pool->deaditemstack;
3367 pool->deaditemstack = dyingitem;
3371 /*****************************************************************************/
3373 /* traversalinit() Prepare to traverse the entire list of items. */
3375 /* This routine is used in conjunction with traverse(). */
3377 /*****************************************************************************/
3379 void traversalinit( pool )
3380 struct memorypool *pool;
3382 unsigned long alignptr;
3384 /* Begin the traversal in the first block. */
3385 pool->pathblock = pool->firstblock;
3386 /* Find the first item in the block. Increment by the size of (VOID *). */
3387 alignptr = (unsigned long) ( pool->pathblock + 1 );
3388 /* Align with item on an `alignbytes'-byte boundary. */
3389 pool->pathitem = (VOID *)
3390 ( alignptr + (unsigned long) pool->alignbytes
3391 - ( alignptr % (unsigned long) pool->alignbytes ) );
3392 /* Set the number of items left in the current block. */
3393 pool->pathitemsleft = pool->itemsperblock;
3396 /*****************************************************************************/
3398 /* traverse() Find the next item in the list. */
3400 /* This routine is used in conjunction with traversalinit(). Be forewarned */
3401 /* that this routine successively returns all items in the list, including */
3402 /* deallocated ones on the deaditemqueue. It's up to you to figure out */
3403 /* which ones are actually dead. Why? I don't want to allocate extra */
3404 /* space just to demarcate dead items. It can usually be done more */
3405 /* space-efficiently by a routine that knows something about the structure */
3408 /*****************************************************************************/
3410 VOID *traverse( pool )
3411 struct memorypool *pool;
3414 unsigned long alignptr;
3416 /* Stop upon exhausting the list of items. */
3417 if ( pool->pathitem == pool->nextitem ) {
3418 return (VOID *) NULL;
3420 /* Check whether any untraversed items remain in the current block. */
3421 if ( pool->pathitemsleft == 0 ) {
3422 /* Find the next block. */
3423 pool->pathblock = (VOID **) *( pool->pathblock );
3424 /* Find the first item in the block. Increment by the size of (VOID *). */
3425 alignptr = (unsigned long) ( pool->pathblock + 1 );
3426 /* Align with item on an `alignbytes'-byte boundary. */
3427 pool->pathitem = (VOID *)
3428 ( alignptr + (unsigned long) pool->alignbytes
3429 - ( alignptr % (unsigned long) pool->alignbytes ) );
3430 /* Set the number of items left in the current block. */
3431 pool->pathitemsleft = pool->itemsperblock;
3433 newitem = pool->pathitem;
3434 /* Find the next item in the block. */
3435 if ( pool->itemwordtype == POINTER ) {
3436 pool->pathitem = (VOID *) ( (VOID **) pool->pathitem + pool->itemwords );
3439 pool->pathitem = (VOID *) ( (REAL *) pool->pathitem + pool->itemwords );
3441 pool->pathitemsleft--;
3445 /*****************************************************************************/
3447 /* dummyinit() Initialize the triangle that fills "outer space" and the */
3448 /* omnipresent shell edge. */
3450 /* The triangle that fills "outer space", called `dummytri', is pointed to */
3451 /* by every triangle and shell edge on a boundary (be it outer or inner) of */
3452 /* the triangulation. Also, `dummytri' points to one of the triangles on */
3453 /* the convex hull (until the holes and concavities are carved), making it */
3454 /* possible to find a starting triangle for point location. */
3456 /* The omnipresent shell edge, `dummysh', is pointed to by every triangle */
3457 /* or shell edge that doesn't have a full complement of real shell edges */
3460 /*****************************************************************************/
3462 void dummyinit( trianglewords, shellewords )
3466 unsigned long alignptr;
3468 /* `triwords' and `shwords' are used by the mesh manipulation primitives */
3469 /* to extract orientations of triangles and shell edges from pointers. */
3470 triwords = trianglewords; /* Initialize `triwords' once and for all. */
3471 shwords = shellewords; /* Initialize `shwords' once and for all. */
3473 /* Set up `dummytri', the `triangle' that occupies "outer space". */
3474 dummytribase = (triangle *) malloc( triwords * sizeof( triangle )
3475 + triangles.alignbytes );
3476 if ( dummytribase == (triangle *) NULL ) {
3477 printf( "Error: Out of memory.\n" );
3480 /* Align `dummytri' on a `triangles.alignbytes'-byte boundary. */
3481 alignptr = (unsigned long) dummytribase;
3482 dummytri = (triangle *)
3483 ( alignptr + (unsigned long) triangles.alignbytes
3484 - ( alignptr % (unsigned long) triangles.alignbytes ) );
3485 /* Initialize the three adjoining triangles to be "outer space". These */
3486 /* will eventually be changed by various bonding operations, but their */
3487 /* values don't really matter, as long as they can legally be */
3489 dummytri[0] = (triangle) dummytri;
3490 dummytri[1] = (triangle) dummytri;
3491 dummytri[2] = (triangle) dummytri;
3492 /* Three NULL vertex points. */
3493 dummytri[3] = (triangle) NULL;
3494 dummytri[4] = (triangle) NULL;
3495 dummytri[5] = (triangle) NULL;
3498 /* Set up `dummysh', the omnipresent "shell edge" pointed to by any */
3499 /* triangle side or shell edge end that isn't attached to a real shell */
3501 dummyshbase = (shelle *) malloc( shwords * sizeof( shelle )
3502 + shelles.alignbytes );
3503 if ( dummyshbase == (shelle *) NULL ) {
3504 printf( "Error: Out of memory.\n" );
3507 /* Align `dummysh' on a `shelles.alignbytes'-byte boundary. */
3508 alignptr = (unsigned long) dummyshbase;
3509 dummysh = (shelle *)
3510 ( alignptr + (unsigned long) shelles.alignbytes
3511 - ( alignptr % (unsigned long) shelles.alignbytes ) );
3512 /* Initialize the two adjoining shell edges to be the omnipresent shell */
3513 /* edge. These will eventually be changed by various bonding */
3514 /* operations, but their values don't really matter, as long as they */
3515 /* can legally be dereferenced. */
3516 dummysh[0] = (shelle) dummysh;
3517 dummysh[1] = (shelle) dummysh;
3518 /* Two NULL vertex points. */
3519 dummysh[2] = (shelle) NULL;
3520 dummysh[3] = (shelle) NULL;
3521 /* Initialize the two adjoining triangles to be "outer space". */
3522 dummysh[4] = (shelle) dummytri;
3523 dummysh[5] = (shelle) dummytri;
3524 /* Set the boundary marker to zero. */
3525 *(int *) ( dummysh + 6 ) = 0;
3527 /* Initialize the three adjoining shell edges of `dummytri' to be */
3528 /* the omnipresent shell edge. */
3529 dummytri[6] = (triangle) dummysh;
3530 dummytri[7] = (triangle) dummysh;
3531 dummytri[8] = (triangle) dummysh;
3535 /*****************************************************************************/
3537 /* initializepointpool() Calculate the size of the point data structure */
3538 /* and initialize its memory pool. */
3540 /* This routine also computes the `pointmarkindex' and `point2triindex' */
3541 /* indices used to find values within each point. */
3543 /*****************************************************************************/
3545 void initializepointpool(){
3548 /* The index within each point at which the boundary marker is found. */
3549 /* Ensure the point marker is aligned to a sizeof(int)-byte address. */
3550 pointmarkindex = ( ( mesh_dim + nextras ) * sizeof( REAL ) + sizeof( int ) - 1 )
3552 pointsize = ( pointmarkindex + 1 ) * sizeof( int );
3554 /* The index within each point at which a triangle pointer is found. */
3555 /* Ensure the pointer is aligned to a sizeof(triangle)-byte address. */
3556 point2triindex = ( pointsize + sizeof( triangle ) - 1 ) / sizeof( triangle );
3557 pointsize = ( point2triindex + 1 ) * sizeof( triangle );
3559 /* Initialize the pool of points. */
3560 poolinit( &points, pointsize, POINTPERBLOCK,
3561 ( sizeof( REAL ) >= sizeof( triangle ) ) ? FLOATINGPOINT : POINTER, 0 );
3564 /*****************************************************************************/
3566 /* initializetrisegpools() Calculate the sizes of the triangle and shell */
3567 /* edge data structures and initialize their */
3570 /* This routine also computes the `highorderindex', `elemattribindex', and */
3571 /* `areaboundindex' indices used to find values within each triangle. */
3573 /*****************************************************************************/
3575 void initializetrisegpools(){
3578 /* The index within each triangle at which the extra nodes (above three) */
3579 /* associated with high order elements are found. There are three */
3580 /* pointers to other triangles, three pointers to corners, and possibly */
3581 /* three pointers to shell edges before the extra nodes. */
3582 highorderindex = 6 + ( useshelles * 3 );
3583 /* The number of bytes occupied by a triangle. */
3584 trisize = ( ( order + 1 ) * ( order + 2 ) / 2 + ( highorderindex - 3 ) ) *
3586 /* The index within each triangle at which its attributes are found, */
3587 /* where the index is measured in REALs. */
3588 elemattribindex = ( trisize + sizeof( REAL ) - 1 ) / sizeof( REAL );
3589 /* The index within each triangle at which the maximum area constraint */
3590 /* is found, where the index is measured in REALs. Note that if the */
3591 /* `regionattrib' flag is set, an additional attribute will be added. */
3592 areaboundindex = elemattribindex + eextras + regionattrib;
3593 /* If triangle attributes or an area bound are needed, increase the number */
3594 /* of bytes occupied by a triangle. */
3596 trisize = ( areaboundindex + 1 ) * sizeof( REAL );
3598 else if ( eextras + regionattrib > 0 ) {
3599 trisize = areaboundindex * sizeof( REAL );
3601 /* If a Voronoi diagram or triangle neighbor graph is requested, make */
3602 /* sure there's room to store an integer index in each triangle. This */
3603 /* integer index can occupy the same space as the shell edges or */
3604 /* attributes or area constraint or extra nodes. */
3605 if ( ( voronoi || neighbors ) &&
3606 ( trisize < 6 * sizeof( triangle ) + sizeof( int ) ) ) {
3607 trisize = 6 * sizeof( triangle ) + sizeof( int );
3609 /* Having determined the memory size of a triangle, initialize the pool. */
3610 poolinit( &triangles, trisize, TRIPERBLOCK, POINTER, 4 );
3613 /* Initialize the pool of shell edges. */
3614 poolinit( &shelles, 6 * sizeof( triangle ) + sizeof( int ), SHELLEPERBLOCK,
3617 /* Initialize the "outer space" triangle and omnipresent shell edge. */
3618 dummyinit( triangles.itemwords, shelles.itemwords );
3621 /* Initialize the "outer space" triangle. */
3622 dummyinit( triangles.itemwords, 0 );
3626 /*****************************************************************************/
3628 /* triangledealloc() Deallocate space for a triangle, marking it dead. */
3630 /*****************************************************************************/
3632 void triangledealloc( dyingtriangle )
3633 triangle * dyingtriangle;
3635 /* Set triangle's vertices to NULL. This makes it possible to */
3636 /* detect dead triangles when traversing the list of all triangles. */
3637 dyingtriangle[3] = (triangle) NULL;
3638 dyingtriangle[4] = (triangle) NULL;
3639 dyingtriangle[5] = (triangle) NULL;
3640 pooldealloc( &triangles, (VOID *) dyingtriangle );
3643 /*****************************************************************************/
3645 /* triangletraverse() Traverse the triangles, skipping dead ones. */
3647 /*****************************************************************************/
3649 triangle *triangletraverse(){
3650 triangle *newtriangle;
3653 newtriangle = (triangle *) traverse( &triangles );
3654 if ( newtriangle == (triangle *) NULL ) {
3655 return (triangle *) NULL;
3657 } while ( newtriangle[3] == (triangle) NULL ); /* Skip dead ones. */
3661 /*****************************************************************************/
3663 /* shelledealloc() Deallocate space for a shell edge, marking it dead. */
3665 /*****************************************************************************/
3667 void shelledealloc( dyingshelle )
3668 shelle * dyingshelle;
3670 /* Set shell edge's vertices to NULL. This makes it possible to */
3671 /* detect dead shells when traversing the list of all shells. */
3672 dyingshelle[2] = (shelle) NULL;
3673 dyingshelle[3] = (shelle) NULL;
3674 pooldealloc( &shelles, (VOID *) dyingshelle );
3677 /*****************************************************************************/
3679 /* shelletraverse() Traverse the shell edges, skipping dead ones. */
3681 /*****************************************************************************/
3683 shelle *shelletraverse(){
3687 newshelle = (shelle *) traverse( &shelles );
3688 if ( newshelle == (shelle *) NULL ) {
3689 return (shelle *) NULL;
3691 } while ( newshelle[2] == (shelle) NULL ); /* Skip dead ones. */
3695 /*****************************************************************************/
3697 /* pointdealloc() Deallocate space for a point, marking it dead. */
3699 /*****************************************************************************/
3701 void pointdealloc( dyingpoint )
3704 /* Mark the point as dead. This makes it possible to detect dead points */
3705 /* when traversing the list of all points. */
3706 setpointmark( dyingpoint, DEADPOINT );
3707 pooldealloc( &points, (VOID *) dyingpoint );
3710 /*****************************************************************************/
3712 /* pointtraverse() Traverse the points, skipping dead ones. */
3714 /*****************************************************************************/
3716 point pointtraverse(){
3720 newpoint = (point) traverse( &points );
3721 if ( newpoint == (point) NULL ) {
3722 return (point) NULL;
3724 } while ( pointmark( newpoint ) == DEADPOINT ); /* Skip dead ones. */
3728 /*****************************************************************************/
3730 /* badsegmentdealloc() Deallocate space for a bad segment, marking it */
3733 /*****************************************************************************/
3737 void badsegmentdealloc( dyingseg )
3738 struct edge *dyingseg;
3740 /* Set segment's orientation to -1. This makes it possible to */
3741 /* detect dead segments when traversing the list of all segments. */
3742 dyingseg->shorient = -1;
3743 pooldealloc( &badsegments, (VOID *) dyingseg );
3746 #endif /* not CDT_ONLY */
3748 /*****************************************************************************/
3750 /* badsegmenttraverse() Traverse the bad segments, skipping dead ones. */
3752 /*****************************************************************************/
3756 struct edge *badsegmenttraverse(){
3757 struct edge *newseg;
3760 newseg = (struct edge *) traverse( &badsegments );
3761 if ( newseg == (struct edge *) NULL ) {
3762 return (struct edge *) NULL;
3764 } while ( newseg->shorient == -1 ); /* Skip dead ones. */
3768 #endif /* not CDT_ONLY */
3770 /*****************************************************************************/
3772 /* getpoint() Get a specific point, by number, from the list. */
3774 /* The first point is number 'firstnumber'. */
3776 /* Note that this takes O(n) time (with a small constant, if POINTPERBLOCK */
3777 /* is large). I don't care to take the trouble to make it work in constant */
3780 /*****************************************************************************/
3782 point getpoint( number )
3787 unsigned long alignptr;
3790 getblock = points.firstblock;
3791 current = firstnumber;
3792 /* Find the right block. */
3793 while ( current + points.itemsperblock <= number ) {
3794 getblock = (VOID **) *getblock;
3795 current += points.itemsperblock;
3797 /* Now find the right point. */
3798 alignptr = (unsigned long) ( getblock + 1 );
3799 foundpoint = (point) ( alignptr + (unsigned long) points.alignbytes
3800 - ( alignptr % (unsigned long) points.alignbytes ) );
3801 while ( current < number ) {
3802 foundpoint += points.itemwords;
3808 /*****************************************************************************/
3810 /* triangledeinit() Free all remaining allocated memory. */
3812 /*****************************************************************************/
3814 void triangledeinit(){
3815 pooldeinit( &triangles );
3816 free( dummytribase );
3818 pooldeinit( &shelles );
3819 free( dummyshbase );
3821 pooldeinit( &points );
3824 pooldeinit( &badsegments );
3825 if ( ( minangle > 0.0 ) || vararea || fixedarea ) {
3826 pooldeinit( &badtriangles );
3829 #endif /* not CDT_ONLY */
3834 /********* Memory management routines end here *********/
3836 /********* Constructors begin here *********/
3840 /*****************************************************************************/
3842 /* maketriangle() Create a new triangle with orientation zero. */
3844 /*****************************************************************************/
3846 void maketriangle( newtriedge )
3847 struct triedge *newtriedge;
3851 newtriedge->tri = (triangle *) poolalloc( &triangles );
3852 /* Initialize the three adjoining triangles to be "outer space". */
3853 newtriedge->tri[0] = (triangle) dummytri;
3854 newtriedge->tri[1] = (triangle) dummytri;
3855 newtriedge->tri[2] = (triangle) dummytri;
3856 /* Three NULL vertex points. */
3857 newtriedge->tri[3] = (triangle) NULL;
3858 newtriedge->tri[4] = (triangle) NULL;
3859 newtriedge->tri[5] = (triangle) NULL;
3860 /* Initialize the three adjoining shell edges to be the omnipresent */
3863 newtriedge->tri[6] = (triangle) dummysh;
3864 newtriedge->tri[7] = (triangle) dummysh;
3865 newtriedge->tri[8] = (triangle) dummysh;
3867 for ( i = 0; i < eextras; i++ ) {
3868 setelemattribute( *newtriedge, i, 0.0 );
3871 setareabound( *newtriedge, -1.0 );
3874 newtriedge->orient = 0;
3877 /*****************************************************************************/
3879 /* makeshelle() Create a new shell edge with orientation zero. */
3881 /*****************************************************************************/
3883 void makeshelle( newedge )
3884 struct edge *newedge;
3886 newedge->sh = (shelle *) poolalloc( &shelles );
3887 /* Initialize the two adjoining shell edges to be the omnipresent */
3889 newedge->sh[0] = (shelle) dummysh;
3890 newedge->sh[1] = (shelle) dummysh;
3891 /* Two NULL vertex points. */
3892 newedge->sh[2] = (shelle) NULL;
3893 newedge->sh[3] = (shelle) NULL;
3894 /* Initialize the two adjoining triangles to be "outer space". */
3895 newedge->sh[4] = (shelle) dummytri;
3896 newedge->sh[5] = (shelle) dummytri;
3897 /* Set the boundary marker to zero. */
3898 setmark( *newedge, 0 );
3900 newedge->shorient = 0;
3905 /********* Constructors end here *********/
3907 /********* Determinant evaluation routines begin here *********/
3911 /* The adaptive exact arithmetic geometric predicates implemented herein are */
3912 /* described in detail in my Technical Report CMU-CS-96-140. The complete */
3913 /* reference is given in the header. */
3915 /* Which of the following two methods of finding the absolute values is */
3916 /* fastest is compiler-dependent. A few compilers can inline and optimize */
3917 /* the fabs() call; but most will incur the overhead of a function call, */
3918 /* which is disastrously slow. A faster way on IEEE machines might be to */
3919 /* mask the appropriate bit, but that's difficult to do in C. */
3921 #define Absolute( a ) ( ( a ) >= 0.0 ? ( a ) : -( a ) )
3922 /* #define Absolute(a) fabs(a) */
3924 /* Many of the operations are broken up into two pieces, a main part that */
3925 /* performs an approximate operation, and a "tail" that computes the */
3926 /* roundoff error of that operation. */
3928 /* The operations Fast_Two_Sum(), Fast_Two_Diff(), Two_Sum(), Two_Diff(), */
3929 /* Split(), and Two_Product() are all implemented as described in the */
3930 /* reference. Each of these macros requires certain variables to be */
3931 /* defined in the calling routine. The variables `bvirt', `c', `abig', */
3932 /* `_i', `_j', `_k', `_l', `_m', and `_n' are declared `INEXACT' because */
3933 /* they store the result of an operation that may incur roundoff error. */
3934 /* The input parameter `x' (or the highest numbered `x_' parameter) must */
3935 /* also be declared `INEXACT'. */
3937 #define Fast_Two_Sum_Tail( a, b, x, y ) \
3941 #define Fast_Two_Sum( a, b, x, y ) \
3942 x = (REAL) ( a + b ); \
3943 Fast_Two_Sum_Tail( a, b, x, y )
3945 #define Two_Sum_Tail( a, b, x, y ) \
3946 bvirt = (REAL) ( x - a ); \
3947 avirt = x - bvirt; \
3948 bround = b - bvirt; \
3949 around = a - avirt; \
3952 #define Two_Sum( a, b, x, y ) \
3953 x = (REAL) ( a + b ); \
3954 Two_Sum_Tail( a, b, x, y )
3956 #define Two_Diff_Tail( a, b, x, y ) \
3957 bvirt = (REAL) ( a - x ); \
3958 avirt = x + bvirt; \
3959 bround = bvirt - b; \
3960 around = a - avirt; \
3963 #define Two_Diff( a, b, x, y ) \
3964 x = (REAL) ( a - b ); \
3965 Two_Diff_Tail( a, b, x, y )
3967 #define Split( a, ahi, alo ) \
3968 c = (REAL) ( splitter * a ); \
3969 abig = (REAL) ( c - a ); \
3970 ahi = (REAL)( c - abig ); \
3971 alo = (REAL)( a - ahi )
3973 #define Two_Product_Tail( a, b, x, y ) \
3974 Split( a, ahi, alo ); \
3975 Split( b, bhi, blo ); \
3976 err1 = x - ( ahi * bhi ); \
3977 err2 = err1 - ( alo * bhi ); \
3978 err3 = err2 - ( ahi * blo ); \
3979 y = ( alo * blo ) - err3
3981 #define Two_Product( a, b, x, y ) \
3982 x = (REAL) ( a * b ); \
3983 Two_Product_Tail( a, b, x, y )
3985 /* Two_Product_Presplit() is Two_Product() where one of the inputs has */
3986 /* already been split. Avoids redundant splitting. */
3988 #define Two_Product_Presplit( a, b, bhi, blo, x, y ) \
3989 x = (REAL) ( a * b ); \
3990 Split( a, ahi, alo ); \
3991 err1 = x - ( ahi * bhi ); \
3992 err2 = err1 - ( alo * bhi ); \
3993 err3 = err2 - ( ahi * blo ); \
3994 y = ( alo * blo ) - err3
3996 /* Square() can be done more quickly than Two_Product(). */
3998 #define Square_Tail( a, x, y ) \
3999 Split( a, ahi, alo ); \
4000 err1 = x - ( ahi * ahi ); \
4001 err3 = err1 - ( ( ahi + ahi ) * alo ); \
4002 y = ( alo * alo ) - err3
4004 #define Square( a, x, y ) \
4005 x = (REAL) ( a * a ); \
4006 Square_Tail( a, x, y )
4008 /* Macros for summing expansions of various fixed lengths. These are all */
4009 /* unrolled versions of Expansion_Sum(). */
4011 #define Two_One_Sum( a1, a0, b, x2, x1, x0 ) \
4012 Two_Sum( a0, b, _i, x0 ); \
4013 Two_Sum( a1, _i, x2, x1 )
4015 #define Two_One_Diff( a1, a0, b, x2, x1, x0 ) \
4016 Two_Diff( a0, b, _i, x0 ); \
4017 Two_Sum( a1, _i, x2, x1 )
4019 #define Two_Two_Sum( a1, a0, b1, b0, x3, x2, x1, x0 ) \
4020 Two_One_Sum( a1, a0, b0, _j, _0, x0 ); \
4021 Two_One_Sum( _j, _0, b1, x3, x2, x1 )
4023 #define Two_Two_Diff( a1, a0, b1, b0, x3, x2, x1, x0 ) \
4024 Two_One_Diff( a1, a0, b0, _j, _0, x0 ); \
4025 Two_One_Diff( _j, _0, b1, x3, x2, x1 )
4027 /*****************************************************************************/
4029 /* exactinit() Initialize the variables used for exact arithmetic. */
4031 /* `epsilon' is the largest power of two such that 1.0 + epsilon = 1.0 in */
4032 /* floating-point arithmetic. `epsilon' bounds the relative roundoff */
4033 /* error. It is used for floating-point error analysis. */
4035 /* `splitter' is used to split floating-point numbers into two half- */
4036 /* length significands for exact multiplication. */
4038 /* I imagine that a highly optimizing compiler might be too smart for its */
4039 /* own good, and somehow cause this routine to fail, if it pretends that */
4040 /* floating-point arithmetic is too much like real arithmetic. */
4042 /* Don't change this routine unless you fully understand it. */
4044 /*****************************************************************************/
4048 REAL check, lastcheck;
4056 /* Repeatedly divide `epsilon' by two until it is too small to add to */
4057 /* one without causing roundoff. (Also check if the sum is equal to */
4058 /* the previous sum, for machines that round up instead of using exact */
4059 /* rounding. Not that these routines will work on such machines anyway. */
4063 if ( every_other ) {
4066 every_other = !every_other;
4067 check = (REAL)( 1.0 + epsilon );
4068 } while ( ( check != 1.0 ) && ( check != lastcheck ) );
4070 if ( verbose > 1 ) {
4071 printf( "Floating point roundoff is of magnitude %.17g\n", epsilon );
4072 printf( "Floating point splitter is %.17g\n", splitter );
4074 /* Error bounds for orientation and incircle tests. */
4075 resulterrbound = (REAL)( ( 3.0 + 8.0 * epsilon ) * epsilon );
4076 ccwerrboundA = (REAL)( ( 3.0 + 16.0 * epsilon ) * epsilon );
4077 ccwerrboundB = (REAL)( ( 2.0 + 12.0 * epsilon ) * epsilon );
4078 ccwerrboundC = (REAL)( ( 9.0 + 64.0 * epsilon ) * epsilon * epsilon );
4079 iccerrboundA = (REAL)( ( 10.0 + 96.0 * epsilon ) * epsilon );
4080 iccerrboundB = (REAL)( ( 4.0 + 48.0 * epsilon ) * epsilon );
4081 iccerrboundC = (REAL)( ( 44.0 + 576.0 * epsilon ) * epsilon * epsilon );
4084 /*****************************************************************************/
4086 /* fast_expansion_sum_zeroelim() Sum two expansions, eliminating zero */
4087 /* components from the output expansion. */
4089 /* Sets h = e + f. See my Robust Predicates paper for details. */
4091 /* If round-to-even is used (as with IEEE 754), maintains the strongly */
4092 /* nonoverlapping property. (That is, if e is strongly nonoverlapping, h */
4093 /* will be also.) Does NOT maintain the nonoverlapping or nonadjacent */
4096 /*****************************************************************************/
4098 int fast_expansion_sum_zeroelim( elen, e, flen, f, h ) /* h cannot be e or f. */
4109 REAL avirt, bround, around;
4110 int eindex, findex, hindex;
4115 eindex = findex = 0;
4116 if ( ( fnow > enow ) == ( fnow > -enow ) ) {
4125 if ( ( eindex < elen ) && ( findex < flen ) ) {
4126 if ( ( fnow > enow ) == ( fnow > -enow ) ) {
4127 Fast_Two_Sum( enow, Q, Qnew, hh );
4131 Fast_Two_Sum( fnow, Q, Qnew, hh );
4138 while ( ( eindex < elen ) && ( findex < flen ) ) {
4139 if ( ( fnow > enow ) == ( fnow > -enow ) ) {
4140 Two_Sum( Q, enow, Qnew, hh );
4144 Two_Sum( Q, fnow, Qnew, hh );
4153 while ( eindex < elen ) {
4154 Two_Sum( Q, enow, Qnew, hh );
4161 while ( findex < flen ) {
4162 Two_Sum( Q, fnow, Qnew, hh );
4169 if ( ( Q != 0.0 ) || ( hindex == 0 ) ) {
4175 /*****************************************************************************/
4177 /* scale_expansion_zeroelim() Multiply an expansion by a scalar, */
4178 /* eliminating zero components from the */
4179 /* output expansion. */
4181 /* Sets h = be. See my Robust Predicates paper for details. */
4183 /* Maintains the nonoverlapping property. If round-to-even is used (as */
4184 /* with IEEE 754), maintains the strongly nonoverlapping and nonadjacent */
4185 /* properties as well. (That is, if e has one of these properties, so */
4188 /*****************************************************************************/
4190 int scale_expansion_zeroelim( elen, e, b, h ) /* e and h cannot be the same. */
4196 INEXACT REAL Q, sum;
4198 INEXACT REAL product1;
4203 REAL avirt, bround, around;
4206 REAL ahi, alo, bhi, blo;
4207 REAL err1, err2, err3;
4209 Split( b, bhi, blo );
4210 Two_Product_Presplit( e[0], b, bhi, blo, Q, hh );
4215 for ( eindex = 1; eindex < elen; eindex++ ) {
4217 Two_Product_Presplit( enow, b, bhi, blo, product1, product0 );
4218 Two_Sum( Q, product0, sum, hh );
4222 Fast_Two_Sum( product1, sum, Q, hh );
4227 if ( ( Q != 0.0 ) || ( hindex == 0 ) ) {
4233 /*****************************************************************************/
4235 /* estimate() Produce a one-word estimate of an expansion's value. */
4237 /* See my Robust Predicates paper for details. */
4239 /*****************************************************************************/
4241 REAL estimate( elen, e )
4249 for ( eindex = 1; eindex < elen; eindex++ ) {
4255 /*****************************************************************************/
4257 /* counterclockwise() Return a positive value if the points pa, pb, and */
4258 /* pc occur in counterclockwise order; a negative */
4259 /* value if they occur in clockwise order; and zero */
4260 /* if they are collinear. The result is also a rough */
4261 /* approximation of twice the signed area of the */
4262 /* triangle defined by the three points. */
4264 /* Uses exact arithmetic if necessary to ensure a correct answer. The */
4265 /* result returned is the determinant of a matrix. This determinant is */
4266 /* computed adaptively, in the sense that exact arithmetic is used only to */
4267 /* the degree it is needed to ensure that the returned value has the */
4268 /* correct sign. Hence, this function is usually quite fast, but will run */
4269 /* more slowly when the input points are collinear or nearly so. */
4271 /* See my Robust Predicates paper for details. */
4273 /*****************************************************************************/
4275 REAL counterclockwiseadapt( pa, pb, pc, detsum )
4281 INEXACT REAL acx, acy, bcx, bcy;
4282 REAL acxtail, acytail, bcxtail, bcytail;
4283 INEXACT REAL detleft, detright;
4284 REAL detlefttail, detrighttail;
4286 REAL B[4], C1[8], C2[12], D[16];
4288 int C1length, C2length, Dlength;
4291 INEXACT REAL s1, t1;
4295 REAL avirt, bround, around;
4298 REAL ahi, alo, bhi, blo;
4299 REAL err1, err2, err3;
4300 INEXACT REAL _i, _j;
4303 acx = (REAL) ( pa[0] - pc[0] );
4304 bcx = (REAL) ( pb[0] - pc[0] );
4305 acy = (REAL) ( pa[1] - pc[1] );
4306 bcy = (REAL) ( pb[1] - pc[1] );
4308 Two_Product( acx, bcy, detleft, detlefttail );
4309 Two_Product( acy, bcx, detright, detrighttail );
4311 Two_Two_Diff( detleft, detlefttail, detright, detrighttail,
4312 B3, B[2], B[1], B[0] );
4315 det = estimate( 4, B );
4316 errbound = (REAL)( ccwerrboundB * detsum );
4317 if ( ( det >= errbound ) || ( -det >= errbound ) ) {
4321 Two_Diff_Tail( pa[0], pc[0], acx, acxtail );
4322 Two_Diff_Tail( pb[0], pc[0], bcx, bcxtail );
4323 Two_Diff_Tail( pa[1], pc[1], acy, acytail );
4324 Two_Diff_Tail( pb[1], pc[1], bcy, bcytail );
4326 if ( ( acxtail == 0.0 ) && ( acytail == 0.0 )
4327 && ( bcxtail == 0.0 ) && ( bcytail == 0.0 ) ) {
4331 errbound = (REAL)( ccwerrboundC * detsum + resulterrbound * Absolute( det ) );
4332 det += ( acx * bcytail + bcy * acxtail )
4333 - ( acy * bcxtail + bcx * acytail );
4334 if ( ( det >= errbound ) || ( -det >= errbound ) ) {
4338 Two_Product( acxtail, bcy, s1, s0 );
4339 Two_Product( acytail, bcx, t1, t0 );
4340 Two_Two_Diff( s1, s0, t1, t0, u3, u[2], u[1], u[0] );
4342 C1length = fast_expansion_sum_zeroelim( 4, B, 4, u, C1 );
4344 Two_Product( acx, bcytail, s1, s0 );
4345 Two_Product( acy, bcxtail, t1, t0 );
4346 Two_Two_Diff( s1, s0, t1, t0, u3, u[2], u[1], u[0] );
4348 C2length = fast_expansion_sum_zeroelim( C1length, C1, 4, u, C2 );
4350 Two_Product( acxtail, bcytail, s1, s0 );
4351 Two_Product( acytail, bcxtail, t1, t0 );
4352 Two_Two_Diff( s1, s0, t1, t0, u3, u[2], u[1], u[0] );
4354 Dlength = fast_expansion_sum_zeroelim( C2length, C2, 4, u, D );
4356 return( D[Dlength - 1] );
4359 REAL counterclockwise( pa, pb, pc )
4364 REAL detleft, detright, det;
4365 REAL detsum, errbound;
4367 counterclockcount++;
4369 detleft = ( pa[0] - pc[0] ) * ( pb[1] - pc[1] );
4370 detright = ( pa[1] - pc[1] ) * ( pb[0] - pc[0] );
4371 det = detleft - detright;
4377 if ( detleft > 0.0 ) {
4378 if ( detright <= 0.0 ) {
4382 detsum = detleft + detright;
4385 else if ( detleft < 0.0 ) {
4386 if ( detright >= 0.0 ) {
4390 detsum = -detleft - detright;
4397 errbound = ccwerrboundA * detsum;
4398 if ( ( det >= errbound ) || ( -det >= errbound ) ) {
4402 return counterclockwiseadapt( pa, pb, pc, detsum );
4405 /*****************************************************************************/
4407 /* incircle() Return a positive value if the point pd lies inside the */
4408 /* circle passing through pa, pb, and pc; a negative value if */
4409 /* it lies outside; and zero if the four points are cocircular.*/
4410 /* The points pa, pb, and pc must be in counterclockwise */
4411 /* order, or the sign of the result will be reversed. */
4413 /* Uses exact arithmetic if necessary to ensure a correct answer. The */
4414 /* result returned is the determinant of a matrix. This determinant is */
4415 /* computed adaptively, in the sense that exact arithmetic is used only to */
4416 /* the degree it is needed to ensure that the returned value has the */
4417 /* correct sign. Hence, this function is usually quite fast, but will run */
4418 /* more slowly when the input points are cocircular or nearly so. */
4420 /* See my Robust Predicates paper for details. */
4422 /*****************************************************************************/
4424 REAL incircleadapt( pa, pb, pc, pd, permanent )
4431 INEXACT REAL adx, bdx, cdx, ady, bdy, cdy;
4434 INEXACT REAL bdxcdy1, cdxbdy1, cdxady1, adxcdy1, adxbdy1, bdxady1;
4435 REAL bdxcdy0, cdxbdy0, cdxady0, adxcdy0, adxbdy0, bdxady0;
4436 REAL bc[4], ca[4], ab[4];
4437 INEXACT REAL bc3, ca3, ab3;
4438 REAL axbc[8], axxbc[16], aybc[8], ayybc[16], adet[32];
4439 int axbclen, axxbclen, aybclen, ayybclen, alen;
4440 REAL bxca[8], bxxca[16], byca[8], byyca[16], bdet[32];
4441 int bxcalen, bxxcalen, bycalen, byycalen, blen;
4442 REAL cxab[8], cxxab[16], cyab[8], cyyab[16], cdet[32];
4443 int cxablen, cxxablen, cyablen, cyyablen, clen;
4446 REAL fin1[1152], fin2[1152];
4447 REAL *finnow, *finother, *finswap;
4450 REAL adxtail, bdxtail, cdxtail, adytail, bdytail, cdytail;
4451 INEXACT REAL adxadx1, adyady1, bdxbdx1, bdybdy1, cdxcdx1, cdycdy1;
4452 REAL adxadx0, adyady0, bdxbdx0, bdybdy0, cdxcdx0, cdycdy0;
4453 REAL aa[4], bb[4], cc[4];
4454 INEXACT REAL aa3, bb3, cc3;
4455 INEXACT REAL ti1, tj1;
4458 INEXACT REAL u3, v3;
4459 REAL temp8[8], temp16a[16], temp16b[16], temp16c[16];
4460 REAL temp32a[32], temp32b[32], temp48[48], temp64[64];
4461 int temp8len, temp16alen, temp16blen, temp16clen;
4462 int temp32alen, temp32blen, temp48len, temp64len;
4463 REAL axtbb[8], axtcc[8], aytbb[8], aytcc[8];
4464 int axtbblen, axtcclen, aytbblen, aytcclen;
4465 REAL bxtaa[8], bxtcc[8], bytaa[8], bytcc[8];
4466 int bxtaalen, bxtcclen, bytaalen, bytcclen;
4467 REAL cxtaa[8], cxtbb[8], cytaa[8], cytbb[8];
4468 int cxtaalen, cxtbblen, cytaalen, cytbblen;
4469 REAL axtbc[8], aytbc[8], bxtca[8], bytca[8], cxtab[8], cytab[8];
4470 int axtbclen, aytbclen, bxtcalen, bytcalen, cxtablen, cytablen;
4471 REAL axtbct[16], aytbct[16], bxtcat[16], bytcat[16], cxtabt[16], cytabt[16];
4472 int axtbctlen, aytbctlen, bxtcatlen, bytcatlen, cxtabtlen, cytabtlen;
4473 REAL axtbctt[8], aytbctt[8], bxtcatt[8];
4474 REAL bytcatt[8], cxtabtt[8], cytabtt[8];
4475 int axtbcttlen, aytbcttlen, bxtcattlen, bytcattlen, cxtabttlen, cytabttlen;
4476 REAL abt[8], bct[8], cat[8];
4477 int abtlen, bctlen, catlen;
4478 REAL abtt[4], bctt[4], catt[4];
4479 int abttlen, bcttlen, cattlen;
4480 INEXACT REAL abtt3, bctt3, catt3;
4484 REAL avirt, bround, around;
4487 REAL ahi, alo, bhi, blo;
4488 REAL err1, err2, err3;
4489 INEXACT REAL _i, _j;
4492 adx = (REAL) ( pa[0] - pd[0] );
4493 bdx = (REAL) ( pb[0] - pd[0] );
4494 cdx = (REAL) ( pc[0] - pd[0] );
4495 ady = (REAL) ( pa[1] - pd[1] );
4496 bdy = (REAL) ( pb[1] - pd[1] );
4497 cdy = (REAL) ( pc[1] - pd[1] );
4499 Two_Product( bdx, cdy, bdxcdy1, bdxcdy0 );
4500 Two_Product( cdx, bdy, cdxbdy1, cdxbdy0 );
4501 Two_Two_Diff( bdxcdy1, bdxcdy0, cdxbdy1, cdxbdy0, bc3, bc[2], bc[1], bc[0] );
4503 axbclen = scale_expansion_zeroelim( 4, bc, adx, axbc );
4504 axxbclen = scale_expansion_zeroelim( axbclen, axbc, adx, axxbc );
4505 aybclen = scale_expansion_zeroelim( 4, bc, ady, aybc );
4506 ayybclen = scale_expansion_zeroelim( aybclen, aybc, ady, ayybc );
4507 alen = fast_expansion_sum_zeroelim( axxbclen, axxbc, ayybclen, ayybc, adet );
4509 Two_Product( cdx, ady, cdxady1, cdxady0 );
4510 Two_Product( adx, cdy, adxcdy1, adxcdy0 );
4511 Two_Two_Diff( cdxady1, cdxady0, adxcdy1, adxcdy0, ca3, ca[2], ca[1], ca[0] );
4513 bxcalen = scale_expansion_zeroelim( 4, ca, bdx, bxca );
4514 bxxcalen = scale_expansion_zeroelim( bxcalen, bxca, bdx, bxxca );
4515 bycalen = scale_expansion_zeroelim( 4, ca, bdy, byca );
4516 byycalen = scale_expansion_zeroelim( bycalen, byca, bdy, byyca );
4517 blen = fast_expansion_sum_zeroelim( bxxcalen, bxxca, byycalen, byyca, bdet );
4519 Two_Product( adx, bdy, adxbdy1, adxbdy0 );
4520 Two_Product( bdx, ady, bdxady1, bdxady0 );
4521 Two_Two_Diff( adxbdy1, adxbdy0, bdxady1, bdxady0, ab3, ab[2], ab[1], ab[0] );
4523 cxablen = scale_expansion_zeroelim( 4, ab, cdx, cxab );
4524 cxxablen = scale_expansion_zeroelim( cxablen, cxab, cdx, cxxab );
4525 cyablen = scale_expansion_zeroelim( 4, ab, cdy, cyab );
4526 cyyablen = scale_expansion_zeroelim( cyablen, cyab, cdy, cyyab );
4527 clen = fast_expansion_sum_zeroelim( cxxablen, cxxab, cyyablen, cyyab, cdet );
4529 ablen = fast_expansion_sum_zeroelim( alen, adet, blen, bdet, abdet );
4530 finlength = fast_expansion_sum_zeroelim( ablen, abdet, clen, cdet, fin1 );
4532 det = estimate( finlength, fin1 );
4533 errbound = (REAL)( iccerrboundB * permanent );
4534 if ( ( det >= errbound ) || ( -det >= errbound ) ) {
4538 Two_Diff_Tail( pa[0], pd[0], adx, adxtail );
4539 Two_Diff_Tail( pa[1], pd[1], ady, adytail );
4540 Two_Diff_Tail( pb[0], pd[0], bdx, bdxtail );
4541 Two_Diff_Tail( pb[1], pd[1], bdy, bdytail );
4542 Two_Diff_Tail( pc[0], pd[0], cdx, cdxtail );
4543 Two_Diff_Tail( pc[1], pd[1], cdy, cdytail );
4544 if ( ( adxtail == 0.0 ) && ( bdxtail == 0.0 ) && ( cdxtail == 0.0 )
4545 && ( adytail == 0.0 ) && ( bdytail == 0.0 ) && ( cdytail == 0.0 ) ) {
4549 errbound = (REAL)( iccerrboundC * permanent + resulterrbound * Absolute( det ) );
4550 det += (REAL)( ( ( adx * adx + ady * ady ) * ( ( bdx * cdytail + cdy * bdxtail )
4551 - ( bdy * cdxtail + cdx * bdytail ) )
4552 + 2.0 * ( adx * adxtail + ady * adytail ) * ( bdx * cdy - bdy * cdx ) )
4553 + ( ( bdx * bdx + bdy * bdy ) * ( ( cdx * adytail + ady * cdxtail )
4554 - ( cdy * adxtail + adx * cdytail ) )
4555 + 2.0 * ( bdx * bdxtail + bdy * bdytail ) * ( cdx * ady - cdy * adx ) )
4556 + ( ( cdx * cdx + cdy * cdy ) * ( ( adx * bdytail + bdy * adxtail )
4557 - ( ady * bdxtail + bdx * adytail ) )
4558 + 2.0 * ( cdx * cdxtail + cdy * cdytail ) * ( adx * bdy - ady * bdx ) ) );
4559 if ( ( det >= errbound ) || ( -det >= errbound ) ) {
4566 if ( ( bdxtail != 0.0 ) || ( bdytail != 0.0 )
4567 || ( cdxtail != 0.0 ) || ( cdytail != 0.0 ) ) {
4568 Square( adx, adxadx1, adxadx0 );
4569 Square( ady, adyady1, adyady0 );
4570 Two_Two_Sum( adxadx1, adxadx0, adyady1, adyady0, aa3, aa[2], aa[1], aa[0] );
4573 if ( ( cdxtail != 0.0 ) || ( cdytail != 0.0 )
4574 || ( adxtail != 0.0 ) || ( adytail != 0.0 ) ) {
4575 Square( bdx, bdxbdx1, bdxbdx0 );
4576 Square( bdy, bdybdy1, bdybdy0 );
4577 Two_Two_Sum( bdxbdx1, bdxbdx0, bdybdy1, bdybdy0, bb3, bb[2], bb[1], bb[0] );
4580 if ( ( adxtail != 0.0 ) || ( adytail != 0.0 )
4581 || ( bdxtail != 0.0 ) || ( bdytail != 0.0 ) ) {
4582 Square( cdx, cdxcdx1, cdxcdx0 );
4583 Square( cdy, cdycdy1, cdycdy0 );
4584 Two_Two_Sum( cdxcdx1, cdxcdx0, cdycdy1, cdycdy0, cc3, cc[2], cc[1], cc[0] );
4588 if ( adxtail != 0.0 ) {
4589 axtbclen = scale_expansion_zeroelim( 4, bc, adxtail, axtbc );
4590 temp16alen = scale_expansion_zeroelim( axtbclen, axtbc, 2.0 * adx,
