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); \
855 #define oprevself(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); \
867 #define dnextself(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) \
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) \
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]; \
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]; \
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]; \
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 /*****************************************************************************/
1160 printf("triangle [-pAcevngBPNEIOXzo_lQVh] input_file\n");
1161 #else /* not REDUCED */
1162 printf("triangle [-pAcevngBPNEIOXzo_iFlCQVh] input_file\n");
1163 #endif /* not REDUCED */
1164 #else /* not CDT_ONLY */
1166 printf("triangle [-prq__a__AcevngBPNEIOXzo_YS__lQVh] input_file\n");
1167 #else /* not REDUCED */
1168 printf("triangle [-prq__a__AcevngBPNEIOXzo_YS__iFlsCQVh] input_file\n");
1169 #endif /* not REDUCED */
1170 #endif /* not CDT_ONLY */
1172 printf(" -p Triangulates a Planar Straight Line Graph (.poly file).\n");
1174 printf(" -r Refines a previously generated mesh.\n");
1176 " -q Quality mesh generation. A minimum angle may be specified.\n");
1177 printf(" -a Applies a maximum triangle area constraint.\n");
1178 #endif /* not CDT_ONLY */
1180 " -A Applies attributes to identify elements in certain regions.\n");
1181 printf(" -c Encloses the convex hull with segments.\n");
1182 printf(" -e Generates an edge list.\n");
1183 printf(" -v Generates a Voronoi diagram.\n");
1184 printf(" -n Generates a list of triangle neighbors.\n");
1185 printf(" -g Generates an .off file for Geomview.\n");
1186 printf(" -B Suppresses output of boundary information.\n");
1187 printf(" -P Suppresses output of .poly file.\n");
1188 printf(" -N Suppresses output of .node file.\n");
1189 printf(" -E Suppresses output of .ele file.\n");
1190 printf(" -I Suppresses mesh iteration numbers.\n");
1191 printf(" -O Ignores holes in .poly file.\n");
1192 printf(" -X Suppresses use of exact arithmetic.\n");
1193 printf(" -z Numbers all items starting from zero (rather than one).\n");
1194 printf(" -o2 Generates second-order subparametric elements.\n");
1196 printf(" -Y Suppresses boundary segment splitting.\n");
1197 printf(" -S Specifies maximum number of added Steiner points.\n");
1198 #endif /* not CDT_ONLY */
1200 printf(" -i Uses incremental method, rather than divide-and-conquer.\n");
1201 printf(" -F Uses Fortune's sweepline algorithm, rather than d-and-c.\n");
1202 #endif /* not REDUCED */
1203 printf(" -l Uses vertical cuts only, rather than alternating cuts.\n");
1207 " -s Force segments into mesh by splitting (instead of using CDT).\n");
1208 #endif /* not CDT_ONLY */
1209 printf(" -C Check consistency of final mesh.\n");
1210 #endif /* not REDUCED */
1211 printf(" -Q Quiet: No terminal output except errors.\n");
1212 printf(" -V Verbose: Detailed information on what I'm doing.\n");
1213 printf(" -h Help: Detailed instructions for Triangle.\n");
1217 #endif /* not TRILIBRARY */
1219 /*****************************************************************************/
1221 /* info() Print out complete instructions. */
1223 /*****************************************************************************/
1229 printf("Triangle\n");
1231 "A Two-Dimensional Quality Mesh Generator and Delaunay Triangulator.\n");
1232 printf("Version 1.3\n\n");
1234 "Copyright 1996 Jonathan Richard Shewchuk (bugs/comments to jrs@cs.cmu.edu)\n"
1236 printf("School of Computer Science / Carnegie Mellon University\n");
1237 printf("5000 Forbes Avenue / Pittsburgh, Pennsylvania 15213-3891\n");
1239 "Created as part of the Archimedes project (tools for parallel FEM).\n");
1241 "Supported in part by NSF Grant CMS-9318163 and an NSERC 1967 Scholarship.\n");
1242 printf("There is no warranty whatsoever. Use at your own risk.\n");
1244 printf("This executable is compiled for single precision arithmetic.\n\n\n");
1245 #else /* not SINGLE */
1246 printf("This executable is compiled for double precision arithmetic.\n\n\n");
1247 #endif /* not SINGLE */
1249 "Triangle generates exact Delaunay triangulations, constrained Delaunay\n");
1251 "triangulations, and quality conforming Delaunay triangulations. The latter\n"
1254 "can be generated with no small angles, and are thus suitable for finite\n");
1256 "element analysis. If no command line switches are specified, your .node\n");
1258 "input file will be read, and the Delaunay triangulation will be returned in\n"
1260 printf(".node and .ele output files. The command syntax is:\n\n");
1263 printf("triangle [-pAcevngBPNEIOXzo_lQVh] input_file\n\n");
1264 #else /* not REDUCED */
1265 printf("triangle [-pAcevngBPNEIOXzo_iFlCQVh] input_file\n\n");
1266 #endif /* not REDUCED */
1267 #else /* not CDT_ONLY */
1269 printf("triangle [-prq__a__AcevngBPNEIOXzo_YS__lQVh] input_file\n\n");
1270 #else /* not REDUCED */
1271 printf("triangle [-prq__a__AcevngBPNEIOXzo_YS__iFlsCQVh] input_file\n\n");
1272 #endif /* not REDUCED */
1273 #endif /* not CDT_ONLY */
1275 "Underscores indicate that numbers may optionally follow certain switches;\n");
1277 "do not leave any space between a switch and its numeric parameter.\n");
1279 "input_file must be a file with extension .node, or extension .poly if the\n");
1281 "-p switch is used. If -r is used, you must supply .node and .ele files,\n");
1283 "and possibly a .poly file and .area file as well. The formats of these\n");
1284 printf("files are described below.\n\n");
1285 printf("Command Line Switches:\n\n");
1287 " -p Reads a Planar Straight Line Graph (.poly file), which can specify\n"
1290 " points, segments, holes, and regional attributes and area\n");
1292 " constraints. Will generate a constrained Delaunay triangulation\n");
1294 " fitting the input; or, if -s, -q, or -a is used, a conforming\n");
1296 " Delaunay triangulation. If -p is not used, Triangle reads a .node\n"
1298 printf(" file by default.\n");
1300 " -r Refines a previously generated mesh. The mesh is read from a .node\n"
1303 " file and an .ele file. If -p is also used, a .poly file is read\n");
1305 " and used to constrain edges in the mesh. Further details on\n");
1306 printf(" refinement are given below.\n");
1308 " -q Quality mesh generation by Jim Ruppert's Delaunay refinement\n");
1310 " algorithm. Adds points to the mesh to ensure that no angles\n");
1312 " smaller than 20 degrees occur. An alternative minimum angle may be\n"
1315 " specified after the `q'. If the minimum angle is 20.7 degrees or\n");
1317 " smaller, the triangulation algorithm is theoretically guaranteed to\n"
1320 " terminate (assuming infinite precision arithmetic - Triangle may\n");
1322 " fail to terminate if you run out of precision). In practice, the\n");
1324 " algorithm often succeeds for minimum angles up to 33.8 degrees.\n");
1326 " For highly refined meshes, however, it may be necessary to reduce\n");
1328 " the minimum angle to well below 20 to avoid problems associated\n");
1330 " with insufficient floating-point precision. The specified angle\n");
1331 printf(" may include a decimal point.\n");
1333 " -a Imposes a maximum triangle area. If a number follows the `a', no\n");
1335 " triangle will be generated whose area is larger than that number.\n");
1337 " If no number is specified, an .area file (if -r is used) or .poly\n");
1339 " file (if -r is not used) specifies a number of maximum area\n");
1341 " constraints. An .area file contains a separate area constraint for\n"
1344 " each triangle, and is useful for refining a finite element mesh\n");
1346 " based on a posteriori error estimates. A .poly file can optionally\n"
1349 " contain an area constraint for each segment-bounded region, thereby\n"
