#define ANSI_DECLARATORS /*****************************************************************************/ /* */ /* 888888888 ,o, / 888 */ /* 888 88o88o " o8888o 88o8888o o88888o 888 o88888o */ /* 888 888 888 88b 888 888 888 888 888 d888 88b */ /* 888 888 888 o88^o888 888 888 "88888" 888 8888oo888 */ /* 888 888 888 C888 888 888 888 / 888 q888 */ /* 888 888 888 "88o^888 888 888 Cb 888 "88oooo" */ /* "8oo8D */ /* */ /* A Two-Dimensional Quality Mesh Generator and Delaunay Triangulator. */ /* (triangle.c) */ /* */ /* Version 1.3 */ /* July 19, 1996 */ /* */ /* Copyright 1996 */ /* Jonathan Richard Shewchuk */ /* School of Computer Science */ /* Carnegie Mellon University */ /* 5000 Forbes Avenue */ /* Pittsburgh, Pennsylvania 15213-3891 */ /* jrs@cs.cmu.edu */ /* */ /* This program may be freely redistributed under the condition that the */ /* copyright notices (including this entire header and the copyright */ /* notice printed when the `-h' switch is selected) are not removed, and */ /* no compensation is received. Private, research, and institutional */ /* use is free. You may distribute modified versions of this code UNDER */ /* THE CONDITION THAT THIS CODE AND ANY MODIFICATIONS MADE TO IT IN THE */ /* SAME FILE REMAIN UNDER COPYRIGHT OF THE ORIGINAL AUTHOR, BOTH SOURCE */ /* AND OBJECT CODE ARE MADE FREELY AVAILABLE WITHOUT CHARGE, AND CLEAR */ /* NOTICE IS GIVEN OF THE MODIFICATIONS. Distribution of this code as */ /* part of a commercial system is permissible ONLY BY DIRECT ARRANGEMENT */ /* WITH THE AUTHOR. (If you are not directly supplying this code to a */ /* customer, and you are instead telling them how they can obtain it for */ /* free, then you are not required to make any arrangement with me.) */ /* */ /* Hypertext instructions for Triangle are available on the Web at */ /* */ /* http://www.cs.cmu.edu/~quake/triangle.html */ /* */ /* Some of the references listed below are marked [*]. These are available */ /* for downloading from the Web page */ /* */ /* http://www.cs.cmu.edu/~quake/triangle.research.html */ /* */ /* A paper discussing some aspects of Triangle is available. See Jonathan */ /* Richard Shewchuk, "Triangle: Engineering a 2D Quality Mesh Generator */ /* and Delaunay Triangulator," First Workshop on Applied Computational */ /* Geometry, ACM, May 1996. [*] */ /* */ /* Triangle was created as part of the Archimedes project in the School of */ /* Computer Science at Carnegie Mellon University. Archimedes is a */ /* system for compiling parallel finite element solvers. For further */ /* information, see Anja Feldmann, Omar Ghattas, John R. Gilbert, Gary L. */ /* Miller, David R. O'Hallaron, Eric J. Schwabe, Jonathan R. Shewchuk, */ /* and Shang-Hua Teng, "Automated Parallel Solution of Unstructured PDE */ /* Problems." To appear in Communications of the ACM, we hope. */ /* */ /* The quality mesh generation algorithm is due to Jim Ruppert, "A */ /* Delaunay Refinement Algorithm for Quality 2-Dimensional Mesh */ /* Generation," Journal of Algorithms 18(3):548-585, May 1995. [*] */ /* */ /* My implementation of the divide-and-conquer and incremental Delaunay */ /* triangulation algorithms follows closely the presentation of Guibas */ /* and Stolfi, even though I use a triangle-based data structure instead */ /* of their quad-edge data structure. (In fact, I originally implemented */ /* Triangle using the quad-edge data structure, but switching to a */ /* triangle-based data structure sped Triangle by a factor of two.) The */ /* mesh manipulation primitives and the two aforementioned Delaunay */ /* triangulation algorithms are described by Leonidas J. Guibas and Jorge */ /* Stolfi, "Primitives for the Manipulation of General Subdivisions and */ /* the Computation of Voronoi Diagrams," ACM Transactions on Graphics */ /* 4(2):74-123, April 1985. */ /* */ /* Their O(n log n) divide-and-conquer algorithm is adapted from Der-Tsai */ /* Lee and Bruce J. Schachter, "Two Algorithms for Constructing the */ /* Delaunay Triangulation," International Journal of Computer and */ /* Information Science 9(3):219-242, 1980. The idea to improve the */ /* divide-and-conquer algorithm by alternating between vertical and */ /* horizontal cuts was introduced by Rex A. Dwyer, "A Faster Divide-and- */ /* Conquer Algorithm for Constructing Delaunay Triangulations," */ /* Algorithmica 2(2):137-151, 1987. */ /* */ /* The incremental insertion algorithm was first proposed by C. L. Lawson, */ /* "Software for C1 Surface Interpolation," in Mathematical Software III, */ /* John R. Rice, editor, Academic Press, New York, pp. 161-194, 1977. */ /* For point location, I use the algorithm of Ernst P. Mucke, Isaac */ /* Saias, and Binhai Zhu, "Fast Randomized Point Location Without */ /* Preprocessing in Two- and Three-dimensional Delaunay Triangulations," */ /* Proceedings of the Twelfth Annual Symposium on Computational Geometry, */ /* ACM, May 1996. [*] If I were to randomize the order of point */ /* insertion (I currently don't bother), their result combined with the */ /* result of Leonidas J. Guibas, Donald E. Knuth, and Micha Sharir, */ /* "Randomized Incremental Construction of Delaunay and Voronoi */ /* Diagrams," Algorithmica 7(4):381-413, 1992, would yield an expected */ /* O(n^{4/3}) bound on running time. */ /* */ /* The O(n log n) sweepline Delaunay triangulation algorithm is taken from */ /* Steven Fortune, "A Sweepline Algorithm for Voronoi Diagrams", */ /* Algorithmica 2(2):153-174, 1987. A random sample of edges on the */ /* boundary of the triangulation are maintained in a splay tree for the */ /* purpose of point location. Splay trees are described by Daniel */ /* Dominic Sleator and Robert Endre Tarjan, "Self-Adjusting Binary Search */ /* Trees," Journal of the ACM 32(3):652-686, July 1985. */ /* */ /* The algorithms for exact computation of the signs of determinants are */ /* described in Jonathan Richard Shewchuk, "Adaptive Precision Floating- */ /* Point Arithmetic and Fast Robust Geometric Predicates," Technical */ /* Report CMU-CS-96-140, School of Computer Science, Carnegie Mellon */ /* University, Pittsburgh, Pennsylvania, May 1996. [*] (Submitted to */ /* Discrete & Computational Geometry.) An abbreviated version appears as */ /* Jonathan Richard Shewchuk, "Robust Adaptive Floating-Point Geometric */ /* Predicates," Proceedings of the Twelfth Annual Symposium on Computa- */ /* tional Geometry, ACM, May 1996. [*] Many of the ideas for my exact */ /* arithmetic routines originate with Douglas M. Priest, "Algorithms for */ /* Arbitrary Precision Floating Point Arithmetic," Tenth Symposium on */ /* Computer Arithmetic, 132-143, IEEE Computer Society Press, 1991. [*] */ /* Many of the ideas for the correct evaluation of the signs of */ /* determinants are taken from Steven Fortune and Christopher J. Van Wyk, */ /* "Efficient Exact Arithmetic for Computational Geometry," Proceedings */ /* of the Ninth Annual Symposium on Computational Geometry, ACM, */ /* pp. 163-172, May 1993, and from Steven Fortune, "Numerical Stability */ /* of Algorithms for 2D Delaunay Triangulations," International Journal */ /* of Computational Geometry & Applications 5(1-2):193-213, March-June */ /* 1995. */ /* */ /* For definitions of and results involving Delaunay triangulations, */ /* constrained and conforming versions thereof, and other aspects of */ /* triangular mesh generation, see the excellent survey by Marshall Bern */ /* and David Eppstein, "Mesh Generation and Optimal Triangulation," in */ /* Computing and Euclidean Geometry, Ding-Zhu Du and Frank Hwang, */ /* editors, World Scientific, Singapore, pp. 23-90, 1992. */ /* */ /* The time for incrementally adding PSLG (planar straight line graph) */ /* segments to create a constrained Delaunay triangulation is probably */ /* O(n^2) per segment in the worst case and O(n) per edge in the common */ /* case, where n is the number of triangles that intersect the segment */ /* before it is inserted. This doesn't count point location, which can */ /* be much more expensive. (This note does not apply to conforming */ /* Delaunay triangulations, for which a different method is used to */ /* insert segments.) */ /* */ /* The time for adding segments to a conforming Delaunay triangulation is */ /* not clear, but does not depend upon n alone. In some cases, very */ /* small features (like a point lying next to a segment) can cause a */ /* single segment to be split an arbitrary number of times. Of course, */ /* floating-point precision is a practical barrier to how much this can */ /* happen. */ /* */ /* The time for deleting a point from a Delaunay triangulation is O(n^2) in */ /* the worst case and O(n) in the common case, where n is the degree of */ /* the point being deleted. I could improve this to expected O(n) time */ /* by "inserting" the neighboring vertices in random order, but n is */ /* usually quite small, so it's not worth the bother. (The O(n) time */ /* for random insertion follows from L. Paul Chew, "Building Voronoi */ /* Diagrams for Convex Polygons in Linear Expected Time," Technical */ /* Report PCS-TR90-147, Department of Mathematics and Computer Science, */ /* Dartmouth College, 1990. */ /* */ /* Ruppert's Delaunay refinement algorithm typically generates triangles */ /* at a linear rate (constant time per triangle) after the initial */ /* triangulation is formed. There may be pathological cases where more */ /* time is required, but these never arise in practice. */ /* */ /* The segment intersection formulae are straightforward. If you want to */ /* see them derived, see Franklin Antonio. "Faster Line Segment */ /* Intersection." In Graphics Gems III (David Kirk, editor), pp. 199- */ /* 202. Academic Press, Boston, 1992. */ /* */ /* If you make any improvements to this code, please please please let me */ /* know, so that I may obtain the improvements. Even if you don't change */ /* the code, I'd still love to hear what it's being used for. */ /* */ /* Disclaimer: Neither I nor Carnegie Mellon warrant this code in any way */ /* whatsoever. This code is provided "as-is". Use at your own risk. */ /* */ /*****************************************************************************/ /* For single precision (which will save some memory and reduce paging), */ /* define the symbol SINGLE by using the -DSINGLE compiler switch or by */ /* writing "#define SINGLE" below. */ /* */ /* For double precision (which will allow you to refine meshes to a smaller */ /* edge length), leave SINGLE undefined. */ /* */ /* Double precision uses more memory, but improves the resolution of the */ /* meshes you can generate with Triangle. It also reduces the likelihood */ /* of a floating exception due to overflow. Finally, it is much faster */ /* than single precision on 64-bit architectures like the DEC Alpha. I */ /* recommend double precision unless you want to generate a mesh for which */ /* you do not have enough memory. */ #define SINGLE #ifdef SINGLE #define REAL float #else /* not SINGLE */ #define REAL double #endif /* not SINGLE */ /* If yours is not a Unix system, define the NO_TIMER compiler switch to */ /* remove the Unix-specific timing code. */ #define NO_TIMER /* To insert lots of self-checks for internal errors, define the SELF_CHECK */ /* symbol. This will slow down the program significantly. It is best to */ /* define the symbol using the -DSELF_CHECK compiler switch, but you could */ /* write "#define SELF_CHECK" below. If you are modifying this code, I */ /* recommend you turn self-checks on. */ /* #define SELF_CHECK */ /* To compile Triangle as a callable object library (triangle.o), define the */ /* TRILIBRARY symbol. Read the file triangle.h for details on how to call */ /* the procedure triangulate() that results. */ #define TRILIBRARY /* It is possible to generate a smaller version of Triangle using one or */ /* both of the following symbols. Define the REDUCED symbol to eliminate */ /* all features that are primarily of research interest; specifically, the */ /* -i, -F, -s, and -C switches. Define the CDT_ONLY symbol to eliminate */ /* all meshing algorithms above and beyond constrained Delaunay */ /* triangulation; specifically, the -r, -q, -a, -S, and -s switches. */ /* These reductions are most likely to be useful when generating an object */ /* library (triangle.o) by defining the TRILIBRARY symbol. */ #define REDUCED #define CDT_ONLY /* On some machines, the exact arithmetic routines might be defeated by the */ /* use of internal extended precision floating-point registers. Sometimes */ /* this problem can be fixed by defining certain values to be volatile, */ /* thus forcing them to be stored to memory and rounded off. This isn't */ /* a great solution, though, as it slows Triangle down. */ /* */ /* To try this out, write "#define INEXACT volatile" below. Normally, */ /* however, INEXACT should be defined to be nothing. ("#define INEXACT".) */ #define INEXACT /* Nothing */ /* #define INEXACT volatile */ /* Maximum number of characters in a file name (including the null). */ #define FILENAMESIZE 512 /* Maximum number of characters in a line read from a file (including the */ /* null). */ #define INPUTLINESIZE 512 /* For efficiency, a variety of data structures are allocated in bulk. The */ /* following constants determine how many of each structure is allocated */ /* at once. */ #define TRIPERBLOCK 4092 /* Number of triangles allocated at once. */ #define SHELLEPERBLOCK 508 /* Number of shell edges allocated at once. */ #define POINTPERBLOCK 4092 /* Number of points allocated at once. */ #define VIRUSPERBLOCK 1020 /* Number of virus triangles allocated at once. */ /* Number of encroached segments allocated at once. */ #define BADSEGMENTPERBLOCK 252 /* Number of skinny triangles allocated at once. */ #define BADTRIPERBLOCK 4092 /* Number of splay tree nodes allocated at once. */ #define SPLAYNODEPERBLOCK 508 /* The point marker DEADPOINT is an arbitrary number chosen large enough to */ /* (hopefully) not conflict with user boundary markers. Make sure that it */ /* is small enough to fit into your machine's integer size. */ #define DEADPOINT -1073741824 /* The next line is used to outsmart some very stupid compilers. If your */ /* compiler is smarter, feel free to replace the "int" with "void". */ /* Not that it matters. */ #define VOID int /* Two constants for algorithms based on random sampling. Both constants */ /* have been chosen empirically to optimize their respective algorithms. */ /* Used for the point location scheme of Mucke, Saias, and Zhu, to decide */ /* how large a random sample of triangles to inspect. */ #define SAMPLEFACTOR 11 /* Used in Fortune's sweepline Delaunay algorithm to determine what fraction */ /* of boundary edges should be maintained in the splay tree for point */ /* location on the front. */ #define SAMPLERATE 10 /* A number that speaks for itself, every kissable digit. */ #define PI 3.141592653589793238462643383279502884197169399375105820974944592308 /* Another fave. */ #define SQUAREROOTTWO 1.4142135623730950488016887242096980785696718753769480732 /* And here's one for those of you who are intimidated by math. */ #define ONETHIRD 0.333333333333333333333333333333333333333333333333333333333333 #include #include #include #ifndef NO_TIMER #include #endif /* NO_TIMER */ #ifdef TRILIBRARY #include "triangle.h" #endif /* TRILIBRARY */ /* The following obscenity seems to be necessary to ensure that this program */ /* will port to Dec Alphas running OSF/1, because their stdio.h file commits */ /* the unpardonable sin of including stdlib.h. Hence, malloc(), free(), and */ /* exit() may or may not already be defined at this point. I declare these */ /* functions explicitly because some non-ANSI C compilers lack stdlib.h. */ #ifndef _STDLIB_H_ extern void *malloc(); extern void free(); extern void exit(); extern double strtod(); extern long strtol(); #endif /* _STDLIB_H_ */ /* A few forward declarations. */ void poolrestart(); #ifndef TRILIBRARY char *readline(); char *findfield(); #endif /* not TRILIBRARY */ /* Labels that signify whether a record consists primarily of pointers or of */ /* floating-point words. Used to make decisions about data alignment. */ enum wordtype {POINTER, FLOATINGPOINT}; /* Labels that signify the result of point location. The result of a */ /* search indicates that the point falls in the interior of a triangle, on */ /* an edge, on a vertex, or outside the mesh. */ enum locateresult {INTRIANGLE, ONEDGE, ONVERTEX, OUTSIDE}; /* Labels that signify the result of site insertion. The result indicates */ /* that the point was inserted with complete success, was inserted but */ /* encroaches on a segment, was not inserted because it lies on a segment, */ /* or was not inserted because another point occupies the same location. */ enum insertsiteresult {SUCCESSFULPOINT, ENCROACHINGPOINT, VIOLATINGPOINT, DUPLICATEPOINT}; /* Labels that signify the result of direction finding. The result */ /* indicates that a segment connecting the two query points falls within */ /* the direction triangle, along the left edge of the direction triangle, */ /* or along the right edge of the direction triangle. */ enum finddirectionresult {WITHIN, LEFTCOLLINEAR, RIGHTCOLLINEAR}; /* Labels that signify the result of the circumcenter computation routine. */ /* The return value indicates which edge of the triangle is shortest. */ enum circumcenterresult {OPPOSITEORG, OPPOSITEDEST, OPPOSITEAPEX}; /*****************************************************************************/ /* */ /* The basic mesh data structures */ /* */ /* There are three: points, triangles, and shell edges (abbreviated */ /* `shelle'). These three data structures, linked by pointers, comprise */ /* the mesh. A point simply represents a point in space and its properties.