4593 axtcclen = scale_expansion_zeroelim( 4, cc, adxtail, axtcc );
4594 temp16blen = scale_expansion_zeroelim( axtcclen, axtcc, bdy, temp16b );
4596 axtbblen = scale_expansion_zeroelim( 4, bb, adxtail, axtbb );
4597 temp16clen = scale_expansion_zeroelim( axtbblen, axtbb, -cdy, temp16c );
4599 temp32alen = fast_expansion_sum_zeroelim( temp16alen, temp16a,
4600 temp16blen, temp16b, temp32a );
4601 temp48len = fast_expansion_sum_zeroelim( temp16clen, temp16c,
4602 temp32alen, temp32a, temp48 );
4603 finlength = fast_expansion_sum_zeroelim( finlength, finnow, temp48len,
4605 finswap = finnow; finnow = finother; finother = finswap;
4607 if ( adytail != 0.0 ) {
4608 aytbclen = scale_expansion_zeroelim( 4, bc, adytail, aytbc );
4609 temp16alen = scale_expansion_zeroelim( aytbclen, aytbc, 2.0 * ady,
4612 aytbblen = scale_expansion_zeroelim( 4, bb, adytail, aytbb );
4613 temp16blen = scale_expansion_zeroelim( aytbblen, aytbb, cdx, temp16b );
4615 aytcclen = scale_expansion_zeroelim( 4, cc, adytail, aytcc );
4616 temp16clen = scale_expansion_zeroelim( aytcclen, aytcc, -bdx, temp16c );
4618 temp32alen = fast_expansion_sum_zeroelim( temp16alen, temp16a,
4619 temp16blen, temp16b, temp32a );
4620 temp48len = fast_expansion_sum_zeroelim( temp16clen, temp16c,
4621 temp32alen, temp32a, temp48 );
4622 finlength = fast_expansion_sum_zeroelim( finlength, finnow, temp48len,
4624 finswap = finnow; finnow = finother; finother = finswap;
4626 if ( bdxtail != 0.0 ) {
4627 bxtcalen = scale_expansion_zeroelim( 4, ca, bdxtail, bxtca );
4628 temp16alen = scale_expansion_zeroelim( bxtcalen, bxtca, 2.0 * bdx,
4631 bxtaalen = scale_expansion_zeroelim( 4, aa, bdxtail, bxtaa );
4632 temp16blen = scale_expansion_zeroelim( bxtaalen, bxtaa, cdy, temp16b );
4634 bxtcclen = scale_expansion_zeroelim( 4, cc, bdxtail, bxtcc );
4635 temp16clen = scale_expansion_zeroelim( bxtcclen, bxtcc, -ady, temp16c );
4637 temp32alen = fast_expansion_sum_zeroelim( temp16alen, temp16a,
4638 temp16blen, temp16b, temp32a );
4639 temp48len = fast_expansion_sum_zeroelim( temp16clen, temp16c,
4640 temp32alen, temp32a, temp48 );
4641 finlength = fast_expansion_sum_zeroelim( finlength, finnow, temp48len,
4643 finswap = finnow; finnow = finother; finother = finswap;
4645 if ( bdytail != 0.0 ) {
4646 bytcalen = scale_expansion_zeroelim( 4, ca, bdytail, bytca );
4647 temp16alen = scale_expansion_zeroelim( bytcalen, bytca, 2.0 * bdy,
4650 bytcclen = scale_expansion_zeroelim( 4, cc, bdytail, bytcc );
4651 temp16blen = scale_expansion_zeroelim( bytcclen, bytcc, adx, temp16b );
4653 bytaalen = scale_expansion_zeroelim( 4, aa, bdytail, bytaa );
4654 temp16clen = scale_expansion_zeroelim( bytaalen, bytaa, -cdx, temp16c );
4656 temp32alen = fast_expansion_sum_zeroelim( temp16alen, temp16a,
4657 temp16blen, temp16b, temp32a );
4658 temp48len = fast_expansion_sum_zeroelim( temp16clen, temp16c,
4659 temp32alen, temp32a, temp48 );
4660 finlength = fast_expansion_sum_zeroelim( finlength, finnow, temp48len,
4662 finswap = finnow; finnow = finother; finother = finswap;
4664 if ( cdxtail != 0.0 ) {
4665 cxtablen = scale_expansion_zeroelim( 4, ab, cdxtail, cxtab );
4666 temp16alen = scale_expansion_zeroelim( cxtablen, cxtab, 2.0 * cdx,
4669 cxtbblen = scale_expansion_zeroelim( 4, bb, cdxtail, cxtbb );
4670 temp16blen = scale_expansion_zeroelim( cxtbblen, cxtbb, ady, temp16b );
4672 cxtaalen = scale_expansion_zeroelim( 4, aa, cdxtail, cxtaa );
4673 temp16clen = scale_expansion_zeroelim( cxtaalen, cxtaa, -bdy, temp16c );
4675 temp32alen = fast_expansion_sum_zeroelim( temp16alen, temp16a,
4676 temp16blen, temp16b, temp32a );
4677 temp48len = fast_expansion_sum_zeroelim( temp16clen, temp16c,
4678 temp32alen, temp32a, temp48 );
4679 finlength = fast_expansion_sum_zeroelim( finlength, finnow, temp48len,
4681 finswap = finnow; finnow = finother; finother = finswap;
4683 if ( cdytail != 0.0 ) {
4684 cytablen = scale_expansion_zeroelim( 4, ab, cdytail, cytab );
4685 temp16alen = scale_expansion_zeroelim( cytablen, cytab, 2.0 * cdy,
4688 cytaalen = scale_expansion_zeroelim( 4, aa, cdytail, cytaa );
4689 temp16blen = scale_expansion_zeroelim( cytaalen, cytaa, bdx, temp16b );
4691 cytbblen = scale_expansion_zeroelim( 4, bb, cdytail, cytbb );
4692 temp16clen = scale_expansion_zeroelim( cytbblen, cytbb, -adx, temp16c );
4694 temp32alen = fast_expansion_sum_zeroelim( temp16alen, temp16a,
4695 temp16blen, temp16b, temp32a );
4696 temp48len = fast_expansion_sum_zeroelim( temp16clen, temp16c,
4697 temp32alen, temp32a, temp48 );
4698 finlength = fast_expansion_sum_zeroelim( finlength, finnow, temp48len,
4700 finswap = finnow; finnow = finother; finother = finswap;
4703 if ( ( adxtail != 0.0 ) || ( adytail != 0.0 ) ) {
4704 if ( ( bdxtail != 0.0 ) || ( bdytail != 0.0 )
4705 || ( cdxtail != 0.0 ) || ( cdytail != 0.0 ) ) {
4706 Two_Product( bdxtail, cdy, ti1, ti0 );
4707 Two_Product( bdx, cdytail, tj1, tj0 );
4708 Two_Two_Sum( ti1, ti0, tj1, tj0, u3, u[2], u[1], u[0] );
4711 Two_Product( cdxtail, negate, ti1, ti0 );
4713 Two_Product( cdx, negate, tj1, tj0 );
4714 Two_Two_Sum( ti1, ti0, tj1, tj0, v3, v[2], v[1], v[0] );
4716 bctlen = fast_expansion_sum_zeroelim( 4, u, 4, v, bct );
4718 Two_Product( bdxtail, cdytail, ti1, ti0 );
4719 Two_Product( cdxtail, bdytail, tj1, tj0 );
4720 Two_Two_Diff( ti1, ti0, tj1, tj0, bctt3, bctt[2], bctt[1], bctt[0] );
4731 if ( adxtail != 0.0 ) {
4732 temp16alen = scale_expansion_zeroelim( axtbclen, axtbc, adxtail, temp16a );
4733 axtbctlen = scale_expansion_zeroelim( bctlen, bct, adxtail, axtbct );
4734 temp32alen = scale_expansion_zeroelim( axtbctlen, axtbct, 2.0 * adx,
4736 temp48len = fast_expansion_sum_zeroelim( temp16alen, temp16a,
4737 temp32alen, temp32a, temp48 );
4738 finlength = fast_expansion_sum_zeroelim( finlength, finnow, temp48len,
4740 finswap = finnow; finnow = finother; finother = finswap;
4741 if ( bdytail != 0.0 ) {
4742 temp8len = scale_expansion_zeroelim( 4, cc, adxtail, temp8 );
4743 temp16alen = scale_expansion_zeroelim( temp8len, temp8, bdytail,
4745 finlength = fast_expansion_sum_zeroelim( finlength, finnow, temp16alen,
4746 temp16a, finother );
4747 finswap = finnow; finnow = finother; finother = finswap;
4749 if ( cdytail != 0.0 ) {
4750 temp8len = scale_expansion_zeroelim( 4, bb, -adxtail, temp8 );
4751 temp16alen = scale_expansion_zeroelim( temp8len, temp8, cdytail,
4753 finlength = fast_expansion_sum_zeroelim( finlength, finnow, temp16alen,
4754 temp16a, finother );
4755 finswap = finnow; finnow = finother; finother = finswap;
4758 temp32alen = scale_expansion_zeroelim( axtbctlen, axtbct, adxtail,
4760 axtbcttlen = scale_expansion_zeroelim( bcttlen, bctt, adxtail, axtbctt );
4761 temp16alen = scale_expansion_zeroelim( axtbcttlen, axtbctt, 2.0 * adx,
4763 temp16blen = scale_expansion_zeroelim( axtbcttlen, axtbctt, adxtail,
4765 temp32blen = fast_expansion_sum_zeroelim( temp16alen, temp16a,
4766 temp16blen, temp16b, temp32b );
4767 temp64len = fast_expansion_sum_zeroelim( temp32alen, temp32a,
4768 temp32blen, temp32b, temp64 );
4769 finlength = fast_expansion_sum_zeroelim( finlength, finnow, temp64len,
4771 finswap = finnow; finnow = finother; finother = finswap;
4773 if ( adytail != 0.0 ) {
4774 temp16alen = scale_expansion_zeroelim( aytbclen, aytbc, adytail, temp16a );
4775 aytbctlen = scale_expansion_zeroelim( bctlen, bct, adytail, aytbct );
4776 temp32alen = scale_expansion_zeroelim( aytbctlen, aytbct, 2.0 * ady,
4778 temp48len = fast_expansion_sum_zeroelim( temp16alen, temp16a,
4779 temp32alen, temp32a, temp48 );
4780 finlength = fast_expansion_sum_zeroelim( finlength, finnow, temp48len,
4782 finswap = finnow; finnow = finother; finother = finswap;
4785 temp32alen = scale_expansion_zeroelim( aytbctlen, aytbct, adytail,
4787 aytbcttlen = scale_expansion_zeroelim( bcttlen, bctt, adytail, aytbctt );
4788 temp16alen = scale_expansion_zeroelim( aytbcttlen, aytbctt, 2.0 * ady,
4790 temp16blen = scale_expansion_zeroelim( aytbcttlen, aytbctt, adytail,
4792 temp32blen = fast_expansion_sum_zeroelim( temp16alen, temp16a,
4793 temp16blen, temp16b, temp32b );
4794 temp64len = fast_expansion_sum_zeroelim( temp32alen, temp32a,
4795 temp32blen, temp32b, temp64 );
4796 finlength = fast_expansion_sum_zeroelim( finlength, finnow, temp64len,
4798 finswap = finnow; finnow = finother; finother = finswap;
4801 if ( ( bdxtail != 0.0 ) || ( bdytail != 0.0 ) ) {
4802 if ( ( cdxtail != 0.0 ) || ( cdytail != 0.0 )
4803 || ( adxtail != 0.0 ) || ( adytail != 0.0 ) ) {
4804 Two_Product( cdxtail, ady, ti1, ti0 );
4805 Two_Product( cdx, adytail, tj1, tj0 );
4806 Two_Two_Sum( ti1, ti0, tj1, tj0, u3, u[2], u[1], u[0] );
4809 Two_Product( adxtail, negate, ti1, ti0 );
4811 Two_Product( adx, negate, tj1, tj0 );
4812 Two_Two_Sum( ti1, ti0, tj1, tj0, v3, v[2], v[1], v[0] );
4814 catlen = fast_expansion_sum_zeroelim( 4, u, 4, v, cat );
4816 Two_Product( cdxtail, adytail, ti1, ti0 );
4817 Two_Product( adxtail, cdytail, tj1, tj0 );
4818 Two_Two_Diff( ti1, ti0, tj1, tj0, catt3, catt[2], catt[1], catt[0] );
4829 if ( bdxtail != 0.0 ) {
4830 temp16alen = scale_expansion_zeroelim( bxtcalen, bxtca, bdxtail, temp16a );
4831 bxtcatlen = scale_expansion_zeroelim( catlen, cat, bdxtail, bxtcat );
4832 temp32alen = scale_expansion_zeroelim( bxtcatlen, bxtcat, 2.0 * bdx,
4834 temp48len = fast_expansion_sum_zeroelim( temp16alen, temp16a,
4835 temp32alen, temp32a, temp48 );
4836 finlength = fast_expansion_sum_zeroelim( finlength, finnow, temp48len,
4838 finswap = finnow; finnow = finother; finother = finswap;
4839 if ( cdytail != 0.0 ) {
4840 temp8len = scale_expansion_zeroelim( 4, aa, bdxtail, temp8 );
4841 temp16alen = scale_expansion_zeroelim( temp8len, temp8, cdytail,
4843 finlength = fast_expansion_sum_zeroelim( finlength, finnow, temp16alen,
4844 temp16a, finother );
4845 finswap = finnow; finnow = finother; finother = finswap;
4847 if ( adytail != 0.0 ) {
4848 temp8len = scale_expansion_zeroelim( 4, cc, -bdxtail, temp8 );
4849 temp16alen = scale_expansion_zeroelim( temp8len, temp8, adytail,
4851 finlength = fast_expansion_sum_zeroelim( finlength, finnow, temp16alen,
4852 temp16a, finother );
4853 finswap = finnow; finnow = finother; finother = finswap;
4856 temp32alen = scale_expansion_zeroelim( bxtcatlen, bxtcat, bdxtail,
4858 bxtcattlen = scale_expansion_zeroelim( cattlen, catt, bdxtail, bxtcatt );
4859 temp16alen = scale_expansion_zeroelim( bxtcattlen, bxtcatt, 2.0 * bdx,
4861 temp16blen = scale_expansion_zeroelim( bxtcattlen, bxtcatt, bdxtail,
4863 temp32blen = fast_expansion_sum_zeroelim( temp16alen, temp16a,
4864 temp16blen, temp16b, temp32b );
4865 temp64len = fast_expansion_sum_zeroelim( temp32alen, temp32a,
4866 temp32blen, temp32b, temp64 );
4867 finlength = fast_expansion_sum_zeroelim( finlength, finnow, temp64len,
4869 finswap = finnow; finnow = finother; finother = finswap;
4871 if ( bdytail != 0.0 ) {
4872 temp16alen = scale_expansion_zeroelim( bytcalen, bytca, bdytail, temp16a );
4873 bytcatlen = scale_expansion_zeroelim( catlen, cat, bdytail, bytcat );
4874 temp32alen = scale_expansion_zeroelim( bytcatlen, bytcat, 2.0 * bdy,
4876 temp48len = fast_expansion_sum_zeroelim( temp16alen, temp16a,
4877 temp32alen, temp32a, temp48 );
4878 finlength = fast_expansion_sum_zeroelim( finlength, finnow, temp48len,
4880 finswap = finnow; finnow = finother; finother = finswap;
4883 temp32alen = scale_expansion_zeroelim( bytcatlen, bytcat, bdytail,
4885 bytcattlen = scale_expansion_zeroelim( cattlen, catt, bdytail, bytcatt );
4886 temp16alen = scale_expansion_zeroelim( bytcattlen, bytcatt, 2.0 * bdy,
4888 temp16blen = scale_expansion_zeroelim( bytcattlen, bytcatt, bdytail,
4890 temp32blen = fast_expansion_sum_zeroelim( temp16alen, temp16a,
4891 temp16blen, temp16b, temp32b );
4892 temp64len = fast_expansion_sum_zeroelim( temp32alen, temp32a,
4893 temp32blen, temp32b, temp64 );
4894 finlength = fast_expansion_sum_zeroelim( finlength, finnow, temp64len,
4896 finswap = finnow; finnow = finother; finother = finswap;
4899 if ( ( cdxtail != 0.0 ) || ( cdytail != 0.0 ) ) {
4900 if ( ( adxtail != 0.0 ) || ( adytail != 0.0 )
4901 || ( bdxtail != 0.0 ) || ( bdytail != 0.0 ) ) {
4902 Two_Product( adxtail, bdy, ti1, ti0 );
4903 Two_Product( adx, bdytail, tj1, tj0 );
4904 Two_Two_Sum( ti1, ti0, tj1, tj0, u3, u[2], u[1], u[0] );
4907 Two_Product( bdxtail, negate, ti1, ti0 );
4909 Two_Product( bdx, negate, tj1, tj0 );
4910 Two_Two_Sum( ti1, ti0, tj1, tj0, v3, v[2], v[1], v[0] );
4912 abtlen = fast_expansion_sum_zeroelim( 4, u, 4, v, abt );
4914 Two_Product( adxtail, bdytail, ti1, ti0 );
4915 Two_Product( bdxtail, adytail, tj1, tj0 );
4916 Two_Two_Diff( ti1, ti0, tj1, tj0, abtt3, abtt[2], abtt[1], abtt[0] );
4927 if ( cdxtail != 0.0 ) {
4928 temp16alen = scale_expansion_zeroelim( cxtablen, cxtab, cdxtail, temp16a );
4929 cxtabtlen = scale_expansion_zeroelim( abtlen, abt, cdxtail, cxtabt );
4930 temp32alen = scale_expansion_zeroelim( cxtabtlen, cxtabt, 2.0 * cdx,
4932 temp48len = fast_expansion_sum_zeroelim( temp16alen, temp16a,
4933 temp32alen, temp32a, temp48 );
4934 finlength = fast_expansion_sum_zeroelim( finlength, finnow, temp48len,
4936 finswap = finnow; finnow = finother; finother = finswap;
4937 if ( adytail != 0.0 ) {
4938 temp8len = scale_expansion_zeroelim( 4, bb, cdxtail, temp8 );
4939 temp16alen = scale_expansion_zeroelim( temp8len, temp8, adytail,
4941 finlength = fast_expansion_sum_zeroelim( finlength, finnow, temp16alen,
4942 temp16a, finother );
4943 finswap = finnow; finnow = finother; finother = finswap;
4945 if ( bdytail != 0.0 ) {
4946 temp8len = scale_expansion_zeroelim( 4, aa, -cdxtail, temp8 );
4947 temp16alen = scale_expansion_zeroelim( temp8len, temp8, bdytail,
4949 finlength = fast_expansion_sum_zeroelim( finlength, finnow, temp16alen,
4950 temp16a, finother );
4951 finswap = finnow; finnow = finother; finother = finswap;
4954 temp32alen = scale_expansion_zeroelim( cxtabtlen, cxtabt, cdxtail,
4956 cxtabttlen = scale_expansion_zeroelim( abttlen, abtt, cdxtail, cxtabtt );
4957 temp16alen = scale_expansion_zeroelim( cxtabttlen, cxtabtt, 2.0 * cdx,
4959 temp16blen = scale_expansion_zeroelim( cxtabttlen, cxtabtt, cdxtail,
4961 temp32blen = fast_expansion_sum_zeroelim( temp16alen, temp16a,
4962 temp16blen, temp16b, temp32b );
4963 temp64len = fast_expansion_sum_zeroelim( temp32alen, temp32a,
4964 temp32blen, temp32b, temp64 );
4965 finlength = fast_expansion_sum_zeroelim( finlength, finnow, temp64len,
4967 finswap = finnow; finnow = finother; finother = finswap;
4969 if ( cdytail != 0.0 ) {
4970 temp16alen = scale_expansion_zeroelim( cytablen, cytab, cdytail, temp16a );
4971 cytabtlen = scale_expansion_zeroelim( abtlen, abt, cdytail, cytabt );
4972 temp32alen = scale_expansion_zeroelim( cytabtlen, cytabt, 2.0 * cdy,
4974 temp48len = fast_expansion_sum_zeroelim( temp16alen, temp16a,
4975 temp32alen, temp32a, temp48 );
4976 finlength = fast_expansion_sum_zeroelim( finlength, finnow, temp48len,
4978 finswap = finnow; finnow = finother; finother = finswap;
4981 temp32alen = scale_expansion_zeroelim( cytabtlen, cytabt, cdytail,
4983 cytabttlen = scale_expansion_zeroelim( abttlen, abtt, cdytail, cytabtt );
4984 temp16alen = scale_expansion_zeroelim( cytabttlen, cytabtt, 2.0 * cdy,
4986 temp16blen = scale_expansion_zeroelim( cytabttlen, cytabtt, cdytail,
4988 temp32blen = fast_expansion_sum_zeroelim( temp16alen, temp16a,
4989 temp16blen, temp16b, temp32b );
4990 temp64len = fast_expansion_sum_zeroelim( temp32alen, temp32a,
4991 temp32blen, temp32b, temp64 );
4992 finlength = fast_expansion_sum_zeroelim( finlength, finnow, temp64len,
4994 finswap = finnow; finnow = finother; finother = finswap;
4998 return finnow[finlength - 1];
5001 REAL incircle( pa, pb, pc, pd )
5007 REAL adx, bdx, cdx, ady, bdy, cdy;
5008 REAL bdxcdy, cdxbdy, cdxady, adxcdy, adxbdy, bdxady;
5009 REAL alift, blift, clift;
5011 REAL permanent, errbound;
5015 adx = pa[0] - pd[0];
5016 bdx = pb[0] - pd[0];
5017 cdx = pc[0] - pd[0];
5018 ady = pa[1] - pd[1];
5019 bdy = pb[1] - pd[1];
5020 cdy = pc[1] - pd[1];
5024 alift = adx * adx + ady * ady;
5028 blift = bdx * bdx + bdy * bdy;
5032 clift = cdx * cdx + cdy * cdy;
5034 det = alift * ( bdxcdy - cdxbdy )
5035 + blift * ( cdxady - adxcdy )
5036 + clift * ( adxbdy - bdxady );
5042 permanent = ( Absolute( bdxcdy ) + Absolute( cdxbdy ) ) * alift
5043 + ( Absolute( cdxady ) + Absolute( adxcdy ) ) * blift
5044 + ( Absolute( adxbdy ) + Absolute( bdxady ) ) * clift;
5045 errbound = iccerrboundA * permanent;
5046 if ( ( det > errbound ) || ( -det > errbound ) ) {
5050 return incircleadapt( pa, pb, pc, pd, permanent );
5055 /********* Determinant evaluation routines end here *********/
5057 /*****************************************************************************/
5059 /* triangleinit() Initialize some variables. */
5061 /*****************************************************************************/
5063 void triangleinit(){
5064 points.maxitems = triangles.maxitems = shelles.maxitems = viri.maxitems =
5065 badsegments.maxitems = badtriangles.maxitems = splaynodes.maxitems = 0l;
5066 points.itembytes = triangles.itembytes = shelles.itembytes = viri.itembytes =
5067 badsegments.itembytes = badtriangles.itembytes = splaynodes.itembytes = 0;
5068 recenttri.tri = (triangle *) NULL; /* No triangle has been visited yet. */
5069 samples = 1; /* Point location should take at least one sample. */
5070 checksegments = 0; /* There are no segments in the triangulation yet. */
5071 incirclecount = counterclockcount = hyperbolacount = 0;
5072 circumcentercount = circletopcount = 0;
5075 exactinit(); /* Initialize exact arithmetic constants. */
5078 /*****************************************************************************/
5080 /* randomnation() Generate a random number between 0 and `choices' - 1. */
5082 /* This is a simple linear congruential random number generator. Hence, it */
5083 /* is a bad random number generator, but good enough for most randomized */
5084 /* geometric algorithms. */
5086 /*****************************************************************************/
5088 unsigned long randomnation( choices )
5089 unsigned int choices;
5091 randomseed = ( randomseed * 1366l + 150889l ) % 714025l;
5092 return randomseed / ( 714025l / choices + 1 );
5095 /********* Mesh quality testing routines begin here *********/
5099 /*****************************************************************************/
5101 /* checkmesh() Test the mesh for topological consistency. */
5103 /*****************************************************************************/
5108 struct triedge triangleloop;
5109 struct triedge oppotri, oppooppotri;
5110 point triorg, tridest, triapex;
5111 point oppoorg, oppodest;
5114 triangle ptr; /* Temporary variable used by sym(). */
5116 /* Temporarily turn on exact arithmetic if it's off. */
5117 saveexact = noexact;
5120 printf( " Checking consistency of mesh...\n" );
5123 /* Run through the list of triangles, checking each one. */
5124 traversalinit( &triangles );
5125 triangleloop.tri = triangletraverse();
5126 while ( triangleloop.tri != (triangle *) NULL ) {
5127 /* Check all three edges of the triangle. */
5128 for ( triangleloop.orient = 0; triangleloop.orient < 3;
5129 triangleloop.orient++ ) {
5130 org( triangleloop, triorg );
5131 dest( triangleloop, tridest );
5132 if ( triangleloop.orient == 0 ) { /* Only test for inversion once. */
5133 /* Test if the triangle is flat or inverted. */
5134 apex( triangleloop, triapex );
5135 if ( counterclockwise( triorg, tridest, triapex ) <= 0.0 ) {
5136 printf( " !! !! Inverted " );
5137 printtriangle( &triangleloop );
5141 /* Find the neighboring triangle on this edge. */
5142 sym( triangleloop, oppotri );
5143 if ( oppotri.tri != dummytri ) {
5144 /* Check that the triangle's neighbor knows it's a neighbor. */
5145 sym( oppotri, oppooppotri );
5146 if ( ( triangleloop.tri != oppooppotri.tri )
5147 || ( triangleloop.orient != oppooppotri.orient ) ) {
5148 printf( " !! !! Asymmetric triangle-triangle bond:\n" );
5149 if ( triangleloop.tri == oppooppotri.tri ) {
5150 printf( " (Right triangle, wrong orientation)\n" );
5152 printf( " First " );
5153 printtriangle( &triangleloop );
5154 printf( " Second (nonreciprocating) " );
5155 printtriangle( &oppotri );
5158 /* Check that both triangles agree on the identities */
5159 /* of their shared vertices. */
5160 org( oppotri, oppoorg );
5161 dest( oppotri, oppodest );
5162 if ( ( triorg != oppodest ) || ( tridest != oppoorg ) ) {
5163 printf( " !! !! Mismatched edge coordinates between two triangles:\n"
5165 printf( " First mismatched " );
5166 printtriangle( &triangleloop );
5167 printf( " Second mismatched " );
5168 printtriangle( &oppotri );
5173 triangleloop.tri = triangletraverse();
5175 if ( horrors == 0 ) {
5177 printf( " In my studied opinion, the mesh appears to be consistent.\n" );
5180 else if ( horrors == 1 ) {
5181 printf( " !! !! !! !! Precisely one festering wound discovered.\n" );
5184 printf( " !! !! !! !! %d abominations witnessed.\n", horrors );
5186 /* Restore the status of exact arithmetic. */
5187 noexact = saveexact;
5190 #endif /* not REDUCED */
5192 /*****************************************************************************/
5194 /* checkdelaunay() Ensure that the mesh is (constrained) Delaunay. */
5196 /*****************************************************************************/
5200 void checkdelaunay(){
5201 struct triedge triangleloop;
5202 struct triedge oppotri;
5203 struct edge opposhelle;
5204 point triorg, tridest, triapex;
5206 int shouldbedelaunay;
5209 triangle ptr; /* Temporary variable used by sym(). */
5210 shelle sptr; /* Temporary variable used by tspivot(). */
5212 /* Temporarily turn on exact arithmetic if it's off. */
5213 saveexact = noexact;
5216 printf( " Checking Delaunay property of mesh...\n" );
5219 /* Run through the list of triangles, checking each one. */
5220 traversalinit( &triangles );
5221 triangleloop.tri = triangletraverse();
5222 while ( triangleloop.tri != (triangle *) NULL ) {
5223 /* Check all three edges of the triangle. */
5224 for ( triangleloop.orient = 0; triangleloop.orient < 3;
5225 triangleloop.orient++ ) {
5226 org( triangleloop, triorg );
5227 dest( triangleloop, tridest );
5228 apex( triangleloop, triapex );
5229 sym( triangleloop, oppotri );
5230 apex( oppotri, oppoapex );
5231 /* Only test that the edge is locally Delaunay if there is an */
5232 /* adjoining triangle whose pointer is larger (to ensure that */
5233 /* each pair isn't tested twice). */
5234 shouldbedelaunay = ( oppotri.tri != dummytri )
5235 && ( triapex != (point) NULL ) && ( oppoapex != (point) NULL )
5236 && ( triangleloop.tri < oppotri.tri );
5237 if ( checksegments && shouldbedelaunay ) {
5238 /* If a shell edge separates the triangles, then the edge is */
5239 /* constrained, so no local Delaunay test should be done. */
5240 tspivot( triangleloop, opposhelle );
5241 if ( opposhelle.sh != dummysh ) {
5242 shouldbedelaunay = 0;
5245 if ( shouldbedelaunay ) {
5246 if ( incircle( triorg, tridest, triapex, oppoapex ) > 0.0 ) {
5247 printf( " !! !! Non-Delaunay pair of triangles:\n" );
5248 printf( " First non-Delaunay " );
5249 printtriangle( &triangleloop );
5250 printf( " Second non-Delaunay " );
5251 printtriangle( &oppotri );
5256 triangleloop.tri = triangletraverse();
5258 if ( horrors == 0 ) {
5261 " By virtue of my perceptive intelligence, I declare the mesh Delaunay.\n" );
5264 else if ( horrors == 1 ) {
5266 " !! !! !! !! Precisely one terrifying transgression identified.\n" );
5269 printf( " !! !! !! !! %d obscenities viewed with horror.\n", horrors );
5271 /* Restore the status of exact arithmetic. */
5272 noexact = saveexact;
5275 #endif /* not REDUCED */
5277 /*****************************************************************************/
5279 /* enqueuebadtri() Add a bad triangle to the end of a queue. */
5281 /* The queue is actually a set of 64 queues. I use multiple queues to give */
5282 /* priority to smaller angles. I originally implemented a heap, but the */
5283 /* queues are (to my surprise) much faster. */
5285 /*****************************************************************************/
5289 void enqueuebadtri( instri, angle, insapex, insorg, insdest )
5290 struct triedge *instri;
5296 struct badface *newface;
5299 if ( verbose > 2 ) {
5300 printf( " Queueing bad triangle:\n" );
5301 printf( " (%.12g, %.12g) (%.12g, %.12g) (%.12g, %.12g)\n", insorg[0],
5302 insorg[1], insdest[0], insdest[1], insapex[0], insapex[1] );
5304 /* Allocate space for the bad triangle. */
5305 newface = (struct badface *) poolalloc( &badtriangles );
5306 triedgecopy( *instri, newface->badfacetri );
5307 newface->key = angle;
5308 newface->faceapex = insapex;
5309 newface->faceorg = insorg;
5310 newface->facedest = insdest;
5311 newface->nextface = (struct badface *) NULL;
5312 /* Determine the appropriate queue to put the bad triangle into. */
5313 if ( angle > 0.6 ) {
5314 queuenumber = (int) ( 160.0 * ( angle - 0.6 ) );
5315 if ( queuenumber > 63 ) {
5320 /* It's not a bad angle; put the triangle in the lowest-priority queue. */
5323 /* Add the triangle to the end of a queue. */
5324 *queuetail[queuenumber] = newface;
5325 /* Maintain a pointer to the NULL pointer at the end of the queue. */
5326 queuetail[queuenumber] = &newface->nextface;
5329 #endif /* not CDT_ONLY */
5331 /*****************************************************************************/
5333 /* dequeuebadtri() Remove a triangle from the front of the queue. */
5335 /*****************************************************************************/
5339 struct badface *dequeuebadtri(){
5340 struct badface *result;
5343 /* Look for a nonempty queue. */
5344 for ( queuenumber = 63; queuenumber >= 0; queuenumber-- ) {
5345 result = queuefront[queuenumber];
5346 if ( result != (struct badface *) NULL ) {
5347 /* Remove the triangle from the queue. */
5348 queuefront[queuenumber] = result->nextface;
5349 /* Maintain a pointer to the NULL pointer at the end of the queue. */
5350 if ( queuefront[queuenumber] == (struct badface *) NULL ) {
5351 queuetail[queuenumber] = &queuefront[queuenumber];
5356 return (struct badface *) NULL;
5359 #endif /* not CDT_ONLY */
5361 /*****************************************************************************/
5363 /* checkedge4encroach() Check a segment to see if it is encroached; add */
5364 /* it to the list if it is. */
5366 /* An encroached segment is an unflippable edge that has a point in its */
5367 /* diametral circle (that is, it faces an angle greater than 90 degrees). */
5368 /* This definition is due to Ruppert. */
5370 /* Returns a nonzero value if the edge is encroached. */
5372 /*****************************************************************************/
5376 int checkedge4encroach( testedge )
5377 struct edge *testedge;
5379 struct triedge neighbortri;
5380 struct edge testsym;
5381 struct edge *badedge;
5384 point eorg, edest, eapex;
5385 triangle ptr; /* Temporary variable used by stpivot(). */
5390 sorg( *testedge, eorg );
5391 sdest( *testedge, edest );
5392 /* Check one neighbor of the shell edge. */
5393 stpivot( *testedge, neighbortri );
5394 /* Does the neighbor exist, or is this a boundary edge? */
5395 if ( neighbortri.tri != dummytri ) {
5397 /* Find a vertex opposite this edge. */
5398 apex( neighbortri, eapex );
5399 /* Check whether the vertex is inside the diametral circle of the */
5400 /* shell edge. Pythagoras' Theorem is used to check whether the */
5401 /* angle at the vertex is greater than 90 degrees. */
5402 if ( eapex[0] * ( eorg[0] + edest[0] ) + eapex[1] * ( eorg[1] + edest[1] ) >
5403 eapex[0] * eapex[0] + eorg[0] * edest[0] +
5404 eapex[1] * eapex[1] + eorg[1] * edest[1] ) {
5408 /* Check the other neighbor of the shell edge. */
5409 ssym( *testedge, testsym );
5410 stpivot( testsym, neighbortri );
5411 /* Does the neighbor exist, or is this a boundary edge? */
5412 if ( neighbortri.tri != dummytri ) {
5414 /* Find the other vertex opposite this edge. */
5415 apex( neighbortri, eapex );
5416 /* Check whether the vertex is inside the diametral circle of the */
5417 /* shell edge. Pythagoras' Theorem is used to check whether the */
5418 /* angle at the vertex is greater than 90 degrees. */
5419 if ( eapex[0] * ( eorg[0] + edest[0] ) +
5420 eapex[1] * ( eorg[1] + edest[1] ) >
5421 eapex[0] * eapex[0] + eorg[0] * edest[0] +
5422 eapex[1] * eapex[1] + eorg[1] * edest[1] ) {
5427 if ( addtolist && ( !nobisect || ( ( nobisect == 1 ) && ( sides == 2 ) ) ) ) {
5428 if ( verbose > 2 ) {
5429 printf( " Queueing encroached segment (%.12g, %.12g) (%.12g, %.12g).\n",
5430 eorg[0], eorg[1], edest[0], edest[1] );
5432 /* Add the shell edge to the list of encroached segments. */
5433 /* Be sure to get the orientation right. */
5434 badedge = (struct edge *) poolalloc( &badsegments );
5435 if ( addtolist == 1 ) {
5436 shellecopy( *testedge, *badedge );
5439 shellecopy( testsym, *badedge );
5445 #endif /* not CDT_ONLY */
5447 /*****************************************************************************/
5449 /* testtriangle() Test a face for quality measures. */
5451 /* Tests a triangle to see if it satisfies the minimum angle condition and */
5452 /* the maximum area condition. Triangles that aren't up to spec are added */
5453 /* to the bad triangle queue. */
5455 /*****************************************************************************/
5459 void testtriangle( testtri )
5460 struct triedge *testtri;
5462 struct triedge sametesttri;
5463 struct edge edge1, edge2;
5464 point torg, tdest, tapex;
5466 REAL dxod, dyod, dxda, dyda, dxao, dyao;
5467 REAL dxod2, dyod2, dxda2, dyda2, dxao2, dyao2;
5468 REAL apexlen, orglen, destlen;
5471 shelle sptr; /* Temporary variable used by tspivot(). */
5473 org( *testtri, torg );
5474 dest( *testtri, tdest );
5475 apex( *testtri, tapex );
5476 dxod = torg[0] - tdest[0];
5477 dyod = torg[1] - tdest[1];
5478 dxda = tdest[0] - tapex[0];
5479 dyda = tdest[1] - tapex[1];
5480 dxao = tapex[0] - torg[0];
5481 dyao = tapex[1] - torg[1];
5482 dxod2 = dxod * dxod;
5483 dyod2 = dyod * dyod;
5484 dxda2 = dxda * dxda;
5485 dyda2 = dyda * dyda;
5486 dxao2 = dxao * dxao;
5487 dyao2 = dyao * dyao;
5488 /* Find the lengths of the triangle's three edges. */
5489 apexlen = dxod2 + dyod2;
5490 orglen = dxda2 + dyda2;
5491 destlen = dxao2 + dyao2;
5492 if ( ( apexlen < orglen ) && ( apexlen < destlen ) ) {
5493 /* The edge opposite the apex is shortest. */
5494 /* Find the square of the cosine of the angle at the apex. */
5495 angle = dxda * dxao + dyda * dyao;
5496 angle = angle * angle / ( orglen * destlen );
5497 anglevertex = tapex;
5498 lnext( *testtri, sametesttri );
5499 tspivot( sametesttri, edge1 );
5500 lnextself( sametesttri );
5501 tspivot( sametesttri, edge2 );
5503 else if ( orglen < destlen ) {
5504 /* The edge opposite the origin is shortest. */
5505 /* Find the square of the cosine of the angle at the origin. */
5506 angle = dxod * dxao + dyod * dyao;
5507 angle = angle * angle / ( apexlen * destlen );
5509 tspivot( *testtri, edge1 );
5510 lprev( *testtri, sametesttri );
5511 tspivot( sametesttri, edge2 );
5514 /* The edge opposite the destination is shortest. */
5515 /* Find the square of the cosine of the angle at the destination. */
5516 angle = dxod * dxda + dyod * dyda;
5517 angle = angle * angle / ( apexlen * orglen );
5518 anglevertex = tdest;
5519 tspivot( *testtri, edge1 );
5520 lnext( *testtri, sametesttri );
5521 tspivot( sametesttri, edge2 );
5523 /* Check if both edges that form the angle are segments. */
5524 if ( ( edge1.sh != dummysh ) && ( edge2.sh != dummysh ) ) {
5525 /* The angle is a segment intersection. */
5526 if ( ( angle > 0.9924 ) && !quiet ) { /* Roughly 5 degrees. */
5527 if ( angle > 1.0 ) {
5528 /* Beware of a floating exception in acos(). */
5531 /* Find the actual angle in degrees, for printing. */
5532 angle = acos( sqrt( angle ) ) * ( 180.0 / PI );
5534 "Warning: Small angle (%.4g degrees) between segments at point\n",
5536 printf( " (%.12g, %.12g)\n", anglevertex[0], anglevertex[1] );
5538 /* Don't add this bad triangle to the list; there's nothing that */
5539 /* can be done about a small angle between two segments. */
5542 /* Check whether the angle is smaller than permitted. */
5543 if ( angle > goodangle ) {
5544 /* Add this triangle to the list of bad triangles. */
5545 enqueuebadtri( testtri, angle, tapex, torg, tdest );
5548 if ( vararea || fixedarea ) {
5549 /* Check whether the area is larger than permitted. */
5550 area = 0.5 * ( dxod * dyda - dyod * dxda );
5551 if ( fixedarea && ( area > maxarea ) ) {
5552 /* Add this triangle to the list of bad triangles. */
5553 enqueuebadtri( testtri, angle, tapex, torg, tdest );
5555 else if ( vararea ) {
5556 /* Nonpositive area constraints are treated as unconstrained. */
5557 if ( ( area > areabound( *testtri ) ) && ( areabound( *testtri ) > 0.0 ) ) {
5558 /* Add this triangle to the list of bad triangles. */
5559 enqueuebadtri( testtri, angle, tapex, torg, tdest );
5565 #endif /* not CDT_ONLY */
5569 /********* Mesh quality testing routines end here *********/
5571 /********* Point location routines begin here *********/
5575 /*****************************************************************************/
5577 /* makepointmap() Construct a mapping from points to triangles to improve */
5578 /* the speed of point location for segment insertion. */
5580 /* Traverses all the triangles, and provides each corner of each triangle */
5581 /* with a pointer to that triangle. Of course, pointers will be */
5582 /* overwritten by other pointers because (almost) each point is a corner */
5583 /* of several triangles, but in the end every point will point to some */
5584 /* triangle that contains it. */
5586 /*****************************************************************************/
5588 void makepointmap(){
5589 struct triedge triangleloop;
5593 printf( " Constructing mapping from points to triangles.\n" );
5595 traversalinit( &triangles );
5596 triangleloop.tri = triangletraverse();
5597 while ( triangleloop.tri != (triangle *) NULL ) {
5598 /* Check all three points of the triangle. */
5599 for ( triangleloop.orient = 0; triangleloop.orient < 3;
5600 triangleloop.orient++ ) {
5601 org( triangleloop, triorg );
5602 setpoint2tri( triorg, encode( triangleloop ) );
5604 triangleloop.tri = triangletraverse();
5608 /*****************************************************************************/
5610 /* preciselocate() Find a triangle or edge containing a given point. */
5612 /* Begins its search from `searchtri'. It is important that `searchtri' */
5613 /* be a handle with the property that `searchpoint' is strictly to the left */
5614 /* of the edge denoted by `searchtri', or is collinear with that edge and */
5615 /* does not intersect that edge. (In particular, `searchpoint' should not */
5616 /* be the origin or destination of that edge.) */
5618 /* These conditions are imposed because preciselocate() is normally used in */
5619 /* one of two situations: */
5621 /* (1) To try to find the location to insert a new point. Normally, we */
5622 /* know an edge that the point is strictly to the left of. In the */
5623 /* incremental Delaunay algorithm, that edge is a bounding box edge. */
5624 /* In Ruppert's Delaunay refinement algorithm for quality meshing, */
5625 /* that edge is the shortest edge of the triangle whose circumcenter */
5626 /* is being inserted. */
5628 /* (2) To try to find an existing point. In this case, any edge on the */
5629 /* convex hull is a good starting edge. The possibility that the */
5630 /* vertex one seeks is an endpoint of the starting edge must be */
5631 /* screened out before preciselocate() is called. */
5633 /* On completion, `searchtri' is a triangle that contains `searchpoint'. */
5635 /* This implementation differs from that given by Guibas and Stolfi. It */
5636 /* walks from triangle to triangle, crossing an edge only if `searchpoint' */
5637 /* is on the other side of the line containing that edge. After entering */
5638 /* a triangle, there are two edges by which one can leave that triangle. */
5639 /* If both edges are valid (`searchpoint' is on the other side of both */
5640 /* edges), one of the two is chosen by drawing a line perpendicular to */
5641 /* the entry edge (whose endpoints are `forg' and `fdest') passing through */
5642 /* `fapex'. Depending on which side of this perpendicular `searchpoint' */
5643 /* falls on, an exit edge is chosen. */
5645 /* This implementation is empirically faster than the Guibas and Stolfi */
5646 /* point location routine (which I originally used), which tends to spiral */
5647 /* in toward its target. */
5649 /* Returns ONVERTEX if the point lies on an existing vertex. `searchtri' */
5650 /* is a handle whose origin is the existing vertex. */
5652 /* Returns ONEDGE if the point lies on a mesh edge. `searchtri' is a */
5653 /* handle whose primary edge is the edge on which the point lies. */
5655 /* Returns INTRIANGLE if the point lies strictly within a triangle. */
5656 /* `searchtri' is a handle on the triangle that contains the point. */
5658 /* Returns OUTSIDE if the point lies outside the mesh. `searchtri' is a */
5659 /* handle whose primary edge the point is to the right of. This might */
5660 /* occur when the circumcenter of a triangle falls just slightly outside */
5661 /* the mesh due to floating-point roundoff error. It also occurs when */
5662 /* seeking a hole or region point that a foolish user has placed outside */
5665 /* WARNING: This routine is designed for convex triangulations, and will */
5666 /* not generally work after the holes and concavities have been carved. */
5667 /* However, it can still be used to find the circumcenter of a triangle, as */
5668 /* long as the search is begun from the triangle in question. */
5670 /*****************************************************************************/
5672 enum locateresult preciselocate( searchpoint, searchtri )
5674 struct triedge *searchtri;
5676 struct triedge backtracktri;
5677 point forg, fdest, fapex;
5679 REAL orgorient, destorient;
5681 triangle ptr; /* Temporary variable used by sym(). */
5683 if ( verbose > 2 ) {
5684 printf( " Searching for point (%.12g, %.12g).\n",
5685 searchpoint[0], searchpoint[1] );
5688 org( *searchtri, forg );
5689 dest( *searchtri, fdest );
5690 apex( *searchtri, fapex );
5692 if ( verbose > 2 ) {
5693 printf( " At (%.12g, %.12g) (%.12g, %.12g) (%.12g, %.12g)\n",
5694 forg[0], forg[1], fdest[0], fdest[1], fapex[0], fapex[1] );
5696 /* Check whether the apex is the point we seek. */
5697 if ( ( fapex[0] == searchpoint[0] ) && ( fapex[1] == searchpoint[1] ) ) {
5698 lprevself( *searchtri );
5701 /* Does the point lie on the other side of the line defined by the */
5702 /* triangle edge opposite the triangle's destination? */
5703 destorient = counterclockwise( forg, fapex, searchpoint );
5704 /* Does the point lie on the other side of the line defined by the */
5705 /* triangle edge opposite the triangle's origin? */
5706 orgorient = counterclockwise( fapex, fdest, searchpoint );
5707 if ( destorient > 0.0 ) {