1352 " enforcing triangle densities in a first triangulation. You can\n");
1354 " impose both a fixed area constraint and a varying area constraint\n");
1356 " by invoking the -a switch twice, once with and once without a\n");
1358 " number following. Each area specified may include a decimal point.\n"
1361 " -A Assigns an additional attribute to each triangle that identifies\n");
1363 " what segment-bounded region each triangle belongs to. Attributes\n");
1365 " are assigned to regions by the .poly file. If a region is not\n");
1367 " explicitly marked by the .poly file, triangles in that region are\n");
1369 " assigned an attribute of zero. The -A switch has an effect only\n");
1370 printf(" when the -p switch is used and the -r switch is not.\n");
1372 " -c Creates segments on the convex hull of the triangulation. If you\n");
1374 " are triangulating a point set, this switch causes a .poly file to\n");
1376 " be written, containing all edges in the convex hull. (By default,\n"
1379 " a .poly file is written only if a .poly file is read.) If you are\n"
1382 " triangulating a PSLG, this switch specifies that the interior of\n");
1384 " the convex hull of the PSLG should be triangulated. If you do not\n"
1387 " use this switch when triangulating a PSLG, it is assumed that you\n");
1389 " have identified the region to be triangulated by surrounding it\n");
1391 " with segments of the input PSLG. Beware: if you are not careful,\n"
1394 " this switch can cause the introduction of an extremely thin angle\n");
1396 " between a PSLG segment and a convex hull segment, which can cause\n");
1398 " overrefinement or failure if Triangle runs out of precision. If\n");
1400 " you are refining a mesh, the -c switch works differently; it\n");
1402 " generates the set of boundary edges of the mesh, rather than the\n");
1403 printf(" convex hull.\n");
1405 " -e Outputs (to an .edge file) a list of edges of the triangulation.\n");
1407 " -v Outputs the Voronoi diagram associated with the triangulation.\n");
1408 printf(" Does not attempt to detect degeneracies.\n");
1410 " -n Outputs (to a .neigh file) a list of triangles neighboring each\n");
1411 printf(" triangle.\n");
1413 " -g Outputs the mesh to an Object File Format (.off) file, suitable for\n"
1415 printf(" viewing with the Geometry Center's Geomview package.\n");
1417 " -B No boundary markers in the output .node, .poly, and .edge output\n");
1419 " files. See the detailed discussion of boundary markers below.\n");
1421 " -P No output .poly file. Saves disk space, but you lose the ability\n");
1423 " to impose segment constraints on later refinements of the mesh.\n");
1424 printf(" -N No output .node file.\n");
1425 printf(" -E No output .ele file.\n");
1427 " -I No iteration numbers. Suppresses the output of .node and .poly\n");
1429 " files, so your input files won't be overwritten. (If your input is\n"
1432 " a .poly file only, a .node file will be written.) Cannot be used\n");
1434 " with the -r switch, because that would overwrite your input .ele\n");
1436 " file. Shouldn't be used with the -s, -q, or -a switch if you are\n");
1438 " using a .node file for input, because no .node file will be\n");
1439 printf(" written, so there will be no record of any added points.\n");
1440 printf(" -O No holes. Ignores the holes in the .poly file.\n");
1442 " -X No exact arithmetic. Normally, Triangle uses exact floating-point\n"
1445 " arithmetic for certain tests if it thinks the inexact tests are not\n"
1448 " accurate enough. Exact arithmetic ensures the robustness of the\n");
1450 " triangulation algorithms, despite floating-point roundoff error.\n");
1452 " Disabling exact arithmetic with the -X switch will cause a small\n");
1454 " improvement in speed and create the possibility (albeit small) that\n"
1457 " Triangle will fail to produce a valid mesh. Not recommended.\n");
1459 " -z Numbers all items starting from zero (rather than one). Note that\n"
1462 " this switch is normally overrided by the value used to number the\n");
1464 " first point of the input .node or .poly file. However, this switch\n"
1466 printf(" is useful when calling Triangle from another program.\n");
1468 " -o2 Generates second-order subparametric elements with six nodes each.\n"
1471 " -Y No new points on the boundary. This switch is useful when the mesh\n"
1474 " boundary must be preserved so that it conforms to some adjacent\n");
1476 " mesh. Be forewarned that you will probably sacrifice some of the\n");
1478 " quality of the mesh; Triangle will try, but the resulting mesh may\n"
1481 " contain triangles of poor aspect ratio. Works well if all the\n");
1483 " boundary points are closely spaced. Specify this switch twice\n");
1485 " (`-YY') to prevent all segment splitting, including internal\n");
1486 printf(" boundaries.\n");
1488 " -S Specifies the maximum number of Steiner points (points that are not\n"
1491 " in the input, but are added to meet the constraints of minimum\n");
1493 " angle and maximum area). The default is to allow an unlimited\n");
1495 " number. If you specify this switch with no number after it,\n");
1497 " the limit is set to zero. Triangle always adds points at segment\n");
1499 " intersections, even if it needs to use more points than the limit\n");
1501 " you set. When Triangle inserts segments by splitting (-s), it\n");
1503 " always adds enough points to ensure that all the segments appear in\n"
1506 " the triangulation, again ignoring the limit. Be forewarned that\n");
1508 " the -S switch may result in a conforming triangulation that is not\n"
1511 " truly Delaunay, because Triangle may be forced to stop adding\n");
1513 " points when the mesh is in a state where a segment is non-Delaunay\n"
1516 " and needs to be split. If so, Triangle will print a warning.\n");
1518 " -i Uses an incremental rather than divide-and-conquer algorithm to\n");
1520 " form a Delaunay triangulation. Try it if the divide-and-conquer\n");
1521 printf(" algorithm fails.\n");
1523 " -F Uses Steven Fortune's sweepline algorithm to form a Delaunay\n");
1525 " triangulation. Warning: does not use exact arithmetic for all\n");
1526 printf(" calculations. An exact result is not guaranteed.\n");
1528 " -l Uses only vertical cuts in the divide-and-conquer algorithm. By\n");
1530 " default, Triangle uses alternating vertical and horizontal cuts,\n");
1532 " which usually improve the speed except with point sets that are\n");
1534 " small or short and wide. This switch is primarily of theoretical\n");
1535 printf(" interest.\n");
1537 " -s Specifies that segments should be forced into the triangulation by\n"
1540 " recursively splitting them at their midpoints, rather than by\n");
1542 " generating a constrained Delaunay triangulation. Segment splitting\n"
1545 " is true to Ruppert's original algorithm, but can create needlessly\n"
1547 printf(" small triangles near external small features.\n");
1549 " -C Check the consistency of the final mesh. Uses exact arithmetic for\n"
1552 " checking, even if the -X switch is used. Useful if you suspect\n");
1553 printf(" Triangle is buggy.\n");
1555 " -Q Quiet: Suppresses all explanation of what Triangle is doing, unless\n"
1557 printf(" an error occurs.\n");
1559 " -V Verbose: Gives detailed information about what Triangle is doing.\n");
1561 " Add more `V's for increasing amount of detail. `-V' gives\n");
1563 " information on algorithmic progress and more detailed statistics.\n");
1565 " `-VV' gives point-by-point details, and will print so much that\n");
1567 " Triangle will run much more slowly. `-VVV' gives information only\n"
1569 printf(" a debugger could love.\n");
1570 printf(" -h Help: Displays these instructions.\n");
1572 printf("Definitions:\n");
1575 " A Delaunay triangulation of a point set is a triangulation whose vertices\n"