*/ /* A triangle is a triangle. A shell edge is a special data structure used */ /* to represent impenetrable segments in the mesh (including the outer */ /* boundary, boundaries of holes, and internal boundaries separating two */ /* triangulated regions). Shell edges represent boundaries defined by the */ /* user that triangles may not lie across. */ /* */ /* A triangle consists of a list of three vertices, a list of three */ /* adjoining triangles, a list of three adjoining shell edges (when shell */ /* edges are used), an arbitrary number of optional user-defined floating- */ /* point attributes, and an optional area constraint. The latter is an */ /* upper bound on the permissible area of each triangle in a region, used */ /* for mesh refinement. */ /* */ /* For a triangle on a boundary of the mesh, some or all of the neighboring */ /* triangles may not be present. For a triangle in the interior of the */ /* mesh, often no neighboring shell edges are present. Such absent */ /* triangles and shell edges are never represented by NULL pointers; they */ /* are represented by two special records: `dummytri', the triangle that */ /* fills "outer space", and `dummysh', the omnipresent shell edge. */ /* `dummytri' and `dummysh' are used for several reasons; for instance, */ /* they can be dereferenced and their contents examined without causing the */ /* memory protection exception that would occur if NULL were dereferenced. */ /* */ /* However, it is important to understand that a triangle includes other */ /* information as well. The pointers to adjoining vertices, triangles, and */ /* shell edges are ordered in a way that indicates their geometric relation */ /* to each other. Furthermore, each of these pointers contains orientation */ /* information. Each pointer to an adjoining triangle indicates which face */ /* of that triangle is contacted. Similarly, each pointer to an adjoining */ /* shell edge indicates which side of that shell edge is contacted, and how */ /* the shell edge is oriented relative to the triangle. */ /* */ /* Shell edges are found abutting edges of triangles; either sandwiched */ /* between two triangles, or resting against one triangle on an exterior */ /* boundary or hole boundary. */ /* */ /* A shell edge consists of a list of two vertices, a list of two */ /* adjoining shell edges, and a list of two adjoining triangles. One of */ /* the two adjoining triangles may not be present (though there should */ /* always be one), and neighboring shell edges might not be present. */ /* Shell edges also store a user-defined integer "boundary marker". */ /* Typically, this integer is used to indicate what sort of boundary */ /* conditions are to be applied at that location in a finite element */ /* simulation. */ /* */ /* Like triangles, shell edges maintain information about the relative */ /* orientation of neighboring objects. */ /* */ /* Points are relatively simple. A point is a list of floating point */ /* numbers, starting with the x, and y coordinates, followed by an */ /* arbitrary number of optional user-defined floating-point attributes, */ /* followed by an integer boundary marker. During the segment insertion */ /* phase, there is also a pointer from each point to a triangle that may */ /* contain it. Each pointer is not always correct, but when one is, it */ /* speeds up segment insertion. These pointers are assigned values once */ /* at the beginning of the segment insertion phase, and are not used or */ /* updated at any other time. Edge swapping during segment insertion will */ /* render some of them incorrect. Hence, don't rely upon them for */ /* anything. For the most part, points do not have any information about */ /* what triangles or shell edges they are linked to. */ /* */ /*****************************************************************************/ /*****************************************************************************/ /* */ /* Handles */ /* */ /* The oriented triangle (`triedge') and oriented shell edge (`edge') data */ /* structures defined below do not themselves store any part of the mesh. */ /* The mesh itself is made of `triangle's, `shelle's, and `point's. */ /* */ /* Oriented triangles and oriented shell edges will usually be referred to */ /* as "handles". A handle is essentially a pointer into the mesh; it */ /* allows you to "hold" one particular part of the mesh. Handles are used */ /* to specify the regions in which one is traversing and modifying the mesh.*/ /* A single `triangle' may be held by many handles, or none at all. (The */ /* latter case is not a memory leak, because the triangle is still */ /* connected to other triangles in the mesh.) */ /* */ /* A `triedge' is a handle that holds a triangle. It holds a specific side */ /* of the triangle. An `edge' is a handle that holds a shell edge. It */ /* holds either the left or right side of the edge. */ /* */ /* Navigation about the mesh is accomplished through a set of mesh */ /* manipulation primitives, further below. Many of these primitives take */ /* a handle and produce a new handle that holds the mesh near the first */ /* handle. Other primitives take two handles and glue the corresponding */ /* parts of the mesh together. The exact position of the handles is */ /* important. For instance, when two triangles are glued together by the */ /* bond() primitive, they are glued by the sides on which the handles lie. */ /* */ /* Because points have no information about which triangles they are */ /* attached to, I commonly represent a point by use of a handle whose */ /* origin is the point. A single handle can simultaneously represent a */ /* triangle, an edge, and a point. */ /* */ /*****************************************************************************/ /* The triangle data structure. Each triangle contains three pointers to */ /* adjoining triangles, plus three pointers to vertex points, plus three */ /* pointers to shell edges (defined below; these pointers are usually */ /* `dummysh'). It may or may not also contain user-defined attributes */ /* and/or a floating-point "area constraint". It may also contain extra */ /* pointers for nodes, when the user asks for high-order elements. */ /* Because the size and structure of a `triangle' is not decided until */ /* runtime, I haven't simply defined the type `triangle' to be a struct. */ typedef REAL **triangle; /* Really: typedef triangle *triangle */ /* An oriented triangle: includes a pointer to a triangle and orientation. */ /* The orientation denotes an edge of the triangle. Hence, there are */ /* three possible orientations. By convention, each edge is always */ /* directed to point counterclockwise about the corresponding triangle. */ struct triedge { triangle *tri; int orient; /* Ranges from 0 to 2. */ }; /* The shell data structure. Each shell edge contains two pointers to */ /* adjoining shell edges, plus two pointers to vertex points, plus two */ /* pointers to adjoining triangles, plus one shell marker. */ typedef REAL **shelle; /* Really: typedef shelle *shelle */ /* An oriented shell edge: includes a pointer to a shell edge and an */ /* orientation. The orientation denotes a side of the edge. Hence, there */ /* are two possible orientations. By convention, the edge is always */ /* directed so that the "side" denoted is the right side of the edge. */ struct edge { shelle *sh; int shorient; /* Ranges from 0 to 1. */ }; /* The point data structure. Each point is actually an array of REALs. */ /* The number of REALs is unknown until runtime. An integer boundary */ /* marker, and sometimes a pointer to a triangle, is appended after the */ /* REALs. */ typedef REAL *point; /* A queue used to store encroached segments. Each segment's vertices are */ /* stored so that one can check whether a segment is still the same. */ struct badsegment { struct edge encsegment; /* An encroached segment. */ point segorg, segdest; /* The two vertices. */ struct badsegment *nextsegment; /* Pointer to next encroached segment. */ }; /* A queue used to store bad triangles. The key is the square of the cosine */ /* of the smallest angle of the triangle. Each triangle's vertices are */ /* stored so that one can check whether a triangle is still the same. */ struct badface { struct triedge badfacetri; /* A bad triangle. */ REAL key; /* cos^2 of smallest (apical) angle. */ point faceorg, facedest, faceapex; /* The three vertices. */ struct badface *nextface; /* Pointer to next bad triangle. */ }; /* A node in a heap used to store events for the sweepline Delaunay */ /* algorithm. Nodes do not point directly to their parents or children in */ /* the heap. Instead, each node knows its position in the heap, and can */ /* look up its parent and children in a separate array. The `eventptr' */ /* points either to a `point' or to a triangle (in encoded format, so that */ /* an orientation is included). In the latter case, the origin of the */ /* oriented triangle is the apex of a "circle event" of the sweepline */ /* algorithm. To distinguish site events from circle events, all circle */ /* events are given an invalid (smaller than `xmin') x-coordinate `xkey'. */ struct event { REAL xkey, ykey; /* Coordinates of the event. */ VOID *eventptr; /* Can be a point or the location of a circle event. */ int heapposition; /* Marks this event's position in the heap. */ }; /* A node in the splay tree. Each node holds an oriented ghost triangle */ /* that represents a boundary edge of the growing triangulation. When a */ /* circle event covers two boundary edges with a triangle, so that they */ /* are no longer boundary edges, those edges are not immediately deleted */ /* from the tree; rather, they are lazily deleted when they are next */ /* encountered. (Since only a random sample of boundary edges are kept */ /* in the tree, lazy deletion is faster.) `keydest' is used to verify */ /* that a triangle is still the same as when it entered the splay tree; if */ /* it has been rotated (due to a circle event), it no longer represents a */ /* boundary edge and should be deleted. */ struct splaynode { struct triedge keyedge; /* Lprev of an edge on the front. */ point keydest; /* Used to verify that splay node is still live. */ struct splaynode *lchild, *rchild; /* Children in splay tree. */ }; /* A type used to allocate memory. firstblock is the first block of items. */ /* nowblock is the block from which items are currently being allocated. */ /* nextitem points to the next slab of free memory for an item. */ /* deaditemstack is the head of a linked list (stack) of deallocated items */ /* that can be recycled. unallocateditems is the number of items that */ /* remain to be allocated from nowblock. */ /* */ /* Traversal is the process of walking through the entire list of items, and */ /* is separate from allocation. Note that a traversal will visit items on */ /* the "deaditemstack" stack as well as live items. pathblock points to */ /* the block currently being traversed. pathitem points to the next item */ /* to be traversed. pathitemsleft is the number of items that remain to */ /* be traversed in pathblock. */ /* */ /* itemwordtype is set to POINTER or FLOATINGPOINT, and is used to suggest */ /* what sort of word the record is primarily made up of. alignbytes */ /* determines how new records should be aligned in memory. itembytes and */ /* itemwords are the length of a record in bytes (after rounding up) and */ /* words. itemsperblock is the number of items allocated at once in a */ /* single block. items is the number of currently allocated items. */ /* maxitems is the maximum number of items that have been allocated at */ /* once; it is the current number of items plus the number of records kept */ /* on deaditemstack. */ struct memorypool { VOID **firstblock, **nowblock; VOID *nextitem; VOID *deaditemstack; VOID **pathblock; VOID *pathitem; enum wordtype itemwordtype; int alignbytes; int itembytes, itemwords; int itemsperblock; long items, maxitems; int unallocateditems; int pathitemsleft; }; /* Variables used to allocate memory for triangles, shell edges, points, */ /* viri (triangles being eaten), bad (encroached) segments, bad (skinny */ /* or too large) triangles, and splay tree nodes. */ static struct memorypool triangles; static struct memorypool shelles; static struct memorypool points; static struct memorypool viri; static struct memorypool badsegments; static struct memorypool badtriangles; static struct memorypool splaynodes; /* Variables that maintain the bad triangle queues. The tails are pointers */ /* to the pointers that have to be filled in to enqueue an item. */ static struct badface *queuefront[64]; static struct badface **queuetail[64]; static REAL xmin, xmax, ymin, ymax; /* x and y bounds. */ static REAL xminextreme; /* Nonexistent x value used as a flag in sweepline. */ static int inpoints; /* Number of input points. */ static int inelements; /* Number of input triangles. */ static int insegments; /* Number of input segments. */ static int holes; /* Number of input holes. */ static int regions; /* Number of input regions. */ static long edges; /* Number of output edges. */ static int mesh_dim; /* Dimension (ought to be 2). */ static int nextras; /* Number of attributes per point. */ static int eextras; /* Number of attributes per triangle. */ static long hullsize; /* Number of edges of convex hull. */ static int triwords; /* Total words per triangle. */ static int shwords; /* Total words per shell edge. */ static int pointmarkindex; /* Index to find boundary marker of a point. */ static int point2triindex; /* Index to find a triangle adjacent to a point. */ static int