5708 if ( orgorient > 0.0 ) {
5709 /* Move left if the inner product of (fapex - searchpoint) and */
5710 /* (fdest - forg) is positive. This is equivalent to drawing */
5711 /* a line perpendicular to the line (forg, fdest) passing */
5712 /* through `fapex', and determining which side of this line */
5713 /* `searchpoint' falls on. */
5714 moveleft = ( fapex[0] - searchpoint[0] ) * ( fdest[0] - forg[0] ) +
5715 ( fapex[1] - searchpoint[1] ) * ( fdest[1] - forg[1] ) > 0.0;
5722 if ( orgorient > 0.0 ) {
5726 /* The point we seek must be on the boundary of or inside this */
5728 if ( destorient == 0.0 ) {
5729 lprevself( *searchtri );
5732 if ( orgorient == 0.0 ) {
5733 lnextself( *searchtri );
5740 /* Move to another triangle. Leave a trace `backtracktri' in case */
5741 /* floating-point roundoff or some such bogey causes us to walk */
5742 /* off a boundary of the triangulation. We can just bounce off */
5743 /* the boundary as if it were an elastic band. */
5745 lprev( *searchtri, backtracktri );
5749 lnext( *searchtri, backtracktri );
5752 sym( backtracktri, *searchtri );
5754 /* Check for walking off the edge. */
5755 if ( searchtri->tri == dummytri ) {
5757 triedgecopy( backtracktri, *searchtri );
5761 apex( *searchtri, fapex );
5762 /* Check if the point really is beyond the triangulation boundary. */
5763 destorient = counterclockwise( forg, fapex, searchpoint );
5764 orgorient = counterclockwise( fapex, fdest, searchpoint );
5765 if ( ( orgorient < 0.0 ) && ( destorient < 0.0 ) ) {
5770 apex( *searchtri, fapex );
5775 /*****************************************************************************/
5777 /* locate() Find a triangle or edge containing a given point. */
5779 /* Searching begins from one of: the input `searchtri', a recently */
5780 /* encountered triangle `recenttri', or from a triangle chosen from a */
5781 /* random sample. The choice is made by determining which triangle's */
5782 /* origin is closest to the point we are searcing for. Normally, */
5783 /* `searchtri' should be a handle on the convex hull of the triangulation. */
5785 /* Details on the random sampling method can be found in the Mucke, Saias, */
5786 /* and Zhu paper cited in the header of this code. */
5788 /* On completion, `searchtri' is a triangle that contains `searchpoint'. */
5790 /* Returns ONVERTEX if the point lies on an existing vertex. `searchtri' */
5791 /* is a handle whose origin is the existing vertex. */
5793 /* Returns ONEDGE if the point lies on a mesh edge. `searchtri' is a */
5794 /* handle whose primary edge is the edge on which the point lies. */
5796 /* Returns INTRIANGLE if the point lies strictly within a triangle. */
5797 /* `searchtri' is a handle on the triangle that contains the point. */
5799 /* Returns OUTSIDE if the point lies outside the mesh. `searchtri' is a */
5800 /* handle whose primary edge the point is to the right of. This might */
5801 /* occur when the circumcenter of a triangle falls just slightly outside */
5802 /* the mesh due to floating-point roundoff error. It also occurs when */
5803 /* seeking a hole or region point that a foolish user has placed outside */
5806 /* WARNING: This routine is designed for convex triangulations, and will */
5807 /* not generally work after the holes and concavities have been carved. */
5809 /*****************************************************************************/
5811 enum locateresult locate( searchpoint, searchtri )
5813 struct triedge *searchtri;
5817 struct triedge sampletri;
5819 unsigned long alignptr;
5820 REAL searchdist, dist;
5822 long sampleblocks, samplesperblock, samplenum;
5825 triangle ptr; /* Temporary variable used by sym(). */
5827 if ( verbose > 2 ) {
5828 printf( " Randomly sampling for a triangle near point (%.12g, %.12g).\n",
5829 searchpoint[0], searchpoint[1] );
5831 /* Record the distance from the suggested starting triangle to the */
5832 /* point we seek. */
5833 org( *searchtri, torg );
5834 searchdist = ( searchpoint[0] - torg[0] ) * ( searchpoint[0] - torg[0] )
5835 + ( searchpoint[1] - torg[1] ) * ( searchpoint[1] - torg[1] );
5836 if ( verbose > 2 ) {
5837 printf( " Boundary triangle has origin (%.12g, %.12g).\n",
5841 /* If a recently encountered triangle has been recorded and has not been */
5842 /* deallocated, test it as a good starting point. */
5843 if ( recenttri.tri != (triangle *) NULL ) {
5844 if ( recenttri.tri[3] != (triangle) NULL ) {
5845 org( recenttri, torg );
5846 if ( ( torg[0] == searchpoint[0] ) && ( torg[1] == searchpoint[1] ) ) {
5847 triedgecopy( recenttri, *searchtri );
5850 dist = ( searchpoint[0] - torg[0] ) * ( searchpoint[0] - torg[0] )
5851 + ( searchpoint[1] - torg[1] ) * ( searchpoint[1] - torg[1] );
5852 if ( dist < searchdist ) {
5853 triedgecopy( recenttri, *searchtri );
5855 if ( verbose > 2 ) {
5856 printf( " Choosing recent triangle with origin (%.12g, %.12g).\n",
5863 /* The number of random samples taken is proportional to the cube root of */
5864 /* the number of triangles in the mesh. The next bit of code assumes */
5865 /* that the number of triangles increases monotonically. */
5866 while ( SAMPLEFACTOR * samples * samples * samples < triangles.items ) {
5869 triblocks = ( triangles.maxitems + TRIPERBLOCK - 1 ) / TRIPERBLOCK;
5870 samplesperblock = 1 + ( samples / triblocks );
5871 sampleblocks = samples / samplesperblock;
5872 sampleblock = triangles.firstblock;
5873 sampletri.orient = 0;
5874 for ( i = 0; i < sampleblocks; i++ ) {
5875 alignptr = (unsigned long) ( sampleblock + 1 );
5876 firsttri = (triangle *) ( alignptr + (unsigned long) triangles.alignbytes
5877 - ( alignptr % (unsigned long) triangles.alignbytes ) );
5878 for ( j = 0; j < samplesperblock; j++ ) {
5879 if ( i == triblocks - 1 ) {
5880 samplenum = randomnation( (int)
5881 ( triangles.maxitems - ( i * TRIPERBLOCK ) ) );
5884 samplenum = randomnation( TRIPERBLOCK );
5886 sampletri.tri = (triangle *)
5887 ( firsttri + ( samplenum * triangles.itemwords ) );
5888 if ( sampletri.tri[3] != (triangle) NULL ) {
5889 org( sampletri, torg );
5890 dist = ( searchpoint[0] - torg[0] ) * ( searchpoint[0] - torg[0] )
5891 + ( searchpoint[1] - torg[1] ) * ( searchpoint[1] - torg[1] );
5892 if ( dist < searchdist ) {
5893 triedgecopy( sampletri, *searchtri );
5895 if ( verbose > 2 ) {
5896 printf( " Choosing triangle with origin (%.12g, %.12g).\n",
5902 sampleblock = (VOID **) *sampleblock;
5905 org( *searchtri, torg );
5906 dest( *searchtri, tdest );
5907 /* Check the starting triangle's vertices. */
5908 if ( ( torg[0] == searchpoint[0] ) && ( torg[1] == searchpoint[1] ) ) {
5911 if ( ( tdest[0] == searchpoint[0] ) && ( tdest[1] == searchpoint[1] ) ) {
5912 lnextself( *searchtri );
5915 /* Orient `searchtri' to fit the preconditions of calling preciselocate(). */
5916 ahead = counterclockwise( torg, tdest, searchpoint );
5917 if ( ahead < 0.0 ) {
5918 /* Turn around so that `searchpoint' is to the left of the */
5919 /* edge specified by `searchtri'. */
5920 symself( *searchtri );
5922 else if ( ahead == 0.0 ) {
5923 /* Check if `searchpoint' is between `torg' and `tdest'. */
5924 if ( ( ( torg[0] < searchpoint[0] ) == ( searchpoint[0] < tdest[0] ) )
5925 && ( ( torg[1] < searchpoint[1] ) == ( searchpoint[1] < tdest[1] ) ) ) {
5929 return preciselocate( searchpoint, searchtri );
5934 /********* Point location routines end here *********/
5936 /********* Mesh transformation routines begin here *********/
5940 /*****************************************************************************/
5942 /* insertshelle() Create a new shell edge and insert it between two */
5945 /* The new shell edge is inserted at the edge described by the handle */
5946 /* `tri'. Its vertices are properly initialized. The marker `shellemark' */
5947 /* is applied to the shell edge and, if appropriate, its vertices. */
5949 /*****************************************************************************/
5951 void insertshelle( tri, shellemark )
5952 struct triedge *tri; /* Edge at which to insert the new shell edge. */
5953 int shellemark; /* Marker for the new shell edge. */
5955 struct triedge oppotri;
5956 struct edge newshelle;
5957 point triorg, tridest;
5958 triangle ptr; /* Temporary variable used by sym(). */
5959 shelle sptr; /* Temporary variable used by tspivot(). */
5961 /* Mark points if possible. */
5962 org( *tri, triorg );
5963 dest( *tri, tridest );
5964 if ( pointmark( triorg ) == 0 ) {
5965 setpointmark( triorg, shellemark );
5967 if ( pointmark( tridest ) == 0 ) {
5968 setpointmark( tridest, shellemark );
5970 /* Check if there's already a shell edge here. */
5971 tspivot( *tri, newshelle );
5972 if ( newshelle.sh == dummysh ) {
5973 /* Make new shell edge and initialize its vertices. */
5974 makeshelle( &newshelle );
5975 setsorg( newshelle, tridest );
5976 setsdest( newshelle, triorg );
5977 /* Bond new shell edge to the two triangles it is sandwiched between. */
5978 /* Note that the facing triangle `oppotri' might be equal to */
5979 /* `dummytri' (outer space), but the new shell edge is bonded to it */
5981 tsbond( *tri, newshelle );
5982 sym( *tri, oppotri );
5983 ssymself( newshelle );
5984 tsbond( oppotri, newshelle );
5985 setmark( newshelle, shellemark );
5986 if ( verbose > 2 ) {
5987 printf( " Inserting new " );
5988 printshelle( &newshelle );
5992 if ( mark( newshelle ) == 0 ) {
5993 setmark( newshelle, shellemark );
5998 /*****************************************************************************/
6002 /* A "local transformation" replaces a small set of triangles with another */
6003 /* set of triangles. This may or may not involve inserting or deleting a */
6006 /* The term "casing" is used to describe the set of triangles that are */
6007 /* attached to the triangles being transformed, but are not transformed */
6008 /* themselves. Think of the casing as a fixed hollow structure inside */
6009 /* which all the action happens. A "casing" is only defined relative to */
6010 /* a single transformation; each occurrence of a transformation will */
6011 /* involve a different casing. */
6013 /* A "shell" is similar to a "casing". The term "shell" describes the set */
6014 /* of shell edges (if any) that are attached to the triangles being */
6015 /* transformed. However, I sometimes use "shell" to refer to a single */
6016 /* shell edge, so don't get confused. */
6018 /*****************************************************************************/
6020 /*****************************************************************************/
6022 /* flip() Transform two triangles to two different triangles by flipping */
6023 /* an edge within a quadrilateral. */
6025 /* Imagine the original triangles, abc and bad, oriented so that the */
6026 /* shared edge ab lies in a horizontal plane, with the point b on the left */
6027 /* and the point a on the right. The point c lies below the edge, and the */
6028 /* point d lies above the edge. The `flipedge' handle holds the edge ab */
6029 /* of triangle abc, and is directed left, from vertex a to vertex b. */
6031 /* The triangles abc and bad are deleted and replaced by the triangles cdb */
6032 /* and dca. The triangles that represent abc and bad are NOT deallocated; */
6033 /* they are reused for dca and cdb, respectively. Hence, any handles that */
6034 /* may have held the original triangles are still valid, although not */
6035 /* directed as they were before. */
6037 /* Upon completion of this routine, the `flipedge' handle holds the edge */
6038 /* dc of triangle dca, and is directed down, from vertex d to vertex c. */
6039 /* (Hence, the two triangles have rotated counterclockwise.) */
6041 /* WARNING: This transformation is geometrically valid only if the */
6042 /* quadrilateral adbc is convex. Furthermore, this transformation is */
6043 /* valid only if there is not a shell edge between the triangles abc and */
6044 /* bad. This routine does not check either of these preconditions, and */
6045 /* it is the responsibility of the calling routine to ensure that they are */
6046 /* met. If they are not, the streets shall be filled with wailing and */
6047 /* gnashing of teeth. */
6049 /*****************************************************************************/
6051 void flip( flipedge )
6052 struct triedge *flipedge; /* Handle for the triangle abc. */
6054 struct triedge botleft, botright;
6055 struct triedge topleft, topright;
6057 struct triedge botlcasing, botrcasing;
6058 struct triedge toplcasing, toprcasing;
6059 struct edge botlshelle, botrshelle;
6060 struct edge toplshelle, toprshelle;
6061 point leftpoint, rightpoint, botpoint;
6063 triangle ptr; /* Temporary variable used by sym(). */
6064 shelle sptr; /* Temporary variable used by tspivot(). */
6066 /* Identify the vertices of the quadrilateral. */
6067 org( *flipedge, rightpoint );
6068 dest( *flipedge, leftpoint );
6069 apex( *flipedge, botpoint );
6070 sym( *flipedge, top );
6072 if ( top.tri == dummytri ) {
6073 printf( "Internal error in flip(): Attempt to flip on boundary.\n" );
6074 lnextself( *flipedge );
6077 if ( checksegments ) {
6078 tspivot( *flipedge, toplshelle );
6079 if ( toplshelle.sh != dummysh ) {
6080 printf( "Internal error in flip(): Attempt to flip a segment.\n" );
6081 lnextself( *flipedge );
6085 #endif /* SELF_CHECK */
6086 apex( top, farpoint );
6088 /* Identify the casing of the quadrilateral. */
6089 lprev( top, topleft );
6090 sym( topleft, toplcasing );
6091 lnext( top, topright );
6092 sym( topright, toprcasing );
6093 lnext( *flipedge, botleft );
6094 sym( botleft, botlcasing );
6095 lprev( *flipedge, botright );
6096 sym( botright, botrcasing );
6097 /* Rotate the quadrilateral one-quarter turn counterclockwise. */
6098 bond( topleft, botlcasing );
6099 bond( botleft, botrcasing );
6100 bond( botright, toprcasing );
6101 bond( topright, toplcasing );
6103 if ( checksegments ) {
6104 /* Check for shell edges and rebond them to the quadrilateral. */
6105 tspivot( topleft, toplshelle );
6106 tspivot( botleft, botlshelle );
6107 tspivot( botright, botrshelle );
6108 tspivot( topright, toprshelle );
6109 if ( toplshelle.sh == dummysh ) {
6110 tsdissolve( topright );
6113 tsbond( topright, toplshelle );
6115 if ( botlshelle.sh == dummysh ) {
6116 tsdissolve( topleft );
6119 tsbond( topleft, botlshelle );
6121 if ( botrshelle.sh == dummysh ) {
6122 tsdissolve( botleft );
6125 tsbond( botleft, botrshelle );
6127 if ( toprshelle.sh == dummysh ) {
6128 tsdissolve( botright );
6131 tsbond( botright, toprshelle );
6135 /* New point assignments for the rotated quadrilateral. */
6136 setorg( *flipedge, farpoint );
6137 setdest( *flipedge, botpoint );
6138 setapex( *flipedge, rightpoint );
6139 setorg( top, botpoint );
6140 setdest( top, farpoint );
6141 setapex( top, leftpoint );
6142 if ( verbose > 2 ) {
6143 printf( " Edge flip results in left " );
6144 lnextself( topleft );
6145 printtriangle( &topleft );
6146 printf( " and right " );
6147 printtriangle( flipedge );
6151 /*****************************************************************************/
6153 /* insertsite() Insert a vertex into a Delaunay triangulation, */
6154 /* performing flips as necessary to maintain the Delaunay */
6157 /* The point `insertpoint' is located. If `searchtri.tri' is not NULL, */
6158 /* the search for the containing triangle begins from `searchtri'. If */
6159 /* `searchtri.tri' is NULL, a full point location procedure is called. */
6160 /* If `insertpoint' is found inside a triangle, the triangle is split into */
6161 /* three; if `insertpoint' lies on an edge, the edge is split in two, */
6162 /* thereby splitting the two adjacent triangles into four. Edge flips are */
6163 /* used to restore the Delaunay property. If `insertpoint' lies on an */
6164 /* existing vertex, no action is taken, and the value DUPLICATEPOINT is */
6165 /* returned. On return, `searchtri' is set to a handle whose origin is the */
6166 /* existing vertex. */
6168 /* Normally, the parameter `splitedge' is set to NULL, implying that no */
6169 /* segment should be split. In this case, if `insertpoint' is found to */
6170 /* lie on a segment, no action is taken, and the value VIOLATINGPOINT is */
6171 /* returned. On return, `searchtri' is set to a handle whose primary edge */
6172 /* is the violated segment. */
6174 /* If the calling routine wishes to split a segment by inserting a point in */
6175 /* it, the parameter `splitedge' should be that segment. In this case, */
6176 /* `searchtri' MUST be the triangle handle reached by pivoting from that */
6177 /* segment; no point location is done. */
6179 /* `segmentflaws' and `triflaws' are flags that indicate whether or not */
6180 /* there should be checks for the creation of encroached segments or bad */
6181 /* quality faces. If a newly inserted point encroaches upon segments, */
6182 /* these segments are added to the list of segments to be split if */
6183 /* `segmentflaws' is set. If bad triangles are created, these are added */
6184 /* to the queue if `triflaws' is set. */
6186 /* If a duplicate point or violated segment does not prevent the point */
6187 /* from being inserted, the return value will be ENCROACHINGPOINT if the */
6188 /* point encroaches upon a segment (and checking is enabled), or */
6189 /* SUCCESSFULPOINT otherwise. In either case, `searchtri' is set to a */
6190 /* handle whose origin is the newly inserted vertex. */
6192 /* insertsite() does not use flip() for reasons of speed; some */
6193 /* information can be reused from edge flip to edge flip, like the */
6194 /* locations of shell edges. */
6196 /*****************************************************************************/
6198 enum insertsiteresult insertsite( insertpoint, searchtri, splitedge,
6199 segmentflaws, triflaws )
6201 struct triedge *searchtri;
6202 struct edge *splitedge;
6206 struct triedge horiz;
6208 struct triedge botleft, botright;
6209 struct triedge topleft, topright;
6210 struct triedge newbotleft, newbotright;
6211 struct triedge newtopright;
6212 struct triedge botlcasing, botrcasing;
6213 struct triedge toplcasing, toprcasing;
6214 struct triedge testtri;
6215 struct edge botlshelle, botrshelle;
6216 struct edge toplshelle, toprshelle;
6217 struct edge brokenshelle;
6218 struct edge checkshelle;
6219 struct edge rightedge;
6220 struct edge newedge;
6221 struct edge *encroached;
6223 point leftpoint, rightpoint, botpoint, toppoint, farpoint;
6226 enum insertsiteresult success;
6227 enum locateresult intersect;
6231 triangle ptr; /* Temporary variable used by sym(). */
6232 shelle sptr; /* Temporary variable used by spivot() and tspivot(). */
6234 if ( verbose > 1 ) {
6235 printf( " Inserting (%.12g, %.12g).\n", insertpoint[0], insertpoint[1] );
6237 if ( splitedge == (struct edge *) NULL ) {
6238 /* Find the location of the point to be inserted. Check if a good */
6239 /* starting triangle has already been provided by the caller. */
6240 if ( searchtri->tri == (triangle *) NULL ) {
6241 /* Find a boundary triangle. */
6242 horiz.tri = dummytri;
6245 /* Search for a triangle containing `insertpoint'. */
6246 intersect = locate( insertpoint, &horiz );
6249 /* Start searching from the triangle provided by the caller. */
6250 triedgecopy( *searchtri, horiz );
6251 intersect = preciselocate( insertpoint, &horiz );
6255 /* The calling routine provides the edge in which the point is inserted. */
6256 triedgecopy( *searchtri, horiz );
6259 if ( intersect == ONVERTEX ) {
6260 /* There's already a vertex there. Return in `searchtri' a triangle */
6261 /* whose origin is the existing vertex. */
6262 triedgecopy( horiz, *searchtri );
6263 triedgecopy( horiz, recenttri );
6264 return DUPLICATEPOINT;
6266 if ( ( intersect == ONEDGE ) || ( intersect == OUTSIDE ) ) {
6267 /* The vertex falls on an edge or boundary. */
6268 if ( checksegments && ( splitedge == (struct edge *) NULL ) ) {
6269 /* Check whether the vertex falls on a shell edge. */
6270 tspivot( horiz, brokenshelle );
6271 if ( brokenshelle.sh != dummysh ) {
6272 /* The vertex falls on a shell edge. */
6273 if ( segmentflaws ) {
6274 if ( nobisect == 0 ) {
6275 /* Add the shell edge to the list of encroached segments. */
6276 encroached = (struct edge *) poolalloc( &badsegments );
6277 shellecopy( brokenshelle, *encroached );
6279 else if ( ( nobisect == 1 ) && ( intersect == ONEDGE ) ) {
6280 /* This segment may be split only if it is an internal boundary. */
6281 sym( horiz, testtri );
6282 if ( testtri.tri != dummytri ) {
6283 /* Add the shell edge to the list of encroached segments. */
6284 encroached = (struct edge *) poolalloc( &badsegments );
6285 shellecopy( brokenshelle, *encroached );
6289 /* Return a handle whose primary edge contains the point, */
6290 /* which has not been inserted. */
6291 triedgecopy( horiz, *searchtri );
6292 triedgecopy( horiz, recenttri );
6293 return VIOLATINGPOINT;
6296 /* Insert the point on an edge, dividing one triangle into two (if */
6297 /* the edge lies on a boundary) or two triangles into four. */
6298 lprev( horiz, botright );
6299 sym( botright, botrcasing );
6300 sym( horiz, topright );
6301 /* Is there a second triangle? (Or does this edge lie on a boundary?) */
6302 mirrorflag = topright.tri != dummytri;
6304 lnextself( topright );
6305 sym( topright, toprcasing );
6306 maketriangle( &newtopright );
6309 /* Splitting the boundary edge increases the number of boundary edges. */
6312 maketriangle( &newbotright );
6314 /* Set the vertices of changed and new triangles. */
6315 org( horiz, rightpoint );
6316 dest( horiz, leftpoint );
6317 apex( horiz, botpoint );
6318 setorg( newbotright, botpoint );
6319 setdest( newbotright, rightpoint );
6320 setapex( newbotright, insertpoint );
6321 setorg( horiz, insertpoint );
6322 for ( i = 0; i < eextras; i++ ) {
6323 /* Set the element attributes of a new triangle. */
6324 setelemattribute( newbotright, i, elemattribute( botright, i ) );
6327 /* Set the area constraint of a new triangle. */
6328 setareabound( newbotright, areabound( botright ) );
6331 dest( topright, toppoint );
6332 setorg( newtopright, rightpoint );
6333 setdest( newtopright, toppoint );
6334 setapex( newtopright, insertpoint );
6335 setorg( topright, insertpoint );
6336 for ( i = 0; i < eextras; i++ ) {
6337 /* Set the element attributes of another new triangle. */
6338 setelemattribute( newtopright, i, elemattribute( topright, i ) );
6341 /* Set the area constraint of another new triangle. */
6342 setareabound( newtopright, areabound( topright ) );
6346 /* There may be shell edges that need to be bonded */
6347 /* to the new triangle(s). */
6348 if ( checksegments ) {
6349 tspivot( botright, botrshelle );
6350 if ( botrshelle.sh != dummysh ) {
6351 tsdissolve( botright );
6352 tsbond( newbotright, botrshelle );
6355 tspivot( topright, toprshelle );
6356 if ( toprshelle.sh != dummysh ) {
6357 tsdissolve( topright );
6358 tsbond( newtopright, toprshelle );
6363 /* Bond the new triangle(s) to the surrounding triangles. */
6364 bond( newbotright, botrcasing );
6365 lprevself( newbotright );
6366 bond( newbotright, botright );
6367 lprevself( newbotright );
6369 bond( newtopright, toprcasing );
6370 lnextself( newtopright );
6371 bond( newtopright, topright );
6372 lnextself( newtopright );
6373 bond( newtopright, newbotright );
6376 if ( splitedge != (struct edge *) NULL ) {
6377 /* Split the shell edge into two. */
6378 setsdest( *splitedge, insertpoint );
6379 ssymself( *splitedge );
6380 spivot( *splitedge, rightedge );
6381 insertshelle( &newbotright, mark( *splitedge ) );
6382 tspivot( newbotright, newedge );
6383 sbond( *splitedge, newedge );
6384 ssymself( newedge );
6385 sbond( newedge, rightedge );
6386 ssymself( *splitedge );
6390 if ( counterclockwise( rightpoint, leftpoint, botpoint ) < 0.0 ) {
6391 printf( "Internal error in insertsite():\n" );
6392 printf( " Clockwise triangle prior to edge point insertion (bottom).\n" );
6395 if ( counterclockwise( leftpoint, rightpoint, toppoint ) < 0.0 ) {
6396 printf( "Internal error in insertsite():\n" );
6397 printf( " Clockwise triangle prior to edge point insertion (top).\n" );
6399 if ( counterclockwise( rightpoint, toppoint, insertpoint ) < 0.0 ) {
6400 printf( "Internal error in insertsite():\n" );
6401 printf( " Clockwise triangle after edge point insertion (top right).\n"
6404 if ( counterclockwise( toppoint, leftpoint, insertpoint ) < 0.0 ) {
6405 printf( "Internal error in insertsite():\n" );
6406 printf( " Clockwise triangle after edge point insertion (top left).\n"
6410 if ( counterclockwise( leftpoint, botpoint, insertpoint ) < 0.0 ) {
6411 printf( "Internal error in insertsite():\n" );
6412 printf( " Clockwise triangle after edge point insertion (bottom left).\n"
6415 if ( counterclockwise( botpoint, rightpoint, insertpoint ) < 0.0 ) {
6416 printf( "Internal error in insertsite():\n" );
6418 " Clockwise triangle after edge point insertion (bottom right).\n" );
6420 #endif /* SELF_CHECK */
6421 if ( verbose > 2 ) {
6422 printf( " Updating bottom left " );
6423 printtriangle( &botright );
6425 printf( " Updating top left " );
6426 printtriangle( &topright );
6427 printf( " Creating top right " );
6428 printtriangle( &newtopright );
6430 printf( " Creating bottom right " );
6431 printtriangle( &newbotright );
6434 /* Position `horiz' on the first edge to check for */
6435 /* the Delaunay property. */
6439 /* Insert the point in a triangle, splitting it into three. */
6440 lnext( horiz, botleft );
6441 lprev( horiz, botright );
6442 sym( botleft, botlcasing );
6443 sym( botright, botrcasing );
6444 maketriangle( &newbotleft );
6445 maketriangle( &newbotright );
6447 /* Set the vertices of changed and new triangles. */
6448 org( horiz, rightpoint );
6449 dest( horiz, leftpoint );
6450 apex( horiz, botpoint );
6451 setorg( newbotleft, leftpoint );
6452 setdest( newbotleft, botpoint );
6453 setapex( newbotleft, insertpoint );
6454 setorg( newbotright, botpoint );
6455 setdest( newbotright, rightpoint );
6456 setapex( newbotright, insertpoint );
6457 setapex( horiz, insertpoint );
6458 for ( i = 0; i < eextras; i++ ) {
6459 /* Set the element attributes of the new triangles. */
6460 attrib = elemattribute( horiz, i );
6461 setelemattribute( newbotleft, i, attrib );
6462 setelemattribute( newbotright, i, attrib );
6465 /* Set the area constraint of the new triangles. */
6466 area = areabound( horiz );
6467 setareabound( newbotleft, area );
6468 setareabound( newbotright, area );
6471 /* There may be shell edges that need to be bonded */
6472 /* to the new triangles. */
6473 if ( checksegments ) {
6474 tspivot( botleft, botlshelle );
6475 if ( botlshelle.sh != dummysh ) {
6476 tsdissolve( botleft );
6477 tsbond( newbotleft, botlshelle );
6479 tspivot( botright, botrshelle );
6480 if ( botrshelle.sh != dummysh ) {
6481 tsdissolve( botright );
6482 tsbond( newbotright, botrshelle );
6486 /* Bond the new triangles to the surrounding triangles. */
6487 bond( newbotleft, botlcasing );
6488 bond( newbotright, botrcasing );
6489 lnextself( newbotleft );
6490 lprevself( newbotright );
6491 bond( newbotleft, newbotright );
6492 lnextself( newbotleft );
6493 bond( botleft, newbotleft );
6494 lprevself( newbotright );
6495 bond( botright, newbotright );
6498 if ( counterclockwise( rightpoint, leftpoint, botpoint ) < 0.0 ) {
6499 printf( "Internal error in insertsite():\n" );
6500 printf( " Clockwise triangle prior to point insertion.\n" );
6502 if ( counterclockwise( rightpoint, leftpoint, insertpoint ) < 0.0 ) {
6503 printf( "Internal error in insertsite():\n" );
6504 printf( " Clockwise triangle after point insertion (top).\n" );
6506 if ( counterclockwise( leftpoint, botpoint, insertpoint ) < 0.0 ) {
6507 printf( "Internal error in insertsite():\n" );
6508 printf( " Clockwise triangle after point insertion (left).\n" );
6510 if ( counterclockwise( botpoint, rightpoint, insertpoint ) < 0.0 ) {
6511 printf( "Internal error in insertsite():\n" );
6512 printf( " Clockwise triangle after point insertion (right).\n" );
6514 #endif /* SELF_CHECK */
6515 if ( verbose > 2 ) {
6516 printf( " Updating top " );
6517 printtriangle( &horiz );
6518 printf( " Creating left " );
6519 printtriangle( &newbotleft );
6520 printf( " Creating right " );
6521 printtriangle( &newbotright );
6525 /* The insertion is successful by default, unless an encroached */
6526 /* edge is found. */
6527 success = SUCCESSFULPOINT;
6528 /* Circle around the newly inserted vertex, checking each edge opposite */
6529 /* it for the Delaunay property. Non-Delaunay edges are flipped. */
6530 /* `horiz' is always the edge being checked. `first' marks where to */
6531 /* stop circling. */
6532 org( horiz, first );
6534 dest( horiz, leftpoint );
6535 /* Circle until finished. */
6537 /* By default, the edge will be flipped. */
6539 if ( checksegments ) {
6540 /* Check for a segment, which cannot be flipped. */
6541 tspivot( horiz, checkshelle );
6542 if ( checkshelle.sh != dummysh ) {
6543 /* The edge is a segment and cannot be flipped. */
6546 if ( segmentflaws ) {
6547 /* Does the new point encroach upon this segment? */
6548 if ( checkedge4encroach( &checkshelle ) ) {
6549 success = ENCROACHINGPOINT;
6552 #endif /* not CDT_ONLY */
6556 /* Check if the edge is a boundary edge. */
6558 if ( top.tri == dummytri ) {
6559 /* The edge is a boundary edge and cannot be flipped. */
6563 /* Find the point on the other side of the edge. */
6564 apex( top, farpoint );
6565 /* In the incremental Delaunay triangulation algorithm, any of */
6566 /* `leftpoint', `rightpoint', and `farpoint' could be vertices */
6567 /* of the triangular bounding box. These vertices must be */
6568 /* treated as if they are infinitely distant, even though their */
6569 /* "coordinates" are not. */
6570 if ( ( leftpoint == infpoint1 ) || ( leftpoint == infpoint2 )
6571 || ( leftpoint == infpoint3 ) ) {
6572 /* `leftpoint' is infinitely distant. Check the convexity of */
6573 /* the boundary of the triangulation. 'farpoint' might be */
6574 /* infinite as well, but trust me, this same condition */
6575 /* should be applied. */
6576 doflip = counterclockwise( insertpoint, rightpoint, farpoint ) > 0.0;
6578 else if ( ( rightpoint == infpoint1 ) || ( rightpoint == infpoint2 )
6579 || ( rightpoint == infpoint3 ) ) {
6580 /* `rightpoint' is infinitely distant. Check the convexity of */
6581 /* the boundary of the triangulation. 'farpoint' might be */
6582 /* infinite as well, but trust me, this same condition */
6583 /* should be applied. */
6584 doflip = counterclockwise( farpoint, leftpoint, insertpoint ) > 0.0;
6586 else if ( ( farpoint == infpoint1 ) || ( farpoint == infpoint2 )
6587 || ( farpoint == infpoint3 ) ) {
6588 /* `farpoint' is infinitely distant and cannot be inside */
6589 /* the circumcircle of the triangle `horiz'. */
6593 /* Test whether the edge is locally Delaunay. */
6594 doflip = incircle( leftpoint, insertpoint, rightpoint, farpoint )
6598 /* We made it! Flip the edge `horiz' by rotating its containing */
6599 /* quadrilateral (the two triangles adjacent to `horiz'). */
6600 /* Identify the casing of the quadrilateral. */
6601 lprev( top, topleft );
6602 sym( topleft, toplcasing );
6603 lnext( top, topright );
6604 sym( topright, toprcasing );
6605 lnext( horiz, botleft );
6606 sym( botleft, botlcasing );
6607 lprev( horiz, botright );
6608 sym( botright, botrcasing );
6609 /* Rotate the quadrilateral one-quarter turn counterclockwise. */
6610 bond( topleft, botlcasing );
6611 bond( botleft, botrcasing );
6612 bond( botright, toprcasing );
6613 bond( topright, toplcasing );
6614 if ( checksegments ) {
6615 /* Check for shell edges and rebond them to the quadrilateral. */
6616 tspivot( topleft, toplshelle );
6617 tspivot( botleft, botlshelle );
6618 tspivot( botright, botrshelle );
6619 tspivot( topright, toprshelle );
6620 if ( toplshelle.sh == dummysh ) {
6621 tsdissolve( topright );
6624 tsbond( topright, toplshelle );
6626 if ( botlshelle.sh == dummysh ) {
6627 tsdissolve( topleft );
6630 tsbond( topleft, botlshelle );
6632 if ( botrshelle.sh == dummysh ) {
6633 tsdissolve( botleft );
6636 tsbond( botleft, botrshelle );
6638 if ( toprshelle.sh == dummysh ) {
6639 tsdissolve( botright );
6642 tsbond( botright, toprshelle );
6645 /* New point assignments for the rotated quadrilateral. */
6646 setorg( horiz, farpoint );
6647 setdest( horiz, insertpoint );
6648 setapex( horiz, rightpoint );
6649 setorg( top, insertpoint );
6650 setdest( top, farpoint );
6651 setapex( top, leftpoint );
6652 for ( i = 0; i < eextras; i++ ) {
6653 /* Take the average of the two triangles' attributes. */
6654 attrib = (REAL)( 0.5 * ( elemattribute( top, i ) + elemattribute( horiz, i ) ) );
6655 setelemattribute( top, i, attrib );
6656 setelemattribute( horiz, i, attrib );
6659 if ( ( areabound( top ) <= 0.0 ) || ( areabound( horiz ) <= 0.0 ) ) {
6663 /* Take the average of the two triangles' area constraints. */
6664 /* This prevents small area constraints from migrating a */
6665 /* long, long way from their original location due to flips. */
6666 area = (REAL)( 0.5 * ( areabound( top ) + areabound( horiz ) ) );
6668 setareabound( top, area );
6669 setareabound( horiz, area );
6672 if ( insertpoint != (point) NULL ) {
6673 if ( counterclockwise( leftpoint, insertpoint, rightpoint ) < 0.0 ) {
6674 printf( "Internal error in insertsite():\n" );
6675 printf( " Clockwise triangle prior to edge flip (bottom).\n" );
6677 /* The following test has been removed because constrainededge() */
6678 /* sometimes generates inverted triangles that insertsite() */
6681 if (counterclockwise(rightpoint, farpoint, leftpoint) < 0.0) {
6682 printf("Internal error in insertsite():\n");
6683 printf(" Clockwise triangle prior to edge flip (top).\n");
6686 if ( counterclockwise( farpoint, leftpoint, insertpoint ) < 0.0 ) {
6687 printf( "Internal error in insertsite():\n" );
6688 printf( " Clockwise triangle after edge flip (left).\n" );
6690 if ( counterclockwise( insertpoint, rightpoint, farpoint ) < 0.0 ) {
6691 printf( "Internal error in insertsite():\n" );
6692 printf( " Clockwise triangle after edge flip (right).\n" );
6695 #endif /* SELF_CHECK */
6696 if ( verbose > 2 ) {
6697 printf( " Edge flip results in left " );
6698 lnextself( topleft );
6699 printtriangle( &topleft );
6700 printf( " and right " );
6701 printtriangle( &horiz );
6703 /* On the next iterations, consider the two edges that were */
6704 /* exposed (this is, are now visible to the newly inserted */
6705 /* point) by the edge flip. */
6707 leftpoint = farpoint;
6712 /* The handle `horiz' is accepted as locally Delaunay. */
6715 /* Check the triangle `horiz' for quality. */
6716 testtriangle( &horiz );
6718 #endif /* not CDT_ONLY */
6719 /* Look for the next edge around the newly inserted point. */
6721 sym( horiz, testtri );
6722 /* Check for finishing a complete revolution about the new point, or */
6723 /* falling off the edge of the triangulation. The latter will */
6724 /* happen when a point is inserted at a boundary. */
6725 if ( ( leftpoint == first ) || ( testtri.tri == dummytri ) ) {
6726 /* We're done. Return a triangle whose origin is the new point. */
6727 lnext( horiz, *searchtri );
6728 lnext( horiz, recenttri );
6731 /* Finish finding the next edge around the newly inserted point. */
6732 lnext( testtri, horiz );
6733 rightpoint = leftpoint;
6734 dest( horiz, leftpoint );
6739 /*****************************************************************************/
6741 /* triangulatepolygon() Find the Delaunay triangulation of a polygon that */
6742 /* has a certain "nice" shape. This includes the */
6743 /* polygons that result from deletion of a point or */
6744 /* insertion of a segment. */
6746 /* This is a conceptually difficult routine. The starting assumption is */
6747 /* that we have a polygon with n sides. n - 1 of these sides are currently */
6748 /* represented as edges in the mesh. One side, called the "base", need not */
6751 /* Inside the polygon is a structure I call a "fan", consisting of n - 1 */
6752 /* triangles that share a common origin. For each of these triangles, the */
6753 /* edge opposite the origin is one of the sides of the polygon. The */
6754 /* primary edge of each triangle is the edge directed from the origin to */
6755 /* the destination; note that this is not the same edge that is a side of */
6756 /* the polygon. `firstedge' is the primary edge of the first triangle. */
6757 /* From there, the triangles follow in counterclockwise order about the */
6758 /* polygon, until `lastedge', the primary edge of the last triangle. */
6759 /* `firstedge' and `lastedge' are probably connected to other triangles */
6760 /* beyond the extremes of the fan, but their identity is not important, as */
6761 /* long as the fan remains connected to them. */
6763 /* Imagine the polygon oriented so that its base is at the bottom. This */
6764 /* puts `firstedge' on the far right, and `lastedge' on the far left. */
6765 /* The right vertex of the base is the destination of `firstedge', and the */
6766 /* left vertex of the base is the apex of `lastedge'. */
6768 /* The challenge now is to find the right sequence of edge flips to */
6769 /* transform the fan into a Delaunay triangulation of the polygon. Each */
6770 /* edge flip effectively removes one triangle from the fan, committing it */
6771 /* to the polygon. The resulting polygon has one fewer edge. If `doflip' */
6772 /* is set, the final flip will be performed, resulting in a fan of one */
6773 /* (useless?) triangle. If `doflip' is not set, the final flip is not */
6774 /* performed, resulting in a fan of two triangles, and an unfinished */
6775 /* triangular polygon that is not yet filled out with a single triangle. */
6776 /* On completion of the routine, `lastedge' is the last remaining triangle, */
6777 /* or the leftmost of the last two. */
6779 /* Although the flips are performed in the order described above, the */
6780 /* decisions about what flips to perform are made in precisely the reverse */
6781 /* order. The recursive triangulatepolygon() procedure makes a decision, */
6782 /* uses up to two recursive calls to triangulate the "subproblems" */
6783 /* (polygons with fewer edges), and then performs an edge flip. */
6785 /* The "decision" it makes is which vertex of the polygon should be */
6786 /* connected to the base. This decision is made by testing every possible */
6787 /* vertex. Once the best vertex is found, the two edges that connect this */
6788 /* vertex to the base become the bases for two smaller polygons. These */
6789 /* are triangulated recursively. Unfortunately, this approach can take */
6790 /* O(n^2) time not only in the worst case, but in many common cases. It's */
6791 /* rarely a big deal for point deletion, where n is rarely larger than ten, */
6792 /* but it could be a big deal for segment insertion, especially if there's */
6793 /* a lot of long segments that each cut many triangles. I ought to code */
6794 /* a faster algorithm some time. */
6796 /* The `edgecount' parameter is the number of sides of the polygon, */