1578 " are the point set, having the property that no point in the point set\n");
1580 " falls in the interior of the circumcircle (circle that passes through all\n"
1582 printf(" three vertices) of any triangle in the triangulation.\n\n");
1584 " A Voronoi diagram of a point set is a subdivision of the plane into\n");
1586 " polygonal regions (some of which may be infinite), where each region is\n");
1588 " the set of points in the plane that are closer to some input point than\n");
1590 " to any other input point. (The Voronoi diagram is the geometric dual of\n"
1592 printf(" the Delaunay triangulation.)\n\n");
1594 " A Planar Straight Line Graph (PSLG) is a collection of points and\n");
1596 " segments. Segments are simply edges, whose endpoints are points in the\n");
1598 " PSLG. The file format for PSLGs (.poly files) is described below.\n");
1601 " A constrained Delaunay triangulation of a PSLG is similar to a Delaunay\n");
1603 " triangulation, but each PSLG segment is present as a single edge in the\n");
1605 " triangulation. (A constrained Delaunay triangulation is not truly a\n");
1606 printf(" Delaunay triangulation.)\n\n");
1608 " A conforming Delaunay triangulation of a PSLG is a true Delaunay\n");
1610 " triangulation in which each PSLG segment may have been subdivided into\n");
1612 " several edges by the insertion of additional points. These inserted\n");
1614 " points are necessary to allow the segments to exist in the mesh while\n");
1615 printf(" maintaining the Delaunay property.\n\n");
1616 printf("File Formats:\n\n");
1618 " All files may contain comments prefixed by the character '#'. Points,\n");
1620 " triangles, edges, holes, and maximum area constraints must be numbered\n");
1622 " consecutively, starting from either 1 or 0. Whichever you choose, all\n");
1624 " input files must be consistent; if the nodes are numbered from 1, so must\n"
1627 " be all other objects. Triangle automatically detects your choice while\n");
1629 " reading the .node (or .poly) file. (When calling Triangle from another\n");
1631 " program, use the -z switch if you wish to number objects from zero.)\n");
1632 printf(" Examples of these file formats are given below.\n\n");
1633 printf(" .node files:\n");
1635 " First line: <# of points> <dimension (must be 2)> <# of attributes>\n");
1637 " <# of boundary markers (0 or 1)>\n"
1640 " Remaining lines: <point #> <x> <y> [attributes] [boundary marker]\n");
1643 " The attributes, which are typically floating-point values of physical\n");
1645 " quantities (such as mass or conductivity) associated with the nodes of\n"
1648 " a finite element mesh, are copied unchanged to the output mesh. If -s,\n"
1651 " -q, or -a is selected, each new Steiner point added to the mesh will\n");
1652 printf(" have attributes assigned to it by linear interpolation.\n\n");
1654 " If the fourth entry of the first line is `1', the last column of the\n");
1656 " remainder of the file is assumed to contain boundary markers. Boundary\n"
1659 " markers are used to identify boundary points and points resting on PSLG\n"
1662 " segments; a complete description appears in a section below. The .node\n"
1665 " file produced by Triangle will contain boundary markers in the last\n");
1666 printf(" column unless they are suppressed by the -B switch.\n\n");
1667 printf(" .ele files:\n");
1669 " First line: <# of triangles> <points per triangle> <# of attributes>\n");
1671 " Remaining lines: <triangle #> <point> <point> <point> ... [attributes]\n"
1675 " Points are indices into the corresponding .node file. The first three\n"
1678 " points are the corners, and are listed in counterclockwise order around\n"
1681 " each triangle. (The remaining points, if any, depend on the type of\n");
1683 " finite element used.) The attributes are just like those of .node\n");
1685 " files. Because there is no simple mapping from input to output\n");
1687 " triangles, an attempt is made to interpolate attributes, which may\n");
1689 " result in a good deal of diffusion of attributes among nearby triangles\n"
1692 " as the triangulation is refined. Diffusion does not occur across\n");
1694 " segments, so attributes used to identify segment-bounded regions remain\n"
1697 " intact. In output .ele files, all triangles have three points each\n");
1699 " unless the -o2 switch is used, in which case they have six, and the\n");
1701 " fourth, fifth, and sixth points lie on the midpoints of the edges\n");
1702 printf(" opposite the first, second, and third corners.\n\n");
1703 printf(" .poly files:\n");
1705 " First line: <# of points> <dimension (must be 2)> <# of attributes>\n");
1707 " <# of boundary markers (0 or 1)>\n"
1710 " Following lines: <point #> <x> <y> [attributes] [boundary marker]\n");
1711 printf(" One line: <# of segments> <# of boundary markers (0 or 1)>\n");
1713 " Following lines: <segment #> <endpoint> <endpoint> [boundary marker]\n");
1714 printf(" One line: <# of holes>\n");
1715 printf(" Following lines: <hole #> <x> <y>\n");
1717 " Optional line: <# of regional attributes and/or area constraints>\n");
1719 " Optional following lines: <constraint #> <x> <y> <attrib> <max area>\n");
1722 " A .poly file represents a PSLG, as well as some additional information.\n"
1725 " The first section lists all the points, and is identical to the format\n"
1728 " of .node files. <# of points> may be set to zero to indicate that the\n"
1731 " points are listed in a separate .node file; .poly files produced by\n");
1733 " Triangle always have this format. This has the advantage that a point\n"
1736 " set may easily be triangulated with or without segments. (The same\n");
1738 " effect can be achieved, albeit using more disk space, by making a copy\n"
1741 " of the .poly file with the extension .node; all sections of the file\n");
1742 printf(" but the first are ignored.)\n\n");
1744 " The second section lists the segments. Segments are edges whose\n");
1746 " presence in the triangulation is enforced. Each segment is specified\n");
1748 " by listing the indices of its two endpoints. This means that you must\n"
1751 " include its endpoints in the point list. If -s, -q, and -a are not\n");
1753 " selected, Triangle will produce a constrained Delaunay triangulation,\n");
1755 " in which each segment appears as a single edge in the triangulation.\n");
1757 " If -q or -a is selected, Triangle will produce a conforming Delaunay\n");
1759 " triangulation, in which segments may be subdivided into smaller edges.\n"
1761 printf(" Each segment, like each point, may have a boundary marker.\n\n");
1763 " The third section lists holes (and concavities, if -c is selected) in\n");
1765 " the triangulation. Holes are specified by identifying a point inside\n");
1767 " each hole. After the triangulation is formed, Triangle creates holes\n");
1769 " by eating triangles, spreading out from each hole point until its\n");
1771 " progress is blocked by PSLG segments; you must be careful to enclose\n");
1773 " each hole in segments, or your whole triangulation may be eaten away.\n");
1775 " If the two triangles abutting a segment are eaten, the segment itself\n");
1777 " is also eaten. Do not place a hole directly on a segment; if you do,\n");
1778 printf(" Triangle will choose one side of the segment arbitrarily.\n\n");
1780 " The optional fourth section lists regional attributes (to be assigned\n");
1782 " to all triangles in a region) and regional constraints on the maximum\n");
1784 " triangle area. Triangle will read this section only if the -A switch\n");
1786 " is used or the -a switch is used without a number following it, and the\n"
1789 " -r switch is not used. Regional attributes and area constraints are\n");
1791 " propagated in the same manner as holes; you specify a point for each\n");