highorderindex; /* Index to find extra nodes for high-order elements. */ static int elemattribindex; /* Index to find attributes of a triangle. */ static int areaboundindex; /* Index to find area bound of a triangle. */ static int checksegments; /* Are there segments in the triangulation yet? */ static int readnodefile; /* Has a .node file been read? */ static long samples; /* Number of random samples for point location. */ static unsigned long randomseed; /* Current random number seed. */ static REAL splitter; /* Used to split REAL factors for exact multiplication. */ static REAL epsilon; /* Floating-point machine epsilon. */ static REAL resulterrbound; static REAL ccwerrboundA, ccwerrboundB, ccwerrboundC; static REAL iccerrboundA, iccerrboundB, iccerrboundC; static long incirclecount; /* Number of incircle tests performed. */ static long counterclockcount; /* Number of counterclockwise tests performed. */ static long hyperbolacount; /* Number of right-of-hyperbola tests performed. */ static long circumcentercount; /* Number of circumcenter calculations performed. */ static long circletopcount; /* Number of circle top calculations performed. */ /* Switches for the triangulator. */ /* poly: -p switch. refine: -r switch. */ /* quality: -q switch. */ /* minangle: minimum angle bound, specified after -q switch. */ /* goodangle: cosine squared of minangle. */ /* vararea: -a switch without number. */ /* fixedarea: -a switch with number. */ /* maxarea: maximum area bound, specified after -a switch. */ /* regionattrib: -A switch. convex: -c switch. */ /* firstnumber: inverse of -z switch. All items are numbered starting */ /* from firstnumber. */ /* edgesout: -e switch. voronoi: -v switch. */ /* neighbors: -n switch. geomview: -g switch. */ /* nobound: -B switch. nopolywritten: -P switch. */ /* nonodewritten: -N switch. noelewritten: -E switch. */ /* noiterationnum: -I switch. noholes: -O switch. */ /* noexact: -X switch. */ /* order: element order, specified after -o switch. */ /* nobisect: count of how often -Y switch is selected. */ /* steiner: maximum number of Steiner points, specified after -S switch. */ /* steinerleft: number of Steiner points not yet used. */ /* incremental: -i switch. sweepline: -F switch. */ /* dwyer: inverse of -l switch. */ /* splitseg: -s switch. */ /* docheck: -C switch. */ /* quiet: -Q switch. verbose: count of how often -V switch is selected. */ /* useshelles: -p, -r, -q, or -c switch; determines whether shell edges */ /* are used at all. */ /* */ /* Read the instructions to find out the meaning of these switches. */ static int poly, refine, quality, vararea, fixedarea, regionattrib, convex; static int firstnumber; static int edgesout, voronoi, neighbors, geomview; static int nobound, nopolywritten, nonodewritten, noelewritten, noiterationnum; static int noholes, noexact; static int incremental, sweepline, dwyer; static int splitseg; static int docheck; static int quiet, verbose; static int useshelles; static int order; static int nobisect; static int steiner, steinerleft; static REAL minangle, goodangle; static REAL maxarea; /* Variables for file names. */ #ifndef TRILIBRARY char innodefilename[FILENAMESIZE]; char inelefilename[FILENAMESIZE]; char inpolyfilename[FILENAMESIZE]; char areafilename[FILENAMESIZE]; char outnodefilename[FILENAMESIZE]; char outelefilename[FILENAMESIZE]; char outpolyfilename[FILENAMESIZE]; char edgefilename[FILENAMESIZE]; char vnodefilename[FILENAMESIZE]; char vedgefilename[FILENAMESIZE]; char neighborfilename[FILENAMESIZE]; char offfilename[FILENAMESIZE]; #endif /* not TRILIBRARY */ /* Triangular bounding box points. */ static point infpoint1, infpoint2, infpoint3; /* Pointer to the `triangle' that occupies all of "outer space". */ static triangle *dummytri; static triangle *dummytribase; /* Keep base address so we can free() it later. */ /* Pointer to the omnipresent shell edge. Referenced by any triangle or */ /* shell edge that isn't really connected to a shell edge at that */ /* location. */ static shelle *dummysh; static shelle *dummyshbase; /* Keep base address so we can free() it later. */ /* Pointer to a recently visited triangle. Improves point location if */ /* proximate points are inserted sequentially. */ static struct triedge recenttri; /*****************************************************************************/ /* */ /* Mesh manipulation primitives. Each triangle contains three pointers to */ /* other triangles, with orientations. Each pointer points not to the */ /* first byte of a triangle, but to one of the first three bytes of a */ /* triangle. It is necessary to extract both the triangle itself and the */ /* orientation. To save memory, I keep both pieces of information in one */ /* pointer. To make this possible, I assume that all triangles are aligned */ /* to four-byte boundaries. The `decode' routine below decodes a pointer, */ /* extracting an orientation (in the range 0 to 2) and a pointer to the */ /* beginning of a triangle. The `encode' routine compresses a pointer to a */ /* triangle and an orientation into a single pointer. My assumptions that */ /* triangles are four-byte-aligned and that the `unsigned long' type is */ /* long enough to hold a pointer are two of the few kludges in this program.*/ /* */ /* Shell edges are manipulated similarly. A pointer to a shell edge */ /* carries both an address and an orientation in the range 0 to 1. */ /* */ /* The other primitives take an oriented triangle or oriented shell edge, */ /* and return an oriented triangle or oriented shell edge or point; or they */ /* change the connections in the data structure. */ /* */ /*****************************************************************************/ /********* Mesh manipulation primitives begin here *********/ /** **/ /** **/ /* Fast lookup arrays to speed some of the mesh manipulation primitives. */ int plus1mod3[3] = {1, 2, 0}; int minus1mod3[3] = {2, 0, 1}; /********* Primitives for triangles *********/ /* */ /* */ /* decode() converts a pointer to an oriented triangle. The orientation is */ /* extracted from the two least significant bits of the pointer. */ #define decode( ptr, triedge ) \ ( triedge ).orient = (int) ( (unsigned long) ( ptr ) & (unsigned long) 3l ); \ ( triedge ).tri = (triangle *) \ ( (unsigned long) ( ptr ) ^ (unsigned long) ( triedge ).orient ) /* encode() compresses an oriented triangle into a single pointer. It */ /* relies on the assumption that all triangles are aligned to four-byte */ /* boundaries, so the two least significant bits of (triedge).tri are zero.*/ #define encode( triedge ) \ (triangle) ( (unsigned long) ( triedge ).tri | (unsigned long) ( triedge ).orient ) /* The following edge manipulation primitives are all described by Guibas */ /* and Stolfi. However, they use an edge-based data structure, whereas I */ /* am using a triangle-based data structure. */ /* sym() finds the abutting triangle, on the same edge. Note that the */ /* edge direction is necessarily reversed, because triangle/edge handles */ /* are always directed counterclockwise around the triangle. */ #define sym( triedge1, triedge2 ) \ ptr = ( triedge1 ).tri[( triedge1 ).orient]; \ decode( ptr, triedge2 ); #define symself( triedge ) \ ptr = ( triedge ).tri[( triedge ).orient]; \ decode( ptr, triedge ); /* lnext() finds the next edge (counterclockwise) of a triangle. */ #define lnext( triedge1, triedge2 ) \ ( triedge2 ).tri = ( triedge1 ).tri; \ ( triedge2 ).orient = plus1mod3[( triedge1 ).orient] #define lnextself( triedge ) \ ( triedge ).orient = plus1mod3[( triedge ).orient] /* lprev() finds the previous edge (clockwise) of a triangle. */ #define lprev( triedge1, triedge2 ) \ ( triedge2 ).tri = ( triedge1 ).tri; \ ( triedge2 ).orient = minus1mod3[( triedge1 ).orient] #define lprevself( triedge ) \ ( triedge ).orient = minus1mod3[( triedge ).orient] /* onext() spins counterclockwise around a point; that is, it finds the next */ /* edge with the same origin in the counterclockwise direction. This edge */ /* will be part of a different triangle. */ #define onext( triedge1, triedge2 ) \ lprev( triedge1, triedge2 ); \ symself( triedge2 ); #define onextself( triedge ) \ lprevself( triedge ); \ symself( triedge ); /* oprev() spins clockwise around a point; that is, it finds the next edge */ /* with the same origin in the clockwise direction. This edge will be */ /* part of a different triangle. */ #define oprev( triedge1, triedge2 ) \ sym( triedge1, triedge2 ); \ lnextself( triedge2 ); #define oprevself( triedge ) \ symself( triedge ); \ lnextself( triedge ); /* dnext() spins counterclockwise around a point; that is, it finds the next */ /* edge with the same destination in the counterclockwise direction. This */ /* edge will be part of a different triangle. */ #define dnext( triedge1, triedge2 ) \ sym( triedge1, triedge2 ); \ lprevself( triedge2 ); #define dnextself( triedge ) \ symself( triedge ); \ lprevself( triedge ); /* dprev() spins clockwise around a point; that is, it finds the next edge */ /* with the same destination in the clockwise direction. This edge will */ /* be part of a different triangle. */ #define dprev( triedge1, triedge2 ) \ lnext( triedge1, triedge2 ); \ symself( triedge2 ); #define dprevself( triedge ) \ lnextself( triedge ); \ symself( triedge ); /* rnext() moves one edge counterclockwise about the adjacent triangle. */ /* (It's best understood by reading Guibas and Stolfi. It involves */ /* changing triangles twice.) */ #define rnext( triedge1, triedge2 ) \ sym( triedge1, triedge2 ); \ lnextself( triedge2 ); \ symself( triedge2 ); #define rnextself( triedge ) \ symself( triedge ); \ lnextself( triedge ); \ symself( triedge ); /* rnext() moves one edge clockwise about the adjacent triangle. */ /* (It's best understood by reading Guibas and Stolfi. It involves */ /* changing triangles twice.) */ #define rprev( triedge1, triedge2 ) \ sym( triedge1, triedge2 ); \ lprevself( triedge2 ); \ symself( triedge2 ); #define rprevself( triedge ) \ symself( triedge ); \ lprevself( triedge ); \ symself( triedge ); /* These primitives determine or set the origin, destination, or apex of a */ /* triangle. */ #define org( triedge, pointptr ) \ pointptr = (point) ( triedge ).tri[plus1mod3[( triedge ).orient] + 3] #define dest( triedge, pointptr ) \ pointptr = (point) ( triedge ).tri[minus1mod3[( triedge ).orient] + 3] #define apex( triedge, pointptr ) \ pointptr = (point) ( triedge ).tri[( triedge ).orient + 3] #define setorg( triedge, pointptr ) \ ( triedge ).tri[plus1mod3[( triedge ).orient] + 3] = (triangle) pointptr #define setdest( triedge, pointptr ) \ ( triedge ).tri[minus1mod3[( triedge ).orient] + 3] = (triangle) pointptr #define setapex( triedge, pointptr ) \ ( triedge ).tri[( triedge ).orient + 3] = (triangle) pointptr #define setvertices2null( triedge ) \ ( triedge ).tri[3] = (triangle) NULL; \ ( triedge ).tri[4] = (triangle) NULL; \ ( triedge ).tri[5] = (triangle) NULL; /* Bond two triangles together. */ #define bond( triedge1, triedge2 ) \ ( triedge1 ).tri[( triedge1 ).orient] = encode( triedge2 ); \ ( triedge2 ).tri[( triedge2 ).orient] = encode( triedge1 ) /* Dissolve a bond (from one side). Note that the other triangle will still */ /* think it's connected to this triangle. Usually, however, the other */ /* triangle is being deleted entirely, or bonded to another triangle, so */ /* it doesn't matter. */ #define dissolve( triedge ) \ ( triedge ).tri[( triedge ).orient] = (triangle) dummytri /* Copy a triangle/edge handle. */ #define triedgecopy( triedge1, triedge2 ) \ ( triedge2 ).tri = ( triedge1 ).tri; \ ( triedge2 ).orient = ( triedge1 ).orient /* Test for equality of triangle/edge handles. */ #define triedgeequal( triedge1, triedge2 ) \ ( ( ( triedge1 ).tri == ( triedge2 ).tri ) && \ ( ( triedge1 ).orient == ( triedge2 ).orient ) ) /* Primitives to infect or cure a triangle with the virus. These rely on */ /* the assumption that all shell edges are aligned to four-byte boundaries.*/ #define infect( triedge ) \ ( triedge ).tri[6] = (triangle) \ ( (unsigned long) ( triedge ).tri[6] | (unsigned long) 2l ) #define uninfect( triedge ) \ ( triedge ).tri[6] = (triangle) \ ( (unsigned long) ( triedge ).tri[6] & ~(unsigned long) 2l ) /* Test a triangle for viral infection. */ #define infected( triedge ) \ ( ( (unsigned long) ( triedge ).tri[6] & (unsigned long) 2l ) != 0 ) /* Check or set a triangle's attributes. */ #define elemattribute( triedge, attnum ) \ ( (REAL *) ( triedge ).tri )[elemattribindex + ( attnum )] #define setelemattribute( triedge, attnum, value ) \ ( (REAL *) ( triedge ).tri )[elemattribindex + ( attnum )] = (REAL)value /* Check or set a triangle's maximum area bound. */ #define areabound( triedge ) ( (REAL *) ( triedge ).tri )[areaboundindex] #define setareabound( triedge, value ) \ ( (REAL *) ( triedge ).tri )[areaboundindex] = (REAL)value /********* Primitives for shell edges *********/ /* */ /* */ /* sdecode() converts a pointer to an oriented shell edge. The orientation */ /* is extracted from the least significant bit of the pointer. The two */ /* least significant bits (one for orientation, one for viral infection) */ /* are masked out to produce the real pointer. */ #define sdecode( sptr, edge ) \ ( edge ).shorient = (int) ( (unsigned long) ( sptr ) & (unsigned long) 1l ); \ ( edge ).sh = (shelle *) \ ( (unsigned long) ( sptr ) & ~(unsigned long) 3l ) /* sencode() compresses an oriented shell edge into a single pointer. It */ /* relies on the assumption that all shell edges are aligned to two-byte */ /* boundaries, so the least significant bit of (edge).sh is zero. */ #define sencode( edge ) \ (shelle) ( (unsigned long) ( edge ).sh | (unsigned long) ( edge ).shorient ) /* ssym() toggles the orientation of a shell edge. */ #define ssym( edge1, edge2 ) \ ( edge2 ).sh = ( edge1 ).sh; \ ( edge2 ).shorient = 1 - ( edge1 ).shorient #define ssymself( edge ) \ ( edge ).shorient = 1 - ( edge ).shorient /* spivot() finds the other shell edge (from the same segment) that shares */ /* the same origin. */ #define spivot( edge1, edge2 ) \ sptr = ( edge1 ).sh[( edge1 ).shorient]; \ sdecode( sptr, edge2 ) #define spivotself( edge ) \ sptr = ( edge ).sh[( edge ).shorient]; \ sdecode( sptr, edge ) /* snext() finds the next shell edge (from the same segment) in sequence; */ /* one whose origin is the input shell edge's destination. */ #define snext( edge1, edge2 ) \ sptr = ( edge1 ).sh[1 - ( edge1 ).shorient]; \ sdecode( sptr, edge2 ) #define snextself( edge ) \ sptr = ( edge ).sh[1 - ( edge ).shorient]; \ sdecode( sptr, edge ) /* These primitives determine or set the origin or destination of a shell */ /* edge. */ #define sorg( edge, pointptr ) \ pointptr = (point) ( edge ).sh[2 + ( edge ).shorient] #define sdest( edge, pointptr ) \ pointptr = (point) ( edge ).sh[3 - ( edge ).shorient] #define setsorg( edge, pointptr ) \ ( edge ).sh[2 + ( edge ).shorient] = (shelle) pointptr #define setsdest( edge, pointptr ) \ ( edge ).sh[3 - ( edge ).shorient] = (shelle) pointptr /* These primitives read or set a shell marker. Shell markers are used to */ /* hold user boundary information. */ #define mark( edge ) ( *(int *) ( ( edge ).sh + 6 ) ) #define setmark( edge, value ) \ *(int *) ( ( edge ).sh + 6 ) = value /* Bond two shell edges together. */ #define sbond( edge1, edge2 ) \ ( edge1 ).sh[( edge1 ).shorient] = sencode( edge2 ); \ ( edge2 ).sh[( edge2 ).shorient] = sencode( edge1 ) /* Dissolve a shell edge bond (from one side). Note that the other shell */ /* edge will still think it's connected to this shell edge. */ #define sdissolve( edge ) \ ( edge ).sh[( edge ).shorient] = (shelle) dummysh /* Copy a shell edge. */ #define shellecopy( edge1, edge2 ) \ ( edge2 ).sh = ( edge1 ).sh; \ ( edge2 ).shorient = ( edge1 ).shorient /* Test for equality of shell edges. */ #define shelleequal( edge1, edge2 ) \ ( ( ( edge1 ).sh == ( edge2 ).sh ) && \ ( ( edge1 ).shorient == ( edge2 ).shorient ) ) /********* Primitives for interacting triangles and shell edges *********/ /* */ /* */ /* tspivot() finds a shell edge abutting a triangle. */ #define tspivot( triedge, edge ) \ sptr = (shelle) ( triedge ).tri[6 + ( triedge ).orient]; \ sdecode( sptr, edge ) /* stpivot() finds a triangle abutting a shell edge. It requires that the */ /* variable `ptr' of type `triangle' be defined. */ #define stpivot( edge, triedge ) \ ptr = (triangle) ( edge ).sh[4 + ( edge ).shorient]; \ decode( ptr, triedge ) /* Bond a triangle to a shell edge. */ #define tsbond( triedge, edge ) \ ( triedge ).tri[6 + ( triedge ).orient] = (triangle) sencode( edge ); \ ( edge ).sh[4 + ( edge ).shorient] = (shelle) encode( triedge ) /* Dissolve a bond (from the triangle side). */ #define tsdissolve( triedge ) \ ( triedge ).tri[6 + ( triedge ).orient] = (triangle) dummysh /* Dissolve a bond (from the shell edge side). */ #define stdissolve( edge ) \ ( edge ).sh[4 + ( edge ).shorient] = (shelle) dummytri /********* Primitives for points *********/ /* */ /* */ #define pointmark( pt ) ( (int *) ( pt ) )[pointmarkindex] #define setpointmark( pt, value ) \ ( (int *) ( pt ) )[pointmarkindex] = value #define point2tri( pt ) ( (triangle *) ( pt ) )[point2triindex] #define setpoint2tri( pt, value ) \ ( (triangle *) ( pt ) )[point2triindex] = value /** **/ /** **/ /********* Mesh manipulation primitives end here *********/ /********* User interaction routines begin here *********/ /** **/ /** **/ /*****************************************************************************/ /* */ /* syntax() Print list of command line switches. */ /* */ /*****************************************************************************/ #ifndef TRILIBRARY void syntax(){ #ifdef CDT_ONLY #ifdef REDUCED printf( "triangle [-pAcevngBPNEIOXzo_lQVh] input_file\n" ); #else /* not REDUCED */ printf( "triangle [-pAcevngBPNEIOXzo_iFlCQVh] input_file\n" ); #endif /* not REDUCED */ #else /* not CDT_ONLY */ #ifdef REDUCED printf( "triangle [-prq__a__AcevngBPNEIOXzo_YS__lQVh] input_file\n" ); #else /* not REDUCED */ printf( "triangle [-prq__a__AcevngBPNEIOXzo_YS__iFlsCQVh] input_file\n" ); #endif /* not REDUCED */ #endif /* not CDT_ONLY */ printf( " -p Triangulates a Planar Straight Line Graph (.poly file).\n" ); #ifndef CDT_ONLY printf( " -r Refines a previously generated mesh.\n" ); printf( " -q Quality mesh generation. A minimum angle may be specified.\n" ); printf( " -a Applies a maximum triangle area constraint.\n" ); #endif /* not CDT_ONLY */ printf( " -A Applies attributes to identify elements in certain regions.\n" ); printf( " -c Encloses the convex hull with segments.\n" ); printf( " -e Generates an edge list.\n" ); printf( " -v Generates a Voronoi diagram.\n" ); printf( " -n Generates a list of triangle neighbors.\n" ); printf( " -g Generates an .off file for Geomview.\n" ); printf( " -B Suppresses output of boundary information.\n" ); printf( " -P Suppresses output of .poly file.\n" ); printf( " -N Suppresses output of .node file.\n" ); printf( " -E Suppresses output of .ele file.\n" ); printf( " -I Suppresses mesh iteration numbers.\n" ); printf( " -O Ignores holes in .poly file.\n" ); printf( " -X Suppresses use of exact arithmetic.\n" ); printf( " -z Numbers all items starting from zero (rather than one).\n" ); printf( " -o2 Generates second-order subparametric elements.\n" ); #ifndef CDT_ONLY printf( " -Y Suppresses boundary segment splitting.\n" ); printf( " -S Specifies maximum number of added Steiner points.\n" ); #endif /* not CDT_ONLY */ #ifndef REDUCED printf( " -i Uses incremental method, rather than divide-and-conquer.\n" ); printf( " -F Uses Fortune's sweepline algorithm, rather than d-and-c.\n" ); #endif /* not REDUCED */ printf( " -l Uses vertical cuts only, rather than alternating cuts.\n" ); #ifndef REDUCED #ifndef CDT_ONLY printf( " -s Force segments into mesh by splitting (instead of using CDT).\n" ); #endif /* not CDT_ONLY */ printf( " -C Check consistency of final mesh.\n" ); #endif /* not REDUCED */ printf( " -Q Quiet: No terminal output except errors.\n" ); printf( " -V Verbose: Detailed information on what I'm doing.\n" ); printf( " -h Help: Detailed instructions for Triangle.\n" ); exit( 0 ); } #endif /* not TRILIBRARY */ /*****************************************************************************/ /* */ /* info() Print out complete instructions. */ /* */ /*****************************************************************************/ #ifndef TRILIBRARY void info(){ printf( "Triangle\n" ); printf( "A Two-Dimensional Quality Mesh Generator and Delaunay Triangulator.\n" ); printf( "Version 1.3\n\n" ); printf( "Copyright 1996 Jonathan Richard Shewchuk (bugs/comments to jrs@cs.cmu.edu)\n" ); printf( "School of Computer Science / Carnegie Mellon University\n" ); printf( "5000 Forbes Avenue / Pittsburgh, Pennsylvania 15213-3891\n" ); printf( "Created as part of the Archimedes project (tools for parallel FEM).\n" ); printf( "Supported in part by NSF Grant CMS-9318163 and an NSERC 1967 Scholarship.\n" ); printf( "There is no warranty whatsoever. Use at your own risk.\n" ); #ifdef SINGLE printf( "This executable is compiled for single precision arithmetic.\n\n\n" ); #else /* not SINGLE */ printf( "This executable is compiled for double precision arithmetic.\n\n\n" ); #endif /* not SINGLE */ printf( "Triangle generates exact Delaunay triangulations, constrained Delaunay\n" ); printf( "triangulations, and quality conforming Delaunay triangulations. The latter\n" ); printf( "can be generated with no small angles, and are thus suitable for finite\n" ); printf( "element analysis. If no command line switches are specified, your .node\n" ); printf( "input file will be read, and the Delaunay triangulation will be returned in\n" ); printf( ".node and .ele output files. The command syntax is:\n\n" ); #ifdef CDT_ONLY #ifdef REDUCED printf( "triangle [-pAcevngBPNEIOXzo_lQVh] input_file\n\n" ); #else /* not REDUCED */ printf( "triangle [-pAcevngBPNEIOXzo_iFlCQVh] input_file\n\n" ); #endif /* not REDUCED */ #else /* not CDT_ONLY */ #ifdef REDUCED printf( "triangle [-prq__a__AcevngBPNEIOXzo_YS__lQVh] input_file\n\n" ); #else /* not REDUCED */ printf( "triangle [-prq__a__AcevngBPNEIOXzo_YS__iFlsCQVh] input_file\n\n" ); #endif /* not REDUCED */ #endif /* not CDT_ONLY */ printf( "Underscores indicate that numbers may optionally follow certain switches;\n" ); printf( "do not leave any space between a switch and its numeric parameter.\n" ); printf( "input_file must be a file with extension .node, or extension .poly if the\n" ); printf( "-p switch is used. If -r is used, you must supply .node and .ele files,\n" ); printf( "and possibly a .poly file and .area file as well. The formats of these\n" ); printf( "files are described below.\n\n" ); printf( "Command Line Switches:\n\n" ); printf( " -p Reads a Planar Straight Line Graph (.poly file), which can specify\n" ); printf( " points, segments, holes, and regional attributes and area\n" ); printf( " constraints. Will generate a constrained Delaunay triangulation\n" ); printf( " fitting the input; or, if -s, -q, or -a is used, a conforming\n" ); printf( " Delaunay triangulation. If -p is not used, Triangle reads a .node\n" ); printf( " file by default.\n" ); printf( " -r Refines a previously generated mesh. The mesh is read from a .node\n" ); printf( " file and an .ele file. If -p is also used, a .poly file is read\n" ); printf( " and used to constrain edges in the mesh. Further details on\n" ); printf( " refinement are given below.\n" ); printf( " -q Quality mesh generation by Jim Ruppert's Delaunay refinement\n" ); printf( " algorithm. Adds points to the mesh to ensure that no angles\n" ); printf( " smaller than 20 degrees occur. An alternative minimum angle may be\n" ); printf( " specified after the `q'. If the minimum angle is 20.7 degrees or\n" ); printf( " smaller, the triangulation algorithm is theoretically guaranteed to\n" ); printf( " terminate (assuming infinite precision arithmetic - Triangle may\n" ); printf( " fail to terminate if you run out of precision). In practice, the\n" ); printf( " algorithm often succeeds for minimum angles up to 33.8 degrees.\n" ); printf( " For highly refined meshes, however, it may be necessary to reduce\n" ); printf( " the minimum angle to well below 20 to avoid problems associated\n" ); printf( " with insufficient floating-point precision. The specified angle\n" ); printf( " may include a decimal point.\n" ); printf( " -a Imposes a maximum triangle area. If a number follows the `a', no\n" ); printf( " triangle will be generated whose area is larger than that number.\n" ); printf( " If no number is specified, an .area file (if -r is used) or .poly\n" ); printf( " file (if -r is not used) specifies a number of maximum area\n" ); printf( " constraints. An .area file contains a separate area constraint for\n" ); printf( " each triangle, and is useful for refining a finite element mesh\n" ); printf( " based on a posteriori error estimates. A .poly file can optionally\n" ); printf( " contain an area constraint for each segment-bounded region, thereby\n" ); printf( " enforcing triangle densities in a first triangulation. You can\n" ); printf( " impose both a fixed area constraint and a varying area constraint\n" ); printf( " by invoking the -a switch twice, once with and once without a\n" ); printf( " number following. Each area specified may include a decimal point.\n" ); printf( " -A Assigns an additional attribute to each triangle that identifies\n" ); printf( " what segment-bounded region each triangle belongs to. Attributes\n" ); printf( " are assigned to regions by the .poly file. If a region is not\n" ); printf( " explicitly marked by the .poly file, triangles in that region are\n" ); printf( " assigned an attribute of zero. The -A switch has an effect only\n" ); printf( " when the -p switch is used and the -r switch is not.\n" ); printf( " -c Creates segments on the convex hull of the triangulation. If you\n" ); printf( " are triangulating a point set, this switch causes a .poly file to\n" ); printf( " be written, containing all edges in the convex hull. (By default,\n" ); printf( " a .poly file is written only if a .poly file is read.) If you are\n" ); printf( " triangulating a PSLG, this switch specifies that the interior of\n" ); printf( " the convex hull of the PSLG should be triangulated. If you do not\n" ); printf( " use this switch when triangulating a PSLG, it is assumed that you\n" ); printf( " have identified the region to be triangulated by surrounding it\n" ); printf( " with segments of the input PSLG. Beware: if you are not careful,\n" ); printf( " this switch can cause the introduction of an extremely thin angle\n" ); printf( " between a PSLG segment and a convex hull segment, which can cause\n" ); printf( " overrefinement or failure if Triangle runs out of precision. If\n" ); printf( " you are refining a mesh, the -c switch works differently; it\n" ); printf( " generates the set of boundary edges of the mesh, rather than the\n" ); printf( " convex hull.\n" ); printf( " -e Outputs (to an .edge file) a list of edges of the triangulation.\n" ); printf( " -v Outputs the Voronoi diagram associated with the triangulation.\n" ); printf( " Does not attempt to detect degeneracies.\n" ); printf( " -n Outputs (to a .neigh file) a list of triangles neighboring each\n" ); printf( " triangle.\n" ); printf( " -g Outputs the mesh to an Object File Format (.off) file, suitable for\n" ); printf( " viewing with the Geometry Center's Geomview package.\n" ); printf( " -B No boundary markers in the output .node, .poly, and .edge output\n" ); printf( " files. See the detailed discussion of boundary markers below.\n" ); printf( " -P No output .poly file. Saves disk space, but you lose the ability\n" ); printf( " to impose segment constraints on later refinements of the mesh.\n" ); printf( " -N No output .node file.\n" ); printf( " -E No output .ele file.\n" ); printf( " -I No iteration numbers. Suppresses the output of .node and .poly\n" ); printf( " files, so your input files won't be overwritten. (If your input is\n" ); printf( " a .poly file only, a .node file will be written.) Cannot be used\n" ); printf( " with the -r switch, because that would overwrite your input .ele\n" ); printf( " file. Shouldn't be used with the -s, -q, or -a switch if you are\n" ); printf( " using a .node file for input, because no .node file will be\n" ); printf( " written, so there will be no record of any added points.\n" ); printf( " -O No holes. Ignores the holes in the .poly file.\n" ); printf( " -X No exact arithmetic. Normally, Triangle uses exact floating-point\n" ); printf( " arithmetic for certain tests if it thinks the inexact tests are not\n" ); printf( " accurate enough. Exact arithmetic ensures the robustness of the\n" ); printf( " triangulation algorithms, despite floating-point roundoff error.\n" ); printf( " Disabling exact arithmetic with the -X switch will cause a small\n" ); printf( " improvement in speed and create the possibility (albeit small) that\n" ); printf( " Triangle will fail to produce a valid mesh. Not recommended.\n" ); printf( " -z Numbers all items starting from zero (rather than one). Note that\n" ); printf( " this switch is normally overrided by the value used to number the\n" ); printf( " first point of the input .node or .poly file. However, this switch\n" ); printf( " is useful when calling Triangle from another program.\n" ); printf( " -o2 Generates second-order subparametric elements with six nodes each.\n" ); printf( " -Y No new points on the boundary. This switch is useful when the mesh\n" ); printf( " boundary must be preserved so that it conforms to some adjacent\n" ); printf( " mesh. Be forewarned that you will probably sacrifice some of the\n" ); printf( " quality of the mesh; Triangle will try, but the resulting mesh may\n" ); printf( " contain triangles of poor aspect ratio. Works well if all the\n" ); printf( " boundary points are closely spaced. Specify this switch twice\n" ); printf( " (`-YY') to prevent all segment splitting, including internal\n" ); printf( " boundaries.\n" ); printf( " -S Specifies the maximum number of Steiner points (points that are not\n" ); printf( " in the input, but are added to meet the constraints of minimum\n" ); printf( " angle and maximum area). The default is to allow an unlimited\n" ); printf( " number. If you specify this switch with no number after it,\n" ); printf( " the limit is set to zero. Triangle always adds points at segment\n" ); printf( " intersections, even if it needs to use more points than the limit\n" ); printf( " you set. When Triangle inserts segments by splitting (-s), it\n" ); printf( " always adds enough points to ensure that all the segments appear in\n" ); printf( " the triangulation, again ignoring the limit. Be forewarned that\n" ); printf( " the -S switch may result in a conforming triangulation that is not\n" ); printf( " truly Delaunay, because Triangle may be forced to stop adding\n" ); printf( " points when the mesh is in a state where a segment is non-Delaunay\n" ); printf( " and needs to be split. If so, Triangle will print a warning.\n" ); printf( " -i Uses an incremental rather than divide-and-conquer algorithm to\n" ); printf( " form a Delaunay triangulation. Try it if the divide-and-conquer\n" ); printf( " algorithm fails.\n" ); printf( " -F Uses Steven Fortune's sweepline algorithm to form a Delaunay\n" ); printf( " triangulation. Warning: does not use exact arithmetic for all\n" ); printf( " calculations. An exact result is not guaranteed.