6797 /* including its base. `triflaws' is a flag that determines whether the */
6798 /* new triangles should be tested for quality, and enqueued if they are */
6801 /*****************************************************************************/
6803 void triangulatepolygon( firstedge, lastedge, edgecount, doflip, triflaws )
6804 struct triedge *firstedge;
6805 struct triedge *lastedge;
6810 struct triedge testtri;
6811 struct triedge besttri;
6812 struct triedge tempedge;
6813 point leftbasepoint, rightbasepoint;
6818 triangle ptr; /* Temporary variable used by sym(), onext(), and oprev(). */
6820 /* Identify the base vertices. */
6821 apex( *lastedge, leftbasepoint );
6822 dest( *firstedge, rightbasepoint );
6823 if ( verbose > 2 ) {
6824 printf( " Triangulating interior polygon at edge\n" );
6825 printf( " (%.12g, %.12g) (%.12g, %.12g)\n", leftbasepoint[0],
6826 leftbasepoint[1], rightbasepoint[0], rightbasepoint[1] );
6828 /* Find the best vertex to connect the base to. */
6829 onext( *firstedge, besttri );
6830 dest( besttri, bestpoint );
6831 triedgecopy( besttri, testtri );
6833 for ( i = 2; i <= edgecount - 2; i++ ) {
6834 onextself( testtri );
6835 dest( testtri, testpoint );
6836 /* Is this a better vertex? */
6837 if ( incircle( leftbasepoint, rightbasepoint, bestpoint, testpoint ) > 0.0 ) {
6838 triedgecopy( testtri, besttri );
6839 bestpoint = testpoint;
6843 if ( verbose > 2 ) {
6844 printf( " Connecting edge to (%.12g, %.12g)\n", bestpoint[0],
6847 if ( bestnumber > 1 ) {
6848 /* Recursively triangulate the smaller polygon on the right. */
6849 oprev( besttri, tempedge );
6850 triangulatepolygon( firstedge, &tempedge, bestnumber + 1, 1, triflaws );
6852 if ( bestnumber < edgecount - 2 ) {
6853 /* Recursively triangulate the smaller polygon on the left. */
6854 sym( besttri, tempedge );
6855 triangulatepolygon( &besttri, lastedge, edgecount - bestnumber, 1,
6857 /* Find `besttri' again; it may have been lost to edge flips. */
6858 sym( tempedge, besttri );
6861 /* Do one final edge flip. */
6865 /* Check the quality of the newly committed triangle. */
6866 sym( besttri, testtri );
6867 testtriangle( &testtri );
6869 #endif /* not CDT_ONLY */
6871 /* Return the base triangle. */
6872 triedgecopy( besttri, *lastedge );
6875 /*****************************************************************************/
6877 /* deletesite() Delete a vertex from a Delaunay triangulation, ensuring */
6878 /* that the triangulation remains Delaunay. */
6880 /* The origin of `deltri' is deleted. The union of the triangles adjacent */
6881 /* to this point is a polygon, for which the Delaunay triangulation is */
6882 /* found. Two triangles are removed from the mesh. */
6884 /* Only interior points that do not lie on segments (shell edges) or */
6885 /* boundaries may be deleted. */
6887 /*****************************************************************************/
6891 void deletesite( deltri )
6892 struct triedge *deltri;
6894 struct triedge countingtri;
6895 struct triedge firstedge, lastedge;
6896 struct triedge deltriright;
6897 struct triedge lefttri, righttri;
6898 struct triedge leftcasing, rightcasing;
6899 struct edge leftshelle, rightshelle;
6903 triangle ptr; /* Temporary variable used by sym(), onext(), and oprev(). */
6904 shelle sptr; /* Temporary variable used by tspivot(). */
6906 org( *deltri, delpoint );
6907 if ( verbose > 1 ) {
6908 printf( " Deleting (%.12g, %.12g).\n", delpoint[0], delpoint[1] );
6910 pointdealloc( delpoint );
6912 /* Count the degree of the point being deleted. */
6913 onext( *deltri, countingtri );
6915 while ( !triedgeequal( *deltri, countingtri ) ) {
6917 if ( countingtri.tri == dummytri ) {
6918 printf( "Internal error in deletesite():\n" );
6919 printf( " Attempt to delete boundary point.\n" );
6922 #endif /* SELF_CHECK */
6924 onextself( countingtri );
6928 if ( edgecount < 3 ) {
6929 printf( "Internal error in deletesite():\n Point has degree %d.\n",
6933 #endif /* SELF_CHECK */
6934 if ( edgecount > 3 ) {
6935 /* Triangulate the polygon defined by the union of all triangles */
6936 /* adjacent to the point being deleted. Check the quality of */
6937 /* the resulting triangles. */
6938 onext( *deltri, firstedge );
6939 oprev( *deltri, lastedge );
6940 triangulatepolygon( &firstedge, &lastedge, edgecount, 0, !nobisect );
6942 /* Splice out two triangles. */
6943 lprev( *deltri, deltriright );
6944 dnext( *deltri, lefttri );
6945 sym( lefttri, leftcasing );
6946 oprev( deltriright, righttri );
6947 sym( righttri, rightcasing );
6948 bond( *deltri, leftcasing );
6949 bond( deltriright, rightcasing );
6950 tspivot( lefttri, leftshelle );
6951 if ( leftshelle.sh != dummysh ) {
6952 tsbond( *deltri, leftshelle );
6954 tspivot( righttri, rightshelle );
6955 if ( rightshelle.sh != dummysh ) {
6956 tsbond( deltriright, rightshelle );
6959 /* Set the new origin of `deltri' and check its quality. */
6960 org( lefttri, neworg );
6961 setorg( *deltri, neworg );
6963 testtriangle( deltri );
6966 /* Delete the two spliced-out triangles. */
6967 triangledealloc( lefttri.tri );
6968 triangledealloc( righttri.tri );
6971 #endif /* not CDT_ONLY */
6975 /********* Mesh transformation routines end here *********/
6977 /********* Divide-and-conquer Delaunay triangulation begins here *********/
6981 /*****************************************************************************/
6983 /* The divide-and-conquer bounding box */
6985 /* I originally implemented the divide-and-conquer and incremental Delaunay */
6986 /* triangulations using the edge-based data structure presented by Guibas */
6987 /* and Stolfi. Switching to a triangle-based data structure doubled the */
6988 /* speed. However, I had to think of a few extra tricks to maintain the */
6989 /* elegance of the original algorithms. */
6991 /* The "bounding box" used by my variant of the divide-and-conquer */
6992 /* algorithm uses one triangle for each edge of the convex hull of the */
6993 /* triangulation. These bounding triangles all share a common apical */
6994 /* vertex, which is represented by NULL and which represents nothing. */
6995 /* The bounding triangles are linked in a circular fan about this NULL */
6996 /* vertex, and the edges on the convex hull of the triangulation appear */
6997 /* opposite the NULL vertex. You might find it easiest to imagine that */
6998 /* the NULL vertex is a point in 3D space behind the center of the */
6999 /* triangulation, and that the bounding triangles form a sort of cone. */
7001 /* This bounding box makes it easy to represent degenerate cases. For */
7002 /* instance, the triangulation of two vertices is a single edge. This edge */
7003 /* is represented by two bounding box triangles, one on each "side" of the */
7004 /* edge. These triangles are also linked together in a fan about the NULL */
7007 /* The bounding box also makes it easy to traverse the convex hull, as the */
7008 /* divide-and-conquer algorithm needs to do. */
7010 /*****************************************************************************/
7012 /*****************************************************************************/
7014 /* pointsort() Sort an array of points by x-coordinate, using the */
7015 /* y-coordinate as a secondary key. */
7017 /* Uses quicksort. Randomized O(n log n) time. No, I did not make any of */
7018 /* the usual quicksort mistakes. */
7020 /*****************************************************************************/
7022 void pointsort( sortarray, arraysize )
7028 REAL pivotx, pivoty;
7031 if ( arraysize == 2 ) {
7032 /* Recursive base case. */
7033 if ( ( sortarray[0][0] > sortarray[1][0] ) ||
7034 ( ( sortarray[0][0] == sortarray[1][0] ) &&
7035 ( sortarray[0][1] > sortarray[1][1] ) ) ) {
7036 temp = sortarray[1];
7037 sortarray[1] = sortarray[0];
7038 sortarray[0] = temp;
7042 /* Choose a random pivot to split the array. */
7043 pivot = (int) randomnation( arraysize );
7044 pivotx = sortarray[pivot][0];
7045 pivoty = sortarray[pivot][1];
7046 /* Split the array. */
7049 while ( left < right ) {
7050 /* Search for a point whose x-coordinate is too large for the left. */
7053 } while ( ( left <= right ) && ( ( sortarray[left][0] < pivotx ) ||
7054 ( ( sortarray[left][0] == pivotx ) &&
7055 ( sortarray[left][1] < pivoty ) ) ) );
7056 /* Search for a point whose x-coordinate is too small for the right. */
7059 } while ( ( left <= right ) && ( ( sortarray[right][0] > pivotx ) ||
7060 ( ( sortarray[right][0] == pivotx ) &&
7061 ( sortarray[right][1] > pivoty ) ) ) );
7062 if ( left < right ) {
7063 /* Swap the left and right points. */
7064 temp = sortarray[left];
7065 sortarray[left] = sortarray[right];
7066 sortarray[right] = temp;
7070 /* Recursively sort the left subset. */
7071 pointsort( sortarray, left );
7073 if ( right < arraysize - 2 ) {
7074 /* Recursively sort the right subset. */
7075 pointsort( &sortarray[right + 1], arraysize - right - 1 );
7079 /*****************************************************************************/
7081 /* pointmedian() An order statistic algorithm, almost. Shuffles an array */
7082 /* of points so that the first `median' points occur */
7083 /* lexicographically before the remaining points. */
7085 /* Uses the x-coordinate as the primary key if axis == 0; the y-coordinate */
7086 /* if axis == 1. Very similar to the pointsort() procedure, but runs in */
7087 /* randomized linear time. */
7089 /*****************************************************************************/
7091 void pointmedian( sortarray, arraysize, median, axis )
7099 REAL pivot1, pivot2;
7102 if ( arraysize == 2 ) {
7103 /* Recursive base case. */
7104 if ( ( sortarray[0][axis] > sortarray[1][axis] ) ||
7105 ( ( sortarray[0][axis] == sortarray[1][axis] ) &&
7106 ( sortarray[0][1 - axis] > sortarray[1][1 - axis] ) ) ) {
7107 temp = sortarray[1];
7108 sortarray[1] = sortarray[0];
7109 sortarray[0] = temp;
7113 /* Choose a random pivot to split the array. */
7114 pivot = (int) randomnation( arraysize );
7115 pivot1 = sortarray[pivot][axis];
7116 pivot2 = sortarray[pivot][1 - axis];
7117 /* Split the array. */
7120 while ( left < right ) {
7121 /* Search for a point whose x-coordinate is too large for the left. */
7124 } while ( ( left <= right ) && ( ( sortarray[left][axis] < pivot1 ) ||
7125 ( ( sortarray[left][axis] == pivot1 ) &&
7126 ( sortarray[left][1 - axis] < pivot2 ) ) ) );
7127 /* Search for a point whose x-coordinate is too small for the right. */
7130 } while ( ( left <= right ) && ( ( sortarray[right][axis] > pivot1 ) ||
7131 ( ( sortarray[right][axis] == pivot1 ) &&
7132 ( sortarray[right][1 - axis] > pivot2 ) ) ) );
7133 if ( left < right ) {
7134 /* Swap the left and right points. */
7135 temp = sortarray[left];
7136 sortarray[left] = sortarray[right];
7137 sortarray[right] = temp;
7140 /* Unlike in pointsort(), at most one of the following */
7141 /* conditionals is true. */
7142 if ( left > median ) {
7143 /* Recursively shuffle the left subset. */
7144 pointmedian( sortarray, left, median, axis );
7146 if ( right < median - 1 ) {
7147 /* Recursively shuffle the right subset. */
7148 pointmedian( &sortarray[right + 1], arraysize - right - 1,
7149 median - right - 1, axis );
7153 /*****************************************************************************/
7155 /* alternateaxes() Sorts the points as appropriate for the divide-and- */
7156 /* conquer algorithm with alternating cuts. */
7158 /* Partitions by x-coordinate if axis == 0; by y-coordinate if axis == 1. */
7159 /* For the base case, subsets containing only two or three points are */
7160 /* always sorted by x-coordinate. */
7162 /*****************************************************************************/
7164 void alternateaxes( sortarray, arraysize, axis )
7171 divider = arraysize >> 1;
7172 if ( arraysize <= 3 ) {
7173 /* Recursive base case: subsets of two or three points will be */
7174 /* handled specially, and should always be sorted by x-coordinate. */
7177 /* Partition with a horizontal or vertical cut. */
7178 pointmedian( sortarray, arraysize, divider, axis );
7179 /* Recursively partition the subsets with a cross cut. */
7180 if ( arraysize - divider >= 2 ) {
7181 if ( divider >= 2 ) {
7182 alternateaxes( sortarray, divider, 1 - axis );
7184 alternateaxes( &sortarray[divider], arraysize - divider, 1 - axis );
7188 /*****************************************************************************/
7190 /* mergehulls() Merge two adjacent Delaunay triangulations into a */
7191 /* single Delaunay triangulation. */
7193 /* This is similar to the algorithm given by Guibas and Stolfi, but uses */
7194 /* a triangle-based, rather than edge-based, data structure. */
7196 /* The algorithm walks up the gap between the two triangulations, knitting */
7197 /* them together. As they are merged, some of their bounding triangles */
7198 /* are converted into real triangles of the triangulation. The procedure */
7199 /* pulls each hull's bounding triangles apart, then knits them together */
7200 /* like the teeth of two gears. The Delaunay property determines, at each */
7201 /* step, whether the next "tooth" is a bounding triangle of the left hull */
7202 /* or the right. When a bounding triangle becomes real, its apex is */
7203 /* changed from NULL to a real point. */
7205 /* Only two new triangles need to be allocated. These become new bounding */
7206 /* triangles at the top and bottom of the seam. They are used to connect */
7207 /* the remaining bounding triangles (those that have not been converted */
7208 /* into real triangles) into a single fan. */
7210 /* On entry, `farleft' and `innerleft' are bounding triangles of the left */
7211 /* triangulation. The origin of `farleft' is the leftmost vertex, and */
7212 /* the destination of `innerleft' is the rightmost vertex of the */
7213 /* triangulation. Similarly, `innerright' and `farright' are bounding */
7214 /* triangles of the right triangulation. The origin of `innerright' and */
7215 /* destination of `farright' are the leftmost and rightmost vertices. */
7217 /* On completion, the origin of `farleft' is the leftmost vertex of the */
7218 /* merged triangulation, and the destination of `farright' is the rightmost */
7221 /*****************************************************************************/
7223 void mergehulls( farleft, innerleft, innerright, farright, axis )
7224 struct triedge *farleft;
7225 struct triedge *innerleft;
7226 struct triedge *innerright;
7227 struct triedge *farright;
7230 struct triedge leftcand, rightcand;
7231 struct triedge baseedge;
7232 struct triedge nextedge;
7233 struct triedge sidecasing, topcasing, outercasing;
7234 struct triedge checkedge;
7235 point innerleftdest;
7236 point innerrightorg;
7237 point innerleftapex, innerrightapex;
7238 point farleftpt, farrightpt;
7239 point farleftapex, farrightapex;
7240 point lowerleft, lowerright;
7241 point upperleft, upperright;
7246 int leftfinished, rightfinished;
7247 triangle ptr; /* Temporary variable used by sym(). */
7249 dest( *innerleft, innerleftdest );
7250 apex( *innerleft, innerleftapex );
7251 org( *innerright, innerrightorg );
7252 apex( *innerright, innerrightapex );
7253 /* Special treatment for horizontal cuts. */
7254 if ( dwyer && ( axis == 1 ) ) {
7255 org( *farleft, farleftpt );
7256 apex( *farleft, farleftapex );
7257 dest( *farright, farrightpt );
7258 apex( *farright, farrightapex );
7259 /* The pointers to the extremal points are shifted to point to the */
7260 /* topmost and bottommost point of each hull, rather than the */
7261 /* leftmost and rightmost points. */
7262 while ( farleftapex[1] < farleftpt[1] ) {
7263 lnextself( *farleft );
7264 symself( *farleft );
7265 farleftpt = farleftapex;
7266 apex( *farleft, farleftapex );
7268 sym( *innerleft, checkedge );
7269 apex( checkedge, checkvertex );
7270 while ( checkvertex[1] > innerleftdest[1] ) {
7271 lnext( checkedge, *innerleft );
7272 innerleftapex = innerleftdest;
7273 innerleftdest = checkvertex;
7274 sym( *innerleft, checkedge );
7275 apex( checkedge, checkvertex );
7277 while ( innerrightapex[1] < innerrightorg[1] ) {
7278 lnextself( *innerright );
7279 symself( *innerright );
7280 innerrightorg = innerrightapex;
7281 apex( *innerright, innerrightapex );
7283 sym( *farright, checkedge );
7284 apex( checkedge, checkvertex );
7285 while ( checkvertex[1] > farrightpt[1] ) {
7286 lnext( checkedge, *farright );
7287 farrightapex = farrightpt;
7288 farrightpt = checkvertex;
7289 sym( *farright, checkedge );
7290 apex( checkedge, checkvertex );
7293 /* Find a line tangent to and below both hulls. */
7296 /* Make innerleftdest the "bottommost" point of the left hull. */
7297 if ( counterclockwise( innerleftdest, innerleftapex, innerrightorg ) > 0.0 ) {
7298 lprevself( *innerleft );
7299 symself( *innerleft );
7300 innerleftdest = innerleftapex;
7301 apex( *innerleft, innerleftapex );
7304 /* Make innerrightorg the "bottommost" point of the right hull. */
7305 if ( counterclockwise( innerrightapex, innerrightorg, innerleftdest ) > 0.0 ) {
7306 lnextself( *innerright );
7307 symself( *innerright );
7308 innerrightorg = innerrightapex;
7309 apex( *innerright, innerrightapex );
7312 } while ( changemade );
7313 /* Find the two candidates to be the next "gear tooth". */
7314 sym( *innerleft, leftcand );
7315 sym( *innerright, rightcand );
7316 /* Create the bottom new bounding triangle. */
7317 maketriangle( &baseedge );
7318 /* Connect it to the bounding boxes of the left and right triangulations. */
7319 bond( baseedge, *innerleft );
7320 lnextself( baseedge );
7321 bond( baseedge, *innerright );
7322 lnextself( baseedge );
7323 setorg( baseedge, innerrightorg );
7324 setdest( baseedge, innerleftdest );
7325 /* Apex is intentionally left NULL. */
7326 if ( verbose > 2 ) {
7327 printf( " Creating base bounding " );
7328 printtriangle( &baseedge );
7330 /* Fix the extreme triangles if necessary. */
7331 org( *farleft, farleftpt );
7332 if ( innerleftdest == farleftpt ) {
7333 lnext( baseedge, *farleft );
7335 dest( *farright, farrightpt );
7336 if ( innerrightorg == farrightpt ) {
7337 lprev( baseedge, *farright );
7339 /* The vertices of the current knitting edge. */
7340 lowerleft = innerleftdest;
7341 lowerright = innerrightorg;
7342 /* The candidate vertices for knitting. */
7343 apex( leftcand, upperleft );
7344 apex( rightcand, upperright );
7345 /* Walk up the gap between the two triangulations, knitting them together. */
7347 /* Have we reached the top? (This isn't quite the right question, */
7348 /* because even though the left triangulation might seem finished now, */
7349 /* moving up on the right triangulation might reveal a new point of */
7350 /* the left triangulation. And vice-versa.) */
7351 leftfinished = counterclockwise( upperleft, lowerleft, lowerright ) <= 0.0;
7352 rightfinished = counterclockwise( upperright, lowerleft, lowerright ) <= 0.0;
7353 if ( leftfinished && rightfinished ) {
7354 /* Create the top new bounding triangle. */
7355 maketriangle( &nextedge );
7356 setorg( nextedge, lowerleft );
7357 setdest( nextedge, lowerright );
7358 /* Apex is intentionally left NULL. */
7359 /* Connect it to the bounding boxes of the two triangulations. */
7360 bond( nextedge, baseedge );
7361 lnextself( nextedge );
7362 bond( nextedge, rightcand );
7363 lnextself( nextedge );
7364 bond( nextedge, leftcand );
7365 if ( verbose > 2 ) {
7366 printf( " Creating top bounding " );
7367 printtriangle( &baseedge );
7369 /* Special treatment for horizontal cuts. */
7370 if ( dwyer && ( axis == 1 ) ) {
7371 org( *farleft, farleftpt );
7372 apex( *farleft, farleftapex );
7373 dest( *farright, farrightpt );
7374 apex( *farright, farrightapex );
7375 sym( *farleft, checkedge );
7376 apex( checkedge, checkvertex );
7377 /* The pointers to the extremal points are restored to the leftmost */
7378 /* and rightmost points (rather than topmost and bottommost). */
7379 while ( checkvertex[0] < farleftpt[0] ) {
7380 lprev( checkedge, *farleft );
7381 farleftapex = farleftpt;
7382 farleftpt = checkvertex;
7383 sym( *farleft, checkedge );
7384 apex( checkedge, checkvertex );
7386 while ( farrightapex[0] > farrightpt[0] ) {
7387 lprevself( *farright );
7388 symself( *farright );
7389 farrightpt = farrightapex;
7390 apex( *farright, farrightapex );
7395 /* Consider eliminating edges from the left triangulation. */
7396 if ( !leftfinished ) {
7397 /* What vertex would be exposed if an edge were deleted? */
7398 lprev( leftcand, nextedge );
7399 symself( nextedge );
7400 apex( nextedge, nextapex );
7401 /* If nextapex is NULL, then no vertex would be exposed; the */
7402 /* triangulation would have been eaten right through. */
7403 if ( nextapex != (point) NULL ) {
7404 /* Check whether the edge is Delaunay. */
7405 badedge = incircle( lowerleft, lowerright, upperleft, nextapex ) > 0.0;
7407 /* Eliminate the edge with an edge flip. As a result, the */
7408 /* left triangulation will have one more boundary triangle. */
7409 lnextself( nextedge );
7410 sym( nextedge, topcasing );
7411 lnextself( nextedge );
7412 sym( nextedge, sidecasing );
7413 bond( nextedge, topcasing );
7414 bond( leftcand, sidecasing );
7415 lnextself( leftcand );
7416 sym( leftcand, outercasing );
7417 lprevself( nextedge );
7418 bond( nextedge, outercasing );
7419 /* Correct the vertices to reflect the edge flip. */
7420 setorg( leftcand, lowerleft );
7421 setdest( leftcand, NULL );
7422 setapex( leftcand, nextapex );
7423 setorg( nextedge, NULL );
7424 setdest( nextedge, upperleft );
7425 setapex( nextedge, nextapex );
7426 /* Consider the newly exposed vertex. */
7427 upperleft = nextapex;
7428 /* What vertex would be exposed if another edge were deleted? */
7429 triedgecopy( sidecasing, nextedge );
7430 apex( nextedge, nextapex );
7431 if ( nextapex != (point) NULL ) {
7432 /* Check whether the edge is Delaunay. */
7433 badedge = incircle( lowerleft, lowerright, upperleft, nextapex )
7437 /* Avoid eating right through the triangulation. */
7443 /* Consider eliminating edges from the right triangulation. */
7444 if ( !rightfinished ) {
7445 /* What vertex would be exposed if an edge were deleted? */
7446 lnext( rightcand, nextedge );
7447 symself( nextedge );
7448 apex( nextedge, nextapex );
7449 /* If nextapex is NULL, then no vertex would be exposed; the */
7450 /* triangulation would have been eaten right through. */
7451 if ( nextapex != (point) NULL ) {
7452 /* Check whether the edge is Delaunay. */
7453 badedge = incircle( lowerleft, lowerright, upperright, nextapex ) > 0.0;
7455 /* Eliminate the edge with an edge flip. As a result, the */
7456 /* right triangulation will have one more boundary triangle. */
7457 lprevself( nextedge );
7458 sym( nextedge, topcasing );
7459 lprevself( nextedge );
7460 sym( nextedge, sidecasing );
7461 bond( nextedge, topcasing );
7462 bond( rightcand, sidecasing );
7463 lprevself( rightcand );
7464 sym( rightcand, outercasing );
7465 lnextself( nextedge );
7466 bond( nextedge, outercasing );
7467 /* Correct the vertices to reflect the edge flip. */
7468 setorg( rightcand, NULL );
7469 setdest( rightcand, lowerright );
7470 setapex( rightcand, nextapex );
7471 setorg( nextedge, upperright );
7472 setdest( nextedge, NULL );
7473 setapex( nextedge, nextapex );
7474 /* Consider the newly exposed vertex. */
7475 upperright = nextapex;
7476 /* What vertex would be exposed if another edge were deleted? */
7477 triedgecopy( sidecasing, nextedge );
7478 apex( nextedge, nextapex );
7479 if ( nextapex != (point) NULL ) {
7480 /* Check whether the edge is Delaunay. */
7481 badedge = incircle( lowerleft, lowerright, upperright, nextapex )
7485 /* Avoid eating right through the triangulation. */
7491 if ( leftfinished || ( !rightfinished &&
7492 ( incircle( upperleft, lowerleft, lowerright, upperright ) > 0.0 ) ) ) {
7493 /* Knit the triangulations, adding an edge from `lowerleft' */
7494 /* to `upperright'. */
7495 bond( baseedge, rightcand );
7496 lprev( rightcand, baseedge );
7497 setdest( baseedge, lowerleft );
7498 lowerright = upperright;
7499 sym( baseedge, rightcand );
7500 apex( rightcand, upperright );
7503 /* Knit the triangulations, adding an edge from `upperleft' */
7504 /* to `lowerright'. */
7505 bond( baseedge, leftcand );
7506 lnext( leftcand, baseedge );
7507 setorg( baseedge, lowerright );
7508 lowerleft = upperleft;
7509 sym( baseedge, leftcand );
7510 apex( leftcand, upperleft );
7512 if ( verbose > 2 ) {
7513 printf( " Connecting " );
7514 printtriangle( &baseedge );
7519 /*****************************************************************************/
7521 /* divconqrecurse() Recursively form a Delaunay triangulation by the */
7522 /* divide-and-conquer method. */
7524 /* Recursively breaks down the problem into smaller pieces, which are */
7525 /* knitted together by mergehulls(). The base cases (problems of two or */
7526 /* three points) are handled specially here. */
7528 /* On completion, `farleft' and `farright' are bounding triangles such that */
7529 /* the origin of `farleft' is the leftmost vertex (breaking ties by */
7530 /* choosing the highest leftmost vertex), and the destination of */
7531 /* `farright' is the rightmost vertex (breaking ties by choosing the */
7532 /* lowest rightmost vertex). */
7534 /*****************************************************************************/
7536 void divconqrecurse( sortarray, vertices, axis, farleft, farright )
7540 struct triedge *farleft;
7541 struct triedge *farright;
7543 struct triedge midtri, tri1, tri2, tri3;
7544 struct triedge innerleft, innerright;
7548 if ( verbose > 2 ) {
7549 printf( " Triangulating %d points.\n", vertices );
7551 if ( vertices == 2 ) {
7552 /* The triangulation of two vertices is an edge. An edge is */
7553 /* represented by two bounding triangles. */
7554 maketriangle( farleft );
7555 setorg( *farleft, sortarray[0] );
7556 setdest( *farleft, sortarray[1] );
7557 /* The apex is intentionally left NULL. */
7558 maketriangle( farright );
7559 setorg( *farright, sortarray[1] );
7560 setdest( *farright, sortarray[0] );
7561 /* The apex is intentionally left NULL. */
7562 bond( *farleft, *farright );
7563 lprevself( *farleft );
7564 lnextself( *farright );
7565 bond( *farleft, *farright );
7566 lprevself( *farleft );
7567 lnextself( *farright );
7568 bond( *farleft, *farright );
7569 if ( verbose > 2 ) {
7570 printf( " Creating " );
7571 printtriangle( farleft );
7572 printf( " Creating " );
7573 printtriangle( farright );
7575 /* Ensure that the origin of `farleft' is sortarray[0]. */
7576 lprev( *farright, *farleft );
7579 else if ( vertices == 3 ) {
7580 /* The triangulation of three vertices is either a triangle (with */
7581 /* three bounding triangles) or two edges (with four bounding */
7582 /* triangles). In either case, four triangles are created. */
7583 maketriangle( &midtri );
7584 maketriangle( &tri1 );
7585 maketriangle( &tri2 );
7586 maketriangle( &tri3 );
7587 area = counterclockwise( sortarray[0], sortarray[1], sortarray[2] );
7588 if ( area == 0.0 ) {
7589 /* Three collinear points; the triangulation is two edges. */
7590 setorg( midtri, sortarray[0] );
7591 setdest( midtri, sortarray[1] );
7592 setorg( tri1, sortarray[1] );
7593 setdest( tri1, sortarray[0] );
7594 setorg( tri2, sortarray[2] );
7595 setdest( tri2, sortarray[1] );
7596 setorg( tri3, sortarray[1] );
7597 setdest( tri3, sortarray[2] );
7598 /* All apices are intentionally left NULL. */
7599 bond( midtri, tri1 );
7601 lnextself( midtri );
7605 bond( midtri, tri3 );
7607 lnextself( midtri );
7611 bond( midtri, tri1 );
7613 /* Ensure that the origin of `farleft' is sortarray[0]. */
7614 triedgecopy( tri1, *farleft );
7615 /* Ensure that the destination of `farright' is sortarray[2]. */
7616 triedgecopy( tri2, *farright );
7619 /* The three points are not collinear; the triangulation is one */
7620 /* triangle, namely `midtri'. */
7621 setorg( midtri, sortarray[0] );
7622 setdest( tri1, sortarray[0] );
7623 setorg( tri3, sortarray[0] );
7624 /* Apices of tri1, tri2, and tri3 are left NULL. */
7626 /* The vertices are in counterclockwise order. */
7627 setdest( midtri, sortarray[1] );
7628 setorg( tri1, sortarray[1] );
7629 setdest( tri2, sortarray[1] );
7630 setapex( midtri, sortarray[2] );
7631 setorg( tri2, sortarray[2] );
7632 setdest( tri3, sortarray[2] );
7635 /* The vertices are in clockwise order. */
7636 setdest( midtri, sortarray[2] );
7637 setorg( tri1, sortarray[2] );
7638 setdest( tri2, sortarray[2] );
7639 setapex( midtri, sortarray[1] );
7640 setorg( tri2, sortarray[1] );
7641 setdest( tri3, sortarray[1] );
7643 /* The topology does not depend on how the vertices are ordered. */
7644 bond( midtri, tri1 );
7645 lnextself( midtri );
7646 bond( midtri, tri2 );
7647 lnextself( midtri );
7648 bond( midtri, tri3 );
7658 /* Ensure that the origin of `farleft' is sortarray[0]. */
7659 triedgecopy( tri1, *farleft );
7660 /* Ensure that the destination of `farright' is sortarray[2]. */
7662 triedgecopy( tri2, *farright );
7665 lnext( *farleft, *farright );
7668 if ( verbose > 2 ) {
7669 printf( " Creating " );
7670 printtriangle( &midtri );
7671 printf( " Creating " );
7672 printtriangle( &tri1 );
7673 printf( " Creating " );
7674 printtriangle( &tri2 );
7675 printf( " Creating " );
7676 printtriangle( &tri3 );
7681 /* Split the vertices in half. */
7682 divider = vertices >> 1;
7683 /* Recursively triangulate each half. */
7684 divconqrecurse( sortarray, divider, 1 - axis, farleft, &innerleft );
7685 divconqrecurse( &sortarray[divider], vertices - divider, 1 - axis,
7686 &innerright, farright );
7687 if ( verbose > 1 ) {
7688 printf( " Joining triangulations with %d and %d vertices.\n", divider,
7689 vertices - divider );
7691 /* Merge the two triangulations into one. */
7692 mergehulls( farleft, &innerleft, &innerright, farright, axis );
7696 long removeghosts( startghost )
7697 struct triedge *startghost;
7699 struct triedge searchedge;
7700 struct triedge dissolveedge;
7701 struct triedge deadtri;
7704 triangle ptr; /* Temporary variable used by sym(). */
7707 printf( " Removing ghost triangles.\n" );
7709 /* Find an edge on the convex hull to start point location from. */
7710 lprev( *startghost, searchedge );
7711 symself( searchedge );
7712 dummytri[0] = encode( searchedge );
7713 /* Remove the bounding box and count the convex hull edges. */
7714 triedgecopy( *startghost, dissolveedge );
7718 lnext( dissolveedge, deadtri );
7719 lprevself( dissolveedge );
7720 symself( dissolveedge );
7721 /* If no PSLG is involved, set the boundary markers of all the points */
7722 /* on the convex hull. If a PSLG is used, this step is done later. */
7724 /* Watch out for the case where all the input points are collinear. */
7725 if ( dissolveedge.tri != dummytri ) {
7726 org( dissolveedge, markorg );
7727 if ( pointmark( markorg ) == 0 ) {
7728 setpointmark( markorg, 1 );
7732 /* Remove a bounding triangle from a convex hull triangle. */
7733 dissolve( dissolveedge );
7734 /* Find the next bounding triangle. */
7735 sym( deadtri, dissolveedge );
7736 /* Delete the bounding triangle. */
7737 triangledealloc( deadtri.tri );
7738 } while ( !triedgeequal( dissolveedge, *startghost ) );
7742 /*****************************************************************************/
7744 /* divconqdelaunay() Form a Delaunay triangulation by the divide-and- */
7745 /* conquer method. */
7747 /* Sorts the points, calls a recursive procedure to triangulate them, and */
7748 /* removes the bounding box, setting boundary markers as appropriate. */
7750 /*****************************************************************************/
7752 long divconqdelaunay(){
7754 struct triedge hullleft, hullright;
7758 /* Allocate an array of pointers to points for sorting. */
7759 sortarray = (point *) malloc( inpoints * sizeof( point ) );
7760 if ( sortarray == (point *) NULL ) {
7761 printf( "Error: Out of memory.\n" );
7764 traversalinit( &points );
7765 for ( i = 0; i < inpoints; i++ ) {
7766 sortarray[i] = pointtraverse();
7769 printf( " Sorting points.\n" );
7771 /* Sort the points. */
7772 pointsort( sortarray, inpoints );
7773 /* Discard duplicate points, which can really mess up the algorithm. */
7775 for ( j = 1; j < inpoints; j++ ) {
7776 if ( ( sortarray[i][0] == sortarray[j][0] )
7777 && ( sortarray[i][1] == sortarray[j][1] ) ) {
7780 "Warning: A duplicate point at (%.12g, %.12g) appeared and was ignored.\n",
7781 sortarray[j][0], sortarray[j][1] );
7783 /* Commented out - would eliminate point from output .node file, but causes
7784 a failure if some segment has this point as an endpoint.
7785 setpointmark(sortarray[j], DEADPOINT);
7790 sortarray[i] = sortarray[j];
7795 /* Re-sort the array of points to accommodate alternating cuts. */
7797 if ( i - divider >= 2 ) {
7798 if ( divider >= 2 ) {
7799 alternateaxes( sortarray, divider, 1 );
7801 alternateaxes( &sortarray[divider], i - divider, 1 );
7805 printf( " Forming triangulation.\n" );
7807 /* Form the Delaunay triangulation. */
7808 divconqrecurse( sortarray, i, 0, &hullleft, &hullright );
7811 return removeghosts( &hullleft );
7816 /********* Divide-and-conquer Delaunay triangulation ends here *********/
7818 /********* Incremental Delaunay triangulation begins here *********/
7822 /*****************************************************************************/
7824 /* boundingbox() Form an "infinite" bounding triangle to insert points */
7827 /* The points at "infinity" are assigned finite coordinates, which are used */
7828 /* by the point location routines, but (mostly) ignored by the Delaunay */
7829 /* edge flip routines. */
7831 /*****************************************************************************/
7836 struct triedge inftri; /* Handle for the triangular bounding box. */
7840 printf( " Creating triangular bounding box.\n" );
7842 /* Find the width (or height, whichever is larger) of the triangulation. */
7843 width = xmax - xmin;
7844 if ( ymax - ymin > width ) {
7845 width = ymax - ymin;
7847 if ( width == 0.0 ) {
7850 /* Create the vertices of the bounding box. */
7851 infpoint1 = (point) malloc( points.itembytes );
7852 infpoint2 = (point) malloc( points.itembytes );
7853 infpoint3 = (point) malloc( points.itembytes );
7854 if ( ( infpoint1 == (point) NULL ) || ( infpoint2 == (point) NULL )
7855 || ( infpoint3 == (point) NULL ) ) {
7856 printf( "Error: Out of memory.\n" );
7859 infpoint1[0] = xmin - 50.0 * width;
7860 infpoint1[1] = ymin - 40.0 * width;
7861 infpoint2[0] = xmax + 50.0 * width;
7862 infpoint2[1] = ymin - 40.0 * width;
7863 infpoint3[0] = 0.5 * ( xmin + xmax );
7864 infpoint3[1] = ymax + 60.0 * width;
7866 /* Create the bounding box. */
7867 maketriangle( &inftri );
7868 setorg( inftri, infpoint1 );
7869 setdest( inftri, infpoint2 );
7870 setapex( inftri, infpoint3 );
7871 /* Link dummytri to the bounding box so we can always find an */
7872 /* edge to begin searching (point location) from. */
7873 dummytri[0] = (triangle) inftri.tri;
7874 if ( verbose > 2 ) {
7875 printf( " Creating " );
7876 printtriangle( &inftri );
7880 #endif /* not REDUCED */
7882 /*****************************************************************************/
7884 /* removebox() Remove the "infinite" bounding triangle, setting boundary */
7885 /* markers as appropriate. */
7887 /* The triangular bounding box has three boundary triangles (one for each */
7888 /* side of the bounding box), and a bunch of triangles fanning out from */
7889 /* the three bounding box vertices (one triangle for each edge of the */
7890 /* convex hull of the inner mesh). This routine removes these triangles. */
7892 /*****************************************************************************/
7897 struct triedge deadtri;
7898 struct triedge searchedge;
7899 struct triedge checkedge;
7900 struct triedge nextedge, finaledge, dissolveedge;
7903 triangle ptr; /* Temporary variable used by sym(). */
7906 printf( " Removing triangular bounding box.\n" );
7908 /* Find a boundary triangle. */
7909 nextedge.tri = dummytri;
7910 nextedge.orient = 0;
7911 symself( nextedge );
7912 /* Mark a place to stop. */
7913 lprev( nextedge, finaledge );
7914 lnextself( nextedge );
7915 symself( nextedge );
7916 /* Find a triangle (on the boundary of the point set) that isn't */
7917 /* a bounding box triangle. */
7918 lprev( nextedge, searchedge );
7919 symself( searchedge );
7920 /* Check whether nextedge is another boundary triangle */
7921 /* adjacent to the first one. */
7922 lnext( nextedge, checkedge );
7923 symself( checkedge );
7924 if ( checkedge.tri == dummytri ) {
7925 /* Go on to the next triangle. There are only three boundary */
7926 /* triangles, and this next triangle cannot be the third one, */
7927 /* so it's safe to stop here. */
7928 lprevself( searchedge );
7929 symself( searchedge );
7931 /* Find a new boundary edge to search from, as the current search */
7932 /* edge lies on a bounding box triangle and will be deleted. */
7933 dummytri[0] = encode( searchedge );
7935 while ( !triedgeequal( nextedge, finaledge ) ) {
7937 lprev( nextedge, dissolveedge );
7938 symself( dissolveedge );
7939 /* If not using a PSLG, the vertices should be marked now. */
7940 /* (If using a PSLG, markhull() will do the job.) */
7942 /* Be careful! One must check for the case where all the input */
7943 /* points are collinear, and thus all the triangles are part of */
7944 /* the bounding box. Otherwise, the setpointmark() call below */
7945 /* will cause a bad pointer reference. */
7946 if ( dissolveedge.tri != dummytri ) {
7947 org( dissolveedge, markorg );
7948 if ( pointmark( markorg ) == 0 ) {
7949 setpointmark( markorg, 1 );
7953 /* Disconnect the bounding box triangle from the mesh triangle. */
7954 dissolve( dissolveedge );
7955 lnext( nextedge, deadtri );
7956 sym( deadtri, nextedge );
7957 /* Get rid of the bounding box triangle. */
7958 triangledealloc( deadtri.tri );
7959 /* Do we need to turn the corner? */
7960 if ( nextedge.tri == dummytri ) {
7961 /* Turn the corner. */
7962 triedgecopy( dissolveedge, nextedge );
7965 triangledealloc( finaledge.tri );
7967 free( infpoint1 ); /* Deallocate the bounding box vertices. */
7974 #endif /* not REDUCED */
7976 /*****************************************************************************/
7978 /* incrementaldelaunay() Form a Delaunay triangulation by incrementally */
7979 /* adding vertices. */
7981 /*****************************************************************************/
7985 long incrementaldelaunay(){
7986 struct triedge starttri;
7990 /* Create a triangular bounding box. */
7993 printf( " Incrementally inserting points.\n" );
7995 traversalinit( &points );
7996 pointloop = pointtraverse();
7998 while ( pointloop != (point) NULL ) {
7999 /* Find a boundary triangle to search from. */
8000 starttri.tri = (triangle *) NULL;
8001 if ( insertsite( pointloop, &starttri, (struct edge *) NULL, 0, 0 ) ==
8005 "Warning: A duplicate point at (%.12g, %.12g) appeared and was ignored.\n",
8006 pointloop[0], pointloop[1] );
8008 /* Commented out - would eliminate point from output .node file.