1793 " attribute and/or constraint, and the attribute and/or constraint will\n");
1795 " affect the whole region (bounded by segments) containing the point. If\n"
1798 " two values are written on a line after the x and y coordinate, the\n");
1800 " former is assumed to be a regional attribute (but will only be applied\n"
1803 " if the -A switch is selected), and the latter is assumed to be a\n");
1805 " regional area constraint (but will only be applied if the -a switch is\n"
1808 " selected). You may also specify just one value after the coordinates,\n"
1811 " which can serve as both an attribute and an area constraint, depending\n"
1814 " on the choice of switches. If you are using the -A and -a switches\n");
1816 " simultaneously and wish to assign an attribute to some region without\n");
1817 printf(" imposing an area constraint, use a negative maximum area.\n\n");
1819 " When a triangulation is created from a .poly file, you must either\n");
1821 " enclose the entire region to be triangulated in PSLG segments, or\n");
1823 " use the -c switch, which encloses the convex hull of the input point\n");
1825 " set. If you do not use the -c switch, Triangle will eat all triangles\n"
1828 " on the outer boundary that are not protected by segments; if you are\n");
1830 " not careful, your whole triangulation may be eaten away. If you do\n");
1832 " use the -c switch, you can still produce concavities by appropriate\n");
1833 printf(" placement of holes just inside the convex hull.\n\n");
1835 " An ideal PSLG has no intersecting segments, nor any points that lie\n");
1837 " upon segments (except, of course, the endpoints of each segment.) You\n"
1840 " aren't required to make your .poly files ideal, but you should be aware\n"
1843 " of what can go wrong. Segment intersections are relatively safe -\n");
1845 " Triangle will calculate the intersection points for you and add them to\n"
1848 " the triangulation - as long as your machine's floating-point precision\n"
1851 " doesn't become a problem. You are tempting the fates if you have three\n"
1854 " segments that cross at the same location, and expect Triangle to figure\n"
1857 " out where the intersection point is. Thanks to floating-point roundoff\n"
1860 " error, Triangle will probably decide that the three segments intersect\n"
1863 " at three different points, and you will find a minuscule triangle in\n");
1865 " your output - unless Triangle tries to refine the tiny triangle, uses\n");
1867 " up the last bit of machine precision, and fails to terminate at all.\n");
1869 " You're better off putting the intersection point in the input files,\n");
1871 " and manually breaking up each segment into two. Similarly, if you\n");
1873 " place a point at the middle of a segment, and hope that Triangle will\n");
1875 " break up the segment at that point, you might get lucky. On the other\n"
1878 " hand, Triangle might decide that the point doesn't lie precisely on the\n"
1881 " line, and you'll have a needle-sharp triangle in your output - or a lot\n"
1883 printf(" of tiny triangles if you're generating a quality mesh.\n\n");
1885 " When Triangle reads a .poly file, it also writes a .poly file, which\n");
1887 " includes all edges that are part of input segments. If the -c switch\n");
1889 " is used, the output .poly file will also include all of the edges on\n");
1891 " the convex hull. Hence, the output .poly file is useful for finding\n");
1893 " edges associated with input segments and setting boundary conditions in\n"
1896 " finite element simulations. More importantly, you will need it if you\n"
1899 " plan to refine the output mesh, and don't want segments to be missing\n");
1900 printf(" in later triangulations.\n\n");
1901 printf(" .area files:\n");
1902 printf(" First line: <# of triangles>\n");
1903 printf(" Following lines: <triangle #> <maximum area>\n\n");
1905 " An .area file associates with each triangle a maximum area that is used\n"
1908 " for mesh refinement. As with other file formats, every triangle must\n");
1910 " be represented, and they must be numbered consecutively. A triangle\n");
1912 " may be left unconstrained by assigning it a negative maximum area.\n");
1914 printf(" .edge files:\n");
1915 printf(" First line: <# of edges> <# of boundary markers (0 or 1)>\n");
1917 " Following lines: <edge #> <endpoint> <endpoint> [boundary marker]\n");
1920 " Endpoints are indices into the corresponding .node file. Triangle can\n"
1923 " produce .edge files (use the -e switch), but cannot read them. The\n");
1925 " optional column of boundary markers is suppressed by the -B switch.\n");
1928 " In Voronoi diagrams, one also finds a special kind of edge that is an\n");
1930 " infinite ray with only one endpoint. For these edges, a different\n");
1931 printf(" format is used:\n\n");
1932 printf(" <edge #> <endpoint> -1 <direction x> <direction y>\n\n");
1934 " The `direction' is a floating-point vector that indicates the direction\n"
1936 printf(" of the infinite ray.\n\n");
1937 printf(" .neigh files:\n");
1939 " First line: <# of triangles> <# of neighbors per triangle (always 3)>\n"
1942 " Following lines: <triangle #> <neighbor> <neighbor> <neighbor>\n");
1945 " Neighbors are indices into the corresponding .ele file. An index of -1\n"
1948 " indicates a mesh boundary, and therefore no neighbor. Triangle can\n");
1950 " produce .neigh files (use the -n switch), but cannot read them.\n");
1953 " The first neighbor of triangle i is opposite the first corner of\n");
1954 printf(" triangle i, and so on.\n\n");
1955 printf("Boundary Markers:\n\n");
1957 " Boundary markers are tags used mainly to identify which output points and\n"
1960 " edges are associated with which PSLG segment, and to identify which\n");
1962 " points and edges occur on a boundary of the triangulation. A common use\n"
1965 " is to determine where boundary conditions should be applied to a finite\n");
1967 " element mesh. You can prevent boundary markers from being written into\n");
1968 printf(" files produced by Triangle by using the -B switch.\n\n");
1970 " The boundary marker associated with each segment in an output .poly file\n"
1972 printf(" or edge in an output .edge file is chosen as follows:\n");
1974 " - If an output edge is part or all of a PSLG segment with a nonzero\n");
1976 " boundary marker, then the edge is assigned the same marker.\n");
1978 " - Otherwise, if the edge occurs on a boundary of the triangulation\n");
1980 " (including boundaries of holes), then the edge is assigned the marker\n"
1982 printf(" one (1).\n");
1983 printf(" - Otherwise, the edge is assigned the marker zero (0).\n");
1985 " The boundary marker associated with each point in an output .node file is\n"
1987 printf(" chosen as follows:\n");
1989 " - If a point is assigned a nonzero boundary marker in the input file,\n");
1991 " then it is assigned the same marker in the output .node file.\n");
1993 " - Otherwise, if the point lies on a PSLG segment (including the\n");
1995 " segment's endpoints) with a nonzero boundary marker, then the point\n");
1997 " is assigned the same marker. If the point lies on several such\n");
1998 printf(" segments, one of the markers is chosen arbitrarily.\n");
2000 " - Otherwise, if the point occurs on a boundary of the triangulation,\n");
2001 printf(" then the point is assigned the marker one (1).\n");
2002 printf(" - Otherwise, the point is assigned the marker zero (0).\n");
2005 " If you want Triangle to determine for you which points and edges are on\n");
2007 " the boundary, assign them the boundary marker zero (or use no markers at\n"
2010 " all) in your input files. Alternatively, you can mark some of them and\n");
2011 printf(" leave others marked zero, allowing Triangle to label them.\n\n");
2012 printf("Triangulation Iteration Numbers:\n\n");