\n" ); printf( " -l Uses only vertical cuts in the divide-and-conquer algorithm. By\n" ); printf( " default, Triangle uses alternating vertical and horizontal cuts,\n" ); printf( " which usually improve the speed except with point sets that are\n" ); printf( " small or short and wide. This switch is primarily of theoretical\n" ); printf( " interest.\n" ); printf( " -s Specifies that segments should be forced into the triangulation by\n" ); printf( " recursively splitting them at their midpoints, rather than by\n" ); printf( " generating a constrained Delaunay triangulation. Segment splitting\n" ); printf( " is true to Ruppert's original algorithm, but can create needlessly\n" ); printf( " small triangles near external small features.\n" ); printf( " -C Check the consistency of the final mesh. Uses exact arithmetic for\n" ); printf( " checking, even if the -X switch is used. Useful if you suspect\n" ); printf( " Triangle is buggy.\n" ); printf( " -Q Quiet: Suppresses all explanation of what Triangle is doing, unless\n" ); printf( " an error occurs.\n" ); printf( " -V Verbose: Gives detailed information about what Triangle is doing.\n" ); printf( " Add more `V's for increasing amount of detail. `-V' gives\n" ); printf( " information on algorithmic progress and more detailed statistics.\n" ); printf( " `-VV' gives point-by-point details, and will print so much that\n" ); printf( " Triangle will run much more slowly. `-VVV' gives information only\n" ); printf( " a debugger could love.\n" ); printf( " -h Help: Displays these instructions.\n" ); printf( "\n" ); printf( "Definitions:\n" ); printf( "\n" ); printf( " A Delaunay triangulation of a point set is a triangulation whose vertices\n" ); printf( " are the point set, having the property that no point in the point set\n" ); printf( " falls in the interior of the circumcircle (circle that passes through all\n" ); printf( " three vertices) of any triangle in the triangulation.\n\n" ); printf( " A Voronoi diagram of a point set is a subdivision of the plane into\n" ); printf( " polygonal regions (some of which may be infinite), where each region is\n" ); printf( " the set of points in the plane that are closer to some input point than\n" ); printf( " to any other input point. (The Voronoi diagram is the geometric dual of\n" ); printf( " the Delaunay triangulation.)\n\n" ); printf( " A Planar Straight Line Graph (PSLG) is a collection of points and\n" ); printf( " segments. Segments are simply edges, whose endpoints are points in the\n" ); printf( " PSLG. The file format for PSLGs (.poly files) is described below.\n" ); printf( "\n" ); printf( " A constrained Delaunay triangulation of a PSLG is similar to a Delaunay\n" ); printf( " triangulation, but each PSLG segment is present as a single edge in the\n" ); printf( " triangulation. (A constrained Delaunay triangulation is not truly a\n" ); printf( " Delaunay triangulation.)\n\n" ); printf( " A conforming Delaunay triangulation of a PSLG is a true Delaunay\n" ); printf( " triangulation in which each PSLG segment may have been subdivided into\n" ); printf( " several edges by the insertion of additional points. These inserted\n" ); printf( " points are necessary to allow the segments to exist in the mesh while\n" ); printf( " maintaining the Delaunay property.\n\n" ); printf( "File Formats:\n\n" ); printf( " All files may contain comments prefixed by the character '#'. Points,\n" ); printf( " triangles, edges, holes, and maximum area constraints must be numbered\n" ); printf( " consecutively, starting from either 1 or 0. Whichever you choose, all\n" ); printf( " input files must be consistent; if the nodes are numbered from 1, so must\n" ); printf( " be all other objects. Triangle automatically detects your choice while\n" ); printf( " reading the .node (or .poly) file. (When calling Triangle from another\n" ); printf( " program, use the -z switch if you wish to number objects from zero.)\n" ); printf( " Examples of these file formats are given below.\n\n" ); printf( " .node files:\n" ); printf( " First line: <# of points> <# of attributes>\n" ); printf( " <# of boundary markers (0 or 1)>\n" ); printf( " Remaining lines: [attributes] [boundary marker]\n" ); printf( "\n" ); printf( " The attributes, which are typically floating-point values of physical\n" ); printf( " quantities (such as mass or conductivity) associated with the nodes of\n" ); printf( " a finite element mesh, are copied unchanged to the output mesh. If -s,\n" ); printf( " -q, or -a is selected, each new Steiner point added to the mesh will\n" ); printf( " have attributes assigned to it by linear interpolation.\n\n" ); printf( " If the fourth entry of the first line is `1', the last column of the\n" ); printf( " remainder of the file is assumed to contain boundary markers. Boundary\n" ); printf( " markers are used to identify boundary points and points resting on PSLG\n" ); printf( " segments; a complete description appears in a section below. The .node\n" ); printf( " file produced by Triangle will contain boundary markers in the last\n" ); printf( " column unless they are suppressed by the -B switch.\n\n" ); printf( " .ele files:\n" ); printf( " First line: <# of triangles> <# of attributes>\n" ); printf( " Remaining lines: ... [attributes]\n" ); printf( "\n" ); printf( " Points are indices into the corresponding .node file. The first three\n" ); printf( " points are the corners, and are listed in counterclockwise order around\n" ); printf( " each triangle. (The remaining points, if any, depend on the type of\n" ); printf( " finite element used.) The attributes are just like those of .node\n" ); printf( " files. Because there is no simple mapping from input to output\n" ); printf( " triangles, an attempt is made to interpolate attributes, which may\n" ); printf( " result in a good deal of diffusion of attributes among nearby triangles\n" ); printf( " as the triangulation is refined. Diffusion does not occur across\n" ); printf( " segments, so attributes used to identify segment-bounded regions remain\n" ); printf( " intact. In output .ele files, all triangles have three points each\n" ); printf( " unless the -o2 switch is used, in which case they have six, and the\n" ); printf( " fourth, fifth, and sixth points lie on the midpoints of the edges\n" ); printf( " opposite the first, second, and third corners.\n\n" ); printf( " .poly files:\n" ); printf( " First line: <# of points> <# of attributes>\n" ); printf( " <# of boundary markers (0 or 1)>\n" ); printf( " Following lines: [attributes] [boundary marker]\n" ); printf( " One line: <# of segments> <# of boundary markers (0 or 1)>\n" ); printf( " Following lines: [boundary marker]\n" ); printf( " One line: <# of holes>\n" ); printf( " Following lines: \n" ); printf( " Optional line: <# of regional attributes and/or area constraints>\n" ); printf( " Optional following lines: \n" ); printf( "\n" ); printf( " A .poly file represents a PSLG, as well as some additional information.\n" ); printf( " The first section lists all the points, and is identical to the format\n" ); printf( " of .node files. <# of points> may be set to zero to indicate that the\n" ); printf( " points are listed in a separate .node file; .poly files produced by\n" ); printf( " Triangle always have this format. This has the advantage that a point\n" ); printf( " set may easily be triangulated with or without segments. (The same\n" ); printf( " effect can be achieved, albeit using more disk space, by making a copy\n" ); printf( " of the .poly file with the extension .node; all sections of the file\n" ); printf( " but the first are ignored.)\n\n" ); printf( " The second section lists the segments. Segments are edges whose\n" ); printf( " presence in the triangulation is enforced. Each segment is specified\n" ); printf( " by listing the indices of its two endpoints. This means that you must\n" ); printf( " include its endpoints in the point list. If -s, -q, and -a are not\n" ); printf( " selected, Triangle will produce a constrained Delaunay triangulation,\n" ); printf( " in which each segment appears as a single edge in the triangulation.\n" ); printf( " If -q or -a is selected, Triangle will produce a conforming Delaunay\n" ); printf( " triangulation, in which segments may be subdivided into smaller edges.\n" ); printf( " Each segment, like each point, may have a boundary marker.\n\n" ); printf( " The third section lists holes (and concavities, if -c is selected) in\n" ); printf( " the triangulation. Holes are specified by identifying a point inside\n" ); printf( " each hole. After the triangulation is formed, Triangle creates holes\n" ); printf( " by eating triangles, spreading out from each hole point until its\n" ); printf( " progress is blocked by PSLG segments; you must be careful to enclose\n" ); printf( " each hole in segments, or your whole triangulation may be eaten away.\n" ); printf( " If the two triangles abutting a segment are eaten, the segment itself\n" ); printf( " is also eaten. Do not place a hole directly on a segment; if you do,\n" ); printf( " Triangle will choose one side of the segment arbitrarily.\n\n" ); printf( " The optional fourth section lists regional attributes (to be assigned\n" ); printf( " to all triangles in a region) and regional constraints on the maximum\n" ); printf( " triangle area. Triangle will read this section only if the -A switch\n" ); printf( " is used or the -a switch is used without a number following it, and the\n" ); printf( " -r switch is not used. Regional attributes and area constraints are\n" ); printf( " propagated in the same manner as holes; you specify a point for each\n" ); printf( " attribute and/or constraint, and the attribute and/or constraint will\n" ); printf( " affect the whole region (bounded by segments) containing the point. If\n" ); printf( " two values are written on a line after the x and y coordinate, the\n" ); printf( " former is assumed to be a regional attribute (but will only be applied\n" ); printf( " if the -A switch is selected), and the latter is assumed to be a\n" ); printf( " regional area constraint (but will only be applied if the -a switch is\n" ); printf( " selected). You may also specify just one value after the coordinates,\n" ); printf( " which can serve as both an attribute and an area constraint, depending\n" ); printf( " on the choice of switches. If you are using the -A and -a switches\n" ); printf( " simultaneously and wish to assign an attribute to some region without\n" ); printf( " imposing an area constraint, use a negative maximum area.\n\n" ); printf( " When a triangulation is created from a .poly file, you must either\n" ); printf( " enclose the entire region to be triangulated in PSLG segments, or\n" ); printf( " use the -c switch, which encloses the convex hull of the input point\n" ); printf( " set. If you do not use the -c switch, Triangle will eat all triangles\n" ); printf( " on the outer boundary that are not protected by segments; if you are\n" ); printf( " not careful, your whole triangulation may be eaten away. If you do\n" ); printf( " use the -c switch, you can still produce concavities by appropriate\n" ); printf( " placement of holes just inside the convex hull.\n\n" ); printf( " An ideal PSLG has no intersecting segments, nor any points that lie\n" ); printf( " upon segments (except, of course, the endpoints of each segment.) You\n" ); printf( " aren't required to make your .poly files ideal, but you should be aware\n" ); printf( " of what can go wrong. Segment intersections are relatively safe -\n" ); printf( " Triangle will calculate the intersection points for you and add them to\n" ); printf( " the triangulation - as long as your machine's floating-point precision\n" ); printf( " doesn't become a problem. You are tempting the fates if you have three\n" ); printf( " segments that cross at the same location, and expect Triangle to figure\n" ); printf( " out where the intersection point is. Thanks to floating-point roundoff\n" ); printf( " error, Triangle will probably decide that the three segments intersect\n" ); printf( " at three different points, and you will find a minuscule triangle in\n" ); printf( " your output - unless Triangle tries to refine the tiny triangle, uses\n" ); printf( " up the last bit of machine precision, and fails to terminate at all.\n" ); printf( " You're better off putting the intersection point in the input files,\n" ); printf( " and manually breaking up each segment into two. Similarly, if you\n" ); printf( " place a point at the middle of a segment, and hope that Triangle will\n" ); printf( " break up the segment at that point, you might get lucky. On the other\n" ); printf( " hand, Triangle might decide that the point doesn't lie precisely on the\n" ); printf( " line, and you'll have a needle-sharp triangle in your output - or a lot\n" ); printf( " of tiny triangles if you're generating a quality mesh.\n\n" ); printf( " When Triangle reads a .poly file, it also writes a .poly file, which\n" ); printf( " includes all edges that are part of input segments. If the -c switch\n" ); printf( " is used, the output .poly file will also include all of the edges on\n" ); printf( " the convex hull. Hence, the output .poly file is useful for finding\n" ); printf( " edges associated with input segments and setting boundary conditions in\n" ); printf( " finite element simulations. More importantly, you will need it if you\n" ); printf( " plan to refine the output mesh, and don't want segments to be missing\n" ); printf( " in later triangulations.\n\n" ); printf( " .area files:\n" ); printf( " First line: <# of triangles>\n" ); printf( " Following lines: \n\n" ); printf( " An .area file associates with each triangle a maximum area that is used\n" ); printf( " for mesh refinement. As with other file formats, every triangle must\n" ); printf( " be represented, and they must be numbered consecutively. A triangle\n" ); printf( " may be left unconstrained by assigning it a negative maximum area.\n" ); printf( "\n" ); printf( " .edge files:\n" ); printf( " First line: <# of edges> <# of boundary markers (0 or 1)>\n" ); printf( " Following lines: [boundary marker]\n" ); printf( "\n" ); printf( " Endpoints are indices into the corresponding .node file. Triangle can\n" ); printf( " produce .edge files (use the -e switch), but cannot read them. The\n" ); printf( " optional column of boundary markers is suppressed by the -B switch.\n" ); printf( "\n" ); printf( " In Voronoi diagrams, one also finds a special kind of edge that is an\n" ); printf( " infinite ray with only one endpoint. For these edges, a different\n" ); printf( " format is used:\n\n" ); printf( " -1 \n\n" ); printf( " The `direction' is a floating-point vector that indicates the direction\n" ); printf( " of the infinite ray.\n\n" ); printf( " .neigh files:\n" ); printf( " First line: <# of triangles> <# of neighbors per triangle (always 3)>\n" ); printf( " Following lines: \n" ); printf( "\n" ); printf( " Neighbors are indices into the corresponding .ele file. An index of -1\n" ); printf( " indicates a mesh boundary, and therefore no neighbor. Triangle can\n" ); printf( " produce .neigh files (use the -n switch), but cannot read them.\n" ); printf( "\n" ); printf( " The first neighbor of triangle i is opposite the first corner of\n" ); printf( " triangle i, and so on.