8009 setpointmark(pointloop, DEADPOINT);
8012 pointloop = pointtraverse();
8015 /* Remove the bounding box. */
8019 #endif /* not REDUCED */
8023 /********* Incremental Delaunay triangulation ends here *********/
8025 /********* Sweepline Delaunay triangulation begins here *********/
8031 void eventheapinsert( heap, heapsize, newevent )
8032 struct event **heap;
8034 struct event *newevent;
8036 REAL eventx, eventy;
8041 eventx = newevent->xkey;
8042 eventy = newevent->ykey;
8043 eventnum = heapsize;
8044 notdone = eventnum > 0;
8046 parent = ( eventnum - 1 ) >> 1;
8047 if ( ( heap[parent]->ykey < eventy ) ||
8048 ( ( heap[parent]->ykey == eventy )
8049 && ( heap[parent]->xkey <= eventx ) ) ) {
8053 heap[eventnum] = heap[parent];
8054 heap[eventnum]->heapposition = eventnum;
8057 notdone = eventnum > 0;
8060 heap[eventnum] = newevent;
8061 newevent->heapposition = eventnum;
8064 #endif /* not REDUCED */
8068 void eventheapify( heap, heapsize, eventnum )
8069 struct event **heap;
8073 struct event *thisevent;
8074 REAL eventx, eventy;
8075 int leftchild, rightchild;
8079 thisevent = heap[eventnum];
8080 eventx = thisevent->xkey;
8081 eventy = thisevent->ykey;
8082 leftchild = 2 * eventnum + 1;
8083 notdone = leftchild < heapsize;
8085 if ( ( heap[leftchild]->ykey < eventy ) ||
8086 ( ( heap[leftchild]->ykey == eventy )
8087 && ( heap[leftchild]->xkey < eventx ) ) ) {
8088 smallest = leftchild;
8091 smallest = eventnum;
8093 rightchild = leftchild + 1;
8094 if ( rightchild < heapsize ) {
8095 if ( ( heap[rightchild]->ykey < heap[smallest]->ykey ) ||
8096 ( ( heap[rightchild]->ykey == heap[smallest]->ykey )
8097 && ( heap[rightchild]->xkey < heap[smallest]->xkey ) ) ) {
8098 smallest = rightchild;
8101 if ( smallest == eventnum ) {
8105 heap[eventnum] = heap[smallest];
8106 heap[eventnum]->heapposition = eventnum;
8107 heap[smallest] = thisevent;
8108 thisevent->heapposition = smallest;
8110 eventnum = smallest;
8111 leftchild = 2 * eventnum + 1;
8112 notdone = leftchild < heapsize;
8117 #endif /* not REDUCED */
8121 void eventheapdelete( heap, heapsize, eventnum )
8122 struct event **heap;
8126 struct event *moveevent;
8127 REAL eventx, eventy;
8131 moveevent = heap[heapsize - 1];
8132 if ( eventnum > 0 ) {
8133 eventx = moveevent->xkey;
8134 eventy = moveevent->ykey;
8136 parent = ( eventnum - 1 ) >> 1;
8137 if ( ( heap[parent]->ykey < eventy ) ||
8138 ( ( heap[parent]->ykey == eventy )
8139 && ( heap[parent]->xkey <= eventx ) ) ) {
8143 heap[eventnum] = heap[parent];
8144 heap[eventnum]->heapposition = eventnum;
8147 notdone = eventnum > 0;
8149 } while ( notdone );
8151 heap[eventnum] = moveevent;
8152 moveevent->heapposition = eventnum;
8153 eventheapify( heap, heapsize - 1, eventnum );
8156 #endif /* not REDUCED */
8160 void createeventheap( eventheap, events, freeevents )
8161 struct event ***eventheap;
8162 struct event **events;
8163 struct event **freeevents;
8169 maxevents = ( 3 * inpoints ) / 2;
8170 *eventheap = (struct event **) malloc( maxevents * sizeof( struct event * ) );
8171 if ( *eventheap == (struct event **) NULL ) {
8172 printf( "Error: Out of memory.\n" );
8175 *events = (struct event *) malloc( maxevents * sizeof( struct event ) );
8176 if ( *events == (struct event *) NULL ) {
8177 printf( "Error: Out of memory.\n" );
8180 traversalinit( &points );
8181 for ( i = 0; i < inpoints; i++ ) {
8182 thispoint = pointtraverse();
8183 ( *events )[i].eventptr = (VOID *) thispoint;
8184 ( *events )[i].xkey = thispoint[0];
8185 ( *events )[i].ykey = thispoint[1];
8186 eventheapinsert( *eventheap, i, *events + i );
8188 *freeevents = (struct event *) NULL;
8189 for ( i = maxevents - 1; i >= inpoints; i-- ) {
8190 ( *events )[i].eventptr = (VOID *) *freeevents;
8191 *freeevents = *events + i;
8195 #endif /* not REDUCED */
8199 int rightofhyperbola( fronttri, newsite )
8200 struct triedge *fronttri;
8203 point leftpoint, rightpoint;
8204 REAL dxa, dya, dxb, dyb;
8208 dest( *fronttri, leftpoint );
8209 apex( *fronttri, rightpoint );
8210 if ( ( leftpoint[1] < rightpoint[1] )
8211 || ( ( leftpoint[1] == rightpoint[1] ) && ( leftpoint[0] < rightpoint[0] ) ) ) {
8212 if ( newsite[0] >= rightpoint[0] ) {
8217 if ( newsite[0] <= leftpoint[0] ) {
8221 dxa = leftpoint[0] - newsite[0];
8222 dya = leftpoint[1] - newsite[1];
8223 dxb = rightpoint[0] - newsite[0];
8224 dyb = rightpoint[1] - newsite[1];
8225 return dya * ( dxb * dxb + dyb * dyb ) > dyb * ( dxa * dxa + dya * dya );
8228 #endif /* not REDUCED */
8232 REAL circletop( pa, pb, pc, ccwabc )
8238 REAL xac, yac, xbc, ybc, xab, yab;
8239 REAL aclen2, bclen2, ablen2;
8243 xac = pa[0] - pc[0];
8244 yac = pa[1] - pc[1];
8245 xbc = pb[0] - pc[0];
8246 ybc = pb[1] - pc[1];
8247 xab = pa[0] - pb[0];
8248 yab = pa[1] - pb[1];
8249 aclen2 = xac * xac + yac * yac;
8250 bclen2 = xbc * xbc + ybc * ybc;
8251 ablen2 = xab * xab + yab * yab;
8252 return pc[1] + ( xac * bclen2 - xbc * aclen2 + sqrt( aclen2 * bclen2 * ablen2 ) )
8256 #endif /* not REDUCED */
8260 void check4deadevent( checktri, freeevents, eventheap, heapsize )
8261 struct triedge *checktri;
8262 struct event **freeevents;
8263 struct event **eventheap;
8266 struct event *deadevent;
8270 org( *checktri, eventpoint );
8271 if ( eventpoint != (point) NULL ) {
8272 deadevent = (struct event *) eventpoint;
8273 eventnum = deadevent->heapposition;
8274 deadevent->eventptr = (VOID *) *freeevents;
8275 *freeevents = deadevent;
8276 eventheapdelete( eventheap, *heapsize, eventnum );
8278 setorg( *checktri, NULL );
8282 #endif /* not REDUCED */
8286 struct splaynode *splay( splaytree, searchpoint, searchtri )
8287 struct splaynode *splaytree;
8289 struct triedge *searchtri;
8291 struct splaynode *child, *grandchild;
8292 struct splaynode *lefttree, *righttree;
8293 struct splaynode *leftright;
8295 int rightofroot, rightofchild;
8297 if ( splaytree == (struct splaynode *) NULL ) {
8298 return (struct splaynode *) NULL;
8300 dest( splaytree->keyedge, checkpoint );
8301 if ( checkpoint == splaytree->keydest ) {
8302 rightofroot = rightofhyperbola( &splaytree->keyedge, searchpoint );
8303 if ( rightofroot ) {
8304 triedgecopy( splaytree->keyedge, *searchtri );
8305 child = splaytree->rchild;
8308 child = splaytree->lchild;
8310 if ( child == (struct splaynode *) NULL ) {
8313 dest( child->keyedge, checkpoint );
8314 if ( checkpoint != child->keydest ) {
8315 child = splay( child, searchpoint, searchtri );
8316 if ( child == (struct splaynode *) NULL ) {
8317 if ( rightofroot ) {
8318 splaytree->rchild = (struct splaynode *) NULL;
8321 splaytree->lchild = (struct splaynode *) NULL;
8326 rightofchild = rightofhyperbola( &child->keyedge, searchpoint );
8327 if ( rightofchild ) {
8328 triedgecopy( child->keyedge, *searchtri );
8329 grandchild = splay( child->rchild, searchpoint, searchtri );
8330 child->rchild = grandchild;
8333 grandchild = splay( child->lchild, searchpoint, searchtri );
8334 child->lchild = grandchild;
8336 if ( grandchild == (struct splaynode *) NULL ) {
8337 if ( rightofroot ) {
8338 splaytree->rchild = child->lchild;
8339 child->lchild = splaytree;
8342 splaytree->lchild = child->rchild;
8343 child->rchild = splaytree;
8347 if ( rightofchild ) {
8348 if ( rightofroot ) {
8349 splaytree->rchild = child->lchild;
8350 child->lchild = splaytree;
8353 splaytree->lchild = grandchild->rchild;
8354 grandchild->rchild = splaytree;
8356 child->rchild = grandchild->lchild;
8357 grandchild->lchild = child;
8360 if ( rightofroot ) {
8361 splaytree->rchild = grandchild->lchild;
8362 grandchild->lchild = splaytree;
8365 splaytree->lchild = child->rchild;
8366 child->rchild = splaytree;
8368 child->lchild = grandchild->rchild;
8369 grandchild->rchild = child;
8374 lefttree = splay( splaytree->lchild, searchpoint, searchtri );
8375 righttree = splay( splaytree->rchild, searchpoint, searchtri );
8377 pooldealloc( &splaynodes, (VOID *) splaytree );
8378 if ( lefttree == (struct splaynode *) NULL ) {
8381 else if ( righttree == (struct splaynode *) NULL ) {
8384 else if ( lefttree->rchild == (struct splaynode *) NULL ) {
8385 lefttree->rchild = righttree->lchild;
8386 righttree->lchild = lefttree;
8389 else if ( righttree->lchild == (struct splaynode *) NULL ) {
8390 righttree->lchild = lefttree->rchild;
8391 lefttree->rchild = righttree;
8395 /* printf("Holy Toledo!!!\n"); */
8396 leftright = lefttree->rchild;
8397 while ( leftright->rchild != (struct splaynode *) NULL ) {
8398 leftright = leftright->rchild;
8400 leftright->rchild = righttree;
8406 #endif /* not REDUCED */
8410 struct splaynode *splayinsert( splayroot, newkey, searchpoint )
8411 struct splaynode *splayroot;
8412 struct triedge *newkey;
8415 struct splaynode *newsplaynode;
8417 newsplaynode = (struct splaynode *) poolalloc( &splaynodes );
8418 triedgecopy( *newkey, newsplaynode->keyedge );
8419 dest( *newkey, newsplaynode->keydest );
8420 if ( splayroot == (struct splaynode *) NULL ) {
8421 newsplaynode->lchild = (struct splaynode *) NULL;
8422 newsplaynode->rchild = (struct splaynode *) NULL;
8424 else if ( rightofhyperbola( &splayroot->keyedge, searchpoint ) ) {
8425 newsplaynode->lchild = splayroot;
8426 newsplaynode->rchild = splayroot->rchild;
8427 splayroot->rchild = (struct splaynode *) NULL;
8430 newsplaynode->lchild = splayroot->lchild;
8431 newsplaynode->rchild = splayroot;
8432 splayroot->lchild = (struct splaynode *) NULL;
8434 return newsplaynode;
8437 #endif /* not REDUCED */
8441 struct splaynode *circletopinsert( splayroot, newkey, pa, pb, pc, topy )
8442 struct splaynode *splayroot;
8443 struct triedge *newkey;
8450 REAL xac, yac, xbc, ybc;
8451 REAL aclen2, bclen2;
8452 REAL searchpoint[2];
8453 struct triedge dummytri;
8455 ccwabc = counterclockwise( pa, pb, pc );
8456 xac = pa[0] - pc[0];
8457 yac = pa[1] - pc[1];
8458 xbc = pb[0] - pc[0];
8459 ybc = pb[1] - pc[1];
8460 aclen2 = xac * xac + yac * yac;
8461 bclen2 = xbc * xbc + ybc * ybc;
8462 searchpoint[0] = pc[0] - ( yac * bclen2 - ybc * aclen2 ) / ( 2.0 * ccwabc );
8463 searchpoint[1] = topy;
8464 return splayinsert( splay( splayroot, (point) searchpoint, &dummytri ), newkey,
8465 (point) searchpoint );
8468 #endif /* not REDUCED */
8472 struct splaynode *frontlocate( splayroot, bottommost, searchpoint, searchtri,
8474 struct splaynode *splayroot;
8475 struct triedge *bottommost;
8477 struct triedge *searchtri;
8481 triangle ptr; /* Temporary variable used by onext(). */
8483 triedgecopy( *bottommost, *searchtri );
8484 splayroot = splay( splayroot, searchpoint, searchtri );
8487 while ( !farrightflag && rightofhyperbola( searchtri, searchpoint ) ) {
8488 onextself( *searchtri );
8489 farrightflag = triedgeequal( *searchtri, *bottommost );
8491 *farright = farrightflag;
8495 #endif /* not REDUCED */
8499 long sweeplinedelaunay(){
8500 struct event **eventheap;
8501 struct event *events;
8502 struct event *freeevents;
8503 struct event *nextevent;
8504 struct event *newevent;
8505 struct splaynode *splayroot;
8506 struct triedge bottommost;
8507 struct triedge searchtri;
8508 struct triedge fliptri;
8509 struct triedge lefttri, righttri, farlefttri, farrighttri;
8510 struct triedge inserttri;
8511 point firstpoint, secondpoint;
8512 point nextpoint, lastpoint;
8514 point leftpoint, midpoint, rightpoint;
8515 REAL lefttest, righttest;
8517 int check4events, farrightflag;
8518 triangle ptr; /* Temporary variable used by sym(), onext(), and oprev(). */
8520 poolinit( &splaynodes, sizeof( struct splaynode ), SPLAYNODEPERBLOCK, POINTER,
8522 splayroot = (struct splaynode *) NULL;
8525 printf( " Placing points in event heap.\n" );
8527 createeventheap( &eventheap, &events, &freeevents );
8528 heapsize = inpoints;
8531 printf( " Forming triangulation.\n" );
8533 maketriangle( &lefttri );
8534 maketriangle( &righttri );
8535 bond( lefttri, righttri );
8536 lnextself( lefttri );
8537 lprevself( righttri );
8538 bond( lefttri, righttri );
8539 lnextself( lefttri );
8540 lprevself( righttri );
8541 bond( lefttri, righttri );
8542 firstpoint = (point) eventheap[0]->eventptr;
8543 eventheap[0]->eventptr = (VOID *) freeevents;
8544 freeevents = eventheap[0];
8545 eventheapdelete( eventheap, heapsize, 0 );
8548 if ( heapsize == 0 ) {
8549 printf( "Error: Input points are all identical.\n" );
8552 secondpoint = (point) eventheap[0]->eventptr;
8553 eventheap[0]->eventptr = (VOID *) freeevents;
8554 freeevents = eventheap[0];
8555 eventheapdelete( eventheap, heapsize, 0 );
8557 if ( ( firstpoint[0] == secondpoint[0] )
8558 && ( firstpoint[1] == secondpoint[1] ) ) {
8560 "Warning: A duplicate point at (%.12g, %.12g) appeared and was ignored.\n",
8561 secondpoint[0], secondpoint[1] );
8562 /* Commented out - would eliminate point from output .node file.
8563 setpointmark(secondpoint, DEADPOINT);
8566 } while ( ( firstpoint[0] == secondpoint[0] )
8567 && ( firstpoint[1] == secondpoint[1] ) );
8568 setorg( lefttri, firstpoint );
8569 setdest( lefttri, secondpoint );
8570 setorg( righttri, secondpoint );
8571 setdest( righttri, firstpoint );
8572 lprev( lefttri, bottommost );
8573 lastpoint = secondpoint;
8574 while ( heapsize > 0 ) {
8575 nextevent = eventheap[0];
8576 eventheapdelete( eventheap, heapsize, 0 );
8579 if ( nextevent->xkey < xmin ) {
8580 decode( nextevent->eventptr, fliptri );
8581 oprev( fliptri, farlefttri );
8582 check4deadevent( &farlefttri, &freeevents, eventheap, &heapsize );
8583 onext( fliptri, farrighttri );
8584 check4deadevent( &farrighttri, &freeevents, eventheap, &heapsize );
8586 if ( triedgeequal( farlefttri, bottommost ) ) {
8587 lprev( fliptri, bottommost );
8590 setapex( fliptri, NULL );
8591 lprev( fliptri, lefttri );
8592 lnext( fliptri, righttri );
8593 sym( lefttri, farlefttri );
8595 if ( randomnation( SAMPLERATE ) == 0 ) {
8597 dest( fliptri, leftpoint );
8598 apex( fliptri, midpoint );
8599 org( fliptri, rightpoint );
8600 splayroot = circletopinsert( splayroot, &lefttri, leftpoint, midpoint,
8601 rightpoint, nextevent->ykey );
8605 nextpoint = (point) nextevent->eventptr;
8606 if ( ( nextpoint[0] == lastpoint[0] ) && ( nextpoint[1] == lastpoint[1] ) ) {
8608 "Warning: A duplicate point at (%.12g, %.12g) appeared and was ignored.\n",
8609 nextpoint[0], nextpoint[1] );
8610 /* Commented out - would eliminate point from output .node file.
8611 setpointmark(nextpoint, DEADPOINT);
8616 lastpoint = nextpoint;
8618 splayroot = frontlocate( splayroot, &bottommost, nextpoint, &searchtri,
8621 triedgecopy(bottommost, searchtri);
8623 while (!farrightflag && rightofhyperbola(&searchtri, nextpoint)) {
8624 onextself(searchtri);
8625 farrightflag = triedgeequal(searchtri, bottommost);
8629 check4deadevent( &searchtri, &freeevents, eventheap, &heapsize );
8631 triedgecopy( searchtri, farrighttri );
8632 sym( searchtri, farlefttri );
8633 maketriangle( &lefttri );
8634 maketriangle( &righttri );
8635 dest( farrighttri, connectpoint );
8636 setorg( lefttri, connectpoint );
8637 setdest( lefttri, nextpoint );
8638 setorg( righttri, nextpoint );
8639 setdest( righttri, connectpoint );
8640 bond( lefttri, righttri );
8641 lnextself( lefttri );
8642 lprevself( righttri );
8643 bond( lefttri, righttri );
8644 lnextself( lefttri );
8645 lprevself( righttri );
8646 bond( lefttri, farlefttri );
8647 bond( righttri, farrighttri );
8648 if ( !farrightflag && triedgeequal( farrighttri, bottommost ) ) {
8649 triedgecopy( lefttri, bottommost );
8652 if ( randomnation( SAMPLERATE ) == 0 ) {
8653 splayroot = splayinsert( splayroot, &lefttri, nextpoint );
8655 else if ( randomnation( SAMPLERATE ) == 0 ) {
8656 lnext( righttri, inserttri );
8657 splayroot = splayinsert( splayroot, &inserttri, nextpoint );
8661 nextevent->eventptr = (VOID *) freeevents;
8662 freeevents = nextevent;
8664 if ( check4events ) {
8665 apex( farlefttri, leftpoint );
8666 dest( lefttri, midpoint );
8667 apex( lefttri, rightpoint );
8668 lefttest = counterclockwise( leftpoint, midpoint, rightpoint );
8669 if ( lefttest > 0.0 ) {
8670 newevent = freeevents;
8671 freeevents = (struct event *) freeevents->eventptr;
8672 newevent->xkey = xminextreme;
8673 newevent->ykey = circletop( leftpoint, midpoint, rightpoint,
8675 newevent->eventptr = (VOID *) encode( lefttri );
8676 eventheapinsert( eventheap, heapsize, newevent );
8678 setorg( lefttri, newevent );
8680 apex( righttri, leftpoint );
8681 org( righttri, midpoint );
8682 apex( farrighttri, rightpoint );
8683 righttest = counterclockwise( leftpoint, midpoint, rightpoint );
8684 if ( righttest > 0.0 ) {
8685 newevent = freeevents;
8686 freeevents = (struct event *) freeevents->eventptr;
8687 newevent->xkey = xminextreme;
8688 newevent->ykey = circletop( leftpoint, midpoint, rightpoint,
8690 newevent->eventptr = (VOID *) encode( farrighttri );
8691 eventheapinsert( eventheap, heapsize, newevent );
8693 setorg( farrighttri, newevent );
8698 pooldeinit( &splaynodes );
8699 lprevself( bottommost );
8700 return removeghosts( &bottommost );
8703 #endif /* not REDUCED */
8707 /********* Sweepline Delaunay triangulation ends here *********/
8709 /********* General mesh construction routines begin here *********/
8713 /*****************************************************************************/
8715 /* delaunay() Form a Delaunay triangulation. */
8717 /*****************************************************************************/
8721 initializetrisegpools();
8726 "Constructing Delaunay triangulation by divide-and-conquer method.\n" );
8728 return divconqdelaunay();
8729 #else /* not REDUCED */
8731 printf( "Constructing Delaunay triangulation " );
8732 if ( incremental ) {
8733 printf( "by incremental method.\n" );
8735 else if ( sweepline ) {
8736 printf( "by sweepline method.\n" );
8739 printf( "by divide-and-conquer method.\n" );
8742 if ( incremental ) {
8743 return incrementaldelaunay();
8745 else if ( sweepline ) {
8746 return sweeplinedelaunay();
8749 return divconqdelaunay();
8751 #endif /* not REDUCED */
8754 /*****************************************************************************/
8756 /* reconstruct() Reconstruct a triangulation from its .ele (and possibly */
8757 /* .poly) file. Used when the -r switch is used. */
8759 /* Reads an .ele file and reconstructs the original mesh. If the -p switch */
8760 /* is used, this procedure will also read a .poly file and reconstruct the */
8761 /* shell edges of the original mesh. If the -a switch is used, this */
8762 /* procedure will also read an .area file and set a maximum area constraint */
8763 /* on each triangle. */
8765 /* Points that are not corners of triangles, such as nodes on edges of */
8766 /* subparametric elements, are discarded. */
8768 /* This routine finds the adjacencies between triangles (and shell edges) */
8769 /* by forming one stack of triangles for each vertex. Each triangle is on */
8770 /* three different stacks simultaneously. Each triangle's shell edge */
8771 /* pointers are used to link the items in each stack. This memory-saving */
8772 /* feature makes the code harder to read. The most important thing to keep */
8773 /* in mind is that each triangle is removed from a stack precisely when */
8774 /* the corresponding pointer is adjusted to refer to a shell edge rather */
8775 /* than the next triangle of the stack. */
8777 /*****************************************************************************/
8783 int reconstruct( trianglelist, triangleattriblist, trianglearealist, elements,
8784 corners, attribs, segmentlist, segmentmarkerlist,
8787 REAL *triangleattriblist;
8788 REAL *trianglearealist;
8793 int *segmentmarkerlist;
8794 int numberofsegments;
8796 #else /* not TRILIBRARY */
8798 long reconstruct( elefilename, areafilename, polyfilename, polyfile )
8804 #endif /* not TRILIBRARY */
8810 #else /* not TRILIBRARY */
8813 char inputline[INPUTLINESIZE];
8816 #endif /* not TRILIBRARY */
8817 struct triedge triangleloop;
8818 struct triedge triangleleft;
8819 struct triedge checktri;
8820 struct triedge checkleft;
8821 struct triedge checkneighbor;
8822 struct edge shelleloop;
8823 triangle *vertexarray;
8827 point checkdest, checkapex;
8840 int elementnumber, segmentnumber;
8842 triangle ptr; /* Temporary variable used by sym(). */
8845 inelements = elements;
8846 incorners = corners;
8847 if ( incorners < 3 ) {
8848 printf( "Error: Triangles must have at least 3 points.\n" );
8852 #else /* not TRILIBRARY */
8853 /* Read the triangles from an .ele file. */
8855 printf( "Opening %s.\n", elefilename );
8857 elefile = fopen( elefilename, "r" );
8858 if ( elefile == (FILE *) NULL ) {
8859 printf( " Error: Cannot access file %s.\n", elefilename );
8862 /* Read number of triangles, number of points per triangle, and */
8863 /* number of triangle attributes from .ele file. */
8864 stringptr = readline( inputline, elefile, elefilename );
8865 inelements = (int) strtol( stringptr, &stringptr, 0 );
8866 stringptr = findfield( stringptr );
8867 if ( *stringptr == '\0' ) {
8871 incorners = (int) strtol( stringptr, &stringptr, 0 );
8872 if ( incorners < 3 ) {
8873 printf( "Error: Triangles in %s must have at least 3 points.\n",
8878 stringptr = findfield( stringptr );
8879 if ( *stringptr == '\0' ) {
8883 eextras = (int) strtol( stringptr, &stringptr, 0 );
8885 #endif /* not TRILIBRARY */
8887 initializetrisegpools();
8889 /* Create the triangles. */
8890 for ( elementnumber = 1; elementnumber <= inelements; elementnumber++ ) {
8891 maketriangle( &triangleloop );
8892 /* Mark the triangle as living. */
8893 triangleloop.tri[3] = (triangle) triangleloop.tri;
8898 insegments = numberofsegments;
8899 segmentmarkers = segmentmarkerlist != (int *) NULL;
8900 #else /* not TRILIBRARY */
8901 /* Read number of segments and number of segment */
8902 /* boundary markers from .poly file. */
8903 stringptr = readline( inputline, polyfile, inpolyfilename );
8904 insegments = (int) strtol( stringptr, &stringptr, 0 );
8905 stringptr = findfield( stringptr );
8906 if ( *stringptr == '\0' ) {
8910 segmentmarkers = (int) strtol( stringptr, &stringptr, 0 );
8912 #endif /* not TRILIBRARY */
8914 /* Create the shell edges. */
8915 for ( segmentnumber = 1; segmentnumber <= insegments; segmentnumber++ ) {
8916 makeshelle( &shelleloop );
8917 /* Mark the shell edge as living. */
8918 shelleloop.sh[2] = (shelle) shelleloop.sh;
8925 #else /* not TRILIBRARY */
8927 /* Open an .area file, check for consistency with the .ele file. */
8929 printf( "Opening %s.\n", areafilename );
8931 areafile = fopen( areafilename, "r" );
8932 if ( areafile == (FILE *) NULL ) {
8933 printf( " Error: Cannot access file %s.\n", areafilename );
8936 stringptr = readline( inputline, areafile, areafilename );
8937 areaelements = (int) strtol( stringptr, &stringptr, 0 );
8938 if ( areaelements != inelements ) {
8939 printf( "Error: %s and %s disagree on number of triangles.\n",
8940 elefilename, areafilename );
8944 #endif /* not TRILIBRARY */
8947 printf( "Reconstructing mesh.\n" );
8949 /* Allocate a temporary array that maps each point to some adjacent */
8950 /* triangle. I took care to allocate all the permanent memory for */
8951 /* triangles and shell edges first. */
8952 vertexarray = (triangle *) malloc( points.items * sizeof( triangle ) );
8953 if ( vertexarray == (triangle *) NULL ) {
8954 printf( "Error: Out of memory.\n" );
8957 /* Each point is initially unrepresented. */
8958 for ( i = 0; i < points.items; i++ ) {
8959 vertexarray[i] = (triangle) dummytri;
8963 printf( " Assembling triangles.\n" );
8965 /* Read the triangles from the .ele file, and link */
8966 /* together those that share an edge. */
8967 traversalinit( &triangles );
8968 triangleloop.tri = triangletraverse();
8969 elementnumber = firstnumber;
8970 while ( triangleloop.tri != (triangle *) NULL ) {
8972 /* Copy the triangle's three corners. */
8973 for ( j = 0; j < 3; j++ ) {
8974 corner[j] = trianglelist[pointindex++];
8975 if ( ( corner[j] < firstnumber ) || ( corner[j] >= firstnumber + inpoints ) ) {
8976 printf( "Error: Triangle %d has an invalid vertex index.\n",
8981 #else /* not TRILIBRARY */
8982 /* Read triangle number and the triangle's three corners. */
8983 stringptr = readline( inputline, elefile, elefilename );
8984 for ( j = 0; j < 3; j++ ) {
8985 stringptr = findfield( stringptr );
8986 if ( *stringptr == '\0' ) {
8987 printf( "Error: Triangle %d is missing point %d in %s.\n",
8988 elementnumber, j + 1, elefilename );
8992 corner[j] = (int) strtol( stringptr, &stringptr, 0 );
8993 if ( ( corner[j] < firstnumber ) ||
8994 ( corner[j] >= firstnumber + inpoints ) ) {
8995 printf( "Error: Triangle %d has an invalid vertex index.\n",
9001 #endif /* not TRILIBRARY */
9003 /* Find out about (and throw away) extra nodes. */
9004 for ( j = 3; j < incorners; j++ ) {
9006 killpointindex = trianglelist[pointindex++];
9007 #else /* not TRILIBRARY */
9008 stringptr = findfield( stringptr );
9009 if ( *stringptr != '\0' ) {
9010 killpointindex = (int) strtol( stringptr, &stringptr, 0 );
9011 #endif /* not TRILIBRARY */
9012 if ( ( killpointindex >= firstnumber ) &&
9013 ( killpointindex < firstnumber + inpoints ) ) {
9014 /* Delete the non-corner point if it's not already deleted. */
9015 killpoint = getpoint( killpointindex );
9016 if ( pointmark( killpoint ) != DEADPOINT ) {
9017 pointdealloc( killpoint );
9022 #endif /* not TRILIBRARY */
9025 /* Read the triangle's attributes. */
9026 for ( j = 0; j < eextras; j++ ) {
9028 setelemattribute( triangleloop, j, triangleattriblist[attribindex++] );
9029 #else /* not TRILIBRARY */
9030 stringptr = findfield( stringptr );
9031 if ( *stringptr == '\0' ) {
9032 setelemattribute( triangleloop, j, 0 );
9035 setelemattribute( triangleloop, j,
9036 (REAL) strtod( stringptr, &stringptr ) );
9038 #endif /* not TRILIBRARY */
9043 area = trianglearealist[elementnumber - firstnumber];
9044 #else /* not TRILIBRARY */
9045 /* Read an area constraint from the .area file. */
9046 stringptr = readline( inputline, areafile, areafilename );
9047 stringptr = findfield( stringptr );
9048 if ( *stringptr == '\0' ) {
9049 area = -1.0; /* No constraint on this triangle. */
9052 area = (REAL) strtod( stringptr, &stringptr );
9054 #endif /* not TRILIBRARY */
9055 setareabound( triangleloop, area );
9058 /* Set the triangle's vertices. */
9059 triangleloop.orient = 0;
9060 setorg( triangleloop, getpoint( corner[0] ) );
9061 setdest( triangleloop, getpoint( corner[1] ) );
9062 setapex( triangleloop, getpoint( corner[2] ) );
9063 /* Try linking the triangle to others that share these vertices. */
9064 for ( triangleloop.orient = 0; triangleloop.orient < 3;
9065 triangleloop.orient++ ) {
9066 /* Take the number for the origin of triangleloop. */
9067 aroundpoint = corner[triangleloop.orient];
9068 /* Look for other triangles having this vertex. */
9069 nexttri = vertexarray[aroundpoint - firstnumber];
9070 /* Link the current triangle to the next one in the stack. */
9071 triangleloop.tri[6 + triangleloop.orient] = nexttri;
9072 /* Push the current triangle onto the stack. */
9073 vertexarray[aroundpoint - firstnumber] = encode( triangleloop );
9074 decode( nexttri, checktri );
9075 if ( checktri.tri != dummytri ) {
9076 dest( triangleloop, tdest );
9077 apex( triangleloop, tapex );
9078 /* Look for other triangles that share an edge. */
9080 dest( checktri, checkdest );
9081 apex( checktri, checkapex );
9082 if ( tapex == checkdest ) {
9083 /* The two triangles share an edge; bond them together. */
9084 lprev( triangleloop, triangleleft );
9085 bond( triangleleft, checktri );
9087 if ( tdest == checkapex ) {
9088 /* The two triangles share an edge; bond them together. */
9089 lprev( checktri, checkleft );
9090 bond( triangleloop, checkleft );
9092 /* Find the next triangle in the stack. */
9093 nexttri = checktri.tri[6 + checktri.orient];
9094 decode( nexttri, checktri );
9095 } while ( checktri.tri != dummytri );
9098 triangleloop.tri = triangletraverse();
9104 #else /* not TRILIBRARY */
9109 #endif /* not TRILIBRARY */
9111 hullsize = 0; /* Prepare to count the boundary edges. */
9114 printf( " Marking segments in triangulation.\n" );
9116 /* Read the segments from the .poly file, and link them */
9117 /* to their neighboring triangles. */
9119 traversalinit( &shelles );
9120 shelleloop.sh = shelletraverse();
9121 segmentnumber = firstnumber;
9122 while ( shelleloop.sh != (shelle *) NULL ) {
9124 end[0] = segmentlist[pointindex++];
9125 end[1] = segmentlist[pointindex++];
9126 if ( segmentmarkers ) {
9127 boundmarker = segmentmarkerlist[segmentnumber - firstnumber];
9129 #else /* not TRILIBRARY */
9130 /* Read the endpoints of each segment, and possibly a boundary marker. */
9131 stringptr = readline( inputline, polyfile, inpolyfilename );
9132 /* Skip the first (segment number) field. */
9133 stringptr = findfield( stringptr );
9134 if ( *stringptr == '\0' ) {
9135 printf( "Error: Segment %d has no endpoints in %s.\n", segmentnumber,
9140 end[0] = (int) strtol( stringptr, &stringptr, 0 );
9142 stringptr = findfield( stringptr );
9143 if ( *stringptr == '\0' ) {
9144 printf( "Error: Segment %d is missing its second endpoint in %s.\n",
9145 segmentnumber, polyfilename );
9149 end[1] = (int) strtol( stringptr, &stringptr, 0 );
9151 if ( segmentmarkers ) {
9152 stringptr = findfield( stringptr );
9153 if ( *stringptr == '\0' ) {
9157 boundmarker = (int) strtol( stringptr, &stringptr, 0 );
9160 #endif /* not TRILIBRARY */
9161 for ( j = 0; j < 2; j++ ) {
9162 if ( ( end[j] < firstnumber ) || ( end[j] >= firstnumber + inpoints ) ) {
9163 printf( "Error: Segment %d has an invalid vertex index.\n",
9169 /* set the shell edge's vertices. */
9170 shelleloop.shorient = 0;
9171 setsorg( shelleloop, getpoint( end[0] ) );
9172 setsdest( shelleloop, getpoint( end[1] ) );
9173 setmark( shelleloop, boundmarker );
9174 /* Try linking the shell edge to triangles that share these vertices. */
9175 for ( shelleloop.shorient = 0; shelleloop.shorient < 2;
9176 shelleloop.shorient++ ) {
9177 /* Take the number for the destination of shelleloop. */
9178 aroundpoint = end[1 - shelleloop.shorient];
9179 /* Look for triangles having this vertex. */
9180 prevlink = &vertexarray[aroundpoint - firstnumber];
9181 nexttri = vertexarray[aroundpoint - firstnumber];
9182 decode( nexttri, checktri );
9183 sorg( shelleloop, shorg );
9185 /* Look for triangles having this edge. Note that I'm only */
9186 /* comparing each triangle's destination with the shell edge; */
9187 /* each triangle's apex is handled through a different vertex. */
9188 /* Because each triangle appears on three vertices' lists, each */
9189 /* occurrence of a triangle on a list can (and does) represent */
9190 /* an edge. In this way, most edges are represented twice, and */
9191 /* every triangle-segment bond is represented once. */
9192 while ( notfound && ( checktri.tri != dummytri ) ) {
9193 dest( checktri, checkdest );
9194 if ( shorg == checkdest ) {
9195 /* We have a match. Remove this triangle from the list. */
9196 *prevlink = checktri.tri[6 + checktri.orient];
9197 /* Bond the shell edge to the triangle. */
9198 tsbond( checktri, shelleloop );
9199 /* Check if this is a boundary edge. */
9200 sym( checktri, checkneighbor );
9201 if ( checkneighbor.tri == dummytri ) {
9202 /* The next line doesn't insert a shell edge (because there's */
9203 /* already one there), but it sets the boundary markers of */
9204 /* the existing shell edge and its vertices. */
9205 insertshelle( &checktri, 1 );
9210 /* Find the next triangle in the stack. */
9211 prevlink = &checktri.tri[6 + checktri.orient];
9212 nexttri = checktri.tri[6 + checktri.orient];
9213 decode( nexttri, checktri );
9216 shelleloop.sh = shelletraverse();
9221 /* Mark the remaining edges as not being attached to any shell edge. */
9222 /* Also, count the (yet uncounted) boundary edges. */
9223 for ( i = 0; i < points.items; i++ ) {
9224 /* Search the stack of triangles adjacent to a point. */
9225 nexttri = vertexarray[i];
9226 decode( nexttri, checktri );
9227 while ( checktri.tri != dummytri ) {
9228 /* Find the next triangle in the stack before this */
9229 /* information gets overwritten. */
9230 nexttri = checktri.tri[6 + checktri.orient];
9231 /* No adjacent shell edge. (This overwrites the stack info.) */
9232 tsdissolve( checktri );
9233 sym( checktri, checkneighbor );
9234 if ( checkneighbor.tri == dummytri ) {
9235 insertshelle( &checktri, 1 );
9238 decode( nexttri, checktri );
9242 free( vertexarray );
9246 #endif /* not CDT_ONLY */
9250 /********* General mesh construction routines end here *********/
9252 /********* Segment (shell edge) insertion begins here *********/
9256 /*****************************************************************************/
9258 /* finddirection() Find the first triangle on the path from one point */
9261 /* Finds the triangle that intersects a line segment drawn from the */
9262 /* origin of `searchtri' to the point `endpoint', and returns the result */
9263 /* in `searchtri'. The origin of `searchtri' does not change, even though */
9264 /* the triangle returned may differ from the one passed in. This routine */
9265 /* is used to find the direction to move in to get from one point to */
9268 /* The return value notes whether the destination or apex of the found */
9269 /* triangle is collinear with the two points in question. */
9271 /*****************************************************************************/
9273 enum finddirectionresult finddirection( searchtri, endpoint )
9274 struct triedge *searchtri;
9277 struct triedge checktri;
9279 point leftpoint, rightpoint;
9280 REAL leftccw, rightccw;
9281 int leftflag, rightflag;
9282 triangle ptr; /* Temporary variable used by onext() and oprev(). */
9284 org( *searchtri, startpoint );
9285 dest( *searchtri, rightpoint );
9286 apex( *searchtri, leftpoint );
9287 /* Is `endpoint' to the left? */
9288 leftccw = counterclockwise( endpoint, startpoint, leftpoint );
9289 leftflag = leftccw > 0.0;
9290 /* Is `endpoint' to the right? */
9291 rightccw = counterclockwise( startpoint, endpoint, rightpoint );
9292 rightflag = rightccw > 0.0;
9293 if ( leftflag && rightflag ) {
9294 /* `searchtri' faces directly away from `endpoint'. We could go */
9295 /* left or right. Ask whether it's a triangle or a boundary */
9297 onext( *searchtri, checktri );
9298 if ( checktri.tri == dummytri ) {
9305 while ( leftflag ) {
9306 /* Turn left until satisfied. */
9307 onextself( *searchtri );
9308 if ( searchtri->tri == dummytri ) {
9309 printf( "Internal error in finddirection(): Unable to find a\n" );
9310 printf( " triangle leading from (%.12g, %.12g) to", startpoint[0],
9312 printf( " (%.12g, %.12g).\n", endpoint[0], endpoint[1] );
9315 apex( *searchtri, leftpoint );
9317 leftccw = counterclockwise( endpoint, startpoint, leftpoint );
9318 leftflag = leftccw > 0.0;
9320 while ( rightflag ) {
9321 /* Turn right until satisfied. */
9322 oprevself( *searchtri );
9323 if ( searchtri->tri == dummytri ) {
9324 printf( "Internal error in finddirection(): Unable to find a\n" );
9325 printf( " triangle leading from (%.12g, %.12g) to", startpoint[0],
9327 printf( " (%.12g, %.12g).\n", endpoint[0], endpoint[1] );
9330 dest( *searchtri, rightpoint );
9332 rightccw = counterclockwise( startpoint, endpoint, rightpoint );
9333 rightflag = rightccw > 0.0;
9335 if ( leftccw == 0.0 ) {
9336 return LEFTCOLLINEAR;
9338 else if ( rightccw == 0.0 ) {
9339 return RIGHTCOLLINEAR;
9346 /*****************************************************************************/
9348 /* segmentintersection() Find the intersection of an existing segment */
9349 /* and a segment that is being inserted. Insert */
9350 /* a point at the intersection, splitting an */
9351 /* existing shell edge. */
9353 /* The segment being inserted connects the apex of splittri to endpoint2. */
9354 /* splitshelle is the shell edge being split, and MUST be opposite */
9355 /* splittri. Hence, the edge being split connects the origin and */
9356 /* destination of splittri. */
9358 /* On completion, splittri is a handle having the newly inserted */
9359 /* intersection point as its origin, and endpoint1 as its destination. */
9361 /*****************************************************************************/
9363 void segmentintersection( splittri, splitshelle, endpoint2 )
9364 struct triedge *splittri;
9365 struct edge *splitshelle;
9370 point leftpoint, rightpoint;
9372 enum insertsiteresult success;
9373 enum finddirectionresult collinear;
9379 triangle ptr; /* Temporary variable used by onext(). */
9381 /* Find the other three segment endpoints. */
9382 apex( *splittri, endpoint1 );
9383 org( *splittri, torg );
9384 dest( *splittri, tdest );
9385 /* Segment intersection formulae; see the Antonio reference. */
9386 tx = tdest[0] - torg[0];
9387 ty = tdest[1] - torg[1];
9388 ex = endpoint2[0] - endpoint1[0];
9389 ey = endpoint2[1] - endpoint1[1];
9390 etx = torg[0] - endpoint2[0];
9391 ety = torg[1] - endpoint2[1];
9392 denom = ty * ex - tx * ey;
9393 if ( denom == 0.0 ) {
9394 printf( "Internal error in segmentintersection():" );
9395 printf( " Attempt to find intersection of parallel segments.\n" );
9398 split = ( ey * etx - ex * ety ) / denom;
9399 /* Create the new point. */
9400 newpoint = (point) poolalloc( &points );
9401 /* Interpolate its coordinate and attributes. */
9402 for ( i = 0; i < 2 + nextras; i++ ) {
9403 newpoint[i] = torg[i] + split * ( tdest[i] - torg[i] );
9405 setpointmark( newpoint, mark( *splitshelle ) );
9406 if ( verbose > 1 ) {
9408 " Splitting edge (%.12g, %.12g) (%.12g, %.12g) at (%.12g, %.12g).\n",
9409 torg[0], torg[1], tdest[0], tdest[1], newpoint[0], newpoint[1] );
9411 /* Insert the intersection point. This should always succeed. */
9412 success = insertsite( newpoint, splittri, splitshelle, 0, 0 );
9413 if ( success != SUCCESSFULPOINT ) {
9414 printf( "Internal error in segmentintersection():\n" );
9415 printf( " Failure to split a segment.\n" );
9418 if ( steinerleft > 0 ) {
9421 /* Inserting the point may have caused edge flips. We wish to rediscover */
9422 /* the edge connecting endpoint1 to the new intersection point. */
9423 collinear = finddirection( splittri, endpoint1 );
9424 dest( *splittri, rightpoint );
9425 apex( *splittri, leftpoint );
9426 if ( ( leftpoint[0] == endpoint1[0] ) && ( leftpoint[1] == endpoint1[1] ) ) {
9427 onextself( *splittri );
9429 else if ( ( rightpoint[0] != endpoint1[0] ) ||
9430 ( rightpoint[1] != endpoint1[1] ) ) {
9431 printf( "Internal error in segmentintersection():\n" );
9432 printf( " Topological inconsistency after splitting a segment.\n" );
9435 /* `splittri' should have destination endpoint1. */
9438 /*****************************************************************************/
9440 /* scoutsegment() Scout the first triangle on the path from one endpoint */
9441 /* to another, and check for completion (reaching the */
9442 /* second endpoint), a collinear point, and the */
9443 /* intersection of two segments. */
9445 /* Returns one if the entire segment is successfully inserted, and zero if */
9446 /* the job must be finished by conformingedge() or constrainededge(). */
9448 /* If the first triangle on the path has the second endpoint as its */
9449 /* destination or apex, a shell edge is inserted and the job is done. */
9451 /* If the first triangle on the path has a destination or apex that lies on */
9452 /* the segment, a shell edge is inserted connecting the first endpoint to */
9453 /* the collinear point, and the search is continued from the collinear */
9456 /* If the first triangle on the path has a shell edge opposite its origin, */
9457 /* then there is a segment that intersects the segment being inserted. */
9458 /* Their intersection point is inserted, splitting the shell edge. */
9460 /* Otherwise, return zero. */
9462 /*****************************************************************************/