2014 " Because Triangle can read and refine its own triangulations, input\n");
2016 " and output files have iteration numbers. For instance, Triangle might\n");
2018 " read the files mesh.3.node, mesh.3.ele, and mesh.3.poly, refine the\n");
2020 " triangulation, and output the files mesh.4.node, mesh.4.ele, and\n");
2021 printf(" mesh.4.poly. Files with no iteration number are treated as if\n");
2023 " their iteration number is zero; hence, Triangle might read the file\n");
2025 " points.node, triangulate it, and produce the files points.1.node and\n");
2026 printf(" points.1.ele.\n\n");
2028 " Iteration numbers allow you to create a sequence of successively finer\n");
2030 " meshes suitable for multigrid methods. They also allow you to produce a\n"
2033 " sequence of meshes using error estimate-driven mesh refinement.\n");
2036 " If you're not using refinement or quality meshing, and you don't like\n");
2038 " iteration numbers, use the -I switch to disable them. This switch will\n");
2040 " also disable output of .node and .poly files to prevent your input files\n"
2043 " from being overwritten. (If the input is a .poly file that contains its\n"
2045 printf(" own points, a .node file will be written.)\n\n");
2046 printf("Examples of How to Use Triangle:\n\n");
2048 " `triangle dots' will read points from dots.node, and write their Delaunay\n"
2051 " triangulation to dots.1.node and dots.1.ele. (dots.1.node will be\n");
2053 " identical to dots.node.) `triangle -I dots' writes the triangulation to\n"
2056 " dots.ele instead. (No additional .node file is needed, so none is\n");
2057 printf(" written.)\n\n");
2059 " `triangle -pe object.1' will read a PSLG from object.1.poly (and possibly\n"
2062 " object.1.node, if the points are omitted from object.1.poly) and write\n");
2063 printf(" their constrained Delaunay triangulation to object.2.node and\n");
2065 " object.2.ele. The segments will be copied to object.2.poly, and all\n");
2066 printf(" edges will be written to object.2.edge.\n\n");
2068 " `triangle -pq31.5a.1 object' will read a PSLG from object.poly (and\n");
2070 " possibly object.node), generate a mesh whose angles are all greater than\n"
2073 " 31.5 degrees and whose triangles all have area smaller than 0.1, and\n");
2075 " write the mesh to object.1.node and object.1.ele. Each segment may have\n"
2078 " been broken up into multiple edges; the resulting constrained edges are\n");
2079 printf(" written to object.1.poly.\n\n");
2081 " Here is a sample file `box.poly' describing a square with a square hole:\n"
2085 " # A box with eight points in 2D, no attributes, one boundary marker.\n");
2086 printf(" 8 2 0 1\n");
2087 printf(" # Outer box has these vertices:\n");
2088 printf(" 1 0 0 0\n");
2089 printf(" 2 0 3 0\n");
2090 printf(" 3 3 0 0\n");
2091 printf(" 4 3 3 33 # A special marker for this point.\n");
2092 printf(" # Inner square has these vertices:\n");
2093 printf(" 5 1 1 0\n");
2094 printf(" 6 1 2 0\n");
2095 printf(" 7 2 1 0\n");
2096 printf(" 8 2 2 0\n");
2097 printf(" # Five segments with boundary markers.\n");
2099 printf(" 1 1 2 5 # Left side of outer box.\n");
2100 printf(" 2 5 7 0 # Segments 2 through 5 enclose the hole.\n");
2101 printf(" 3 7 8 0\n");
2102 printf(" 4 8 6 10\n");
2103 printf(" 5 6 5 0\n");
2104 printf(" # One hole in the middle of the inner square.\n");
2106 printf(" 1 1.5 1.5\n\n");
2108 " Note that some segments are missing from the outer square, so one must\n");
2110 " use the `-c' switch. After `triangle -pqc box.poly', here is the output\n"
2113 " file `box.1.node', with twelve points. The last four points were added\n");
2115 " to meet the angle constraint. Points 1, 2, and 9 have markers from\n");
2117 " segment 1. Points 6 and 8 have markers from segment 4. All the other\n");
2119 " points but 4 have been marked to indicate that they lie on a boundary.\n");
2121 printf(" 12 2 0 1\n");
2122 printf(" 1 0 0 5\n");
2123 printf(" 2 0 3 5\n");
2124 printf(" 3 3 0 1\n");
2125 printf(" 4 3 3 33\n");
2126 printf(" 5 1 1 1\n");
2127 printf(" 6 1 2 10\n");
2128 printf(" 7 2 1 1\n");
2129 printf(" 8 2 2 10\n");
2130 printf(" 9 0 1.5 5\n");
2131 printf(" 10 1.5 0 1\n");
2132 printf(" 11 3 1.5 1\n");
2133 printf(" 12 1.5 3 1\n");
2134 printf(" # Generated by triangle -pqc box.poly\n\n");
2135 printf(" Here is the output file `box.1.ele', with twelve triangles.\n\n");
2136 printf(" 12 3 0\n");
2137 printf(" 1 5 6 9\n");
2138 printf(" 2 10 3 7\n");
2139 printf(" 3 6 8 12\n");
2140 printf(" 4 9 1 5\n");
2141 printf(" 5 6 2 9\n");
2142 printf(" 6 7 3 11\n");
2143 printf(" 7 11 4 8\n");
2144 printf(" 8 7 5 10\n");
2145 printf(" 9 12 2 6\n");
2146 printf(" 10 8 7 11\n");
2147 printf(" 11 5 1 10\n");
2148 printf(" 12 8 4 12\n");
2149 printf(" # Generated by triangle -pqc box.poly\n\n");
2151 " Here is the output file `box.1.poly'. Note that segments have been added\n"
2154 " to represent the convex hull, and some segments have been split by newly\n"
2157 " added points. Note also that <# of points> is set to zero to indicate\n");
2158 printf(" that the points should be read from the .node file.\n\n");
2159 printf(" 0 2 0 1\n");
2161 printf(" 1 1 9 5\n");
2162 printf(" 2 5 7 1\n");
2163 printf(" 3 8 7 1\n");
2164 printf(" 4 6 8 10\n");
2165 printf(" 5 5 6 1\n");
2166 printf(" 6 3 10 1\n");
2167 printf(" 7 4 11 1\n");
2168 printf(" 8 2 12 1\n");
2169 printf(" 9 9 2 5\n");
2170 printf(" 10 10 1 1\n");
2171 printf(" 11 11 3 1\n");
2172 printf(" 12 12 4 1\n");
2174 printf(" 1 1.5 1.5\n");
2175 printf(" # Generated by triangle -pqc box.poly\n\n");
2176 printf("Refinement and Area Constraints:\n\n");
2178 " The -r switch causes a mesh (.node and .ele files) to be read and\n");
2180 " refined. If the -p switch is also used, a .poly file is read and used to\n"
2183 " specify edges that are constrained and cannot be eliminated (although\n");
2185 " they can be divided into smaller edges) by the refinement process.\n");
2188 " When you refine a mesh, you generally want to impose tighter quality\n");
2190 " constraints. One way to accomplish this is to use -q with a larger\n");
2192 " angle, or -a followed by a smaller area than you used to generate the\n");
2194 " mesh you are refining. Another way to do this is to create an .area\n");
2196 " file, which specifies a maximum area for each triangle, and use the -a\n");
2198 " switch (without a number following). Each triangle's area constraint is\n"
2201 " applied to that triangle. Area constraints tend to diffuse as the mesh\n");
2203 " is refined, so if there are large variations in area constraint between\n");
2204 printf(" adjacent triangles, you may not get the results you want.\n\n");
2206 " If you are refining a mesh composed of linear (three-node) elements, the\n"
2209 " output mesh will contain all the nodes present in the input mesh, in the\n"
2212 " same order, with new nodes added at the end of the .node file. However,\n"
2215 " there is no guarantee that each output element is contained in a single\n");
2217 " input element. Often, output elements will overlap two input elements,\n");
2219 " and input edges are not present in the output mesh. Hence, a sequence of\n"
2222 " refined meshes will form a hierarchy of nodes, but not a hierarchy of\n");
2224 " elements. If you a refining a mesh of higher-order elements, the\n");
2226 " hierarchical property applies only to the nodes at the corners of an\n");
2227 printf(" element; other nodes may not be present in the refined mesh.\n\n");
2229 " It is important to understand that maximum area constraints in .poly\n");
2231 " files are handled differently from those in .area files. A maximum area\n"
2234 " in a .poly file applies to the whole (segment-bounded) region in which a\n"
2237 " point falls, whereas a maximum area in an .area file applies to only one\n"