\n\n" ); printf( "Boundary Markers:\n\n" ); printf( " Boundary markers are tags used mainly to identify which output points and\n" ); printf( " edges are associated with which PSLG segment, and to identify which\n" ); printf( " points and edges occur on a boundary of the triangulation. A common use\n" ); printf( " is to determine where boundary conditions should be applied to a finite\n" ); printf( " element mesh. You can prevent boundary markers from being written into\n" ); printf( " files produced by Triangle by using the -B switch.\n\n" ); printf( " The boundary marker associated with each segment in an output .poly file\n" ); printf( " or edge in an output .edge file is chosen as follows:\n" ); printf( " - If an output edge is part or all of a PSLG segment with a nonzero\n" ); printf( " boundary marker, then the edge is assigned the same marker.\n" ); printf( " - Otherwise, if the edge occurs on a boundary of the triangulation\n" ); printf( " (including boundaries of holes), then the edge is assigned the marker\n" ); printf( " one (1).\n" ); printf( " - Otherwise, the edge is assigned the marker zero (0).\n" ); printf( " The boundary marker associated with each point in an output .node file is\n" ); printf( " chosen as follows:\n" ); printf( " - If a point is assigned a nonzero boundary marker in the input file,\n" ); printf( " then it is assigned the same marker in the output .node file.\n" ); printf( " - Otherwise, if the point lies on a PSLG segment (including the\n" ); printf( " segment's endpoints) with a nonzero boundary marker, then the point\n" ); printf( " is assigned the same marker. If the point lies on several such\n" ); printf( " segments, one of the markers is chosen arbitrarily.\n" ); printf( " - Otherwise, if the point occurs on a boundary of the triangulation,\n" ); printf( " then the point is assigned the marker one (1).\n" ); printf( " - Otherwise, the point is assigned the marker zero (0).\n" ); printf( "\n" ); printf( " If you want Triangle to determine for you which points and edges are on\n" ); printf( " the boundary, assign them the boundary marker zero (or use no markers at\n" ); printf( " all) in your input files. Alternatively, you can mark some of them and\n" ); printf( " leave others marked zero, allowing Triangle to label them.\n\n" ); printf( "Triangulation Iteration Numbers:\n\n" ); printf( " Because Triangle can read and refine its own triangulations, input\n" ); printf( " and output files have iteration numbers. For instance, Triangle might\n" ); printf( " read the files mesh.3.node, mesh.3.ele, and mesh.3.poly, refine the\n" ); printf( " triangulation, and output the files mesh.4.node, mesh.4.ele, and\n" ); printf( " mesh.4.poly. Files with no iteration number are treated as if\n" ); printf( " their iteration number is zero; hence, Triangle might read the file\n" ); printf( " points.node, triangulate it, and produce the files points.1.node and\n" ); printf( " points.1.ele.\n\n" ); printf( " Iteration numbers allow you to create a sequence of successively finer\n" ); printf( " meshes suitable for multigrid methods. They also allow you to produce a\n" ); printf( " sequence of meshes using error estimate-driven mesh refinement.\n" ); printf( "\n" ); printf( " If you're not using refinement or quality meshing, and you don't like\n" ); printf( " iteration numbers, use the -I switch to disable them. This switch will\n" ); printf( " also disable output of .node and .poly files to prevent your input files\n" ); printf( " from being overwritten. (If the input is a .poly file that contains its\n" ); printf( " own points, a .node file will be written.)\n\n" ); printf( "Examples of How to Use Triangle:\n\n" ); printf( " `triangle dots' will read points from dots.node, and write their Delaunay\n" ); printf( " triangulation to dots.1.node and dots.1.ele. (dots.1.node will be\n" ); printf( " identical to dots.node.) `triangle -I dots' writes the triangulation to\n" ); printf( " dots.ele instead. (No additional .node file is needed, so none is\n" ); printf( " written.)\n\n" ); printf( " `triangle -pe object.1' will read a PSLG from object.1.poly (and possibly\n" ); printf( " object.1.node, if the points are omitted from object.1.poly) and write\n" ); printf( " their constrained Delaunay triangulation to object.2.node and\n" ); printf( " object.2.ele. The segments will be copied to object.2.poly, and all\n" ); printf( " edges will be written to object.2.edge.\n\n" ); printf( " `triangle -pq31.5a.1 object' will read a PSLG from object.poly (and\n" ); printf( " possibly object.node), generate a mesh whose angles are all greater than\n" ); printf( " 31.5 degrees and whose triangles all have area smaller than 0.1, and\n" ); printf( " write the mesh to object.1.node and object.1.ele. Each segment may have\n" ); printf( " been broken up into multiple edges; the resulting constrained edges are\n" ); printf( " written to object.1.poly.\n\n" ); printf( " Here is a sample file `box.poly' describing a square with a square hole:\n" ); printf( "\n" ); printf( " # A box with eight points in 2D, no attributes, one boundary marker.\n" ); printf( " 8 2 0 1\n" ); printf( " # Outer box has these vertices:\n" ); printf( " 1 0 0 0\n" ); printf( " 2 0 3 0\n" ); printf( " 3 3 0 0\n" ); printf( " 4 3 3 33 # A special marker for this point.\n" ); printf( " # Inner square has these vertices:\n" ); printf( " 5 1 1 0\n" ); printf( " 6 1 2 0\n" ); printf( " 7 2 1 0\n" ); printf( " 8 2 2 0\n" ); printf( " # Five segments with boundary markers.\n" ); printf( " 5 1\n" ); printf( " 1 1 2 5 # Left side of outer box.\n" ); printf( " 2 5 7 0 # Segments 2 through 5 enclose the hole.\n" ); printf( " 3 7 8 0\n" ); printf( " 4 8 6 10\n" ); printf( " 5 6 5 0\n" ); printf( " # One hole in the middle of the inner square.\n" ); printf( " 1\n" ); printf( " 1 1.5 1.5\n\n" ); printf( " Note that some segments are missing from the outer square, so one must\n" ); printf( " use the `-c' switch. After `triangle -pqc box.poly', here is the output\n" ); printf( " file `box.1.node', with twelve points. The last four points were added\n" ); printf( " to meet the angle constraint. Points 1, 2, and 9 have markers from\n" ); printf( " segment 1. Points 6 and 8 have markers from segment 4. All the other\n" ); printf( " points but 4 have been marked to indicate that they lie on a boundary.\n" ); printf( "\n" ); printf( " 12 2 0 1\n" ); printf( " 1 0 0 5\n" ); printf( " 2 0 3 5\n" ); printf( " 3 3 0 1\n" ); printf( " 4 3 3 33\n" ); printf( " 5 1 1 1\n" ); printf( " 6 1 2 10\n" ); printf( " 7 2 1 1\n" ); printf( " 8 2 2 10\n" ); printf( " 9 0 1.5 5\n" ); printf( " 10 1.5 0 1\n" ); printf( " 11 3 1.5 1\n" ); printf( " 12 1.5 3 1\n" ); printf( " # Generated by triangle -pqc box.poly\n\n" ); printf( " Here is the output file `box.1.ele', with twelve triangles.\n\n" ); printf( " 12 3 0\n" ); printf( " 1 5 6 9\n" ); printf( " 2 10 3 7\n" ); printf( " 3 6 8 12\n" ); printf( " 4 9 1 5\n" ); printf( " 5 6 2 9\n" ); printf( " 6 7 3 11\n" ); printf( " 7 11 4 8\n" ); printf( " 8 7 5 10\n" ); printf( " 9 12 2 6\n" ); printf( " 10 8 7 11\n" ); printf( " 11 5 1 10\n" ); printf( " 12 8 4 12\n" ); printf( " # Generated by triangle -pqc box.poly\n\n" ); printf( " Here is the output file `box.1.poly'. Note that segments have been added\n" ); printf( " to represent the convex hull, and some segments have been split by newly\n" ); printf( " added points. Note also that <# of points> is set to zero to indicate\n" ); printf( " that the points should be read from the .node file.\n\n" ); printf( " 0 2 0 1\n" ); printf( " 12 1\n" ); printf( " 1 1 9 5\n" ); printf( " 2 5 7 1\n" ); printf( " 3 8 7 1\n" ); printf( " 4 6 8 10\n" ); printf( " 5 5 6 1\n" ); printf( " 6 3 10 1\n" ); printf( " 7 4 11 1\n" ); printf( " 8 2 12 1\n" ); printf( " 9 9 2 5\n" ); printf( " 10 10 1 1\n" ); printf( " 11 11 3 1\n" ); printf( " 12 12 4 1\n" ); printf( " 1\n" ); printf( " 1 1.5 1.5\n" ); printf( " # Generated by triangle -pqc box.poly\n\n" ); printf( "Refinement and Area Constraints:\n\n" ); printf( " The -r switch causes a mesh (.node and .ele files) to be read and\n" ); printf( " refined. If the -p switch is also used, a .poly file is read and used to\n" ); printf( " specify edges that are constrained and cannot be eliminated (although\n" ); printf( " they can be divided into smaller edges) by the refinement process.\n" ); printf( "\n" ); printf( " When you refine a mesh, you generally want to impose tighter quality\n" ); printf( " constraints. One way to accomplish this is to use -q with a larger\n" ); printf( " angle, or -a followed by a smaller area than you used to generate the\n" ); printf( " mesh you are refining. Another way to do this is to create an .area\n" ); printf( " file, which specifies a maximum area for each triangle, and use the -a\n" ); printf( " switch (without a number following). Each triangle's area constraint is\n" ); printf( " applied to that triangle. Area constraints tend to diffuse as the mesh\n" ); printf( " is refined, so if there are large variations in area constraint between\n" ); printf( " adjacent triangles, you may not get the results you want.\n\n" ); printf( " If you are refining a mesh composed of linear (three-node) elements, the\n" ); printf( " output mesh will contain all the nodes present in the input mesh, in the\n" ); printf( " same order, with new nodes added at the end of the .node file. However,\n" ); printf( " there is no guarantee that each output element is contained in a single\n" ); printf( " input element. Often, output elements will overlap two input elements,\n" ); printf( " and input edges are not present in the output mesh. Hence, a sequence of\n" ); printf( " refined meshes will form a hierarchy of nodes, but not a hierarchy of\n" ); printf( " elements. If you a refining a mesh of higher-order elements, the\n" ); printf( " hierarchical property applies only to the nodes at the corners of an\n" ); printf( " element; other nodes may not be present in the refined mesh.\n\n" ); printf( " It is important to understand that maximum area constraints in .poly\n" ); printf( " files are handled differently from those in .area files. A maximum area\n" ); printf( " in a .poly file applies to the whole (segment-bounded) region in which a\n" ); printf( " point falls, whereas a maximum area in an .area file applies to only one\n" ); printf( " triangle. Area constraints in .poly files are used only when a mesh is\n" ); printf( " first generated, whereas area constraints in .area files are used only to\n" ); printf( " refine an existing mesh, and are typically based on a posteriori error\n" ); printf( " estimates resulting from a finite element simulation on that mesh.\n" ); printf( "\n" ); printf( " `triangle -rq25 object.1' will read object.1.node and object.1.ele, then\n" ); printf( " refine the triangulation to enforce a 25 degree minimum angle, and then\n" ); printf( " write the refined triangulation to object.2.node and object.2.ele.\n" ); printf( "\n" ); printf( " `triangle -rpaa6.2 z.3' will read z.3.node, z.3.ele, z.3.poly, and\n" ); printf( " z.3.area. After reconstructing the mesh and its segments, Triangle will\n" ); printf( " refine the mesh so that no triangle has area greater than 6.2, and\n" ); printf( " furthermore the triangles satisfy the maximum area constraints in\n" ); printf( " z.3.area. The output is written to z.4.node, z.4.ele, and z.4.poly.\n" ); printf( "\n" ); printf( " The sequence `triangle -qa1 x', `triangle -rqa.3 x.1', `triangle -rqa.1\n" ); printf( " x.2' creates a sequence of successively finer meshes x.1, x.2, and x.3,\n" ); printf( " suitable for multigrid.\n\n" ); printf( "Convex Hulls and Mesh Boundaries:\n\n" ); printf( " If the input is a point set (rather than a PSLG), Triangle produces its\n" ); printf( " convex hull as a by-product in the output .poly file if you use the -c\n" ); printf( " switch. There are faster algorithms for finding a two-dimensional convex\n" ); printf( " hull than triangulation, of course, but this one comes for free. If the\n" ); printf( " input is an unconstrained mesh (you are using the -r switch but not the\n" ); printf( " -p switch), Triangle produces a list of its boundary edges (including\n" ); printf( " hole boundaries) as a by-product if you use the -c switch.\n\n" ); printf( "Voronoi Diagrams:\n\n" ); printf( " The -v switch produces a Voronoi diagram, in files suffixed .v.node and\n" ); printf( " .v.edge. For example, `triangle -v points' will read points.node,\n" ); printf( " produce its Delaunay triangulation in points.1.node and points.1.ele,\n" ); printf( " and produce its Voronoi diagram in points.1.v.node and points.1.v.edge.\n" ); printf( " The .v.node file contains a list of all Voronoi vertices, and the .v.edge\n" ); printf( " file contains a list of all Voronoi edges, some of which may be infinite\n" ); printf( " rays. (The choice of filenames makes it easy to run the set of Voronoi\n" ); printf( " vertices through Triangle, if so desired.)\n\n" ); printf( " This implementation does not use exact arithmetic to compute the Voronoi\n" ); printf( " vertices, and does not check whether neighboring vertices are identical.\n" ); printf( " Be forewarned that if the Delaunay triangulation is degenerate or\n" ); printf( " near-degenerate, the Voronoi diagram may have duplicate points, crossing\n" ); printf( " edges, or infinite rays whose direction vector is zero. Also, if you\n" ); printf( " generate a constrained (as opposed to conforming) Delaunay triangulation,\n" ); printf( " or if the triangulation has holes, the corresponding Voronoi diagram is\n" ); printf( " likely to have crossing edges and unlikely to make sense.\n\n" ); printf( "Mesh Topology:\n\n" ); printf( " You may wish to know which triangles are adjacent to a certain Delaunay\n" ); printf( " edge in an .edge file, which Voronoi regions are adjacent to a certain\n" ); printf( " Voronoi edge in a .v.edge file, or which Voronoi regions are adjacent to\n" ); printf( " each other. All of this information can be found by cross-referencing\n" ); printf( " output files with the recollection that the Delaunay triangulation and\n" ); printf( " the Voronoi diagrams are planar duals.\n\n" ); printf( " Specifically, edge i of an .edge file is the dual of Voronoi edge i of\n" ); printf( " the corresponding .v.edge file, and is rotated 90 degrees counterclock-\n" ); printf( " wise from the Voronoi edge. Triangle j of an .ele file is the dual of\n" ); printf( " vertex j of the corresponding .v.node file; and Voronoi region k is the\n" ); printf( " dual of point k of the corresponding .node file.\n\n" ); printf( " Hence, to find the triangles adjacent to a Delaunay edge, look at the\n" ); printf( " vertices of the corresponding Voronoi edge; their dual triangles are on\n" ); printf( " the left and right of the Delaunay edge, respectively. To find the\n" ); printf( " Voronoi regions adjacent to a Voronoi edge, look at the endpoints of the\n" ); printf( " corresponding Delaunay edge; their dual regions are on the right and left\n" ); printf( " of the Voronoi edge, respectively. To find which Voronoi regions are\n" ); printf( " adjacent to each other, just read the list of Delaunay edges.\n" ); printf( "\n" ); printf( "Statistics:\n" ); printf( "\n" ); printf( " After generating a mesh, Triangle prints a count of the number of points,\n" ); printf( " triangles, edges, boundary edges, and segments in the output mesh. If\n" ); printf( " you've forgotten the statistics for an existing mesh, the -rNEP switches\n" ); printf( " (or -rpNEP if you've got a .poly file for the existing mesh) will\n" ); printf( " regenerate these statistics without writing any output.\n\n" ); printf( " The -V switch produces extended statistics, including a rough estimate\n" ); printf( " of memory use and a histogram of triangle aspect ratios and angles in the\n" ); printf( " mesh.