9464 int scoutsegment( searchtri, endpoint2, newmark )
9465 struct triedge *searchtri;
9469 struct triedge crosstri;
9470 struct edge crossedge;
9471 point leftpoint, rightpoint;
9473 enum finddirectionresult collinear;
9474 shelle sptr; /* Temporary variable used by tspivot(). */
9476 collinear = finddirection( searchtri, endpoint2 );
9477 dest( *searchtri, rightpoint );
9478 apex( *searchtri, leftpoint );
9479 if ( ( ( leftpoint[0] == endpoint2[0] ) && ( leftpoint[1] == endpoint2[1] ) ) ||
9480 ( ( rightpoint[0] == endpoint2[0] ) && ( rightpoint[1] == endpoint2[1] ) ) ) {
9481 /* The segment is already an edge in the mesh. */
9482 if ( ( leftpoint[0] == endpoint2[0] ) && ( leftpoint[1] == endpoint2[1] ) ) {
9483 lprevself( *searchtri );
9485 /* Insert a shell edge, if there isn't already one there. */
9486 insertshelle( searchtri, newmark );
9489 else if ( collinear == LEFTCOLLINEAR ) {
9490 /* We've collided with a point between the segment's endpoints. */
9491 /* Make the collinear point be the triangle's origin. */
9492 lprevself( *searchtri );
9493 insertshelle( searchtri, newmark );
9494 /* Insert the remainder of the segment. */
9495 return scoutsegment( searchtri, endpoint2, newmark );
9497 else if ( collinear == RIGHTCOLLINEAR ) {
9498 /* We've collided with a point between the segment's endpoints. */
9499 insertshelle( searchtri, newmark );
9500 /* Make the collinear point be the triangle's origin. */
9501 lnextself( *searchtri );
9502 /* Insert the remainder of the segment. */
9503 return scoutsegment( searchtri, endpoint2, newmark );
9506 lnext( *searchtri, crosstri );
9507 tspivot( crosstri, crossedge );
9508 /* Check for a crossing segment. */
9509 if ( crossedge.sh == dummysh ) {
9513 org( *searchtri, endpoint1 );
9514 /* Insert a point at the intersection. */
9515 segmentintersection( &crosstri, &crossedge, endpoint2 );
9516 triedgecopy( crosstri, *searchtri );
9517 insertshelle( searchtri, newmark );
9518 /* Insert the remainder of the segment. */
9519 return scoutsegment( searchtri, endpoint2, newmark );
9524 /*****************************************************************************/
9526 /* conformingedge() Force a segment into a conforming Delaunay */
9527 /* triangulation by inserting a point at its midpoint, */
9528 /* and recursively forcing in the two half-segments if */
9531 /* Generates a sequence of edges connecting `endpoint1' to `endpoint2'. */
9532 /* `newmark' is the boundary marker of the segment, assigned to each new */
9533 /* splitting point and shell edge. */
9535 /* Note that conformingedge() does not always maintain the conforming */
9536 /* Delaunay property. Once inserted, segments are locked into place; */
9537 /* points inserted later (to force other segments in) may render these */
9538 /* fixed segments non-Delaunay. The conforming Delaunay property will be */
9539 /* restored by enforcequality() by splitting encroached segments. */
9541 /*****************************************************************************/
9546 void conformingedge( endpoint1, endpoint2, newmark )
9551 struct triedge searchtri1, searchtri2;
9552 struct edge brokenshelle;
9554 point midpoint1, midpoint2;
9555 enum insertsiteresult success;
9556 int result1, result2;
9558 shelle sptr; /* Temporary variable used by tspivot(). */
9560 if ( verbose > 2 ) {
9561 printf( "Forcing segment into triangulation by recursive splitting:\n" );
9562 printf( " (%.12g, %.12g) (%.12g, %.12g)\n", endpoint1[0], endpoint1[1],
9563 endpoint2[0], endpoint2[1] );
9565 /* Create a new point to insert in the middle of the segment. */
9566 newpoint = (point) poolalloc( &points );
9567 /* Interpolate coordinates and attributes. */
9568 for ( i = 0; i < 2 + nextras; i++ ) {
9569 newpoint[i] = 0.5 * ( endpoint1[i] + endpoint2[i] );
9571 setpointmark( newpoint, newmark );
9572 /* Find a boundary triangle to search from. */
9573 searchtri1.tri = (triangle *) NULL;
9574 /* Attempt to insert the new point. */
9575 success = insertsite( newpoint, &searchtri1, (struct edge *) NULL, 0, 0 );
9576 if ( success == DUPLICATEPOINT ) {
9577 if ( verbose > 2 ) {
9578 printf( " Segment intersects existing point (%.12g, %.12g).\n",
9579 newpoint[0], newpoint[1] );
9581 /* Use the point that's already there. */
9582 pointdealloc( newpoint );
9583 org( searchtri1, newpoint );
9586 if ( success == VIOLATINGPOINT ) {
9587 if ( verbose > 2 ) {
9588 printf( " Two segments intersect at (%.12g, %.12g).\n",
9589 newpoint[0], newpoint[1] );
9591 /* By fluke, we've landed right on another segment. Split it. */
9592 tspivot( searchtri1, brokenshelle );
9593 success = insertsite( newpoint, &searchtri1, &brokenshelle, 0, 0 );
9594 if ( success != SUCCESSFULPOINT ) {
9595 printf( "Internal error in conformingedge():\n" );
9596 printf( " Failure to split a segment.\n" );
9600 /* The point has been inserted successfully. */
9601 if ( steinerleft > 0 ) {
9605 triedgecopy( searchtri1, searchtri2 );
9606 result1 = scoutsegment( &searchtri1, endpoint1, newmark );
9607 result2 = scoutsegment( &searchtri2, endpoint2, newmark );
9609 /* The origin of searchtri1 may have changed if a collision with an */
9610 /* intervening vertex on the segment occurred. */
9611 org( searchtri1, midpoint1 );
9612 conformingedge( midpoint1, endpoint1, newmark );
9615 /* The origin of searchtri2 may have changed if a collision with an */
9616 /* intervening vertex on the segment occurred. */
9617 org( searchtri2, midpoint2 );
9618 conformingedge( midpoint2, endpoint2, newmark );
9622 #endif /* not CDT_ONLY */
9623 #endif /* not REDUCED */
9625 /*****************************************************************************/
9627 /* delaunayfixup() Enforce the Delaunay condition at an edge, fanning out */
9628 /* recursively from an existing point. Pay special */
9629 /* attention to stacking inverted triangles. */
9631 /* This is a support routine for inserting segments into a constrained */
9632 /* Delaunay triangulation. */
9634 /* The origin of fixuptri is treated as if it has just been inserted, and */
9635 /* the local Delaunay condition needs to be enforced. It is only enforced */
9636 /* in one sector, however, that being the angular range defined by */
9639 /* This routine also needs to make decisions regarding the "stacking" of */
9640 /* triangles. (Read the description of constrainededge() below before */
9641 /* reading on here, so you understand the algorithm.) If the position of */
9642 /* the new point (the origin of fixuptri) indicates that the vertex before */
9643 /* it on the polygon is a reflex vertex, then "stack" the triangle by */
9644 /* doing nothing. (fixuptri is an inverted triangle, which is how stacked */
9645 /* triangles are identified.) */
9647 /* Otherwise, check whether the vertex before that was a reflex vertex. */
9648 /* If so, perform an edge flip, thereby eliminating an inverted triangle */
9649 /* (popping it off the stack). The edge flip may result in the creation */
9650 /* of a new inverted triangle, depending on whether or not the new vertex */
9651 /* is visible to the vertex three edges behind on the polygon. */
9653 /* If neither of the two vertices behind the new vertex are reflex */
9654 /* vertices, fixuptri and fartri, the triangle opposite it, are not */
9655 /* inverted; hence, ensure that the edge between them is locally Delaunay. */
9657 /* `leftside' indicates whether or not fixuptri is to the left of the */
9658 /* segment being inserted. (Imagine that the segment is pointing up from */
9659 /* endpoint1 to endpoint2.) */
9661 /*****************************************************************************/
9663 void delaunayfixup( fixuptri, leftside )
9664 struct triedge *fixuptri;
9667 struct triedge neartri;
9668 struct triedge fartri;
9669 struct edge faredge;
9670 point nearpoint, leftpoint, rightpoint, farpoint;
9671 triangle ptr; /* Temporary variable used by sym(). */
9672 shelle sptr; /* Temporary variable used by tspivot(). */
9674 lnext( *fixuptri, neartri );
9675 sym( neartri, fartri );
9676 /* Check if the edge opposite the origin of fixuptri can be flipped. */
9677 if ( fartri.tri == dummytri ) {
9680 tspivot( neartri, faredge );
9681 if ( faredge.sh != dummysh ) {
9684 /* Find all the relevant vertices. */
9685 apex( neartri, nearpoint );
9686 org( neartri, leftpoint );
9687 dest( neartri, rightpoint );
9688 apex( fartri, farpoint );
9689 /* Check whether the previous polygon vertex is a reflex vertex. */
9691 if ( counterclockwise( nearpoint, leftpoint, farpoint ) <= 0.0 ) {
9692 /* leftpoint is a reflex vertex too. Nothing can */
9693 /* be done until a convex section is found. */
9698 if ( counterclockwise( farpoint, rightpoint, nearpoint ) <= 0.0 ) {
9699 /* rightpoint is a reflex vertex too. Nothing can */
9700 /* be done until a convex section is found. */
9704 if ( counterclockwise( rightpoint, leftpoint, farpoint ) > 0.0 ) {
9705 /* fartri is not an inverted triangle, and farpoint is not a reflex */
9706 /* vertex. As there are no reflex vertices, fixuptri isn't an */
9707 /* inverted triangle, either. Hence, test the edge between the */
9708 /* triangles to ensure it is locally Delaunay. */
9709 if ( incircle( leftpoint, farpoint, rightpoint, nearpoint ) <= 0.0 ) {
9712 /* Not locally Delaunay; go on to an edge flip. */
9713 } /* else fartri is inverted; remove it from the stack by flipping. */
9715 lprevself( *fixuptri ); /* Restore the origin of fixuptri after the flip. */
9716 /* Recursively process the two triangles that result from the flip. */
9717 delaunayfixup( fixuptri, leftside );
9718 delaunayfixup( &fartri, leftside );
9721 /*****************************************************************************/
9723 /* constrainededge() Force a segment into a constrained Delaunay */
9724 /* triangulation by deleting the triangles it */
9725 /* intersects, and triangulating the polygons that */
9726 /* form on each side of it. */
9728 /* Generates a single edge connecting `endpoint1' to `endpoint2'. The */
9729 /* triangle `starttri' has `endpoint1' as its origin. `newmark' is the */
9730 /* boundary marker of the segment. */
9732 /* To insert a segment, every triangle whose interior intersects the */
9733 /* segment is deleted. The union of these deleted triangles is a polygon */
9734 /* (which is not necessarily monotone, but is close enough), which is */
9735 /* divided into two polygons by the new segment. This routine's task is */
9736 /* to generate the Delaunay triangulation of these two polygons. */
9738 /* You might think of this routine's behavior as a two-step process. The */
9739 /* first step is to walk from endpoint1 to endpoint2, flipping each edge */
9740 /* encountered. This step creates a fan of edges connected to endpoint1, */
9741 /* including the desired edge to endpoint2. The second step enforces the */
9742 /* Delaunay condition on each side of the segment in an incremental manner: */
9743 /* proceeding along the polygon from endpoint1 to endpoint2 (this is done */
9744 /* independently on each side of the segment), each vertex is "enforced" */
9745 /* as if it had just been inserted, but affecting only the previous */
9746 /* vertices. The result is the same as if the vertices had been inserted */
9747 /* in the order they appear on the polygon, so the result is Delaunay. */
9749 /* In truth, constrainededge() interleaves these two steps. The procedure */
9750 /* walks from endpoint1 to endpoint2, and each time an edge is encountered */
9751 /* and flipped, the newly exposed vertex (at the far end of the flipped */
9752 /* edge) is "enforced" upon the previously flipped edges, usually affecting */
9753 /* only one side of the polygon (depending upon which side of the segment */
9754 /* the vertex falls on). */
9756 /* The algorithm is complicated by the need to handle polygons that are not */
9757 /* convex. Although the polygon is not necessarily monotone, it can be */
9758 /* triangulated in a manner similar to the stack-based algorithms for */
9759 /* monotone polygons. For each reflex vertex (local concavity) of the */
9760 /* polygon, there will be an inverted triangle formed by one of the edge */
9761 /* flips. (An inverted triangle is one with negative area - that is, its */
9762 /* vertices are arranged in clockwise order - and is best thought of as a */
9763 /* wrinkle in the fabric of the mesh.) Each inverted triangle can be */
9764 /* thought of as a reflex vertex pushed on the stack, waiting to be fixed */
9767 /* A reflex vertex is popped from the stack when a vertex is inserted that */
9768 /* is visible to the reflex vertex. (However, if the vertex behind the */
9769 /* reflex vertex is not visible to the reflex vertex, a new inverted */
9770 /* triangle will take its place on the stack.) These details are handled */
9771 /* by the delaunayfixup() routine above. */
9773 /*****************************************************************************/
9775 void constrainededge( starttri, endpoint2, newmark )
9776 struct triedge *starttri;
9780 struct triedge fixuptri, fixuptri2;
9781 struct edge fixupedge;
9787 triangle ptr; /* Temporary variable used by sym() and oprev(). */
9788 shelle sptr; /* Temporary variable used by tspivot(). */
9790 org( *starttri, endpoint1 );
9791 lnext( *starttri, fixuptri );
9793 /* `collision' indicates whether we have found a point directly */
9794 /* between endpoint1 and endpoint2. */
9798 org( fixuptri, farpoint );
9799 /* `farpoint' is the extreme point of the polygon we are "digging" */
9800 /* to get from endpoint1 to endpoint2. */
9801 if ( ( farpoint[0] == endpoint2[0] ) && ( farpoint[1] == endpoint2[1] ) ) {
9802 oprev( fixuptri, fixuptri2 );
9803 /* Enforce the Delaunay condition around endpoint2. */
9804 delaunayfixup( &fixuptri, 0 );
9805 delaunayfixup( &fixuptri2, 1 );
9809 /* Check whether farpoint is to the left or right of the segment */
9810 /* being inserted, to decide which edge of fixuptri to dig */
9812 area = counterclockwise( endpoint1, endpoint2, farpoint );
9813 if ( area == 0.0 ) {
9814 /* We've collided with a point between endpoint1 and endpoint2. */
9816 oprev( fixuptri, fixuptri2 );
9817 /* Enforce the Delaunay condition around farpoint. */
9818 delaunayfixup( &fixuptri, 0 );
9819 delaunayfixup( &fixuptri2, 1 );
9823 if ( area > 0.0 ) { /* farpoint is to the left of the segment. */
9824 oprev( fixuptri, fixuptri2 );
9825 /* Enforce the Delaunay condition around farpoint, on the */
9826 /* left side of the segment only. */
9827 delaunayfixup( &fixuptri2, 1 );
9828 /* Flip the edge that crosses the segment. After the edge is */
9829 /* flipped, one of its endpoints is the fan vertex, and the */
9830 /* destination of fixuptri is the fan vertex. */
9831 lprevself( fixuptri );
9833 else { /* farpoint is to the right of the segment. */
9834 delaunayfixup( &fixuptri, 0 );
9835 /* Flip the edge that crosses the segment. After the edge is */
9836 /* flipped, one of its endpoints is the fan vertex, and the */
9837 /* destination of fixuptri is the fan vertex. */
9838 oprevself( fixuptri );
9840 /* Check for two intersecting segments. */
9841 tspivot( fixuptri, fixupedge );
9842 if ( fixupedge.sh == dummysh ) {
9843 flip( &fixuptri ); /* May create an inverted triangle on the left. */
9846 /* We've collided with a segment between endpoint1 and endpoint2. */
9848 /* Insert a point at the intersection. */
9849 segmentintersection( &fixuptri, &fixupedge, endpoint2 );
9855 /* Insert a shell edge to make the segment permanent. */
9856 insertshelle( &fixuptri, newmark );
9857 /* If there was a collision with an interceding vertex, install another */
9858 /* segment connecting that vertex with endpoint2. */
9860 /* Insert the remainder of the segment. */
9861 if ( !scoutsegment( &fixuptri, endpoint2, newmark ) ) {
9862 constrainededge( &fixuptri, endpoint2, newmark );
9867 /*****************************************************************************/
9869 /* insertsegment() Insert a PSLG segment into a triangulation. */
9871 /*****************************************************************************/
9873 void insertsegment( endpoint1, endpoint2, newmark )
9878 struct triedge searchtri1, searchtri2;
9879 triangle encodedtri;
9881 triangle ptr; /* Temporary variable used by sym(). */
9883 if ( verbose > 1 ) {
9884 printf( " Connecting (%.12g, %.12g) to (%.12g, %.12g).\n",
9885 endpoint1[0], endpoint1[1], endpoint2[0], endpoint2[1] );
9888 /* Find a triangle whose origin is the segment's first endpoint. */
9889 checkpoint = (point) NULL;
9890 encodedtri = point2tri( endpoint1 );
9891 if ( encodedtri != (triangle) NULL ) {
9892 decode( encodedtri, searchtri1 );
9893 org( searchtri1, checkpoint );
9895 if ( checkpoint != endpoint1 ) {
9896 /* Find a boundary triangle to search from. */
9897 searchtri1.tri = dummytri;
9898 searchtri1.orient = 0;
9899 symself( searchtri1 );
9900 /* Search for the segment's first endpoint by point location. */
9901 if ( locate( endpoint1, &searchtri1 ) != ONVERTEX ) {
9903 "Internal error in insertsegment(): Unable to locate PSLG point\n" );
9904 printf( " (%.12g, %.12g) in triangulation.\n",
9905 endpoint1[0], endpoint1[1] );
9909 /* Remember this triangle to improve subsequent point location. */
9910 triedgecopy( searchtri1, recenttri );
9911 /* Scout the beginnings of a path from the first endpoint */
9912 /* toward the second. */
9913 if ( scoutsegment( &searchtri1, endpoint2, newmark ) ) {
9914 /* The segment was easily inserted. */
9917 /* The first endpoint may have changed if a collision with an intervening */
9918 /* vertex on the segment occurred. */
9919 org( searchtri1, endpoint1 );
9921 /* Find a triangle whose origin is the segment's second endpoint. */
9922 checkpoint = (point) NULL;
9923 encodedtri = point2tri( endpoint2 );
9924 if ( encodedtri != (triangle) NULL ) {
9925 decode( encodedtri, searchtri2 );
9926 org( searchtri2, checkpoint );
9928 if ( checkpoint != endpoint2 ) {
9929 /* Find a boundary triangle to search from. */
9930 searchtri2.tri = dummytri;
9931 searchtri2.orient = 0;
9932 symself( searchtri2 );
9933 /* Search for the segment's second endpoint by point location. */
9934 if ( locate( endpoint2, &searchtri2 ) != ONVERTEX ) {
9936 "Internal error in insertsegment(): Unable to locate PSLG point\n" );
9937 printf( " (%.12g, %.12g) in triangulation.\n",
9938 endpoint2[0], endpoint2[1] );
9942 /* Remember this triangle to improve subsequent point location. */
9943 triedgecopy( searchtri2, recenttri );
9944 /* Scout the beginnings of a path from the second endpoint */
9945 /* toward the first. */
9946 if ( scoutsegment( &searchtri2, endpoint1, newmark ) ) {
9947 /* The segment was easily inserted. */
9950 /* The second endpoint may have changed if a collision with an intervening */
9951 /* vertex on the segment occurred. */
9952 org( searchtri2, endpoint2 );
9957 /* Insert vertices to force the segment into the triangulation. */
9958 conformingedge( endpoint1, endpoint2, newmark );
9961 #endif /* not CDT_ONLY */
9962 #endif /* not REDUCED */
9963 /* Insert the segment directly into the triangulation. */
9964 constrainededge( &searchtri1, endpoint2, newmark );
9968 #endif /* not CDT_ONLY */
9969 #endif /* not REDUCED */
9972 /*****************************************************************************/
9974 /* markhull() Cover the convex hull of a triangulation with shell edges. */
9976 /*****************************************************************************/
9979 struct triedge hulltri;
9980 struct triedge nexttri;
9981 struct triedge starttri;
9982 triangle ptr; /* Temporary variable used by sym() and oprev(). */
9984 /* Find a triangle handle on the hull. */
9985 hulltri.tri = dummytri;
9988 /* Remember where we started so we know when to stop. */
9989 triedgecopy( hulltri, starttri );
9990 /* Go once counterclockwise around the convex hull. */
9992 /* Create a shell edge if there isn't already one here. */
9993 insertshelle( &hulltri, 1 );
9994 /* To find the next hull edge, go clockwise around the next vertex. */
9995 lnextself( hulltri );
9996 oprev( hulltri, nexttri );
9997 while ( nexttri.tri != dummytri ) {
9998 triedgecopy( nexttri, hulltri );
9999 oprev( hulltri, nexttri );
10001 } while ( !triedgeequal( hulltri, starttri ) );
10004 /*****************************************************************************/
10006 /* formskeleton() Create the shell edges of a triangulation, including */
10007 /* PSLG edges and edges on the convex hull. */
10009 /* The PSLG edges are read from a .poly file. The return value is the */
10010 /* number of segments in the file. */
10012 /*****************************************************************************/
10016 int formskeleton( segmentlist, segmentmarkerlist, numberofsegments )
10018 int *segmentmarkerlist;
10019 int numberofsegments;
10021 #else /* not TRILIBRARY */
10023 int formskeleton( polyfile, polyfilename )
10025 char *polyfilename;
10027 #endif /* not TRILIBRARY */
10031 char polyfilename[6];
10033 #else /* not TRILIBRARY */
10034 char inputline[INPUTLINESIZE];
10036 #endif /* not TRILIBRARY */
10037 point endpoint1, endpoint2;
10039 int segmentmarkers;
10046 printf( "Inserting segments into Delaunay triangulation.\n" );
10049 strcpy( polyfilename, "input" );
10050 segments = numberofsegments;
10051 segmentmarkers = segmentmarkerlist != (int *) NULL;
10053 #else /* not TRILIBRARY */
10054 /* Read the segments from a .poly file. */
10055 /* Read number of segments and number of boundary markers. */
10056 stringptr = readline( inputline, polyfile, polyfilename );
10057 segments = (int) strtol( stringptr, &stringptr, 0 );
10058 stringptr = findfield( stringptr );
10059 if ( *stringptr == '\0' ) {
10060 segmentmarkers = 0;
10063 segmentmarkers = (int) strtol( stringptr, &stringptr, 0 );
10065 #endif /* not TRILIBRARY */
10066 /* If segments are to be inserted, compute a mapping */
10067 /* from points to triangles. */
10068 if ( segments > 0 ) {
10070 printf( " Inserting PSLG segments.\n" );
10076 /* Read and insert the segments. */
10077 for ( i = 1; i <= segments; i++ ) {
10079 end1 = segmentlist[index++];
10080 end2 = segmentlist[index++];
10081 if ( segmentmarkers ) {
10082 boundmarker = segmentmarkerlist[i - 1];
10084 #else /* not TRILIBRARY */
10085 stringptr = readline( inputline, polyfile, inpolyfilename );
10086 stringptr = findfield( stringptr );
10087 if ( *stringptr == '\0' ) {
10088 printf( "Error: Segment %d has no endpoints in %s.\n", i,
10093 end1 = (int) strtol( stringptr, &stringptr, 0 );
10095 stringptr = findfield( stringptr );
10096 if ( *stringptr == '\0' ) {
10097 printf( "Error: Segment %d is missing its second endpoint in %s.\n", i,
10102 end2 = (int) strtol( stringptr, &stringptr, 0 );
10104 if ( segmentmarkers ) {
10105 stringptr = findfield( stringptr );
10106 if ( *stringptr == '\0' ) {
10110 boundmarker = (int) strtol( stringptr, &stringptr, 0 );
10113 #endif /* not TRILIBRARY */
10114 if ( ( end1 < firstnumber ) || ( end1 >= firstnumber + inpoints ) ) {
10116 printf( "Warning: Invalid first endpoint of segment %d in %s.\n", i,
10120 else if ( ( end2 < firstnumber ) || ( end2 >= firstnumber + inpoints ) ) {
10122 printf( "Warning: Invalid second endpoint of segment %d in %s.\n", i,
10127 endpoint1 = getpoint( end1 );
10128 endpoint2 = getpoint( end2 );
10129 if ( ( endpoint1[0] == endpoint2[0] ) && ( endpoint1[1] == endpoint2[1] ) ) {
10131 printf( "Warning: Endpoints of segment %d are coincident in %s.\n",
10136 insertsegment( endpoint1, endpoint2, boundmarker );
10144 if ( convex || !poly ) {
10145 /* Enclose the convex hull with shell edges. */
10147 printf( " Enclosing convex hull with segments.\n" );
10156 /********* Segment (shell edge) insertion ends here *********/
10158 /********* Carving out holes and concavities begins here *********/
10162 /*****************************************************************************/
10164 /* infecthull() Virally infect all of the triangles of the convex hull */
10165 /* that are not protected by shell edges. Where there are */
10166 /* shell edges, set boundary markers as appropriate. */
10168 /*****************************************************************************/
10171 struct triedge hulltri;
10172 struct triedge nexttri;
10173 struct triedge starttri;
10174 struct edge hulledge;
10175 triangle **deadtri;
10177 triangle ptr; /* Temporary variable used by sym(). */
10178 shelle sptr; /* Temporary variable used by tspivot(). */
10181 printf( " Marking concavities (external triangles) for elimination.\n" );
10183 /* Find a triangle handle on the hull. */
10184 hulltri.tri = dummytri;
10185 hulltri.orient = 0;
10186 symself( hulltri );
10187 /* Remember where we started so we know when to stop. */
10188 triedgecopy( hulltri, starttri );
10189 /* Go once counterclockwise around the convex hull. */
10191 /* Ignore triangles that are already infected. */
10192 if ( !infected( hulltri ) ) {
10193 /* Is the triangle protected by a shell edge? */
10194 tspivot( hulltri, hulledge );
10195 if ( hulledge.sh == dummysh ) {
10196 /* The triangle is not protected; infect it. */
10198 deadtri = (triangle **) poolalloc( &viri );
10199 *deadtri = hulltri.tri;
10202 /* The triangle is protected; set boundary markers if appropriate. */
10203 if ( mark( hulledge ) == 0 ) {
10204 setmark( hulledge, 1 );
10205 org( hulltri, horg );
10206 dest( hulltri, hdest );
10207 if ( pointmark( horg ) == 0 ) {
10208 setpointmark( horg, 1 );
10210 if ( pointmark( hdest ) == 0 ) {
10211 setpointmark( hdest, 1 );
10216 /* To find the next hull edge, go clockwise around the next vertex. */
10217 lnextself( hulltri );
10218 oprev( hulltri, nexttri );
10219 while ( nexttri.tri != dummytri ) {
10220 triedgecopy( nexttri, hulltri );
10221 oprev( hulltri, nexttri );
10223 } while ( !triedgeequal( hulltri, starttri ) );
10226 /*****************************************************************************/
10228 /* plague() Spread the virus from all infected triangles to any neighbors */
10229 /* not protected by shell edges. Delete all infected triangles. */
10231 /* This is the procedure that actually creates holes and concavities. */
10233 /* This procedure operates in two phases. The first phase identifies all */
10234 /* the triangles that will die, and marks them as infected. They are */
10235 /* marked to ensure that each triangle is added to the virus pool only */
10236 /* once, so the procedure will terminate. */
10238 /* The second phase actually eliminates the infected triangles. It also */
10239 /* eliminates orphaned points. */
10241 /*****************************************************************************/
10244 struct triedge testtri;
10245 struct triedge neighbor;
10246 triangle **virusloop;
10247 triangle **deadtri;
10248 struct edge neighborshelle;
10251 point deadorg, deaddest, deadapex;
10253 triangle ptr; /* Temporary variable used by sym() and onext(). */
10254 shelle sptr; /* Temporary variable used by tspivot(). */
10257 printf( " Marking neighbors of marked triangles.\n" );
10259 /* Loop through all the infected triangles, spreading the virus to */
10260 /* their neighbors, then to their neighbors' neighbors. */
10261 traversalinit( &viri );
10262 virusloop = (triangle **) traverse( &viri );
10263 while ( virusloop != (triangle **) NULL ) {
10264 testtri.tri = *virusloop;
10265 /* A triangle is marked as infected by messing with one of its shell */
10266 /* edges, setting it to an illegal value. Hence, we have to */
10267 /* temporarily uninfect this triangle so that we can examine its */
10268 /* adjacent shell edges. */
10269 uninfect( testtri );
10270 if ( verbose > 2 ) {
10271 /* Assign the triangle an orientation for convenience in */
10272 /* checking its points. */
10273 testtri.orient = 0;
10274 org( testtri, deadorg );
10275 dest( testtri, deaddest );
10276 apex( testtri, deadapex );
10277 printf( " Checking (%.12g, %.12g) (%.12g, %.12g) (%.12g, %.12g)\n",
10278 deadorg[0], deadorg[1], deaddest[0], deaddest[1],
10279 deadapex[0], deadapex[1] );
10281 /* Check each of the triangle's three neighbors. */
10282 for ( testtri.orient = 0; testtri.orient < 3; testtri.orient++ ) {
10283 /* Find the neighbor. */
10284 sym( testtri, neighbor );
10285 /* Check for a shell between the triangle and its neighbor. */
10286 tspivot( testtri, neighborshelle );
10287 /* Check if the neighbor is nonexistent or already infected. */
10288 if ( ( neighbor.tri == dummytri ) || infected( neighbor ) ) {
10289 if ( neighborshelle.sh != dummysh ) {
10290 /* There is a shell edge separating the triangle from its */
10291 /* neighbor, but both triangles are dying, so the shell */
10292 /* edge dies too. */
10293 shelledealloc( neighborshelle.sh );
10294 if ( neighbor.tri != dummytri ) {
10295 /* Make sure the shell edge doesn't get deallocated again */
10296 /* later when the infected neighbor is visited. */
10297 uninfect( neighbor );
10298 tsdissolve( neighbor );
10299 infect( neighbor );
10303 else { /* The neighbor exists and is not infected. */
10304 if ( neighborshelle.sh == dummysh ) {
10305 /* There is no shell edge protecting the neighbor, so */
10306 /* the neighbor becomes infected. */
10307 if ( verbose > 2 ) {
10308 org( neighbor, deadorg );
10309 dest( neighbor, deaddest );
10310 apex( neighbor, deadapex );
10312 " Marking (%.12g, %.12g) (%.12g, %.12g) (%.12g, %.12g)\n",
10313 deadorg[0], deadorg[1], deaddest[0], deaddest[1],
10314 deadapex[0], deadapex[1] );
10316 infect( neighbor );
10317 /* Ensure that the neighbor's neighbors will be infected. */
10318 deadtri = (triangle **) poolalloc( &viri );
10319 *deadtri = neighbor.tri;
10321 else { /* The neighbor is protected by a shell edge. */
10322 /* Remove this triangle from the shell edge. */
10323 stdissolve( neighborshelle );
10324 /* The shell edge becomes a boundary. Set markers accordingly. */
10325 if ( mark( neighborshelle ) == 0 ) {
10326 setmark( neighborshelle, 1 );
10328 org( neighbor, norg );
10329 dest( neighbor, ndest );
10330 if ( pointmark( norg ) == 0 ) {
10331 setpointmark( norg, 1 );
10333 if ( pointmark( ndest ) == 0 ) {
10334 setpointmark( ndest, 1 );
10339 /* Remark the triangle as infected, so it doesn't get added to the */
10340 /* virus pool again. */
10342 virusloop = (triangle **) traverse( &viri );
10346 printf( " Deleting marked triangles.\n" );
10348 traversalinit( &viri );
10349 virusloop = (triangle **) traverse( &viri );
10350 while ( virusloop != (triangle **) NULL ) {
10351 testtri.tri = *virusloop;
10353 /* Check each of the three corners of the triangle for elimination. */
10354 /* This is done by walking around each point, checking if it is */
10355 /* still connected to at least one live triangle. */
10356 for ( testtri.orient = 0; testtri.orient < 3; testtri.orient++ ) {
10357 org( testtri, testpoint );
10358 /* Check if the point has already been tested. */
10359 if ( testpoint != (point) NULL ) {
10361 /* Mark the corner of the triangle as having been tested. */
10362 setorg( testtri, NULL );
10363 /* Walk counterclockwise about the point. */
10364 onext( testtri, neighbor );
10365 /* Stop upon reaching a boundary or the starting triangle. */
10366 while ( ( neighbor.tri != dummytri )
10367 && ( !triedgeequal( neighbor, testtri ) ) ) {
10368 if ( infected( neighbor ) ) {
10369 /* Mark the corner of this triangle as having been tested. */
10370 setorg( neighbor, NULL );
10373 /* A live triangle. The point survives. */
10376 /* Walk counterclockwise about the point. */
10377 onextself( neighbor );
10379 /* If we reached a boundary, we must walk clockwise as well. */
10380 if ( neighbor.tri == dummytri ) {
10381 /* Walk clockwise about the point. */
10382 oprev( testtri, neighbor );
10383 /* Stop upon reaching a boundary. */
10384 while ( neighbor.tri != dummytri ) {
10385 if ( infected( neighbor ) ) {
10386 /* Mark the corner of this triangle as having been tested. */
10387 setorg( neighbor, NULL );
10390 /* A live triangle. The point survives. */
10393 /* Walk clockwise about the point. */
10394 oprevself( neighbor );
10398 if ( verbose > 1 ) {
10399 printf( " Deleting point (%.12g, %.12g)\n",
10400 testpoint[0], testpoint[1] );
10402 pointdealloc( testpoint );
10407 /* Record changes in the number of boundary edges, and disconnect */
10408 /* dead triangles from their neighbors. */
10409 for ( testtri.orient = 0; testtri.orient < 3; testtri.orient++ ) {
10410 sym( testtri, neighbor );
10411 if ( neighbor.tri == dummytri ) {
10412 /* There is no neighboring triangle on this edge, so this edge */
10413 /* is a boundary edge. This triangle is being deleted, so this */
10414 /* boundary edge is deleted. */
10418 /* Disconnect the triangle from its neighbor. */
10419 dissolve( neighbor );
10420 /* There is a neighboring triangle on this edge, so this edge */
10421 /* becomes a boundary edge when this triangle is deleted. */
10425 /* Return the dead triangle to the pool of triangles. */
10426 triangledealloc( testtri.tri );
10427 virusloop = (triangle **) traverse( &viri );
10429 /* Empty the virus pool. */
10430 poolrestart( &viri );
10433 /*****************************************************************************/
10435 /* regionplague() Spread regional attributes and/or area constraints */
10436 /* (from a .poly file) throughout the mesh. */
10438 /* This procedure operates in two phases. The first phase spreads an */
10439 /* attribute and/or an area constraint through a (segment-bounded) region. */
10440 /* The triangles are marked to ensure that each triangle is added to the */
10441 /* virus pool only once, so the procedure will terminate. */
10443 /* The second phase uninfects all infected triangles, returning them to */
10446 /*****************************************************************************/
10448 void regionplague( attribute, area )
10452 struct triedge testtri;
10453 struct triedge neighbor;
10454 triangle **virusloop;
10455 triangle **regiontri;
10456 struct edge neighborshelle;
10457 point regionorg, regiondest, regionapex;
10458 triangle ptr; /* Temporary variable used by sym() and onext(). */
10459 shelle sptr; /* Temporary variable used by tspivot(). */
10461 if ( verbose > 1 ) {
10462 printf( " Marking neighbors of marked triangles.\n" );
10464 /* Loop through all the infected triangles, spreading the attribute */
10465 /* and/or area constraint to their neighbors, then to their neighbors' */
10467 traversalinit( &viri );
10468 virusloop = (triangle **) traverse( &viri );
10469 while ( virusloop != (triangle **) NULL ) {
10470 testtri.tri = *virusloop;
10471 /* A triangle is marked as infected by messing with one of its shell */
10472 /* edges, setting it to an illegal value. Hence, we have to */
10473 /* temporarily uninfect this triangle so that we can examine its */
10474 /* adjacent shell edges. */
10475 uninfect( testtri );
10476 if ( regionattrib ) {
10477 /* Set an attribute. */
10478 setelemattribute( testtri, eextras, attribute );
10481 /* Set an area constraint. */
10482 setareabound( testtri, area );
10484 if ( verbose > 2 ) {
10485 /* Assign the triangle an orientation for convenience in */
10486 /* checking its points. */
10487 testtri.orient = 0;
10488 org( testtri, regionorg );
10489 dest( testtri, regiondest );
10490 apex( testtri, regionapex );
10491 printf( " Checking (%.12g, %.12g) (%.12g, %.12g) (%.12g, %.12g)\n",
10492 regionorg[0], regionorg[1], regiondest[0], regiondest[1],
10493 regionapex[0], regionapex[1] );
10495 /* Check each of the triangle's three neighbors. */
10496 for ( testtri.orient = 0; testtri.orient < 3; testtri.orient++ ) {
10497 /* Find the neighbor. */
10498 sym( testtri, neighbor );
10499 /* Check for a shell between the triangle and its neighbor. */
10500 tspivot( testtri, neighborshelle );
10501 /* Make sure the neighbor exists, is not already infected, and */
10502 /* isn't protected by a shell edge. */
10503 if ( ( neighbor.tri != dummytri ) && !infected( neighbor )
10504 && ( neighborshelle.sh == dummysh ) ) {
10505 if ( verbose > 2 ) {
10506 org( neighbor, regionorg );
10507 dest( neighbor, regiondest );
10508 apex( neighbor, regionapex );
10509 printf( " Marking (%.12g, %.12g) (%.12g, %.12g) (%.12g, %.12g)\n",
10510 regionorg[0], regionorg[1], regiondest[0], regiondest[1],
10511 regionapex[0], regionapex[1] );
10513 /* Infect the neighbor. */
10514 infect( neighbor );
10515 /* Ensure that the neighbor's neighbors will be infected. */
10516 regiontri = (triangle **) poolalloc( &viri );
10517 *regiontri = neighbor.tri;
10520 /* Remark the triangle as infected, so it doesn't get added to the */
10521 /* virus pool again. */
10523 virusloop = (triangle **) traverse( &viri );
10526 /* Uninfect all triangles. */
10527 if ( verbose > 1 ) {
10528 printf( " Unmarking marked triangles.\n" );
10530 traversalinit( &viri );
10531 virusloop = (triangle **) traverse( &viri );
10532 while ( virusloop != (triangle **) NULL ) {
10533 testtri.tri = *virusloop;
10534 uninfect( testtri );
10535 virusloop = (triangle **) traverse( &viri );
10537 /* Empty the virus pool. */
10538 poolrestart( &viri );
10541 /*****************************************************************************/
10543 /* carveholes() Find the holes and infect them. Find the area */
10544 /* constraints and infect them. Infect the convex hull. */
10545 /* Spread the infection and kill triangles. Spread the */
10546 /* area constraints. */
10548 /* This routine mainly calls other routines to carry out all these */
10551 /*****************************************************************************/
10553 void carveholes( holelist, holes, regionlist, regions )
10559 struct triedge searchtri;
10560 struct triedge triangleloop;
10561 struct triedge *regiontris;
10562 triangle **holetri;
10563 triangle **regiontri;
10564 point searchorg, searchdest;
10565 enum locateresult intersect;
10567 triangle ptr; /* Temporary variable used by sym(). */
10569 if ( !( quiet || ( noholes && convex ) ) ) {
10570 printf( "Removing unwanted triangles.\n" );
10571 if ( verbose && ( holes > 0 ) ) {
10572 printf( " Marking holes for elimination.\n" );
10576 if ( regions > 0 ) {
10577 /* Allocate storage for the triangles in which region points fall. */
10578 regiontris = (struct triedge *) malloc( regions * sizeof( struct triedge ) );
10579 if ( regiontris == (struct triedge *) NULL ) {
10580 printf( "Error: Out of memory.\n" );
10585 if ( ( ( holes > 0 ) && !noholes ) || !convex || ( regions > 0 ) ) {
10586 /* Initialize a pool of viri to be used for holes, concavities, */
10587 /* regional attributes, and/or regional area constraints. */
10588 poolinit( &viri, sizeof( triangle * ), VIRUSPERBLOCK, POINTER, 0 );
10592 /* Mark as infected any unprotected triangles on the boundary. */
10593 /* This is one way by which concavities are created. */
10597 if ( ( holes > 0 ) && !noholes ) {
10598 /* Infect each triangle in which a hole lies. */
10599 for ( i = 0; i < 2 * holes; i += 2 ) {
10600 /* Ignore holes that aren't within the bounds of the mesh. */
10601 if ( ( holelist[i] >= xmin ) && ( holelist[i] <= xmax )
10602 && ( holelist[i + 1] >= ymin ) && ( holelist[i + 1] <= ymax ) ) {
10603 /* Start searching from some triangle on the outer boundary. */
10604 searchtri.tri = dummytri;
10605 searchtri.orient = 0;
10606 symself( searchtri );
10607 /* Ensure that the hole is to the left of this boundary edge; */
10608 /* otherwise, locate() will falsely report that the hole */
10609 /* falls within the starting triangle. */
10610 org( searchtri, searchorg );
10611 dest( searchtri, searchdest );
10612 if ( counterclockwise( searchorg, searchdest, &holelist[i] ) > 0.0 ) {
10613 /* Find a triangle that contains the hole. */
10614 intersect = locate( &holelist[i], &searchtri );
10615 if ( ( intersect != OUTSIDE ) && ( !infected( searchtri ) ) ) {