2240 " triangle. Area constraints in .poly files are used only when a mesh is\n");
2242 " first generated, whereas area constraints in .area files are used only to\n"
2245 " refine an existing mesh, and are typically based on a posteriori error\n");
2247 " estimates resulting from a finite element simulation on that mesh.\n");
2250 " `triangle -rq25 object.1' will read object.1.node and object.1.ele, then\n"
2253 " refine the triangulation to enforce a 25 degree minimum angle, and then\n");
2255 " write the refined triangulation to object.2.node and object.2.ele.\n");
2258 " `triangle -rpaa6.2 z.3' will read z.3.node, z.3.ele, z.3.poly, and\n");
2260 " z.3.area. After reconstructing the mesh and its segments, Triangle will\n"
2263 " refine the mesh so that no triangle has area greater than 6.2, and\n");
2265 " furthermore the triangles satisfy the maximum area constraints in\n");
2267 " z.3.area. The output is written to z.4.node, z.4.ele, and z.4.poly.\n");
2270 " The sequence `triangle -qa1 x', `triangle -rqa.3 x.1', `triangle -rqa.1\n");
2272 " x.2' creates a sequence of successively finer meshes x.1, x.2, and x.3,\n");
2273 printf(" suitable for multigrid.\n\n");
2274 printf("Convex Hulls and Mesh Boundaries:\n\n");
2276 " If the input is a point set (rather than a PSLG), Triangle produces its\n");
2278 " convex hull as a by-product in the output .poly file if you use the -c\n");
2280 " switch. There are faster algorithms for finding a two-dimensional convex\n"
2283 " hull than triangulation, of course, but this one comes for free. If the\n"
2286 " input is an unconstrained mesh (you are using the -r switch but not the\n");
2288 " -p switch), Triangle produces a list of its boundary edges (including\n");
2289 printf(" hole boundaries) as a by-product if you use the -c switch.\n\n");
2290 printf("Voronoi Diagrams:\n\n");
2292 " The -v switch produces a Voronoi diagram, in files suffixed .v.node and\n");
2294 " .v.edge. For example, `triangle -v points' will read points.node,\n");
2296 " produce its Delaunay triangulation in points.1.node and points.1.ele,\n");
2298 " and produce its Voronoi diagram in points.1.v.node and points.1.v.edge.\n");
2300 " The .v.node file contains a list of all Voronoi vertices, and the .v.edge\n"
2303 " file contains a list of all Voronoi edges, some of which may be infinite\n"
2306 " rays. (The choice of filenames makes it easy to run the set of Voronoi\n");
2307 printf(" vertices through Triangle, if so desired.)\n\n");
2309 " This implementation does not use exact arithmetic to compute the Voronoi\n"
2312 " vertices, and does not check whether neighboring vertices are identical.\n"
2315 " Be forewarned that if the Delaunay triangulation is degenerate or\n");
2317 " near-degenerate, the Voronoi diagram may have duplicate points, crossing\n"
2320 " edges, or infinite rays whose direction vector is zero. Also, if you\n");
2322 " generate a constrained (as opposed to conforming) Delaunay triangulation,\n"
2325 " or if the triangulation has holes, the corresponding Voronoi diagram is\n");
2326 printf(" likely to have crossing edges and unlikely to make sense.\n\n");
2327 printf("Mesh Topology:\n\n");
2329 " You may wish to know which triangles are adjacent to a certain Delaunay\n");
2331 " edge in an .edge file, which Voronoi regions are adjacent to a certain\n");
2333 " Voronoi edge in a .v.edge file, or which Voronoi regions are adjacent to\n"
2336 " each other. All of this information can be found by cross-referencing\n");
2338 " output files with the recollection that the Delaunay triangulation and\n");
2339 printf(" the Voronoi diagrams are planar duals.\n\n");
2341 " Specifically, edge i of an .edge file is the dual of Voronoi edge i of\n");
2343 " the corresponding .v.edge file, and is rotated 90 degrees counterclock-\n");
2345 " wise from the Voronoi edge. Triangle j of an .ele file is the dual of\n");
2347 " vertex j of the corresponding .v.node file; and Voronoi region k is the\n");
2348 printf(" dual of point k of the corresponding .node file.\n\n");
2350 " Hence, to find the triangles adjacent to a Delaunay edge, look at the\n");
2352 " vertices of the corresponding Voronoi edge; their dual triangles are on\n");
2354 " the left and right of the Delaunay edge, respectively. To find the\n");
2356 " Voronoi regions adjacent to a Voronoi edge, look at the endpoints of the\n"
2359 " corresponding Delaunay edge; their dual regions are on the right and left\n"
2362 " of the Voronoi edge, respectively. To find which Voronoi regions are\n");
2363 printf(" adjacent to each other, just read the list of Delaunay edges.\n");
2365 printf("Statistics:\n");
2368 " After generating a mesh, Triangle prints a count of the number of points,\n"
2371 " triangles, edges, boundary edges, and segments in the output mesh. If\n");
2373 " you've forgotten the statistics for an existing mesh, the -rNEP switches\n"
2376 " (or -rpNEP if you've got a .poly file for the existing mesh) will\n");
2377 printf(" regenerate these statistics without writing any output.\n\n");
2379 " The -V switch produces extended statistics, including a rough estimate\n");
2381 " of memory use and a histogram of triangle aspect ratios and angles in the\n"
2383 printf(" mesh.\n\n");
2384 printf("Exact Arithmetic:\n\n");
2386 " Triangle uses adaptive exact arithmetic to perform what computational\n");
2388 " geometers call the `orientation' and `incircle' tests. If the floating-\n"
2391 " point arithmetic of your machine conforms to the IEEE 754 standard (as\n");
2393 " most workstations do), and does not use extended precision internal\n");
2395 " registers, then your output is guaranteed to be an absolutely true\n");
2396 printf(" Delaunay or conforming Delaunay triangulation, roundoff error\n");
2398 " notwithstanding. The word `adaptive' implies that these arithmetic\n");
2400 " routines compute the result only to the precision necessary to guarantee\n"
2403 " correctness, so they are usually nearly as fast as their approximate\n");
2405 " counterparts. The exact tests can be disabled with the -X switch. On\n");
2407 " most inputs, this switch will reduce the computation time by about eight\n"
2410 " percent - it's not worth the risk. There are rare difficult inputs\n");
2412 " (having many collinear and cocircular points), however, for which the\n");
2414 " difference could be a factor of two. These are precisely the inputs most\n"
2416 printf(" likely to cause errors if you use the -X switch.\n\n");
2418 " Unfortunately, these routines don't solve every numerical problem. Exact\n"
2421 " arithmetic is not used to compute the positions of points, because the\n");
2423 " bit complexity of point coordinates would grow without bound. Hence,\n");
2425 " segment intersections aren't computed exactly; in very unusual cases,\n");
2427 " roundoff error in computing an intersection point might actually lead to\n"
2430 " an inverted triangle and an invalid triangulation. (This is one reason\n");
2432 " to compute your own intersection points in your .poly files.) Similarly,\n"
2435 " exact arithmetic is not used to compute the vertices of the Voronoi\n");
2436 printf(" diagram.\n\n");
2438 " Underflow and overflow can also cause difficulties; the exact arithmetic\n"
2441 " routines do not ameliorate out-of-bounds exponents, which can arise\n");
2443 " during the orientation and incircle tests. As a rule of thumb, you\n");
2445 " should ensure that your input values are within a range such that their\n");
2447 " third powers can be taken without underflow or overflow. Underflow can\n");
2449 " silently prevent the tests from being performed exactly, while overflow\n");
2450 printf(" will typically cause a floating exception.\n\n");