\n\n" ); printf( "Exact Arithmetic:\n\n" ); printf( " Triangle uses adaptive exact arithmetic to perform what computational\n" ); printf( " geometers call the `orientation' and `incircle' tests. If the floating-\n" ); printf( " point arithmetic of your machine conforms to the IEEE 754 standard (as\n" ); printf( " most workstations do), and does not use extended precision internal\n" ); printf( " registers, then your output is guaranteed to be an absolutely true\n" ); printf( " Delaunay or conforming Delaunay triangulation, roundoff error\n" ); printf( " notwithstanding. The word `adaptive' implies that these arithmetic\n" ); printf( " routines compute the result only to the precision necessary to guarantee\n" ); printf( " correctness, so they are usually nearly as fast as their approximate\n" ); printf( " counterparts. The exact tests can be disabled with the -X switch. On\n" ); printf( " most inputs, this switch will reduce the computation time by about eight\n" ); printf( " percent - it's not worth the risk. There are rare difficult inputs\n" ); printf( " (having many collinear and cocircular points), however, for which the\n" ); printf( " difference could be a factor of two. These are precisely the inputs most\n" ); printf( " likely to cause errors if you use the -X switch.\n\n" ); printf( " Unfortunately, these routines don't solve every numerical problem. Exact\n" ); printf( " arithmetic is not used to compute the positions of points, because the\n" ); printf( " bit complexity of point coordinates would grow without bound. Hence,\n" ); printf( " segment intersections aren't computed exactly; in very unusual cases,\n" ); printf( " roundoff error in computing an intersection point might actually lead to\n" ); printf( " an inverted triangle and an invalid triangulation. (This is one reason\n" ); printf( " to compute your own intersection points in your .poly files.) Similarly,\n" ); printf( " exact arithmetic is not used to compute the vertices of the Voronoi\n" ); printf( " diagram.\n\n" ); printf( " Underflow and overflow can also cause difficulties; the exact arithmetic\n" ); printf( " routines do not ameliorate out-of-bounds exponents, which can arise\n" ); printf( " during the orientation and incircle tests. As a rule of thumb, you\n" ); printf( " should ensure that your input values are within a range such that their\n" ); printf( " third powers can be taken without underflow or overflow. Underflow can\n" ); printf( " silently prevent the tests from being performed exactly, while overflow\n" ); printf( " will typically cause a floating exception.\n\n" ); printf( "Calling Triangle from Another Program:\n\n" ); printf( " Read the file triangle.h for details.\n\n" ); printf( "Troubleshooting:\n\n" ); printf( " Please read this section before mailing me bugs.\n\n" ); printf( " `My output mesh has no triangles!'\n\n" ); printf( " If you're using a PSLG, you've probably failed to specify a proper set\n" ); printf( " of bounding segments, or forgotten to use the -c switch. Or you may\n" ); printf( " have placed a hole badly. To test these possibilities, try again with\n" ); printf( " the -c and -O switches. Alternatively, all your input points may be\n" ); printf( " collinear, in which case you can hardly expect to triangulate them.\n" ); printf( "\n" ); printf( " `Triangle doesn't terminate, or just crashes.'\n" ); printf( "\n" ); printf( " Bad things can happen when triangles get so small that the distance\n" ); printf( " between their vertices isn't much larger than the precision of your\n" ); printf( " machine's arithmetic. If you've compiled Triangle for single-precision\n" ); printf( " arithmetic, you might do better by recompiling it for double-precision.\n" ); printf( " Then again, you might just have to settle for more lenient constraints\n" ); printf( " on the minimum angle and the maximum area than you had planned.\n" ); printf( "\n" ); printf( " You can minimize precision problems by ensuring that the origin lies\n" ); printf( " inside your point set, or even inside the densest part of your\n" ); printf( " mesh. On the other hand, if you're triangulating an object whose x\n" ); printf( " coordinates all fall between 6247133 and 6247134, you're not leaving\n" ); printf( " much floating-point precision for Triangle to work with.\n\n" ); printf( " Precision problems can occur covertly if the input PSLG contains two\n" ); printf( " segments that meet (or intersect) at a very small angle, or if such an\n" ); printf( " angle is introduced by the -c switch, which may occur if a point lies\n" ); printf( " ever-so-slightly inside the convex hull, and is connected by a PSLG\n" ); printf( " segment to a point on the convex hull. If you don't realize that a\n" ); printf( " small angle is being formed, you might never discover why Triangle is\n" ); printf( " crashing. To check for this possibility, use the -S switch (with an\n" ); printf( " appropriate limit on the number of Steiner points, found by trial-and-\n" ); printf( " error) to stop Triangle early, and view the output .poly file with\n" ); printf( " Show Me (described below). Look carefully for small angles between\n" ); printf( " segments; zoom in closely, as such segments might look like a single\n" ); printf( " segment from a distance.\n\n" ); printf( " If some of the input values are too large, Triangle may suffer a\n" ); printf( " floating exception due to overflow when attempting to perform an\n" ); printf( " orientation or incircle test. (Read the section on exact arithmetic\n" ); printf( " above.) Again, I recommend compiling Triangle for double (rather\n" ); printf( " than single) precision arithmetic.\n\n" ); printf( " `The numbering of the output points doesn't match the input points.'\n" ); printf( "\n" ); printf( " You may have eaten some of your input points with a hole, or by placing\n" ); printf( " them outside the area enclosed by segments.\n\n" ); printf( " `Triangle executes without incident, but when I look at the resulting\n" ); printf( " mesh, it has overlapping triangles or other geometric inconsistencies.'\n" ); printf( "\n" ); printf( " If you select the -X switch, Triangle's divide-and-conquer Delaunay\n" ); printf( " triangulation algorithm occasionally makes mistakes due to floating-\n" ); printf( " point roundoff error. Although these errors are rare, don't use the -X\n" ); printf( " switch. If you still have problems, please report the bug.\n" ); printf( "\n" ); printf( " Strange things can happen if you've taken liberties with your PSLG. Do\n" ); printf( " you have a point lying in the middle of a segment? Triangle sometimes\n" ); printf( " copes poorly with that sort of thing. Do you want to lay out a collinear\n" ); printf( " row of evenly spaced, segment-connected points? Have you simply defined\n" ); printf( " one long segment connecting the leftmost point to the rightmost point,\n" ); printf( " and a bunch of points lying along it? This method occasionally works,\n" ); printf( " especially with horizontal and vertical lines, but often it doesn't, and\n" ); printf( " you'll have to connect each adjacent pair of points with a separate\n" ); printf( " segment. If you don't like it, tough.\n\n" ); printf( " Furthermore, if you have segments that intersect other than at their\n" ); printf( " endpoints, try not to let the intersections fall extremely close to PSLG\n" ); printf( " points or each other.\n\n" ); printf( " If you have problems refining a triangulation not produced by Triangle:\n" ); printf( " Are you sure the triangulation is geometrically valid? Is it formatted\n" ); printf( " correctly for Triangle? Are the triangles all listed so the first three\n" ); printf( " points are their corners in counterclockwise order?\n\n" ); printf( "Show Me:\n\n" ); printf( " Triangle comes with a separate program named `Show Me', whose primary\n" ); printf( " purpose is to draw meshes on your screen or in PostScript. Its secondary\n" ); printf( " purpose is to check the validity of your input files, and do so more\n" ); printf( " thoroughly than Triangle does. Show Me requires that you have the X\n" ); printf( " Windows system. If you didn't receive Show Me with Triangle, complain to\n" ); printf( " whomever you obtained Triangle from, then send me mail.\n\n" ); printf( "Triangle on the Web:\n\n" ); printf( " To see an illustrated, updated version of these instructions, check out\n" ); printf( "\n" ); printf( " http://www.cs.cmu.edu/~quake/triangle.html\n" ); printf( "\n" ); printf( "A Brief Plea:\n" ); printf( "\n" ); printf( " If you use Triangle, and especially if you use it to accomplish real\n" ); printf( " work, I would like very much to hear from you. A short letter or email\n" ); printf( " (to jrs@cs.cmu.edu) describing how you use Triangle will mean a lot to\n" ); printf( " me. The more people I know are using this program, the more easily I can\n" ); printf( " justify spending time on improvements and on the three-dimensional\n" ); printf( " successor to Triangle, which in turn will benefit you. Also, I can put\n" ); printf( " you on a list to receive email whenever a new version of Triangle is\n" ); printf( " available.\n\n" ); printf( " If you use a mesh generated by Triangle in a publication, please include\n" ); printf( " an acknowledgment as well.\n\n" ); printf( "Research credit:\n\n" ); printf( " Of course, I can take credit for only a fraction of the ideas that made\n" ); printf( " this mesh generator possible. Triangle owes its existence to the efforts\n" ); printf( " of many fine computational geometers and other researchers, including\n" ); printf( " Marshall Bern, L. Paul Chew, Boris Delaunay, Rex A. Dwyer, David\n" ); printf( " Eppstein, Steven Fortune, Leonidas J. Guibas, Donald E. Knuth, C. L.\n" ); printf( " Lawson, Der-Tsai Lee, Ernst P. Mucke, Douglas M. Priest, Jim Ruppert,\n" ); printf( " Isaac Saias, Bruce J. Schachter, Micha Sharir, Jorge Stolfi, Christopher\n" ); printf( " J. Van Wyk, David F. Watson, and Binhai Zhu. See the comments at the\n" ); printf( " beginning of the source code for references.\n\n" ); exit( 0 ); } #endif /* not TRILIBRARY */ /*****************************************************************************/ /* */ /* internalerror() Ask the user to send me the defective product. Exit. */ /* */ /*****************************************************************************/ void internalerror(){ printf( " Please report this bug to jrs@cs.cmu.edu\n" ); printf( " Include the message above, your input data set, and the exact\n" ); printf( " command line you used to run Triangle.\n" ); exit( 1 ); } /*****************************************************************************/ /* */ /* parsecommandline() Read the command line, identify switches, and set */ /* up options and file names. */ /* */ /* The effects of this routine are felt entirely through global variables. */ /* */ /*****************************************************************************/ void parsecommandline( argc, argv ) int argc; char **argv; { #ifdef TRILIBRARY #define STARTINDEX 0 #else /* not TRILIBRARY */ #define STARTINDEX 1 int increment; int meshnumber; #endif /* not TRILIBRARY */ int i, j; #ifndef CDT_ONLY int k; char workstring[FILENAMESIZE]; #endif poly = refine = quality = vararea = fixedarea = regionattrib = convex = 0; firstnumber = 1; edgesout = voronoi = neighbors = geomview = 0; nobound = nopolywritten = nonodewritten = noelewritten = noiterationnum = 0; noholes = noexact = 0; incremental = sweepline = 0; dwyer = 1; splitseg = 0; docheck = 0; nobisect = 0; steiner = -1; order = 1; minangle = 0.0; maxarea = -1.0; quiet = verbose = 0; #ifndef TRILIBRARY innodefilename[0] = '\0'; #endif /* not TRILIBRARY */ for ( i = STARTINDEX; i < argc; i++ ) { #ifndef TRILIBRARY if ( argv[i][0] == '-' ) { #endif /* not TRILIBRARY */ for ( j = STARTINDEX; argv[i][j] != '\0'; j++ ) { if ( argv[i][j] == 'p' ) { poly = 1; } #ifndef CDT_ONLY if ( argv[i][j] == 'r' ) { refine = 1; } if ( argv[i][j] == 'q' ) { quality = 1; if ( ( ( argv[i][j + 1] >= '0' ) && ( argv[i][j + 1] <= '9' ) ) || ( argv[i][j + 1] == '.' ) ) { k = 0; while ( ( ( argv[i][j + 1] >= '0' ) && ( argv[i][j + 1] <= '9' ) ) || ( argv[i][j + 1] == '.' ) ) { j++; workstring[k] = argv[i][j]; k++; } workstring[k] = '\0'; minangle = (REAL) strtod( workstring, (char **) NULL ); } else { minangle = 20.0; } } if ( argv[i][j] == 'a' ) { quality = 1; if ( ( ( argv[i][j + 1] >= '0' ) && ( argv[i][j + 1] <= '9' ) ) || ( argv[i][j + 1] == '.' ) ) { fixedarea = 1; k = 0; while ( ( ( argv[i][j + 1] >= '0' ) && ( argv[i][j + 1] <= '9' ) ) || ( argv[i][j + 1] == '.' ) ) { j++; workstring[k] = argv[i][j]; k++; } workstring[k] = '\0'; maxarea = (REAL) strtod( workstring, (char **) NULL ); if ( maxarea <= 0.0 ) { printf( "Error: Maximum area must be greater than zero.\n" ); exit( 1 ); } } else { vararea = 1; } } #endif /* not CDT_ONLY */ if ( argv[i][j] == 'A' ) { regionattrib = 1; } if ( argv[i][j] == 'c' ) { convex = 1; } if ( argv[i][j] == 'z' ) { firstnumber = 0; } if ( argv[i][j] == 'e' ) { edgesout = 1; } if ( argv[i][j] == 'v' ) { voronoi = 1; } if ( argv[i][j] == 'n' ) { neighbors = 1; } if ( argv[i][j] == 'g' ) { geomview = 1; } if ( argv[i][j] == 'B' ) { nobound = 1; } if ( argv[i][j] == 'P' ) { nopolywritten = 1; } if ( argv[i][j] == 'N' ) { nonodewritten = 1; } if ( argv[i][j] == 'E' ) { noelewritten = 1; } #ifndef TRILIBRARY if ( argv[i][j] == 'I' ) { noiterationnum = 1; } #endif /* not TRILIBRARY */ if ( argv[i][j] == 'O' ) { noholes = 1; } if ( argv[i][j] == 'X' ) { noexact = 1; } if ( argv[i][j] == 'o' ) { if ( argv[i][j + 1] == '2' ) { j++; order = 2; } } #ifndef CDT_ONLY if ( argv[i][j] == 'Y' ) { nobisect++; } if ( argv[i][j] == 'S' ) { steiner = 0; while ( ( argv[i][j + 1] >= '0' ) && ( argv[i][j + 1] <= '9' ) ) { j++; steiner = steiner * 10 + (int) ( argv[i][j] - '0' ); } } #endif /* not CDT_ONLY */ #ifndef REDUCED if ( argv[i][j] == 'i' ) { incremental = 1; } if ( argv[i][j] == 'F' ) { sweepline = 1; } #endif /* not REDUCED */ if ( argv[i][j] == 'l' ) { dwyer = 0; } #ifndef REDUCED #ifndef CDT_ONLY if ( argv[i][j] == 's' ) { splitseg = 1; } #endif /* not CDT_ONLY */ if ( argv[i][j] == 'C' ) { docheck = 1; } #endif /* not REDUCED */ if ( argv[i][j] == 'Q' ) { quiet = 1; } if ( argv[i][j] == 'V' ) { verbose++; } #ifndef TRILIBRARY if ( ( argv[i][j] == 'h' ) || ( argv[i][j] == 'H' ) || ( argv[i][j] == '?' ) ) { info(); } #endif /* not TRILIBRARY */ } #ifndef TRILIBRARY } else { strncpy( innodefilename, argv[i], FILENAMESIZE - 1 ); innodefilename[FILENAMESIZE - 1] = '\0'; } #endif /* not TRILIBRARY */ } #ifndef TRILIBRARY if ( innodefilename[0] == '\0' ) { syntax(); } if ( !strcmp( &innodefilename[strlen( innodefilename ) - 5], ".node" ) ) { innodefilename[strlen( innodefilename ) - 5] = '\0'; } if ( !strcmp( &innodefilename[strlen( innodefilename ) - 5], ".poly" ) ) { innodefilename[strlen( innodefilename ) - 5] = '\0'; poly = 1; } #ifndef CDT_ONLY if ( !strcmp( &innodefilename[strlen( innodefilename ) - 4], ".ele" ) ) { innodefilename[strlen( innodefilename ) - 4] = '\0'; refine = 1; } if ( !strcmp( &innodefilename[strlen( innodefilename ) - 5], ".area" ) ) { innodefilename[strlen( innodefilename ) - 5] = '\0'; refine = 1; quality = 1; vararea = 1; } #endif /* not CDT_ONLY */ #endif /* not TRILIBRARY */ steinerleft = steiner; useshelles = poly || refine || quality || convex; goodangle = (REAL)cos( minangle * PI / 180.0 ); goodangle *= goodangle; if ( refine && noiterationnum ) { printf( "Error: You cannot use the -I switch when refining a triangulation.