10616 /* Infect the triangle. This is done by marking the triangle */
10617 /* as infect and including the triangle in the virus pool. */
10618 infect( searchtri );
10619 holetri = (triangle **) poolalloc( &viri );
10620 *holetri = searchtri.tri;
10627 /* Now, we have to find all the regions BEFORE we carve the holes, because */
10628 /* locate() won't work when the triangulation is no longer convex. */
10629 /* (Incidentally, this is the reason why regional attributes and area */
10630 /* constraints can't be used when refining a preexisting mesh, which */
10631 /* might not be convex; they can only be used with a freshly */
10632 /* triangulated PSLG.) */
10633 if ( regions > 0 ) {
10634 /* Find the starting triangle for each region. */
10635 for ( i = 0; i < regions; i++ ) {
10636 regiontris[i].tri = dummytri;
10637 /* Ignore region points that aren't within the bounds of the mesh. */
10638 if ( ( regionlist[4 * i] >= xmin ) && ( regionlist[4 * i] <= xmax ) &&
10639 ( regionlist[4 * i + 1] >= ymin ) && ( regionlist[4 * i + 1] <= ymax ) ) {
10640 /* Start searching from some triangle on the outer boundary. */
10641 searchtri.tri = dummytri;
10642 searchtri.orient = 0;
10643 symself( searchtri );
10644 /* Ensure that the region point is to the left of this boundary */
10645 /* edge; otherwise, locate() will falsely report that the */
10646 /* region point falls within the starting triangle. */
10647 org( searchtri, searchorg );
10648 dest( searchtri, searchdest );
10649 if ( counterclockwise( searchorg, searchdest, ®ionlist[4 * i] ) >
10651 /* Find a triangle that contains the region point. */
10652 intersect = locate( ®ionlist[4 * i], &searchtri );
10653 if ( ( intersect != OUTSIDE ) && ( !infected( searchtri ) ) ) {
10654 /* Record the triangle for processing after the */
10655 /* holes have been carved. */
10656 triedgecopy( searchtri, regiontris[i] );
10663 if ( viri.items > 0 ) {
10664 /* Carve the holes and concavities. */
10667 /* The virus pool should be empty now. */
10669 if ( regions > 0 ) {
10671 if ( regionattrib ) {
10673 printf( "Spreading regional attributes and area constraints.\n" );
10676 printf( "Spreading regional attributes.\n" );
10680 printf( "Spreading regional area constraints.\n" );
10683 if ( regionattrib && !refine ) {
10684 /* Assign every triangle a regional attribute of zero. */
10685 traversalinit( &triangles );
10686 triangleloop.orient = 0;
10687 triangleloop.tri = triangletraverse();
10688 while ( triangleloop.tri != (triangle *) NULL ) {
10689 setelemattribute( triangleloop, eextras, 0.0 );
10690 triangleloop.tri = triangletraverse();
10693 for ( i = 0; i < regions; i++ ) {
10694 if ( regiontris[i].tri != dummytri ) {
10695 /* Make sure the triangle under consideration still exists. */
10696 /* It may have been eaten by the virus. */
10697 if ( regiontris[i].tri[3] != (triangle) NULL ) {
10698 /* Put one triangle in the virus pool. */
10699 infect( regiontris[i] );
10700 regiontri = (triangle **) poolalloc( &viri );
10701 *regiontri = regiontris[i].tri;
10702 /* Apply one region's attribute and/or area constraint. */
10703 regionplague( regionlist[4 * i + 2], regionlist[4 * i + 3] );
10704 /* The virus pool should be empty now. */
10708 if ( regionattrib && !refine ) {
10709 /* Note the fact that each triangle has an additional attribute. */
10714 /* Free up memory. */
10715 if ( ( ( holes > 0 ) && !noholes ) || !convex || ( regions > 0 ) ) {
10716 pooldeinit( &viri );
10718 if ( regions > 0 ) {
10719 free( regiontris );
10725 /********* Carving out holes and concavities ends here *********/
10727 /********* Mesh quality maintenance begins here *********/
10731 /*****************************************************************************/
10733 /* tallyencs() Traverse the entire list of shell edges, check each edge */
10734 /* to see if it is encroached. If so, add it to the list. */
10736 /*****************************************************************************/
10741 struct edge edgeloop;
10744 traversalinit( &shelles );
10745 edgeloop.shorient = 0;
10746 edgeloop.sh = shelletraverse();
10747 while ( edgeloop.sh != (shelle *) NULL ) {
10748 /* If the segment is encroached, add it to the list. */
10749 dummy = checkedge4encroach( &edgeloop );
10750 edgeloop.sh = shelletraverse();
10754 #endif /* not CDT_ONLY */
10756 /*****************************************************************************/
10758 /* precisionerror() Print an error message for precision problems. */
10760 /*****************************************************************************/
10764 void precisionerror(){
10765 printf( "Try increasing the area criterion and/or reducing the minimum\n" );
10766 printf( " allowable angle so that tiny triangles are not created.\n" );
10768 printf( "Alternatively, try recompiling me with double precision\n" );
10769 printf( " arithmetic (by removing \"#define SINGLE\" from the\n" );
10770 printf( " source file or \"-DSINGLE\" from the makefile).\n" );
10771 #endif /* SINGLE */
10774 #endif /* not CDT_ONLY */
10776 /*****************************************************************************/
10778 /* repairencs() Find and repair all the encroached segments. */
10780 /* Encroached segments are repaired by splitting them by inserting a point */
10781 /* at or near their centers. */
10783 /* `flaws' is a flag that specifies whether one should take note of new */
10784 /* encroached segments and bad triangles that result from inserting points */
10785 /* to repair existing encroached segments. */
10787 /* When a segment is split, the two resulting subsegments are always */
10788 /* tested to see if they are encroached upon, regardless of the value */
10791 /*****************************************************************************/
10795 void repairencs( flaws )
10798 struct triedge enctri;
10799 struct triedge testtri;
10800 struct edge *encloop;
10801 struct edge testsh;
10804 enum insertsiteresult success;
10805 REAL segmentlength, nearestpoweroftwo;
10807 int acuteorg, acutedest;
10810 triangle ptr; /* Temporary variable used by stpivot(). */
10811 shelle sptr; /* Temporary variable used by snext(). */
10813 while ( ( badsegments.items > 0 ) && ( steinerleft != 0 ) ) {
10814 traversalinit( &badsegments );
10815 encloop = badsegmenttraverse();
10816 while ( ( encloop != (struct edge *) NULL ) && ( steinerleft != 0 ) ) {
10817 /* To decide where to split a segment, we need to know if the */
10818 /* segment shares an endpoint with an adjacent segment. */
10819 /* The concern is that, if we simply split every encroached */
10820 /* segment in its center, two adjacent segments with a small */
10821 /* angle between them might lead to an infinite loop; each */
10822 /* point added to split one segment will encroach upon the */
10823 /* other segment, which must then be split with a point that */
10824 /* will encroach upon the first segment, and so on forever. */
10825 /* To avoid this, imagine a set of concentric circles, whose */
10826 /* radii are powers of two, about each segment endpoint. */
10827 /* These concentric circles determine where the segment is */
10828 /* split. (If both endpoints are shared with adjacent */
10829 /* segments, split the segment in the middle, and apply the */
10830 /* concentric shells for later splittings.) */
10832 /* Is the origin shared with another segment? */
10833 stpivot( *encloop, enctri );
10834 lnext( enctri, testtri );
10835 tspivot( testtri, testsh );
10836 acuteorg = testsh.sh != dummysh;
10837 /* Is the destination shared with another segment? */
10838 lnextself( testtri );
10839 tspivot( testtri, testsh );
10840 acutedest = testsh.sh != dummysh;
10841 /* Now, check the other side of the segment, if there's a triangle */
10843 sym( enctri, testtri );
10844 if ( testtri.tri != dummytri ) {
10845 /* Is the destination shared with another segment? */
10846 lnextself( testtri );
10847 tspivot( testtri, testsh );
10848 acutedest = acutedest || ( testsh.sh != dummysh );
10849 /* Is the origin shared with another segment? */
10850 lnextself( testtri );
10851 tspivot( testtri, testsh );
10852 acuteorg = acuteorg || ( testsh.sh != dummysh );
10855 sorg( *encloop, eorg );
10856 sdest( *encloop, edest );
10857 /* Use the concentric circles if exactly one endpoint is shared */
10858 /* with another adjacent segment. */
10859 if ( acuteorg ^ acutedest ) {
10860 segmentlength = sqrt( ( edest[0] - eorg[0] ) * ( edest[0] - eorg[0] )
10861 + ( edest[1] - eorg[1] ) * ( edest[1] - eorg[1] ) );
10862 /* Find the power of two nearest the segment's length. */
10863 nearestpoweroftwo = 1.0;
10864 while ( segmentlength > SQUAREROOTTWO * nearestpoweroftwo ) {
10865 nearestpoweroftwo *= 2.0;
10867 while ( segmentlength < ( 0.5 * SQUAREROOTTWO ) * nearestpoweroftwo ) {
10868 nearestpoweroftwo *= 0.5;
10870 /* Where do we split the segment? */
10871 split = 0.5 * nearestpoweroftwo / segmentlength;
10873 split = 1.0 - split;
10877 /* If we're not worried about adjacent segments, split */
10878 /* this segment in the middle. */
10882 /* Create the new point. */
10883 newpoint = (point) poolalloc( &points );
10884 /* Interpolate its coordinate and attributes. */
10885 for ( i = 0; i < 2 + nextras; i++ ) {
10886 newpoint[i] = ( 1.0 - split ) * eorg[i] + split * edest[i];
10888 setpointmark( newpoint, mark( *encloop ) );
10889 if ( verbose > 1 ) {
10891 " Splitting edge (%.12g, %.12g) (%.12g, %.12g) at (%.12g, %.12g).\n",
10892 eorg[0], eorg[1], edest[0], edest[1], newpoint[0], newpoint[1] );
10894 /* Check whether the new point lies on an endpoint. */
10895 if ( ( ( newpoint[0] == eorg[0] ) && ( newpoint[1] == eorg[1] ) )
10896 || ( ( newpoint[0] == edest[0] ) && ( newpoint[1] == edest[1] ) ) ) {
10897 printf( "Error: Ran out of precision at (%.12g, %.12g).\n",
10898 newpoint[0], newpoint[1] );
10899 printf( "I attempted to split a segment to a smaller size than can\n" );
10900 printf( " be accommodated by the finite precision of floating point\n"
10902 printf( " arithmetic.\n" );
10906 /* Insert the splitting point. This should always succeed. */
10907 success = insertsite( newpoint, &enctri, encloop, flaws, flaws );
10908 if ( ( success != SUCCESSFULPOINT ) && ( success != ENCROACHINGPOINT ) ) {
10909 printf( "Internal error in repairencs():\n" );
10910 printf( " Failure to split a segment.\n" );
10913 if ( steinerleft > 0 ) {
10916 /* Check the two new subsegments to see if they're encroached. */
10917 dummy = checkedge4encroach( encloop );
10918 snextself( *encloop );
10919 dummy = checkedge4encroach( encloop );
10921 badsegmentdealloc( encloop );
10922 encloop = badsegmenttraverse();
10927 #endif /* not CDT_ONLY */
10929 /*****************************************************************************/
10931 /* tallyfaces() Test every triangle in the mesh for quality measures. */
10933 /*****************************************************************************/
10938 struct triedge triangleloop;
10941 printf( " Making a list of bad triangles.\n" );
10943 traversalinit( &triangles );
10944 triangleloop.orient = 0;
10945 triangleloop.tri = triangletraverse();
10946 while ( triangleloop.tri != (triangle *) NULL ) {
10947 /* If the triangle is bad, enqueue it. */
10948 testtriangle( &triangleloop );
10949 triangleloop.tri = triangletraverse();
10953 #endif /* not CDT_ONLY */
10955 /*****************************************************************************/
10957 /* findcircumcenter() Find the circumcenter of a triangle. */
10959 /* The result is returned both in terms of x-y coordinates and xi-eta */
10960 /* coordinates. The xi-eta coordinate system is defined in terms of the */
10961 /* triangle: the origin of the triangle is the origin of the coordinate */
10962 /* system; the destination of the triangle is one unit along the xi axis; */
10963 /* and the apex of the triangle is one unit along the eta axis. */
10965 /* The return value indicates which edge of the triangle is shortest. */
10967 /*****************************************************************************/
10969 enum circumcenterresult findcircumcenter( torg, tdest, tapex, circumcenter,
10974 point circumcenter;
10978 REAL xdo, ydo, xao, yao, xad, yad;
10979 REAL dodist, aodist, addist;
10983 circumcentercount++;
10985 /* Compute the circumcenter of the triangle. */
10986 xdo = tdest[0] - torg[0];
10987 ydo = tdest[1] - torg[1];
10988 xao = tapex[0] - torg[0];
10989 yao = tapex[1] - torg[1];
10990 dodist = xdo * xdo + ydo * ydo;
10991 aodist = xao * xao + yao * yao;
10993 denominator = (REAL)( 0.5 / ( xdo * yao - xao * ydo ) );
10996 /* Use the counterclockwise() routine to ensure a positive (and */
10997 /* reasonably accurate) result, avoiding any possibility of */
10998 /* division by zero. */
10999 denominator = (REAL)( 0.5 / counterclockwise( tdest, tapex, torg ) );
11000 /* Don't count the above as an orientation test. */
11001 counterclockcount--;
11003 circumcenter[0] = torg[0] - ( ydo * aodist - yao * dodist ) * denominator;
11004 circumcenter[1] = torg[1] + ( xdo * aodist - xao * dodist ) * denominator;
11006 /* To interpolate point attributes for the new point inserted at */
11007 /* the circumcenter, define a coordinate system with a xi-axis, */
11008 /* directed from the triangle's origin to its destination, and */
11009 /* an eta-axis, directed from its origin to its apex. */
11010 /* Calculate the xi and eta coordinates of the circumcenter. */
11011 dx = circumcenter[0] - torg[0];
11012 dy = circumcenter[1] - torg[1];
11013 *xi = (REAL)( ( dx * yao - xao * dy ) * ( 2.0 * denominator ) );
11014 *eta = (REAL)( ( xdo * dy - dx * ydo ) * ( 2.0 * denominator ) );
11016 xad = tapex[0] - tdest[0];
11017 yad = tapex[1] - tdest[1];
11018 addist = xad * xad + yad * yad;
11019 if ( ( addist < dodist ) && ( addist < aodist ) ) {
11020 return OPPOSITEORG;
11022 else if ( dodist < aodist ) {
11023 return OPPOSITEAPEX;
11026 return OPPOSITEDEST;
11030 /*****************************************************************************/
11032 /* splittriangle() Inserts a point at the circumcenter of a triangle. */
11033 /* Deletes the newly inserted point if it encroaches upon */
11036 /*****************************************************************************/
11040 void splittriangle( badtri )
11041 struct badface *badtri;
11043 point borg, bdest, bapex;
11046 enum insertsiteresult success;
11047 enum circumcenterresult shortedge;
11051 org( badtri->badfacetri, borg );
11052 dest( badtri->badfacetri, bdest );
11053 apex( badtri->badfacetri, bapex );
11054 /* Make sure that this triangle is still the same triangle it was */
11055 /* when it was tested and determined to be of bad quality. */
11056 /* Subsequent transformations may have made it a different triangle. */
11057 if ( ( borg == badtri->faceorg ) && ( bdest == badtri->facedest ) &&
11058 ( bapex == badtri->faceapex ) ) {
11059 if ( verbose > 1 ) {
11060 printf( " Splitting this triangle at its circumcenter:\n" );
11061 printf( " (%.12g, %.12g) (%.12g, %.12g) (%.12g, %.12g)\n", borg[0],
11062 borg[1], bdest[0], bdest[1], bapex[0], bapex[1] );
11065 /* Create a new point at the triangle's circumcenter. */
11066 newpoint = (point) poolalloc( &points );
11067 shortedge = findcircumcenter( borg, bdest, bapex, newpoint, &xi, &eta );
11068 /* Check whether the new point lies on a triangle vertex. */
11069 if ( ( ( newpoint[0] == borg[0] ) && ( newpoint[1] == borg[1] ) )
11070 || ( ( newpoint[0] == bdest[0] ) && ( newpoint[1] == bdest[1] ) )
11071 || ( ( newpoint[0] == bapex[0] ) && ( newpoint[1] == bapex[1] ) ) ) {
11073 printf( "Warning: New point (%.12g, %.12g) falls on existing vertex.\n"
11074 , newpoint[0], newpoint[1] );
11077 pointdealloc( newpoint );
11080 for ( i = 2; i < 2 + nextras; i++ ) {
11081 /* Interpolate the point attributes at the circumcenter. */
11082 newpoint[i] = borg[i] + xi * ( bdest[i] - borg[i] )
11083 + eta * ( bapex[i] - borg[i] );
11085 /* The new point must be in the interior, and have a marker of zero. */
11086 setpointmark( newpoint, 0 );
11087 /* Ensure that the handle `badtri->badfacetri' represents the shortest */
11088 /* edge of the triangle. This ensures that the circumcenter must */
11089 /* fall to the left of this edge, so point location will work. */
11090 if ( shortedge == OPPOSITEORG ) {
11091 lnextself( badtri->badfacetri );
11093 else if ( shortedge == OPPOSITEDEST ) {
11094 lprevself( badtri->badfacetri );
11096 /* Insert the circumcenter, searching from the edge of the triangle, */
11097 /* and maintain the Delaunay property of the triangulation. */
11098 success = insertsite( newpoint, &( badtri->badfacetri ),
11099 (struct edge *) NULL, 1, 1 );
11100 if ( success == SUCCESSFULPOINT ) {
11101 if ( steinerleft > 0 ) {
11105 else if ( success == ENCROACHINGPOINT ) {
11106 /* If the newly inserted point encroaches upon a segment, delete it. */
11107 deletesite( &( badtri->badfacetri ) );
11109 else if ( success == VIOLATINGPOINT ) {
11110 /* Failed to insert the new point, but some segment was */
11111 /* marked as being encroached. */
11112 pointdealloc( newpoint );
11114 else { /* success == DUPLICATEPOINT */
11115 /* Failed to insert the new point because a vertex is already there. */
11118 "Warning: New point (%.12g, %.12g) falls on existing vertex.\n"
11119 , newpoint[0], newpoint[1] );
11122 pointdealloc( newpoint );
11127 printf( " The new point is at the circumcenter of triangle\n" );
11128 printf( " (%.12g, %.12g) (%.12g, %.12g) (%.12g, %.12g)\n",
11129 borg[0], borg[1], bdest[0], bdest[1], bapex[0], bapex[1] );
11131 printf( "This probably means that I am trying to refine triangles\n" );
11132 printf( " to a smaller size than can be accommodated by the finite\n" );
11133 printf( " precision of floating point arithmetic. (You can be\n" );
11134 printf( " sure of this if I fail to terminate.)\n" );
11138 /* Return the bad triangle to the pool. */
11139 pooldealloc( &badtriangles, (VOID *) badtri );
11142 #endif /* not CDT_ONLY */
11144 /*****************************************************************************/
11146 /* enforcequality() Remove all the encroached edges and bad triangles */
11147 /* from the triangulation. */
11149 /*****************************************************************************/
11153 void enforcequality(){
11157 printf( "Adding Steiner points to enforce quality.\n" );
11159 /* Initialize the pool of encroached segments. */
11160 poolinit( &badsegments, sizeof( struct edge ), BADSEGMENTPERBLOCK, POINTER, 0 );
11162 printf( " Looking for encroached segments.\n" );
11164 /* Test all segments to see if they're encroached. */
11166 if ( verbose && ( badsegments.items > 0 ) ) {
11167 printf( " Splitting encroached segments.\n" );
11169 /* Note that steinerleft == -1 if an unlimited number */
11170 /* of Steiner points is allowed. */
11171 while ( ( badsegments.items > 0 ) && ( steinerleft != 0 ) ) {
11172 /* Fix the segments without noting newly encroached segments or */
11173 /* bad triangles. The reason we don't want to note newly */
11174 /* encroached segments is because some encroached segments are */
11175 /* likely to be noted multiple times, and would then be blindly */
11176 /* split multiple times. I should fix that some time. */
11178 /* Now, find all the segments that became encroached while adding */
11179 /* points to split encroached segments. */
11182 /* At this point, if we haven't run out of Steiner points, the */
11183 /* triangulation should be (conforming) Delaunay. */
11185 /* Next, we worry about enforcing triangle quality. */
11186 if ( ( minangle > 0.0 ) || vararea || fixedarea ) {
11187 /* Initialize the pool of bad triangles. */
11188 poolinit( &badtriangles, sizeof( struct badface ), BADTRIPERBLOCK, POINTER,
11190 /* Initialize the queues of bad triangles. */
11191 for ( i = 0; i < 64; i++ ) {
11192 queuefront[i] = (struct badface *) NULL;
11193 queuetail[i] = &queuefront[i];
11195 /* Test all triangles to see if they're bad. */
11198 printf( " Splitting bad triangles.\n" );
11200 while ( ( badtriangles.items > 0 ) && ( steinerleft != 0 ) ) {
11201 /* Fix one bad triangle by inserting a point at its circumcenter. */
11202 splittriangle( dequeuebadtri() );
11203 /* Fix any encroached segments that may have resulted. Record */
11204 /* any new bad triangles or encroached segments that result. */
11205 if ( badsegments.items > 0 ) {
11210 /* At this point, if we haven't run out of Steiner points, the */
11211 /* triangulation should be (conforming) Delaunay and have no */
11212 /* low-quality triangles. */
11214 /* Might we have run out of Steiner points too soon? */
11215 if ( !quiet && ( badsegments.items > 0 ) && ( steinerleft == 0 ) ) {
11216 printf( "\nWarning: I ran out of Steiner points, but the mesh has\n" );
11217 if ( badsegments.items == 1 ) {
11218 printf( " an encroached segment, and therefore might not be truly\n" );
11221 printf( " %ld encroached segments, and therefore might not be truly\n",
11222 badsegments.items );
11224 printf( " Delaunay. If the Delaunay property is important to you,\n" );
11225 printf( " try increasing the number of Steiner points (controlled by\n" );
11226 printf( " the -S switch) slightly and try again.\n\n" );
11230 #endif /* not CDT_ONLY */
11234 /********* Mesh quality maintenance ends here *********/
11236 /*****************************************************************************/
11238 /* highorder() Create extra nodes for quadratic subparametric elements. */
11240 /*****************************************************************************/
11243 struct triedge triangleloop, trisym;
11244 struct edge checkmark;
11248 triangle ptr; /* Temporary variable used by sym(). */
11249 shelle sptr; /* Temporary variable used by tspivot(). */
11252 printf( "Adding vertices for second-order triangles.\n" );
11254 /* The following line ensures that dead items in the pool of nodes */
11255 /* cannot be allocated for the extra nodes associated with high */
11256 /* order elements. This ensures that the primary nodes (at the */
11257 /* corners of elements) will occur earlier in the output files, and */
11258 /* have lower indices, than the extra nodes. */
11259 points.deaditemstack = (VOID *) NULL;
11261 traversalinit( &triangles );
11262 triangleloop.tri = triangletraverse();
11263 /* To loop over the set of edges, loop over all triangles, and look at */
11264 /* the three edges of each triangle. If there isn't another triangle */
11265 /* adjacent to the edge, operate on the edge. If there is another */
11266 /* adjacent triangle, operate on the edge only if the current triangle */
11267 /* has a smaller pointer than its neighbor. This way, each edge is */
11268 /* considered only once. */
11269 while ( triangleloop.tri != (triangle *) NULL ) {
11270 for ( triangleloop.orient = 0; triangleloop.orient < 3;
11271 triangleloop.orient++ ) {
11272 sym( triangleloop, trisym );
11273 if ( ( triangleloop.tri < trisym.tri ) || ( trisym.tri == dummytri ) ) {
11274 org( triangleloop, torg );
11275 dest( triangleloop, tdest );
11276 /* Create a new node in the middle of the edge. Interpolate */
11277 /* its attributes. */
11278 newpoint = (point) poolalloc( &points );
11279 for ( i = 0; i < 2 + nextras; i++ ) {
11280 newpoint[i] = (REAL)( 0.5 * ( torg[i] + tdest[i] ) );
11282 /* Set the new node's marker to zero or one, depending on */
11283 /* whether it lies on a boundary. */
11284 setpointmark( newpoint, trisym.tri == dummytri );
11285 if ( useshelles ) {
11286 tspivot( triangleloop, checkmark );
11287 /* If this edge is a segment, transfer the marker to the new node. */
11288 if ( checkmark.sh != dummysh ) {
11289 setpointmark( newpoint, mark( checkmark ) );
11292 if ( verbose > 1 ) {
11293 printf( " Creating (%.12g, %.12g).\n", newpoint[0], newpoint[1] );
11295 /* Record the new node in the (one or two) adjacent elements. */
11296 triangleloop.tri[highorderindex + triangleloop.orient] =
11297 (triangle) newpoint;
11298 if ( trisym.tri != dummytri ) {
11299 trisym.tri[highorderindex + trisym.orient] = (triangle) newpoint;
11303 triangleloop.tri = triangletraverse();
11307 /********* File I/O routines begin here *********/
11311 /*****************************************************************************/
11313 /* readline() Read a nonempty line from a file. */
11315 /* A line is considered "nonempty" if it contains something that looks like */
11318 /*****************************************************************************/
11322 char *readline( string, infile, infilename )
11329 /* Search for something that looks like a number. */
11331 result = fgets( string, INPUTLINESIZE, infile );
11332 if ( result == (char *) NULL ) {
11333 printf( " Error: Unexpected end of file in %s.\n", infilename );
11336 /* Skip anything that doesn't look like a number, a comment, */
11337 /* or the end of a line. */
11338 while ( ( *result != '\0' ) && ( *result != '#' )
11339 && ( *result != '.' ) && ( *result != '+' ) && ( *result != '-' )
11340 && ( ( *result < '0' ) || ( *result > '9' ) ) ) {
11343 /* If it's a comment or end of line, read another line and try again. */
11344 } while ( ( *result == '#' ) || ( *result == '\0' ) );
11348 #endif /* not TRILIBRARY */
11350 /*****************************************************************************/
11352 /* findfield() Find the next field of a string. */
11354 /* Jumps past the current field by searching for whitespace, then jumps */
11355 /* past the whitespace to find the next field. */
11357 /*****************************************************************************/
11361 char *findfield( string )
11367 /* Skip the current field. Stop upon reaching whitespace. */
11368 while ( ( *result != '\0' ) && ( *result != '#' )
11369 && ( *result != ' ' ) && ( *result != '\t' ) ) {
11372 /* Now skip the whitespace and anything else that doesn't look like a */
11373 /* number, a comment, or the end of a line. */
11374 while ( ( *result != '\0' ) && ( *result != '#' )
11375 && ( *result != '.' ) && ( *result != '+' ) && ( *result != '-' )
11376 && ( ( *result < '0' ) || ( *result > '9' ) ) ) {
11379 /* Check for a comment (prefixed with `#'). */
11380 if ( *result == '#' ) {
11386 #endif /* not TRILIBRARY */
11388 /*****************************************************************************/
11390 /* readnodes() Read the points from a file, which may be a .node or .poly */
11393 /*****************************************************************************/
11397 void readnodes( nodefilename, polyfilename, polyfile )
11398 char *nodefilename;
11399 char *polyfilename;
11404 char inputline[INPUTLINESIZE];
11414 /* Read the points from a .poly file. */
11416 printf( "Opening %s.\n", polyfilename );
11418 *polyfile = fopen( polyfilename, "r" );
11419 if ( *polyfile == (FILE *) NULL ) {
11420 printf( " Error: Cannot access file %s.\n", polyfilename );
11423 /* Read number of points, number of dimensions, number of point */
11424 /* attributes, and number of boundary markers. */
11425 stringptr = readline( inputline, *polyfile, polyfilename );
11426 inpoints = (int) strtol( stringptr, &stringptr, 0 );
11427 stringptr = findfield( stringptr );
11428 if ( *stringptr == '\0' ) {
11432 mesh_dim = (int) strtol( stringptr, &stringptr, 0 );
11434 stringptr = findfield( stringptr );
11435 if ( *stringptr == '\0' ) {
11439 nextras = (int) strtol( stringptr, &stringptr, 0 );
11441 stringptr = findfield( stringptr );
11442 if ( *stringptr == '\0' ) {
11446 nodemarkers = (int) strtol( stringptr, &stringptr, 0 );
11448 if ( inpoints > 0 ) {
11449 infile = *polyfile;
11450 infilename = polyfilename;
11454 /* If the .poly file claims there are zero points, that means that */
11455 /* the points should be read from a separate .node file. */
11457 infilename = innodefilename;
11462 infilename = innodefilename;
11463 *polyfile = (FILE *) NULL;
11466 if ( readnodefile ) {
11467 /* Read the points from a .node file. */
11469 printf( "Opening %s.\n", innodefilename );
11471 infile = fopen( innodefilename, "r" );
11472 if ( infile == (FILE *) NULL ) {
11473 printf( " Error: Cannot access file %s.\n", innodefilename );
11476 /* Read number of points, number of dimensions, number of point */
11477 /* attributes, and number of boundary markers. */
11478 stringptr = readline( inputline, infile, innodefilename );
11479 inpoints = (int) strtol( stringptr, &stringptr, 0 );
11480 stringptr = findfield( stringptr );
11481 if ( *stringptr == '\0' ) {
11485 mesh_dim = (int) strtol( stringptr, &stringptr, 0 );
11487 stringptr = findfield( stringptr );
11488 if ( *stringptr == '\0' ) {
11492 nextras = (int) strtol( stringptr, &stringptr, 0 );
11494 stringptr = findfield( stringptr );
11495 if ( *stringptr == '\0' ) {
11499 nodemarkers = (int) strtol( stringptr, &stringptr, 0 );
11503 if ( inpoints < 3 ) {
11504 printf( "Error: Input must have at least three input points.\n" );
11507 if ( mesh_dim != 2 ) {
11508 printf( "Error: Triangle only works with two-dimensional meshes.\n" );
11512 initializepointpool();
11514 /* Read the points. */
11515 for ( i = 0; i < inpoints; i++ ) {
11516 pointloop = (point) poolalloc( &points );
11517 stringptr = readline( inputline, infile, infilename );
11519 firstnode = (int) strtol( stringptr, &stringptr, 0 );
11520 if ( ( firstnode == 0 ) || ( firstnode == 1 ) ) {
11521 firstnumber = firstnode;
11524 stringptr = findfield( stringptr );
11525 if ( *stringptr == '\0' ) {
11526 printf( "Error: Point %d has no x coordinate.\n", firstnumber + i );
11529 x = (REAL) strtod( stringptr, &stringptr );
11530 stringptr = findfield( stringptr );
11531 if ( *stringptr == '\0' ) {
11532 printf( "Error: Point %d has no y coordinate.\n", firstnumber + i );
11535 y = (REAL) strtod( stringptr, &stringptr );
11538 /* Read the point attributes. */
11539 for ( j = 2; j < 2 + nextras; j++ ) {
11540 stringptr = findfield( stringptr );
11541 if ( *stringptr == '\0' ) {
11542 pointloop[j] = 0.0;
11545 pointloop[j] = (REAL) strtod( stringptr, &stringptr );
11548 if ( nodemarkers ) {
11549 /* Read a point marker. */
11550 stringptr = findfield( stringptr );
11551 if ( *stringptr == '\0' ) {
11552 setpointmark( pointloop, 0 );
11555 currentmarker = (int) strtol( stringptr, &stringptr, 0 );
11556 setpointmark( pointloop, currentmarker );
11560 /* If no markers are specified in the file, they default to zero. */
11561 setpointmark( pointloop, 0 );
11563 /* Determine the smallest and largest x and y coordinates. */
11569 xmin = ( x < xmin ) ? x : xmin;
11570 xmax = ( x > xmax ) ? x : xmax;
11571 ymin = ( y < ymin ) ? y : ymin;
11572 ymax = ( y > ymax ) ? y : ymax;
11575 if ( readnodefile ) {
11579 /* Nonexistent x value used as a flag to mark circle events in sweepline */
11580 /* Delaunay algorithm. */
11581 xminextreme = 10 * xmin - 9 * xmax;
11584 #endif /* not TRILIBRARY */
11586 /*****************************************************************************/
11588 /* transfernodes() Read the points from memory. */
11590 /*****************************************************************************/
11594 void transfernodes( pointlist, pointattriblist, pointmarkerlist, numberofpoints,
11595 numberofpointattribs )
11597 REAL *pointattriblist;
11598 int *pointmarkerlist;
11599 int numberofpoints;
11600 int numberofpointattribs;
11608 inpoints = numberofpoints;
11610 nextras = numberofpointattribs;
11612 if ( inpoints < 3 ) {
11613 printf( "Error: Input must have at least three input points.\n" );
11617 initializepointpool();
11619 /* Read the points. */
11622 for ( i = 0; i < inpoints; i++ ) {
11623 pointloop = (point) poolalloc( &points );
11624 /* Read the point coordinates. */
11625 x = pointloop[0] = pointlist[coordindex++];
11626 y = pointloop[1] = pointlist[coordindex++];
11627 /* Read the point attributes. */
11628 for ( j = 0; j < numberofpointattribs; j++ ) {
11629 pointloop[2 + j] = pointattriblist[attribindex++];
11631 if ( pointmarkerlist != (int *) NULL ) {
11632 /* Read a point marker. */
11633 setpointmark( pointloop, pointmarkerlist[i] );
11636 /* If no markers are specified, they default to zero. */
11637 setpointmark( pointloop, 0 );
11641 /* Determine the smallest and largest x and y coordinates. */
11647 xmin = ( x < xmin ) ? x : xmin;
11648 xmax = ( x > xmax ) ? x : xmax;
11649 ymin = ( y < ymin ) ? y : ymin;
11650 ymax = ( y > ymax ) ? y : ymax;
11654 /* Nonexistent x value used as a flag to mark circle events in sweepline */
11655 /* Delaunay algorithm. */
11656 xminextreme = 10 * xmin - 9 * xmax;
11659 #endif /* TRILIBRARY */
11661 /*****************************************************************************/
11663 /* readholes() Read the holes, and possibly regional attributes and area */
11664 /* constraints, from a .poly file. */
11666 /*****************************************************************************/
11670 void readholes( polyfile, polyfilename, hlist, holes, rlist, regions )
11672 char *polyfilename;
11680 char inputline[INPUTLINESIZE];
11685 /* Read the holes. */
11686 stringptr = readline( inputline, polyfile, polyfilename );
11687 *holes = (int) strtol( stringptr, &stringptr, 0 );
11688 if ( *holes > 0 ) {
11689 holelist = (REAL *) malloc( 2 * *holes * sizeof( REAL ) );
11691 if ( holelist == (REAL *) NULL ) {
11692 printf( "Error: Out of memory.\n" );
11695 for ( i = 0; i < 2 * *holes; i += 2 ) {
11696 stringptr = readline( inputline, polyfile, polyfilename );
11697 stringptr = findfield( stringptr );
11698 if ( *stringptr == '\0' ) {
11699 printf( "Error: Hole %d has no x coordinate.\n",
11700 firstnumber + ( i >> 1 ) );
11704 holelist[i] = (REAL) strtod( stringptr, &stringptr );
11706 stringptr = findfield( stringptr );
11707 if ( *stringptr == '\0' ) {
11708 printf( "Error: Hole %d has no y coordinate.\n",
11709 firstnumber + ( i >> 1 ) );
11713 holelist[i + 1] = (REAL) strtod( stringptr, &stringptr );
11718 *hlist = (REAL *) NULL;
11722 if ( ( regionattrib || vararea ) && !refine ) {
11723 /* Read the area constraints. */
11724 stringptr = readline( inputline, polyfile, polyfilename );
11725 *regions = (int) strtol( stringptr, &stringptr, 0 );
11726 if ( *regions > 0 ) {
11727 regionlist = (REAL *) malloc( 4 * *regions * sizeof( REAL ) );
11728 *rlist = regionlist;
11729 if ( regionlist == (REAL *) NULL ) {
11730 printf( "Error: Out of memory.\n" );
11734 for ( i = 0; i < *regions; i++ ) {
11735 stringptr = readline( inputline, polyfile, polyfilename );
11736 stringptr = findfield( stringptr );
11737 if ( *stringptr == '\0' ) {
11738 printf( "Error: Region %d has no x coordinate.\n",
11743 regionlist[index++] = (REAL) strtod( stringptr, &stringptr );
11745 stringptr = findfield( stringptr );
11746 if ( *stringptr == '\0' ) {
11747 printf( "Error: Region %d has no y coordinate.\n",
11752 regionlist[index++] = (REAL) strtod( stringptr, &stringptr );
11754 stringptr = findfield( stringptr );
11755 if ( *stringptr == '\0' ) {
11757 "Error: Region %d has no region attribute or area constraint.\n",
11762 regionlist[index++] = (REAL) strtod( stringptr, &stringptr );
11764 stringptr = findfield( stringptr );
11765 if ( *stringptr == '\0' ) {
11766 regionlist[index] = regionlist[index - 1];
11769 regionlist[index] = (REAL) strtod( stringptr, &stringptr );
11776 /* Set `*regions' to zero to avoid an accidental free() later. */
11778 *rlist = (REAL *) NULL;
11780 #endif /* not CDT_ONLY */
11782 fclose( polyfile );
11785 #endif /* not TRILIBRARY */
11787 /*****************************************************************************/
11789 /* finishfile() Write the command line to the output file so the user */
11790 /* can remember how the file was generated. Close the file. */
11792 /*****************************************************************************/
11796 void finishfile( outfile, argc, argv )
11803 fprintf( outfile, "# Generated by" );
11804 for ( i = 0; i < argc; i++ ) {
11805 fprintf( outfile, " " );
11806 fputs( argv[i], outfile );
11808 fprintf( outfile, "\n" );
11812 #endif /* not TRILIBRARY */
11814 /*****************************************************************************/
11816 /* writenodes() Number the points and write them to a .node file. */
11818 /* To save memory, the point numbers are written over the shell markers */
11819 /* after the points are written to a file. */
11821 /*****************************************************************************/
11825 void writenodes( pointlist, pointattriblist, pointmarkerlist )
11827 REAL **pointattriblist;
11828 int **pointmarkerlist;
11830 #else /* not TRILIBRARY */
11832 void writenodes( nodefilename, argc, argv )
11833 char *nodefilename;
11837 #endif /* not TRILIBRARY */
11846 #else /* not TRILIBRARY */
11848 #endif /* not TRILIBRARY */
11855 printf( "Writing points.\n" );
11857 /* Allocate memory for output points if necessary. */
11858 if ( *pointlist == (REAL *) NULL ) {
11859 *pointlist = (REAL *) malloc( points.items * 2 * sizeof( REAL ) );
11860 if ( *pointlist == (REAL *) NULL ) {
11861 printf( "Error: Out of memory.\n" );
11865 /* Allocate memory for output point attributes if necessary. */
11866 if ( ( nextras > 0 ) && ( *pointattriblist == (REAL *) NULL ) ) {
11867 *pointattriblist = (REAL *) malloc( points.items * nextras * sizeof( REAL ) );
11868 if ( *pointattriblist == (REAL *) NULL ) {
11869 printf( "Error: Out of memory.\n" );
11873 /* Allocate memory for output point markers if necessary. */
11874 if ( !nobound && ( *pointmarkerlist == (int *) NULL ) ) {
11875 *pointmarkerlist = (int *) malloc( points.items * sizeof( int ) );
11876 if ( *pointmarkerlist == (int *) NULL ) {
11877 printf( "Error: Out of memory.\n" );
11881 plist = *pointlist;
11882 palist = *pointattriblist;
11883 pmlist = *pointmarkerlist;
11886 #else /* not TRILIBRARY */
11888 printf( "Writing %s.\n", nodefilename );
11890 outfile = fopen( nodefilename, "w" );
11891 if ( outfile == (FILE *) NULL ) {
11892 printf( " Error: Cannot create file %s.\n", nodefilename );
11895 /* Number of points, number of dimensions, number of point attributes, */
11896 /* and number of boundary markers (zero or one). */
11897 fprintf( outfile, "%ld %d %d %d\n", points.items, mesh_dim, nextras,
11899 #endif /* not TRILIBRARY */
11901 traversalinit( &points );
11902 pointloop = pointtraverse();
11903 pointnumber = firstnumber;
11904 while ( pointloop != (point) NULL ) {
11906 /* X and y coordinates. */
11907 plist[coordindex++] = pointloop[0];
11908 plist[coordindex++] = pointloop[1];
11909 /* Point attributes. */
11910 for ( i = 0; i < nextras; i++ ) {
11911 palist[attribindex++] = pointloop[2 + i];
11914 /* Copy the boundary marker. */
11915 pmlist[pointnumber - firstnumber] = pointmark( pointloop );
11917 #else /* not TRILIBRARY */
11918 /* Point number, x and y coordinates. */
11919 fprintf( outfile, "%4d %.17g %.17g", pointnumber, pointloop[0],
11921 for ( i = 0; i < nextras; i++ ) {
11922 /* Write an attribute. */
11923 fprintf( outfile, " %.17g", pointloop[i + 2] );
11926 fprintf( outfile, "\n" );
11929 /* Write the boundary marker. */
11930 fprintf( outfile, " %d\n", pointmark( pointloop ) );
11932 #endif /* not TRILIBRARY */
11934 setpointmark( pointloop, pointnumber );
11935 pointloop = pointtraverse();
11940 finishfile( outfile, argc, argv );
11941 #endif /* not TRILIBRARY */
11944 /*****************************************************************************/
11946 /* numbernodes() Number the points. */
11948 /* Each point is assigned a marker equal to its number. */
11950 /* Used when writenodes() is not called because no .node file is written. */
11952 /*****************************************************************************/
11954 void numbernodes(){
11958 traversalinit( &points );
11959 pointloop = pointtraverse();