2451 printf("Calling Triangle from Another Program:\n\n");
2452 printf(" Read the file triangle.h for details.\n\n");
2453 printf("Troubleshooting:\n\n");
2454 printf(" Please read this section before mailing me bugs.\n\n");
2455 printf(" `My output mesh has no triangles!'\n\n");
2457 " If you're using a PSLG, you've probably failed to specify a proper set\n"
2460 " of bounding segments, or forgotten to use the -c switch. Or you may\n");
2462 " have placed a hole badly. To test these possibilities, try again with\n"
2465 " the -c and -O switches. Alternatively, all your input points may be\n");
2467 " collinear, in which case you can hardly expect to triangulate them.\n");
2469 printf(" `Triangle doesn't terminate, or just crashes.'\n");
2472 " Bad things can happen when triangles get so small that the distance\n");
2474 " between their vertices isn't much larger than the precision of your\n");
2476 " machine's arithmetic. If you've compiled Triangle for single-precision\n"
2479 " arithmetic, you might do better by recompiling it for double-precision.\n"
2482 " Then again, you might just have to settle for more lenient constraints\n"
2485 " on the minimum angle and the maximum area than you had planned.\n");
2488 " You can minimize precision problems by ensuring that the origin lies\n");
2490 " inside your point set, or even inside the densest part of your\n");
2492 " mesh. On the other hand, if you're triangulating an object whose x\n");
2494 " coordinates all fall between 6247133 and 6247134, you're not leaving\n");
2495 printf(" much floating-point precision for Triangle to work with.\n\n");
2497 " Precision problems can occur covertly if the input PSLG contains two\n");
2499 " segments that meet (or intersect) at a very small angle, or if such an\n"
2502 " angle is introduced by the -c switch, which may occur if a point lies\n");
2504 " ever-so-slightly inside the convex hull, and is connected by a PSLG\n");
2506 " segment to a point on the convex hull. If you don't realize that a\n");
2508 " small angle is being formed, you might never discover why Triangle is\n");
2510 " crashing. To check for this possibility, use the -S switch (with an\n");
2512 " appropriate limit on the number of Steiner points, found by trial-and-\n"
2515 " error) to stop Triangle early, and view the output .poly file with\n");
2517 " Show Me (described below). Look carefully for small angles between\n");
2519 " segments; zoom in closely, as such segments might look like a single\n");
2520 printf(" segment from a distance.\n\n");
2522 " If some of the input values are too large, Triangle may suffer a\n");
2524 " floating exception due to overflow when attempting to perform an\n");
2526 " orientation or incircle test. (Read the section on exact arithmetic\n");
2528 " above.) Again, I recommend compiling Triangle for double (rather\n");
2529 printf(" than single) precision arithmetic.\n\n");
2531 " `The numbering of the output points doesn't match the input points.'\n");
2534 " You may have eaten some of your input points with a hole, or by placing\n"
2536 printf(" them outside the area enclosed by segments.\n\n");
2538 " `Triangle executes without incident, but when I look at the resulting\n");
2540 " mesh, it has overlapping triangles or other geometric inconsistencies.'\n");
2543 " If you select the -X switch, Triangle's divide-and-conquer Delaunay\n");
2545 " triangulation algorithm occasionally makes mistakes due to floating-\n");
2547 " point roundoff error. Although these errors are rare, don't use the -X\n"
2549 printf(" switch. If you still have problems, please report the bug.\n");
2552 " Strange things can happen if you've taken liberties with your PSLG. Do\n");
2554 " you have a point lying in the middle of a segment? Triangle sometimes\n");
2556 " copes poorly with that sort of thing. Do you want to lay out a collinear\n"
2559 " row of evenly spaced, segment-connected points? Have you simply defined\n"
2562 " one long segment connecting the leftmost point to the rightmost point,\n");
2564 " and a bunch of points lying along it? This method occasionally works,\n");
2566 " especially with horizontal and vertical lines, but often it doesn't, and\n"
2569 " you'll have to connect each adjacent pair of points with a separate\n");
2570 printf(" segment. If you don't like it, tough.\n\n");
2572 " Furthermore, if you have segments that intersect other than at their\n");
2574 " endpoints, try not to let the intersections fall extremely close to PSLG\n"
2576 printf(" points or each other.\n\n");
2578 " If you have problems refining a triangulation not produced by Triangle:\n");
2580 " Are you sure the triangulation is geometrically valid? Is it formatted\n");
2582 " correctly for Triangle? Are the triangles all listed so the first three\n"
2584 printf(" points are their corners in counterclockwise order?\n\n");
2585 printf("Show Me:\n\n");
2587 " Triangle comes with a separate program named `Show Me', whose primary\n");
2589 " purpose is to draw meshes on your screen or in PostScript. Its secondary\n"
2592 " purpose is to check the validity of your input files, and do so more\n");
2594 " thoroughly than Triangle does. Show Me requires that you have the X\n");
2596 " Windows system. If you didn't receive Show Me with Triangle, complain to\n"
2598 printf(" whomever you obtained Triangle from, then send me mail.\n\n");
2599 printf("Triangle on the Web:\n\n");
2601 " To see an illustrated, updated version of these instructions, check out\n");
2603 printf(" http://www.cs.cmu.edu/~quake/triangle.html\n");
2605 printf("A Brief Plea:\n");
2608 " If you use Triangle, and especially if you use it to accomplish real\n");
2610 " work, I would like very much to hear from you. A short letter or email\n");
2612 " (to jrs@cs.cmu.edu) describing how you use Triangle will mean a lot to\n");
2614 " me. The more people I know are using this program, the more easily I can\n"
2617 " justify spending time on improvements and on the three-dimensional\n");
2619 " successor to Triangle, which in turn will benefit you. Also, I can put\n");
2621 " you on a list to receive email whenever a new version of Triangle is\n");
2622 printf(" available.\n\n");
2624 " If you use a mesh generated by Triangle in a publication, please include\n"
2626 printf(" an acknowledgment as well.\n\n");
2627 printf("Research credit:\n\n");
2629 " Of course, I can take credit for only a fraction of the ideas that made\n");
2631 " this mesh generator possible. Triangle owes its existence to the efforts\n"
2634 " of many fine computational geometers and other researchers, including\n");
2636 " Marshall Bern, L. Paul Chew, Boris Delaunay, Rex A. Dwyer, David\n");
2638 " Eppstein, Steven Fortune, Leonidas J. Guibas, Donald E. Knuth, C. L.\n");
2640 " Lawson, Der-Tsai Lee, Ernst P. Mucke, Douglas M. Priest, Jim Ruppert,\n");
2642 " Isaac Saias, Bruce J. Schachter, Micha Sharir, Jorge Stolfi, Christopher\n"
2645 " J. Van Wyk, David F. Watson, and Binhai Zhu. See the comments at the\n");
2646 printf(" beginning of the source code for references.\n\n");
2650 #endif /* not TRILIBRARY */
2652 /*****************************************************************************/
2654 /* internalerror() Ask the user to send me the defective product. Exit. */
2656 /*****************************************************************************/
2658 void internalerror()
2660 printf(" Please report this bug to jrs@cs.cmu.edu\n");
2661 printf(" Include the message above, your input data set, and the exact\n");
2662 printf(" command line you used to run Triangle.\n");
2666 /*****************************************************************************/
2668 /* parsecommandline() Read the command line, identify switches, and set */
2669 /* up options and file names. */