11960 pointnumber = firstnumber;
11961 while ( pointloop != (point) NULL ) {
11962 setpointmark( pointloop, pointnumber );
11963 pointloop = pointtraverse();
11968 /*****************************************************************************/
11970 /* writeelements() Write the triangles to an .ele file. */
11972 /*****************************************************************************/
11976 void writeelements( trianglelist, triangleattriblist )
11977 int **trianglelist;
11978 REAL **triangleattriblist;
11980 #else /* not TRILIBRARY */
11982 void writeelements( elefilename, argc, argv )
11987 #endif /* not TRILIBRARY */
11995 #else /* not TRILIBRARY */
11997 #endif /* not TRILIBRARY */
11998 struct triedge triangleloop;
12000 point mid1, mid2, mid3;
12006 printf( "Writing triangles.\n" );
12008 /* Allocate memory for output triangles if necessary. */
12009 if ( *trianglelist == (int *) NULL ) {
12010 *trianglelist = (int *) malloc( triangles.items *
12011 ( ( order + 1 ) * ( order + 2 ) / 2 ) * sizeof( int ) );
12012 if ( *trianglelist == (int *) NULL ) {
12013 printf( "Error: Out of memory.\n" );
12017 /* Allocate memory for output triangle attributes if necessary. */
12018 if ( ( eextras > 0 ) && ( *triangleattriblist == (REAL *) NULL ) ) {
12019 *triangleattriblist = (REAL *) malloc( triangles.items * eextras *
12021 if ( *triangleattriblist == (REAL *) NULL ) {
12022 printf( "Error: Out of memory.\n" );
12026 tlist = *trianglelist;
12027 talist = *triangleattriblist;
12030 #else /* not TRILIBRARY */
12032 printf( "Writing %s.\n", elefilename );
12034 outfile = fopen( elefilename, "w" );
12035 if ( outfile == (FILE *) NULL ) {
12036 printf( " Error: Cannot create file %s.\n", elefilename );
12039 /* Number of triangles, points per triangle, attributes per triangle. */
12040 fprintf( outfile, "%ld %d %d\n", triangles.items,
12041 ( order + 1 ) * ( order + 2 ) / 2, eextras );
12042 #endif /* not TRILIBRARY */
12044 traversalinit( &triangles );
12045 triangleloop.tri = triangletraverse();
12046 triangleloop.orient = 0;
12047 elementnumber = firstnumber;
12048 while ( triangleloop.tri != (triangle *) NULL ) {
12049 org( triangleloop, p1 );
12050 dest( triangleloop, p2 );
12051 apex( triangleloop, p3 );
12052 if ( order == 1 ) {
12054 tlist[pointindex++] = pointmark( p1 );
12055 tlist[pointindex++] = pointmark( p2 );
12056 tlist[pointindex++] = pointmark( p3 );
12057 #else /* not TRILIBRARY */
12058 /* Triangle number, indices for three points. */
12059 fprintf( outfile, "%4d %4d %4d %4d", elementnumber,
12060 pointmark( p1 ), pointmark( p2 ), pointmark( p3 ) );
12061 #endif /* not TRILIBRARY */
12064 mid1 = (point) triangleloop.tri[highorderindex + 1];
12065 mid2 = (point) triangleloop.tri[highorderindex + 2];
12066 mid3 = (point) triangleloop.tri[highorderindex];
12068 tlist[pointindex++] = pointmark( p1 );
12069 tlist[pointindex++] = pointmark( p2 );
12070 tlist[pointindex++] = pointmark( p3 );
12071 tlist[pointindex++] = pointmark( mid1 );
12072 tlist[pointindex++] = pointmark( mid2 );
12073 tlist[pointindex++] = pointmark( mid3 );
12074 #else /* not TRILIBRARY */
12075 /* Triangle number, indices for six points. */
12076 fprintf( outfile, "%4d %4d %4d %4d %4d %4d %4d", elementnumber,
12077 pointmark( p1 ), pointmark( p2 ), pointmark( p3 ), pointmark( mid1 ),
12078 pointmark( mid2 ), pointmark( mid3 ) );
12079 #endif /* not TRILIBRARY */
12083 for ( i = 0; i < eextras; i++ ) {
12084 talist[attribindex++] = elemattribute( triangleloop, i );
12086 #else /* not TRILIBRARY */
12087 for ( i = 0; i < eextras; i++ ) {
12088 fprintf( outfile, " %.17g", elemattribute( triangleloop, i ) );
12090 fprintf( outfile, "\n" );
12091 #endif /* not TRILIBRARY */
12093 triangleloop.tri = triangletraverse();
12098 finishfile( outfile, argc, argv );
12099 #endif /* not TRILIBRARY */
12102 /*****************************************************************************/
12104 /* writepoly() Write the segments and holes to a .poly file. */
12106 /*****************************************************************************/
12110 void writepoly( segmentlist, segmentmarkerlist )
12112 int **segmentmarkerlist;
12114 #else /* not TRILIBRARY */
12116 void writepoly( polyfilename, holelist, holes, regionlist, regions, argc, argv )
12117 char *polyfilename;
12125 #endif /* not TRILIBRARY */
12132 #else /* not TRILIBRARY */
12135 #endif /* not TRILIBRARY */
12136 struct edge shelleloop;
12137 point endpoint1, endpoint2;
12142 printf( "Writing segments.\n" );
12144 /* Allocate memory for output segments if necessary. */
12145 if ( *segmentlist == (int *) NULL ) {
12146 *segmentlist = (int *) malloc( shelles.items * 2 * sizeof( int ) );
12147 if ( *segmentlist == (int *) NULL ) {
12148 printf( "Error: Out of memory.\n" );
12152 /* Allocate memory for output segment markers if necessary. */
12153 if ( !nobound && ( *segmentmarkerlist == (int *) NULL ) ) {
12154 *segmentmarkerlist = (int *) malloc( shelles.items * sizeof( int ) );
12155 if ( *segmentmarkerlist == (int *) NULL ) {
12156 printf( "Error: Out of memory.\n" );
12160 slist = *segmentlist;
12161 smlist = *segmentmarkerlist;
12163 #else /* not TRILIBRARY */
12165 printf( "Writing %s.\n", polyfilename );
12167 outfile = fopen( polyfilename, "w" );
12168 if ( outfile == (FILE *) NULL ) {
12169 printf( " Error: Cannot create file %s.\n", polyfilename );
12172 /* The zero indicates that the points are in a separate .node file. */
12173 /* Followed by number of dimensions, number of point attributes, */
12174 /* and number of boundary markers (zero or one). */
12175 fprintf( outfile, "%d %d %d %d\n", 0, mesh_dim, nextras, 1 - nobound );
12176 /* Number of segments, number of boundary markers (zero or one). */
12177 fprintf( outfile, "%ld %d\n", shelles.items, 1 - nobound );
12178 #endif /* not TRILIBRARY */
12180 traversalinit( &shelles );
12181 shelleloop.sh = shelletraverse();
12182 shelleloop.shorient = 0;
12183 shellenumber = firstnumber;
12184 while ( shelleloop.sh != (shelle *) NULL ) {
12185 sorg( shelleloop, endpoint1 );
12186 sdest( shelleloop, endpoint2 );
12188 /* Copy indices of the segment's two endpoints. */
12189 slist[index++] = pointmark( endpoint1 );
12190 slist[index++] = pointmark( endpoint2 );
12192 /* Copy the boundary marker. */
12193 smlist[shellenumber - firstnumber] = mark( shelleloop );
12195 #else /* not TRILIBRARY */
12196 /* Segment number, indices of its two endpoints, and possibly a marker. */
12198 fprintf( outfile, "%4d %4d %4d\n", shellenumber,
12199 pointmark( endpoint1 ), pointmark( endpoint2 ) );
12202 fprintf( outfile, "%4d %4d %4d %4d\n", shellenumber,
12203 pointmark( endpoint1 ), pointmark( endpoint2 ), mark( shelleloop ) );
12205 #endif /* not TRILIBRARY */
12207 shelleloop.sh = shelletraverse();
12213 fprintf( outfile, "%d\n", holes );
12215 for ( i = 0; i < holes; i++ ) {
12216 /* Hole number, x and y coordinates. */
12217 fprintf( outfile, "%4d %.17g %.17g\n", firstnumber + i,
12218 holelist[2 * i], holelist[2 * i + 1] );
12221 if ( regions > 0 ) {
12222 fprintf( outfile, "%d\n", regions );
12223 for ( i = 0; i < regions; i++ ) {
12224 /* Region number, x and y coordinates, attribute, maximum area. */
12225 fprintf( outfile, "%4d %.17g %.17g %.17g %.17g\n", firstnumber + i,
12226 regionlist[4 * i], regionlist[4 * i + 1],
12227 regionlist[4 * i + 2], regionlist[4 * i + 3] );
12230 #endif /* not CDT_ONLY */
12232 finishfile( outfile, argc, argv );
12233 #endif /* not TRILIBRARY */
12236 /*****************************************************************************/
12238 /* writeedges() Write the edges to a .edge file. */
12240 /*****************************************************************************/
12244 void writeedges( edgelist, edgemarkerlist )
12246 int **edgemarkerlist;
12248 #else /* not TRILIBRARY */
12250 void writeedges( edgefilename, argc, argv )
12251 char *edgefilename;
12255 #endif /* not TRILIBRARY */
12262 #else /* not TRILIBRARY */
12264 #endif /* not TRILIBRARY */
12265 struct triedge triangleloop, trisym;
12266 struct edge checkmark;
12269 triangle ptr; /* Temporary variable used by sym(). */
12270 shelle sptr; /* Temporary variable used by tspivot(). */
12274 printf( "Writing edges.\n" );
12276 /* Allocate memory for edges if necessary. */
12277 if ( *edgelist == (int *) NULL ) {
12278 *edgelist = (int *) malloc( edges * 2 * sizeof( int ) );
12279 if ( *edgelist == (int *) NULL ) {
12280 printf( "Error: Out of memory.\n" );
12284 /* Allocate memory for edge markers if necessary. */
12285 if ( !nobound && ( *edgemarkerlist == (int *) NULL ) ) {
12286 *edgemarkerlist = (int *) malloc( edges * sizeof( int ) );
12287 if ( *edgemarkerlist == (int *) NULL ) {
12288 printf( "Error: Out of memory.\n" );
12293 emlist = *edgemarkerlist;
12295 #else /* not TRILIBRARY */
12297 printf( "Writing %s.\n", edgefilename );
12299 outfile = fopen( edgefilename, "w" );
12300 if ( outfile == (FILE *) NULL ) {
12301 printf( " Error: Cannot create file %s.\n", edgefilename );
12304 /* Number of edges, number of boundary markers (zero or one). */
12305 fprintf( outfile, "%ld %d\n", edges, 1 - nobound );
12306 #endif /* not TRILIBRARY */
12308 traversalinit( &triangles );
12309 triangleloop.tri = triangletraverse();
12310 edgenumber = firstnumber;
12311 /* To loop over the set of edges, loop over all triangles, and look at */
12312 /* the three edges of each triangle. If there isn't another triangle */
12313 /* adjacent to the edge, operate on the edge. If there is another */
12314 /* adjacent triangle, operate on the edge only if the current triangle */
12315 /* has a smaller pointer than its neighbor. This way, each edge is */
12316 /* considered only once. */
12317 while ( triangleloop.tri != (triangle *) NULL ) {
12318 for ( triangleloop.orient = 0; triangleloop.orient < 3;
12319 triangleloop.orient++ ) {
12320 sym( triangleloop, trisym );
12321 if ( ( triangleloop.tri < trisym.tri ) || ( trisym.tri == dummytri ) ) {
12322 org( triangleloop, p1 );
12323 dest( triangleloop, p2 );
12325 elist[index++] = pointmark( p1 );
12326 elist[index++] = pointmark( p2 );
12327 #endif /* TRILIBRARY */
12330 /* Edge number, indices of two endpoints. */
12331 fprintf( outfile, "%4d %d %d\n", edgenumber,
12332 pointmark( p1 ), pointmark( p2 ) );
12333 #endif /* not TRILIBRARY */
12336 /* Edge number, indices of two endpoints, and a boundary marker. */
12337 /* If there's no shell edge, the boundary marker is zero. */
12338 if ( useshelles ) {
12339 tspivot( triangleloop, checkmark );
12340 if ( checkmark.sh == dummysh ) {
12342 emlist[edgenumber - firstnumber] = 0;
12343 #else /* not TRILIBRARY */
12344 fprintf( outfile, "%4d %d %d %d\n", edgenumber,
12345 pointmark( p1 ), pointmark( p2 ), 0 );
12346 #endif /* not TRILIBRARY */
12350 emlist[edgenumber - firstnumber] = mark( checkmark );
12351 #else /* not TRILIBRARY */
12352 fprintf( outfile, "%4d %d %d %d\n", edgenumber,
12353 pointmark( p1 ), pointmark( p2 ), mark( checkmark ) );
12354 #endif /* not TRILIBRARY */
12359 emlist[edgenumber - firstnumber] = trisym.tri == dummytri;
12360 #else /* not TRILIBRARY */
12361 fprintf( outfile, "%4d %d %d %d\n", edgenumber,
12362 pointmark( p1 ), pointmark( p2 ), trisym.tri == dummytri );
12363 #endif /* not TRILIBRARY */
12369 triangleloop.tri = triangletraverse();
12373 finishfile( outfile, argc, argv );
12374 #endif /* not TRILIBRARY */
12377 /*****************************************************************************/
12379 /* writevoronoi() Write the Voronoi diagram to a .v.node and .v.edge */
12382 /* The Voronoi diagram is the geometric dual of the Delaunay triangulation. */
12383 /* Hence, the Voronoi vertices are listed by traversing the Delaunay */
12384 /* triangles, and the Voronoi edges are listed by traversing the Delaunay */
12387 /* WARNING: In order to assign numbers to the Voronoi vertices, this */
12388 /* procedure messes up the shell edges or the extra nodes of every */
12389 /* element. Hence, you should call this procedure last. */
12391 /*****************************************************************************/
12395 void writevoronoi( vpointlist, vpointattriblist, vpointmarkerlist, vedgelist,
12396 vedgemarkerlist, vnormlist )
12397 REAL * *vpointlist;
12398 REAL **vpointattriblist;
12399 int **vpointmarkerlist;
12401 int **vedgemarkerlist;
12404 #else /* not TRILIBRARY */
12406 void writevoronoi( vnodefilename, vedgefilename, argc, argv )
12407 char *vnodefilename;
12408 char *vedgefilename;
12412 #endif /* not TRILIBRARY */
12422 #else /* not TRILIBRARY */
12424 #endif /* not TRILIBRARY */
12425 struct triedge triangleloop, trisym;
12426 point torg, tdest, tapex;
12427 REAL circumcenter[2];
12429 int vnodenumber, vedgenumber;
12432 triangle ptr; /* Temporary variable used by sym(). */
12436 printf( "Writing Voronoi vertices.\n" );
12438 /* Allocate memory for Voronoi vertices if necessary. */
12439 if ( *vpointlist == (REAL *) NULL ) {
12440 *vpointlist = (REAL *) malloc( triangles.items * 2 * sizeof( REAL ) );
12441 if ( *vpointlist == (REAL *) NULL ) {
12442 printf( "Error: Out of memory.\n" );
12446 /* Allocate memory for Voronoi vertex attributes if necessary. */
12447 if ( *vpointattriblist == (REAL *) NULL ) {
12448 *vpointattriblist = (REAL *) malloc( triangles.items * nextras *
12450 if ( *vpointattriblist == (REAL *) NULL ) {
12451 printf( "Error: Out of memory.\n" );
12455 *vpointmarkerlist = (int *) NULL;
12456 plist = *vpointlist;
12457 palist = *vpointattriblist;
12460 #else /* not TRILIBRARY */
12462 printf( "Writing %s.\n", vnodefilename );
12464 outfile = fopen( vnodefilename, "w" );
12465 if ( outfile == (FILE *) NULL ) {
12466 printf( " Error: Cannot create file %s.\n", vnodefilename );
12469 /* Number of triangles, two dimensions, number of point attributes, */
12470 /* zero markers. */
12471 fprintf( outfile, "%ld %d %d %d\n", triangles.items, 2, nextras, 0 );
12472 #endif /* not TRILIBRARY */
12474 traversalinit( &triangles );
12475 triangleloop.tri = triangletraverse();
12476 triangleloop.orient = 0;
12477 vnodenumber = firstnumber;
12478 while ( triangleloop.tri != (triangle *) NULL ) {
12479 org( triangleloop, torg );
12480 dest( triangleloop, tdest );
12481 apex( triangleloop, tapex );
12482 findcircumcenter( torg, tdest, tapex, circumcenter, &xi, &eta );
12484 /* X and y coordinates. */
12485 plist[coordindex++] = circumcenter[0];
12486 plist[coordindex++] = circumcenter[1];
12487 for ( i = 2; i < 2 + nextras; i++ ) {
12488 /* Interpolate the point attributes at the circumcenter. */
12489 palist[attribindex++] = torg[i] + xi * ( tdest[i] - torg[i] )
12490 + eta * ( tapex[i] - torg[i] );
12492 #else /* not TRILIBRARY */
12493 /* Voronoi vertex number, x and y coordinates. */
12494 fprintf( outfile, "%4d %.17g %.17g", vnodenumber, circumcenter[0],
12496 for ( i = 2; i < 2 + nextras; i++ ) {
12497 /* Interpolate the point attributes at the circumcenter. */
12498 fprintf( outfile, " %.17g", torg[i] + xi * ( tdest[i] - torg[i] )
12499 + eta * ( tapex[i] - torg[i] ) );
12501 fprintf( outfile, "\n" );
12502 #endif /* not TRILIBRARY */
12504 *(int *) ( triangleloop.tri + 6 ) = vnodenumber;
12505 triangleloop.tri = triangletraverse();
12510 finishfile( outfile, argc, argv );
12511 #endif /* not TRILIBRARY */
12515 printf( "Writing Voronoi edges.\n" );
12517 /* Allocate memory for output Voronoi edges if necessary. */
12518 if ( *vedgelist == (int *) NULL ) {
12519 *vedgelist = (int *) malloc( edges * 2 * sizeof( int ) );
12520 if ( *vedgelist == (int *) NULL ) {
12521 printf( "Error: Out of memory.\n" );
12525 *vedgemarkerlist = (int *) NULL;
12526 /* Allocate memory for output Voronoi norms if necessary. */
12527 if ( *vnormlist == (REAL *) NULL ) {
12528 *vnormlist = (REAL *) malloc( edges * 2 * sizeof( REAL ) );
12529 if ( *vnormlist == (REAL *) NULL ) {
12530 printf( "Error: Out of memory.\n" );
12534 elist = *vedgelist;
12535 normlist = *vnormlist;
12537 #else /* not TRILIBRARY */
12539 printf( "Writing %s.\n", vedgefilename );
12541 outfile = fopen( vedgefilename, "w" );
12542 if ( outfile == (FILE *) NULL ) {
12543 printf( " Error: Cannot create file %s.\n", vedgefilename );
12546 /* Number of edges, zero boundary markers. */
12547 fprintf( outfile, "%ld %d\n", edges, 0 );
12548 #endif /* not TRILIBRARY */
12550 traversalinit( &triangles );
12551 triangleloop.tri = triangletraverse();
12552 vedgenumber = firstnumber;
12553 /* To loop over the set of edges, loop over all triangles, and look at */
12554 /* the three edges of each triangle. If there isn't another triangle */
12555 /* adjacent to the edge, operate on the edge. If there is another */
12556 /* adjacent triangle, operate on the edge only if the current triangle */
12557 /* has a smaller pointer than its neighbor. This way, each edge is */
12558 /* considered only once. */
12559 while ( triangleloop.tri != (triangle *) NULL ) {
12560 for ( triangleloop.orient = 0; triangleloop.orient < 3;
12561 triangleloop.orient++ ) {
12562 sym( triangleloop, trisym );
12563 if ( ( triangleloop.tri < trisym.tri ) || ( trisym.tri == dummytri ) ) {
12564 /* Find the number of this triangle (and Voronoi vertex). */
12565 p1 = *(int *) ( triangleloop.tri + 6 );
12566 if ( trisym.tri == dummytri ) {
12567 org( triangleloop, torg );
12568 dest( triangleloop, tdest );
12570 /* Copy an infinite ray. Index of one endpoint, and -1. */
12571 elist[coordindex] = p1;
12572 normlist[coordindex++] = tdest[1] - torg[1];
12573 elist[coordindex] = -1;
12574 normlist[coordindex++] = torg[0] - tdest[0];
12575 #else /* not TRILIBRARY */
12576 /* Write an infinite ray. Edge number, index of one endpoint, -1, */
12577 /* and x and y coordinates of a vector representing the */
12578 /* direction of the ray. */
12579 fprintf( outfile, "%4d %d %d %.17g %.17g\n", vedgenumber,
12580 p1, -1, tdest[1] - torg[1], torg[0] - tdest[0] );
12581 #endif /* not TRILIBRARY */
12584 /* Find the number of the adjacent triangle (and Voronoi vertex). */
12585 p2 = *(int *) ( trisym.tri + 6 );
12586 /* Finite edge. Write indices of two endpoints. */
12588 elist[coordindex] = p1;
12589 normlist[coordindex++] = 0.0;
12590 elist[coordindex] = p2;
12591 normlist[coordindex++] = 0.0;
12592 #else /* not TRILIBRARY */
12593 fprintf( outfile, "%4d %d %d\n", vedgenumber, p1, p2 );
12594 #endif /* not TRILIBRARY */
12599 triangleloop.tri = triangletraverse();
12603 finishfile( outfile, argc, argv );
12604 #endif /* not TRILIBRARY */
12609 void writeneighbors( neighborlist )
12610 int **neighborlist;
12612 #else /* not TRILIBRARY */
12614 void writeneighbors( neighborfilename, argc, argv )
12615 char *neighborfilename;
12619 #endif /* not TRILIBRARY */
12625 #else /* not TRILIBRARY */
12627 #endif /* not TRILIBRARY */
12628 struct triedge triangleloop, trisym;
12630 int neighbor1, neighbor2, neighbor3;
12631 triangle ptr; /* Temporary variable used by sym(). */
12635 printf( "Writing neighbors.\n" );
12637 /* Allocate memory for neighbors if necessary. */
12638 if ( *neighborlist == (int *) NULL ) {
12639 *neighborlist = (int *) malloc( triangles.items * 3 * sizeof( int ) );
12640 if ( *neighborlist == (int *) NULL ) {
12641 printf( "Error: Out of memory.\n" );
12645 nlist = *neighborlist;
12647 #else /* not TRILIBRARY */
12649 printf( "Writing %s.\n", neighborfilename );
12651 outfile = fopen( neighborfilename, "w" );
12652 if ( outfile == (FILE *) NULL ) {
12653 printf( " Error: Cannot create file %s.\n", neighborfilename );
12656 /* Number of triangles, three edges per triangle. */
12657 fprintf( outfile, "%ld %d\n", triangles.items, 3 );
12658 #endif /* not TRILIBRARY */
12660 traversalinit( &triangles );
12661 triangleloop.tri = triangletraverse();
12662 triangleloop.orient = 0;
12663 elementnumber = firstnumber;
12664 while ( triangleloop.tri != (triangle *) NULL ) {
12665 *(int *) ( triangleloop.tri + 6 ) = elementnumber;
12666 triangleloop.tri = triangletraverse();
12669 *(int *) ( dummytri + 6 ) = -1;
12671 traversalinit( &triangles );
12672 triangleloop.tri = triangletraverse();
12673 elementnumber = firstnumber;
12674 while ( triangleloop.tri != (triangle *) NULL ) {
12675 triangleloop.orient = 1;
12676 sym( triangleloop, trisym );
12677 neighbor1 = *(int *) ( trisym.tri + 6 );
12678 triangleloop.orient = 2;
12679 sym( triangleloop, trisym );
12680 neighbor2 = *(int *) ( trisym.tri + 6 );
12681 triangleloop.orient = 0;
12682 sym( triangleloop, trisym );
12683 neighbor3 = *(int *) ( trisym.tri + 6 );
12685 nlist[index++] = neighbor1;
12686 nlist[index++] = neighbor2;
12687 nlist[index++] = neighbor3;
12688 #else /* not TRILIBRARY */
12689 /* Triangle number, neighboring triangle numbers. */
12690 fprintf( outfile, "%4d %d %d %d\n", elementnumber,
12691 neighbor1, neighbor2, neighbor3 );
12692 #endif /* not TRILIBRARY */
12694 triangleloop.tri = triangletraverse();
12699 finishfile( outfile, argc, argv );
12700 #endif /* TRILIBRARY */
12703 /*****************************************************************************/
12705 /* writeoff() Write the triangulation to an .off file. */
12707 /* OFF stands for the Object File Format, a format used by the Geometry */
12708 /* Center's Geomview package. */
12710 /*****************************************************************************/
12714 void writeoff( offfilename, argc, argv )
12720 struct triedge triangleloop;
12725 printf( "Writing %s.\n", offfilename );
12727 outfile = fopen( offfilename, "w" );
12728 if ( outfile == (FILE *) NULL ) {
12729 printf( " Error: Cannot create file %s.\n", offfilename );
12732 /* Number of points, triangles, and edges. */
12733 fprintf( outfile, "OFF\n%ld %ld %ld\n", points.items, triangles.items,
12736 /* Write the points. */
12737 traversalinit( &points );
12738 pointloop = pointtraverse();
12739 while ( pointloop != (point) NULL ) {
12740 /* The "0.0" is here because the OFF format uses 3D coordinates. */
12741 fprintf( outfile, " %.17g %.17g %.17g\n", pointloop[0],
12742 pointloop[1], 0.0 );
12743 pointloop = pointtraverse();
12746 /* Write the triangles. */
12747 traversalinit( &triangles );
12748 triangleloop.tri = triangletraverse();
12749 triangleloop.orient = 0;
12750 while ( triangleloop.tri != (triangle *) NULL ) {
12751 org( triangleloop, p1 );
12752 dest( triangleloop, p2 );
12753 apex( triangleloop, p3 );
12754 /* The "3" means a three-vertex polygon. */
12755 fprintf( outfile, " 3 %4d %4d %4d\n", pointmark( p1 ) - 1,
12756 pointmark( p2 ) - 1, pointmark( p3 ) - 1 );
12757 triangleloop.tri = triangletraverse();
12759 finishfile( outfile, argc, argv );
12762 #endif /* not TRILIBRARY */
12766 /********* File I/O routines end here *********/
12768 /*****************************************************************************/
12770 /* quality_statistics() Print statistics about the quality of the mesh. */
12772 /*****************************************************************************/
12774 void quality_statistics(){
12775 struct triedge triangleloop;
12777 REAL cossquaretable[8];
12778 REAL ratiotable[16];
12780 REAL edgelength[3];
12784 REAL shortest, longest;
12786 REAL smallestarea, biggestarea;
12787 REAL triminaltitude2;
12791 REAL smallestangle, biggestangle;
12792 REAL radconst, degconst;
12793 int angletable[18];
12794 int aspecttable[16];
12800 printf( "Mesh quality statistics:\n\n" );
12801 radconst = (REAL)( PI / 18.0 );
12802 degconst = (REAL)( 180.0 / PI );
12803 for ( i = 0; i < 8; i++ ) {
12804 cossquaretable[i] = (REAL)( cos( radconst * (REAL) ( i + 1 ) ) );
12805 cossquaretable[i] = cossquaretable[i] * cossquaretable[i];
12807 for ( i = 0; i < 18; i++ ) {
12811 ratiotable[0] = 1.5; ratiotable[1] = 2.0;
12812 ratiotable[2] = 2.5; ratiotable[3] = 3.0;
12813 ratiotable[4] = 4.0; ratiotable[5] = 6.0;
12814 ratiotable[6] = 10.0; ratiotable[7] = 15.0;
12815 ratiotable[8] = 25.0; ratiotable[9] = 50.0;
12816 ratiotable[10] = 100.0; ratiotable[11] = 300.0;
12817 ratiotable[12] = 1000.0; ratiotable[13] = 10000.0;
12818 ratiotable[14] = 100000.0; ratiotable[15] = 0.0;
12819 for ( i = 0; i < 16; i++ ) {
12820 aspecttable[i] = 0;
12824 minaltitude = xmax - xmin + ymax - ymin;
12825 minaltitude = minaltitude * minaltitude;
12826 shortest = minaltitude;
12828 smallestarea = minaltitude;
12831 smallestangle = 0.0;
12832 biggestangle = 2.0;
12835 traversalinit( &triangles );
12836 triangleloop.tri = triangletraverse();
12837 triangleloop.orient = 0;
12838 while ( triangleloop.tri != (triangle *) NULL ) {
12839 org( triangleloop, p[0] );
12840 dest( triangleloop, p[1] );
12841 apex( triangleloop, p[2] );
12844 for ( i = 0; i < 3; i++ ) {
12847 dx[i] = p[j][0] - p[k][0];
12848 dy[i] = p[j][1] - p[k][1];
12849 edgelength[i] = dx[i] * dx[i] + dy[i] * dy[i];
12850 if ( edgelength[i] > trilongest2 ) {
12851 trilongest2 = edgelength[i];
12853 if ( edgelength[i] > longest ) {
12854 longest = edgelength[i];
12856 if ( edgelength[i] < shortest ) {
12857 shortest = edgelength[i];
12861 triarea = counterclockwise( p[0], p[1], p[2] );
12862 if ( triarea < smallestarea ) {
12863 smallestarea = triarea;
12865 if ( triarea > biggestarea ) {
12866 biggestarea = triarea;
12868 triminaltitude2 = triarea * triarea / trilongest2;
12869 if ( triminaltitude2 < minaltitude ) {
12870 minaltitude = triminaltitude2;
12872 triaspect2 = trilongest2 / triminaltitude2;
12873 if ( triaspect2 > worstaspect ) {
12874 worstaspect = triaspect2;
12877 while ( ( triaspect2 > ratiotable[aspectindex] * ratiotable[aspectindex] )
12878 && ( aspectindex < 15 ) ) {
12881 aspecttable[aspectindex]++;
12883 for ( i = 0; i < 3; i++ ) {
12886 dotproduct = dx[j] * dx[k] + dy[j] * dy[k];
12887 cossquare = dotproduct * dotproduct / ( edgelength[j] * edgelength[k] );
12889 for ( ii = 7; ii >= 0; ii-- ) {
12890 if ( cossquare > cossquaretable[ii] ) {
12894 if ( dotproduct <= 0.0 ) {
12895 angletable[tendegree]++;
12896 if ( cossquare > smallestangle ) {
12897 smallestangle = cossquare;
12899 if ( acutebiggest && ( cossquare < biggestangle ) ) {
12900 biggestangle = cossquare;
12904 angletable[17 - tendegree]++;
12905 if ( acutebiggest || ( cossquare > biggestangle ) ) {
12906 biggestangle = cossquare;
12911 triangleloop.tri = triangletraverse();
12914 shortest = (REAL)sqrt( shortest );
12915 longest = (REAL)sqrt( longest );
12916 minaltitude = (REAL)sqrt( minaltitude );
12917 worstaspect = (REAL)sqrt( worstaspect );
12918 smallestarea *= 2.0;
12919 biggestarea *= 2.0;
12920 if ( smallestangle >= 1.0 ) {
12921 smallestangle = 0.0;
12924 smallestangle = (REAL)( degconst * acos( sqrt( smallestangle ) ) );
12926 if ( biggestangle >= 1.0 ) {
12927 biggestangle = 180.0;
12930 if ( acutebiggest ) {
12931 biggestangle = (REAL)( degconst * acos( sqrt( biggestangle ) ) );
12934 biggestangle = (REAL)( 180.0 - degconst * acos( sqrt( biggestangle ) ) );
12938 printf( " Smallest area: %16.5g | Largest area: %16.5g\n",
12939 smallestarea, biggestarea );
12940 printf( " Shortest edge: %16.5g | Longest edge: %16.5g\n",
12941 shortest, longest );
12942 printf( " Shortest altitude: %12.5g | Largest aspect ratio: %8.5g\n\n",
12943 minaltitude, worstaspect );
12944 printf( " Aspect ratio histogram:\n" );
12945 printf( " 1.1547 - %-6.6g : %8d | %6.6g - %-6.6g : %8d\n",
12946 ratiotable[0], aspecttable[0], ratiotable[7], ratiotable[8],
12948 for ( i = 1; i < 7; i++ ) {
12949 printf( " %6.6g - %-6.6g : %8d | %6.6g - %-6.6g : %8d\n",
12950 ratiotable[i - 1], ratiotable[i], aspecttable[i],
12951 ratiotable[i + 7], ratiotable[i + 8], aspecttable[i + 8] );
12953 printf( " %6.6g - %-6.6g : %8d | %6.6g - : %8d\n",
12954 ratiotable[6], ratiotable[7], aspecttable[7], ratiotable[14],
12957 " (Triangle aspect ratio is longest edge divided by shortest altitude)\n\n" );
12958 printf( " Smallest angle: %15.5g | Largest angle: %15.5g\n\n",
12959 smallestangle, biggestangle );
12960 printf( " Angle histogram:\n" );
12961 for ( i = 0; i < 9; i++ ) {
12962 printf( " %3d - %3d degrees: %8d | %3d - %3d degrees: %8d\n",
12963 i * 10, i * 10 + 10, angletable[i],
12964 i * 10 + 90, i * 10 + 100, angletable[i + 9] );
12969 /*****************************************************************************/
12971 /* statistics() Print all sorts of cool facts. */
12973 /*****************************************************************************/
12976 printf( "\nStatistics:\n\n" );
12977 printf( " Input points: %d\n", inpoints );
12979 printf( " Input triangles: %d\n", inelements );
12982 printf( " Input segments: %d\n", insegments );
12984 printf( " Input holes: %d\n", holes );
12988 printf( "\n Mesh points: %ld\n", points.items );
12989 printf( " Mesh triangles: %ld\n", triangles.items );
12990 printf( " Mesh edges: %ld\n", edges );
12991 if ( poly || refine ) {
12992 printf( " Mesh boundary edges: %ld\n", hullsize );
12993 printf( " Mesh segments: %ld\n\n", shelles.items );
12996 printf( " Mesh convex hull edges: %ld\n\n", hullsize );
12999 quality_statistics();
13000 printf( "Memory allocation statistics:\n\n" );
13001 printf( " Maximum number of points: %ld\n", points.maxitems );
13002 printf( " Maximum number of triangles: %ld\n", triangles.maxitems );
13003 if ( shelles.maxitems > 0 ) {
13004 printf( " Maximum number of segments: %ld\n", shelles.maxitems );
13006 if ( viri.maxitems > 0 ) {
13007 printf( " Maximum number of viri: %ld\n", viri.maxitems );
13009 if ( badsegments.maxitems > 0 ) {
13010 printf( " Maximum number of encroached segments: %ld\n",
13011 badsegments.maxitems );
13013 if ( badtriangles.maxitems > 0 ) {
13014 printf( " Maximum number of bad triangles: %ld\n",
13015 badtriangles.maxitems );
13017 if ( splaynodes.maxitems > 0 ) {
13018 printf( " Maximum number of splay tree nodes: %ld\n",
13019 splaynodes.maxitems );
13021 printf( " Approximate heap memory use (bytes): %ld\n\n",
13022 points.maxitems * points.itembytes
13023 + triangles.maxitems * triangles.itembytes
13024 + shelles.maxitems * shelles.itembytes
13025 + viri.maxitems * viri.itembytes
13026 + badsegments.maxitems * badsegments.itembytes
13027 + badtriangles.maxitems * badtriangles.itembytes
13028 + splaynodes.maxitems * splaynodes.itembytes );
13030 printf( "Algorithmic statistics:\n\n" );
13031 printf( " Number of incircle tests: %ld\n", incirclecount );
13032 printf( " Number of orientation tests: %ld\n", counterclockcount );
13033 if ( hyperbolacount > 0 ) {
13034 printf( " Number of right-of-hyperbola tests: %ld\n",
13037 if ( circumcentercount > 0 ) {
13038 printf( " Number of circumcenter computations: %ld\n",
13039 circumcentercount );
13041 if ( circletopcount > 0 ) {
13042 printf( " Number of circle top computations: %ld\n",
13049 /*****************************************************************************/
13051 /* main() or triangulate() Gosh, do everything. */
13053 /* The sequence is roughly as follows. Many of these steps can be skipped, */
13054 /* depending on the command line switches. */
13056 /* - Initialize constants and parse the command line. */
13057 /* - Read the points from a file and either */
13058 /* - triangulate them (no -r), or */
13059 /* - read an old mesh from files and reconstruct it (-r). */
13060 /* - Insert the PSLG segments (-p), and possibly segments on the convex */
13062 /* - Read the holes (-p), regional attributes (-pA), and regional area */
13063 /* constraints (-pa). Carve the holes and concavities, and spread the */
13064 /* regional attributes and area constraints. */
13065 /* - Enforce the constraints on minimum angle (-q) and maximum area (-a). */
13066 /* Also enforce the conforming Delaunay property (-q and -a). */
13067 /* - Compute the number of edges in the resulting mesh. */
13068 /* - Promote the mesh's linear triangles to higher order elements (-o). */
13069 /* - Write the output files and print the statistics. */
13070 /* - Check the consistency and Delaunay property of the mesh (-C). */
13072 /*****************************************************************************/
13076 void triangulate( triswitches, in, out, vorout )
13078 struct triangulateio *in;
13079 struct triangulateio *out;
13080 struct triangulateio *vorout;
13082 #else /* not TRILIBRARY */
13084 int main( argc, argv )
13088 #endif /* not TRILIBRARY */
13091 REAL *holearray; /* Array of holes. */
13092 REAL *regionarray; /* Array of regional attributes and area constraints. */
13095 #endif /* not TRILIBRARY */
13097 /* Variables for timing the performance of Triangle. The types are */
13098 /* defined in sys/time.h. */
13099 struct timeval tv0, tv1, tv2, tv3, tv4, tv5, tv6;
13100 struct timezone tz;
13101 #endif /* NO_TIMER */
13104 gettimeofday( &tv0, &tz );
13105 #endif /* NO_TIMER */
13109 parsecommandline( 1, &triswitches );
13110 #else /* not TRILIBRARY */
13111 parsecommandline( argc, argv );
13112 #endif /* not TRILIBRARY */
13115 transfernodes( in->pointlist, in->pointattributelist, in->pointmarkerlist,
13116 in->numberofpoints, in->numberofpointattributes );
13117 #else /* not TRILIBRARY */
13118 readnodes( innodefilename, inpolyfilename, &polyfile );
13119 #endif /* not TRILIBRARY */
13123 gettimeofday( &tv1, &tz );
13125 #endif /* NO_TIMER */
13128 hullsize = delaunay(); /* Triangulate the points. */
13129 #else /* not CDT_ONLY */
13131 /* Read and reconstruct a mesh. */
13133 hullsize = reconstruct( in->trianglelist, in->triangleattributelist,
13134 in->trianglearealist, in->numberoftriangles,
13135 in->numberofcorners, in->numberoftriangleattributes,
13136 in->segmentlist, in->segmentmarkerlist,
13137 in->numberofsegments );
13138 #else /* not TRILIBRARY */
13139 hullsize = reconstruct( inelefilename, areafilename, inpolyfilename,
13141 #endif /* not TRILIBRARY */
13144 hullsize = delaunay(); /* Triangulate the points. */
13146 #endif /* not CDT_ONLY */
13150 gettimeofday( &tv2, &tz );
13152 printf( "Mesh reconstruction" );
13155 printf( "Delaunay" );
13157 printf( " milliseconds: %ld\n", 1000l * ( tv2.tv_sec - tv1.tv_sec )
13158 + ( tv2.tv_usec - tv1.tv_usec ) / 1000l );
13160 #endif /* NO_TIMER */
13162 /* Ensure that no point can be mistaken for a triangular bounding */
13163 /* box point in insertsite(). */
13164 infpoint1 = (point) NULL;
13165 infpoint2 = (point) NULL;
13166 infpoint3 = (point) NULL;
13168 if ( useshelles ) {
13169 checksegments = 1; /* Segments will be introduced next. */
13171 /* Insert PSLG segments and/or convex hull segments. */
13173 insegments = formskeleton( in->segmentlist, in->segmentmarkerlist,
13174 in->numberofsegments );
13175 #else /* not TRILIBRARY */
13176 insegments = formskeleton( polyfile, inpolyfilename );
13177 #endif /* not TRILIBRARY */
13183 gettimeofday( &tv3, &tz );
13184 if ( useshelles && !refine ) {
13185 printf( "Segment milliseconds: %ld\n",
13186 1000l * ( tv3.tv_sec - tv2.tv_sec )
13187 + ( tv3.tv_usec - tv2.tv_usec ) / 1000l );
13190 #endif /* NO_TIMER */
13194 holearray = in->holelist;
13195 holes = in->numberofholes;
13196 regionarray = in->regionlist;
13197 regions = in->numberofregions;
13198 #else /* not TRILIBRARY */
13199 readholes( polyfile, inpolyfilename, &holearray, &holes,
13200 ®ionarray, ®ions );
13201 #endif /* not TRILIBRARY */
13203 /* Carve out holes and concavities. */
13204 carveholes( holearray, holes, regionarray, regions );
13208 /* Without a PSLG, there can be no holes or regional attributes */
13209 /* or area constraints. The following are set to zero to avoid */
13210 /* an accidental free() later. */
13217 gettimeofday( &tv4, &tz );
13218 if ( poly && !refine ) {
13219 printf( "Hole milliseconds: %ld\n", 1000l * ( tv4.tv_sec - tv3.tv_sec )
13220 + ( tv4.tv_usec - tv3.tv_usec ) / 1000l );
13223 #endif /* NO_TIMER */
13227 enforcequality(); /* Enforce angle and area constraints. */
13229 #endif /* not CDT_ONLY */
13233 gettimeofday( &tv5, &tz );
13236 printf( "Quality milliseconds: %ld\n",
13237 1000l * ( tv5.tv_sec - tv4.tv_sec )
13238 + ( tv5.tv_usec - tv4.tv_usec ) / 1000l );
13240 #endif /* not CDT_ONLY */
13242 #endif /* NO_TIMER */
13244 /* Compute the number of edges. */
13245 edges = ( 3l * triangles.items + hullsize ) / 2l;
13248 highorder(); /* Promote elements to higher polynomial order. */
13255 out->numberofpoints = points.items;
13256 out->numberofpointattributes = nextras;
13257 out->numberoftriangles = triangles.items;
13258 out->numberofcorners = ( order + 1 ) * ( order + 2 ) / 2;
13259 out->numberoftriangleattributes = eextras;
13260 out->numberofedges = edges;
13261 if ( useshelles ) {
13262 out->numberofsegments = shelles.items;
13265 out->numberofsegments = hullsize;
13267 if ( vorout != (struct triangulateio *) NULL ) {
13268 vorout->numberofpoints = triangles.items;
13269 vorout->numberofpointattributes = nextras;
13270 vorout->numberofedges = edges;
13272 #endif /* TRILIBRARY */
13273 /* If not using iteration numbers, don't write a .node file if one was */
13274 /* read, because the original one would be overwritten! */
13275 if ( nonodewritten || ( noiterationnum && readnodefile ) ) {
13278 printf( "NOT writing points.\n" );
13279 #else /* not TRILIBRARY */
13280 printf( "NOT writing a .node file.\n" );
13281 #endif /* not TRILIBRARY */
13283 numbernodes(); /* We must remember to number the points. */
13287 writenodes( &out->pointlist, &out->pointattributelist,
13288 &out->pointmarkerlist );
13289 #else /* not TRILIBRARY */
13290 writenodes( outnodefilename, argc, argv ); /* Numbers the points too. */
13291 #endif /* TRILIBRARY */
13293 if ( noelewritten ) {
13296 printf( "NOT writing triangles.\n" );
13297 #else /* not TRILIBRARY */
13298 printf( "NOT writing an .ele file.\n" );
13299 #endif /* not TRILIBRARY */
13304 writeelements( &out->trianglelist, &out->triangleattributelist );
13305 #else /* not TRILIBRARY */
13306 writeelements( outelefilename, argc, argv );
13307 #endif /* not TRILIBRARY */
13309 /* The -c switch (convex switch) causes a PSLG to be written */
13310 /* even if none was read. */
13311 if ( poly || convex ) {
13312 /* If not using iteration numbers, don't overwrite the .poly file. */
13313 if ( nopolywritten || noiterationnum ) {
13316 printf( "NOT writing segments.\n" );
13317 #else /* not TRILIBRARY */
13318 printf( "NOT writing a .poly file.\n" );
13319 #endif /* not TRILIBRARY */
13324 writepoly( &out->segmentlist, &out->segmentmarkerlist );
13325 out->numberofholes = holes;
13326 out->numberofregions = regions;
13328 out->holelist = in->holelist;
13329 out->regionlist = in->regionlist;
13332 out->holelist = (REAL *) NULL;
13333 out->regionlist = (REAL *) NULL;
13335 #else /* not TRILIBRARY */
13336 writepoly( outpolyfilename, holearray, holes, regionarray, regions,
13338 #endif /* not TRILIBRARY */
13343 if ( regions > 0 ) {
13344 free( regionarray );
13346 #endif /* not CDT_ONLY */
13351 writeoff( offfilename, argc, argv );
13353 #endif /* not TRILIBRARY */
13356 writeedges( &out->edgelist, &out->edgemarkerlist );
13357 #else /* not TRILIBRARY */
13358 writeedges( edgefilename, argc, argv );
13359 #endif /* not TRILIBRARY */
13363 writevoronoi( &vorout->pointlist, &vorout->pointattributelist,
13364 &vorout->pointmarkerlist, &vorout->edgelist,
13365 &vorout->edgemarkerlist, &vorout->normlist );
13366 #else /* not TRILIBRARY */
13367 writevoronoi( vnodefilename, vedgefilename, argc, argv );
13368 #endif /* not TRILIBRARY */
13372 writeneighbors( &out->neighborlist );
13373 #else /* not TRILIBRARY */
13374 writeneighbors( neighborfilename, argc, argv );
13375 #endif /* not TRILIBRARY */
13380 gettimeofday( &tv6, &tz );
13381 printf( "\nOutput milliseconds: %ld\n",
13382 1000l * ( tv6.tv_sec - tv5.tv_sec )
13383 + ( tv6.tv_usec - tv5.tv_usec ) / 1000l );
13384 printf( "Total running milliseconds: %ld\n",
13385 1000l * ( tv6.tv_sec - tv0.tv_sec )
13386 + ( tv6.tv_usec - tv0.tv_usec ) / 1000l );
13387 #endif /* NO_TIMER */
13397 #endif /* not REDUCED */
13402 #endif /* not TRILIBRARY */
projects / xonotic / netradiant.git / blob