2671 /* The effects of this routine are felt entirely through global variables. */
2673 /*****************************************************************************/
2675 void parsecommandline(argc, argv)
2680 #define STARTINDEX 0
2681 #else /* not TRILIBRARY */
2682 #define STARTINDEX 1
2685 #endif /* not TRILIBRARY */
2689 char workstring[FILENAMESIZE];
2692 poly = refine = quality = vararea = fixedarea = regionattrib = convex = 0;
2694 edgesout = voronoi = neighbors = geomview = 0;
2695 nobound = nopolywritten = nonodewritten = noelewritten = noiterationnum = 0;
2696 noholes = noexact = 0;
2697 incremental = sweepline = 0;
2706 quiet = verbose = 0;
2708 innodefilename[0] = '\0';
2709 #endif /* not TRILIBRARY */
2711 for (i = STARTINDEX; i < argc; i++) {
2713 if (argv[i][0] == '-') {
2714 #endif /* not TRILIBRARY */
2715 for (j = STARTINDEX; argv[i][j] != '\0'; j++) {
2716 if (argv[i][j] == 'p') {
2720 if (argv[i][j] == 'r') {
2723 if (argv[i][j] == 'q') {
2725 if (((argv[i][j + 1] >= '0') && (argv[i][j + 1] <= '9')) ||
2726 (argv[i][j + 1] == '.')) {
2728 while (((argv[i][j + 1] >= '0') && (argv[i][j + 1] <= '9')) ||
2729 (argv[i][j + 1] == '.')) {
2731 workstring[k] = argv[i][j];
2734 workstring[k] = '\0';
2735 minangle = (REAL) strtod(workstring, (char **) NULL);
2740 if (argv[i][j] == 'a') {
2742 if (((argv[i][j + 1] >= '0') && (argv[i][j + 1] <= '9')) ||
2743 (argv[i][j + 1] == '.')) {
2746 while (((argv[i][j + 1] >= '0') && (argv[i][j + 1] <= '9')) ||
2747 (argv[i][j + 1] == '.')) {
2749 workstring[k] = argv[i][j];
2752 workstring[k] = '\0';
2753 maxarea = (REAL) strtod(workstring, (char **) NULL);
2754 if (maxarea <= 0.0) {
2755 printf("Error: Maximum area must be greater than zero.\n");
2762 #endif /* not CDT_ONLY */
2763 if (argv[i][j] == 'A') {
2766 if (argv[i][j] == 'c') {
2769 if (argv[i][j] == 'z') {
2772 if (argv[i][j] == 'e') {
2775 if (argv[i][j] == 'v') {
2778 if (argv[i][j] == 'n') {
2781 if (argv[i][j] == 'g') {
2784 if (argv[i][j] == 'B') {
2787 if (argv[i][j] == 'P') {
2790 if (argv[i][j] == 'N') {
2793 if (argv[i][j] == 'E') {
2797 if (argv[i][j] == 'I') {
2800 #endif /* not TRILIBRARY */
2801 if (argv[i][j] == 'O') {
2804 if (argv[i][j] == 'X') {
2807 if (argv[i][j] == 'o') {
2808 if (argv[i][j + 1] == '2') {
2814 if (argv[i][j] == 'Y') {
2817 if (argv[i][j] == 'S') {
2819 while ((argv[i][j + 1] >= '0') && (argv[i][j + 1] <= '9')) {
2821 steiner = steiner * 10 + (int) (argv[i][j] - '0');
2824 #endif /* not CDT_ONLY */
2826 if (argv[i][j] == 'i') {
2829 if (argv[i][j] == 'F') {
2832 #endif /* not REDUCED */
2833 if (argv[i][j] == 'l') {
2838 if (argv[i][j] == 's') {
2841 #endif /* not CDT_ONLY */
2842 if (argv[i][j] == 'C') {
2845 #endif /* not REDUCED */
2846 if (argv[i][j] == 'Q') {
2849 if (argv[i][j] == 'V') {
2853 if ((argv[i][j] == 'h') || (argv[i][j] == 'H') ||
2854 (argv[i][j] == '?')) {
2857 #endif /* not TRILIBRARY */
2861 strncpy(innodefilename, argv[i], FILENAMESIZE - 1);
2862 innodefilename[FILENAMESIZE - 1] = '\0';
2864 #endif /* not TRILIBRARY */
2867 if (innodefilename[0] == '\0') {
2870 if (!strcmp(&innodefilename[strlen(innodefilename) - 5], ".node")) {
2871 innodefilename[strlen(innodefilename) - 5] = '\0';
2873 if (!strcmp(&innodefilename[strlen(innodefilename) - 5], ".poly")) {
2874 innodefilename[strlen(innodefilename) - 5] = '\0';
2878 if (!strcmp(&innodefilename[strlen(innodefilename) - 4], ".ele")) {
2879 innodefilename[strlen(innodefilename) - 4] = '\0';
2882 if (!strcmp(&innodefilename[strlen(innodefilename) - 5], ".area")) {
2883 innodefilename[strlen(innodefilename) - 5] = '\0';
2888 #endif /* not CDT_ONLY */
2889 #endif /* not TRILIBRARY */
2890 steinerleft = steiner;
2891 useshelles = poly || refine || quality || convex;
2892 goodangle = (REAL)cos(minangle * PI / 180.0);
2893 goodangle *= goodangle;
2894 if (refine && noiterationnum) {
2896 "Error: You cannot use the -I switch when refining a triangulation.\n");
2899 /* Be careful not to allocate space for element area constraints that */
2900 /* will never be assigned any value (other than the default -1.0). */
2901 if (!refine && !poly) {
2904 /* Be careful not to add an extra attribute to each element unless the */
2905 /* input supports it (PSLG in, but not refining a preexisting mesh). */
2906 if (refine || !poly) {
2911 strcpy(inpolyfilename, innodefilename);
2912 strcpy(inelefilename, innodefilename);
2913 strcpy(areafilename, innodefilename);
2915 strcpy(workstring, innodefilename);
2917 while (workstring[j] != '\0') {
2918 if ((workstring[j] == '.') && (workstring[j + 1] != '\0')) {
2924 if (increment > 0) {
2927 if ((workstring[j] >= '0') && (workstring[j] <= '9')) {
2928 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");
2950 } else if (increment == 0) {
2951 strcpy(outnodefilename, innodefilename);
2952 strcpy(outpolyfilename, innodefilename);
2953 strcpy(outelefilename, innodefilename);
2954 strcpy(edgefilename, innodefilename);
2955 strcpy(vnodefilename, innodefilename);
2956 strcpy(vedgefilename, innodefilename);
2957 strcpy(neighborfilename, innodefilename);
2958 strcpy(offfilename, innodefilename);
2959 strcat(outnodefilename, ".1.node");
2960 strcat(outpolyfilename, ".1.poly");
2961 strcat(outelefilename, ".1.ele");
2962 strcat(edgefilename, ".1.edge");
2963 strcat(vnodefilename, ".1.v.node");
2964 strcat(vedgefilename, ".1.v.edge");
2965 strcat(neighborfilename, ".1.neigh");
2966 strcat(offfilename, ".1.off");
2968 workstring[increment] = '%';
2969 workstring[increment + 1] = 'd';
2970 workstring[increment + 2] = '\0';
2971 sprintf(outnodefilename, workstring, meshnumber + 1);
2972 strcpy(outpolyfilename, outnodefilename);
2973 strcpy(outelefilename, outnodefilename);
2974 strcpy(edgefilename, outnodefilename);
2975 strcpy(vnodefilename, outnodefilename);
2976 strcpy(vedgefilename, outnodefilename);
2977 strcpy(neighborfilename, outnodefilename);
2978 strcpy(offfilename, outnodefilename);
2979 strcat(outnodefilename, ".node");
2980 strcat(outpolyfilename, ".poly");
2981 strcat(outelefilename, ".ele");
2982 strcat(edgefilename, ".edge");
2983 strcat(vnodefilename, ".v.node");
2984 strcat(vedgefilename, ".v.edge");
2985 strcat(neighborfilename, ".neigh");
2986 strcat(offfilename, ".off");
2988 strcat(innodefilename, ".node");
2989 strcat(inpolyfilename, ".poly");
2990 strcat(inelefilename, ".ele");
2991 strcat(areafilename, ".area");
2992 #endif /* not TRILIBRARY */
2997 /********* User interaction routines begin here *********/
2999 /********* Debugging routines begin here *********/
3003 /*****************************************************************************/
3005 /* printtriangle() Print out the details of a triangle/edge handle. */
3007 /* I originally wrote this procedure to simplify debugging; it can be */
3008 /* called directly from the debugger, and presents information about a */
3009 /* triangle/edge handle in digestible form. It's also used when the */
3010 /* highest level of verbosity (`-VVV') is specified. */
3012 /*****************************************************************************/
3014 void printtriangle(t)
3017 struct triedge printtri;
3018 struct edge printsh;
3021 printf("triangle x%lx with orientation %d:\n", (unsigned long) t->tri,
3023 decode(t->tri[0], printtri);
3024 if (printtri.tri == dummytri) {
3025 printf(" [0] = Outer space\n");
3027 printf(" [0] = x%lx %d\n", (unsigned long) printtri.tri,
3030 decode(t->tri[1], printtri);
3031 if (printtri.tri == dummytri) {
3032 printf(" [1] = Outer space\n");
3034 printf(" [1] = x%lx %d\n", (unsigned long) printtri.tri,
3037 decode(t->tri[2], printtri);
3038 if (printtri.tri == dummytri) {
3039 printf(" [2] = Outer space\n");
3041 printf(" [2] = x%lx %d\n", (unsigned long) printtri.tri,
3044 org(*t, printpoint);
3045 if (printpoint == (point) NULL)
projects / xonotic / netradiant.git / blob