\n" ); exit( 1 ); } /* Be careful not to allocate space for element area constraints that */ /* will never be assigned any value (other than the default -1.0). */ if ( !refine && !poly ) { vararea = 0; } /* Be careful not to add an extra attribute to each element unless the */ /* input supports it (PSLG in, but not refining a preexisting mesh). */ if ( refine || !poly ) { regionattrib = 0; } #ifndef TRILIBRARY strcpy( inpolyfilename, innodefilename ); strcpy( inelefilename, innodefilename ); strcpy( areafilename, innodefilename ); increment = 0; strcpy( workstring, innodefilename ); j = 1; while ( workstring[j] != '\0' ) { if ( ( workstring[j] == '.' ) && ( workstring[j + 1] != '\0' ) ) { increment = j + 1; } j++; } meshnumber = 0; if ( increment > 0 ) { j = increment; do { if ( ( workstring[j] >= '0' ) && ( workstring[j] <= '9' ) ) { meshnumber = meshnumber * 10 + (int) ( workstring[j] - '0' ); } else { increment = 0; } j++; } while ( workstring[j] != '\0' ); } if ( noiterationnum ) { strcpy( outnodefilename, innodefilename ); strcpy( outelefilename, innodefilename ); strcpy( edgefilename, innodefilename ); strcpy( vnodefilename, innodefilename ); strcpy( vedgefilename, innodefilename ); strcpy( neighborfilename, innodefilename ); strcpy( offfilename, innodefilename ); strcat( outnodefilename, ".node" ); strcat( outelefilename, ".ele" ); strcat( edgefilename, ".edge" ); strcat( vnodefilename, ".v.node" ); strcat( vedgefilename, ".v.edge" ); strcat( neighborfilename, ".neigh" ); strcat( offfilename, ".off" ); } else if ( increment == 0 ) { strcpy( outnodefilename, innodefilename ); strcpy( outpolyfilename, innodefilename ); strcpy( outelefilename, innodefilename ); strcpy( edgefilename, innodefilename ); strcpy( vnodefilename, innodefilename ); strcpy( vedgefilename, innodefilename ); strcpy( neighborfilename, innodefilename ); strcpy( offfilename, innodefilename ); strcat( outnodefilename, ".1.node" ); strcat( outpolyfilename, ".1.poly" ); strcat( outelefilename, ".1.ele" ); strcat( edgefilename, ".1.edge" ); strcat( vnodefilename, ".1.v.node" ); strcat( vedgefilename, ".1.v.edge" ); strcat( neighborfilename, ".1.neigh" ); strcat( offfilename, ".1.off" ); } else { workstring[increment] = '%'; workstring[increment + 1] = 'd'; workstring[increment + 2] = '\0'; sprintf( outnodefilename, workstring, meshnumber + 1 ); strcpy( outpolyfilename, outnodefilename ); strcpy( outelefilename, outnodefilename ); strcpy( edgefilename, outnodefilename ); strcpy( vnodefilename, outnodefilename ); strcpy( vedgefilename, outnodefilename ); strcpy( neighborfilename, outnodefilename ); strcpy( offfilename, outnodefilename ); strcat( outnodefilename, ".node" ); strcat( outpolyfilename, ".poly" ); strcat( outelefilename, ".ele" ); strcat( edgefilename, ".edge" ); strcat( vnodefilename, ".v.node" ); strcat( vedgefilename, ".v.edge" ); strcat( neighborfilename, ".neigh" ); strcat( offfilename, ".off" ); } strcat( innodefilename, ".node" ); strcat( inpolyfilename, ".poly" ); strcat( inelefilename, ".ele" ); strcat( areafilename, ".area" ); #endif /* not TRILIBRARY */ } /** **/ /** **/ /********* User interaction routines begin here *********/ /********* Debugging routines begin here *********/ /** **/ /** **/ /*****************************************************************************/ /* */ /* printtriangle() Print out the details of a triangle/edge handle. */ /* */ /* I originally wrote this procedure to simplify debugging; it can be */ /* called directly from the debugger, and presents information about a */ /* triangle/edge handle in digestible form. It's also used when the */ /* highest level of verbosity (`-VVV') is specified. */ /* */ /*****************************************************************************/ void printtriangle( t ) struct triedge *t; { struct triedge printtri; struct edge printsh; point printpoint; printf( "triangle x%lx with orientation %d:\n", (unsigned long) t->tri, t->orient ); decode( t->tri[0], printtri ); if ( printtri.tri == dummytri ) { printf( " [0] = Outer space\n" ); } else { printf( " [0] = x%lx %d\n", (unsigned long) printtri.tri, printtri.orient ); } decode( t->tri[1], printtri ); if ( printtri.tri == dummytri ) { printf( " [1] = Outer space\n" ); } else { printf( " [1] = x%lx %d\n", (unsigned long) printtri.tri, printtri.orient ); } decode( t->tri[2], printtri ); if ( printtri.tri == dummytri ) { printf( " [2] = Outer space\n" ); } else { printf( " [2] = x%lx %d\n", (unsigned long) printtri.tri, printtri.orient ); } org( *t, printpoint ); if ( printpoint == (point) NULL ) { printf( " Origin[%d] = NULL\n", ( t->orient + 1 ) % 3 + 3 ); } else{ printf( " Origin[%d] = x%lx (%.12g, %.12g)\n", ( t->orient + 1 ) % 3 + 3, (unsigned long) printpoint, printpoint[0], printpoint[1] ); } dest( *t, printpoint ); if ( printpoint == (point) NULL ) { printf( " Dest [%d] = NULL\n", ( t->orient + 2 ) % 3 + 3 ); } else{ printf( " Dest [%d] = x%lx (%.12g, %.12g)\n", ( t->orient + 2 ) % 3 + 3, (unsigned long) printpoint, printpoint[0], printpoint[1] ); } apex( *t, printpoint ); if ( printpoint == (point) NULL ) { printf( " Apex [%d] = NULL\n", t->orient + 3 ); } else{ printf( " Apex [%d] = x%lx (%.12g, %.12g)\n", t->orient + 3, (unsigned long) printpoint, printpoint[0], printpoint[1] ); } if ( useshelles ) { sdecode( t->tri[6], printsh ); if ( printsh.sh != dummysh ) { printf( " [6] = x%lx %d\n", (unsigned long) printsh.sh, printsh.shorient ); } sdecode( t->tri[7], printsh ); if ( printsh.sh != dummysh ) { printf( " [7] = x%lx %d\n", (unsigned long) printsh.sh, printsh.shorient ); } sdecode( t->tri[8], printsh ); if ( printsh.sh != dummysh ) { printf( " [8] = x%lx %d\n", (unsigned long) printsh.sh, printsh.shorient ); } } if ( vararea ) { printf( " Area constraint: %.4g\n", areabound( *t ) ); } } /*****************************************************************************/ /* */ /* printshelle() Print out the details of a shell edge handle. */ /* */ /* I originally wrote this procedure to simplify debugging; it can be */ /* called directly from the debugger, and presents information about a */ /* shell edge handle in digestible form. It's also used when the highest */ /* level of verbosity (`-VVV') is specified. */ /* */ /*****************************************************************************/ void printshelle( s ) struct edge *s; { struct edge printsh; struct triedge printtri; point printpoint; printf( "shell edge x%lx with orientation %d and mark %d:\n", (unsigned long) s->sh, s->shorient, mark( *s ) ); sdecode( s->sh[0], printsh ); if ( printsh.sh == dummysh ) { printf( " [0] = No shell\n" ); } else { printf( " [0] = x%lx %d\n", (unsigned long) printsh.sh, printsh.shorient ); } sdecode( s->sh[1], printsh ); if ( printsh.sh == dummysh ) { printf( " [1] = No shell\n" ); } else { printf( " [1] = x%lx %d\n", (unsigned long) printsh.sh, printsh.shorient ); } sorg( *s, printpoint ); if ( printpoint == (point) NULL ) { printf( " Origin[%d] = NULL\n", 2 + s->shorient ); } else{ printf( " Origin[%d] = x%lx (%.12g, %.12g)\n", 2 + s->shorient, (unsigned long) printpoint, printpoint[0], printpoint[1] ); } sdest( *s, printpoint ); if ( printpoint == (point) NULL ) { printf( " Dest [%d] = NULL\n", 3 - s->shorient ); } else{ printf( " Dest [%d] = x%lx (%.12g, %.12g)\n", 3 - s->shorient, (unsigned long) printpoint, printpoint[0], printpoint[1] ); } decode( s->sh[4], printtri ); if ( printtri.tri == dummytri ) { printf( " [4] = Outer space\n" ); } else { printf( " [4] = x%lx %d\n", (unsigned long) printtri.tri, printtri.orient ); } decode( s->sh[5], printtri ); if ( printtri.tri == dummytri ) { printf( " [5] = Outer space\n" ); } else { printf( " [5] = x%lx %d\n", (unsigned long) printtri.tri, printtri.orient ); } } /** **/ /** **/ /********* Debugging routines end here *********/ /********* Memory management routines begin here *********/ /** **/ /** **/ /*****************************************************************************/ /* */ /* poolinit() Initialize a pool of memory for allocation of items. */ /* */ /* This routine initializes the machinery for allocating items. A `pool' */ /* is created whose records have size at least `bytecount'. Items will be */ /* allocated in `itemcount'-item blocks. Each item is assumed to be a */ /* collection of words, and either pointers or floating-point values are */ /* assumed to be the "primary" word type. (The "primary" word type is used */ /* to determine alignment of items.) If `alignment' isn't zero, all items */ /* will be `alignment'-byte aligned in memory. `alignment' must be either */ /* a multiple or a factor of the primary word size; powers of two are safe. */ /* `alignment' is normally used to create a few unused bits at the bottom */ /* of each item's pointer, in which information may be stored. */ /* */ /* Don't change this routine unless you understand it. */ /* */ /*****************************************************************************/ void poolinit( pool, bytecount, itemcount, wtype, alignment ) struct memorypool *pool; int bytecount; int itemcount; enum wordtype wtype; int alignment; { int wordsize; /* Initialize values in the pool. */ pool->itemwordtype = wtype; wordsize = ( pool->itemwordtype == POINTER ) ? sizeof( VOID * ) : sizeof( REAL ); /* Find the proper alignment, which must be at least as large as: */ /* - The parameter `alignment'. */ /* - The primary word type, to avoid unaligned accesses. */ /* - sizeof(VOID *), so the stack of dead items can be maintained */ /* without unaligned accesses. */ if ( alignment > wordsize ) { pool->alignbytes = alignment; } else { pool->alignbytes = wordsize; } if ( sizeof( VOID * ) > pool->alignbytes ) { pool->alignbytes = sizeof( VOID * ); } pool->itemwords = ( ( bytecount + pool->alignbytes - 1 ) / pool->alignbytes ) * ( pool->alignbytes / wordsize ); pool->itembytes = pool->itemwords * wordsize; pool->itemsperblock = itemcount; /* Allocate a block of items. Space for `itemsperblock' items and one */ /* pointer (to point to the next block) are allocated, as well as space */ /* to ensure alignment of the items. */ pool->firstblock = (VOID **) malloc( pool->itemsperblock * pool->itembytes + sizeof( VOID * ) + pool->alignbytes ); if ( pool->firstblock == (VOID **) NULL ) { printf( "Error: Out of memory.\n" ); exit( 1 ); } /* Set the next block pointer to NULL. */ *( pool->firstblock ) = (VOID *) NULL; poolrestart( pool ); } /*****************************************************************************/ /* */ /* poolrestart() Deallocate all items in a pool. */ /* */ /* The pool is returned to its starting state, except that no memory is */ /* freed to the operating system. Rather, the previously allocated blocks */ /* are ready to be reused. */ /* */ /*****************************************************************************/ void poolrestart( pool ) struct memorypool *pool; { unsigned long alignptr; pool->items = 0; pool->maxitems = 0; /* Set the currently active block. */ pool->nowblock = pool->firstblock; /* Find the first item in the pool. Increment by the size of (VOID *). */ alignptr = (unsigned long) ( pool->nowblock + 1 ); /* Align the item on an `alignbytes'-byte boundary. */ pool->nextitem = (VOID *) ( alignptr + (unsigned long) pool->alignbytes - ( alignptr % (unsigned long) pool->alignbytes ) ); /* There are lots of unallocated items left in this block. */ pool->unallocateditems = pool->itemsperblock; /* The stack of deallocated items is empty. */ pool->deaditemstack = (VOID *) NULL; } /*****************************************************************************/ /* */ /* pooldeinit() Free to the operating system all memory taken by a pool. */ /* */ /*****************************************************************************/ void pooldeinit( pool ) struct memorypool *pool; { while ( pool->firstblock != (VOID **) NULL ) { pool->nowblock = (VOID **) *( pool->firstblock ); free( pool->firstblock ); pool->firstblock = pool->nowblock; } } /*****************************************************************************/ /* */ /* poolalloc() Allocate space for an item. */ /* */ /*****************************************************************************/ VOID *poolalloc( pool ) struct memorypool *pool; { VOID *newitem; VOID **newblock; unsigned long alignptr; /* First check the linked list of dead items. If the list is not */ /* empty, allocate an item from the list rather than a fresh one. */ if ( pool->deaditemstack != (VOID *) NULL ) { newitem = pool->deaditemstack; /* Take first item in list. */ pool->deaditemstack = *(VOID **) pool->deaditemstack; } else { /* Check if there are any free items left in the current block. */ if ( pool->unallocateditems == 0 ) { /* Check if another block must be allocated. */ if ( *( pool->nowblock ) == (VOID *) NULL ) { /* Allocate a new block of items, pointed to by the previous block. */ newblock = (VOID **) malloc( pool->itemsperblock * pool->itembytes + sizeof( VOID * ) + pool->alignbytes ); if ( newblock == (VOID **) NULL ) { printf( "Error: Out of memory.\n" ); exit( 1 ); } *( pool->nowblock ) = (VOID *) newblock; /* The next block pointer is NULL. */ *newblock = (VOID *) NULL; } /* Move to the new block. */ pool->nowblock = (VOID **) *( pool->nowblock ); /* Find the first item in the block. */ /* Increment by the size of (VOID *). */ alignptr = (unsigned long) ( pool->nowblock + 1 ); /* Align the item on an `alignbytes'-byte boundary. */ pool->nextitem = (VOID *) ( alignptr + (unsigned long) pool->alignbytes - ( alignptr % (unsigned long) pool->alignbytes ) ); /* There are lots of unallocated items left in this block. */ pool->unallocateditems = pool->itemsperblock; } /* Allocate a new item. */ newitem = pool->nextitem; /* Advance `nextitem' pointer to next free item in block. */ if ( pool->itemwordtype == POINTER ) { pool->nextitem = (VOID *) ( (VOID **) pool->nextitem + pool->itemwords ); } else { pool->nextitem = (VOID *) ( (REAL *) pool->nextitem + pool->itemwords ); } pool->unallocateditems--; pool->maxitems++; } pool->items++; return newitem; } /*****************************************************************************/ /* */ /* pooldealloc() Deallocate space for an item. */ /* */ /* The deallocated space is stored in a queue for later reuse. */ /* */ /*****************************************************************************/ void pooldealloc( pool, dyingitem ) struct memorypool *pool; VOID *dyingitem; { /* Push freshly killed item onto stack. */ *( (VOID **) dyingitem ) = pool->deaditemstack; pool->deaditemstack = dyingitem; pool->items--; } /*****************************************************************************/ /* */ /* traversalinit() Prepare to traverse the entire list of items. */ /* */ /* This routine is used in conjunction with traverse(). */ /* */ /*****************************************************************************/ void traversalinit( pool ) struct memorypool *pool; { unsigned long alignptr; /* Begin the traversal in the first block. */ pool->pathblock = pool->firstblock; /* Find the first item in the block. Increment by the size of (VOID *). */ alignptr = (unsigned long) ( pool->pathblock + 1 ); /* Align with item on an `alignbytes'-byte boundary. */ pool->pathitem = (VOID *) ( alignptr + (unsigned long) pool->alignbytes - ( alignptr % (unsigned long) pool->alignbytes ) ); /* Set the number of items left in the current block. */ pool->pathitemsleft = pool->itemsperblock; } /*****************************************************************************/ /* */ /* traverse() Find the next item in the list. */ /* */ /* This routine is used in conjunction with traversalinit(). Be forewarned */ /* that this routine successively returns all items in the list, including */ /* deallocated ones on the deaditemqueue. It's up to you to figure out */ /* which ones are actually dead. Why? I don't want to allocate extra */ /* space just to demarcate dead items. It can usually be done more */ /* space-efficiently by a routine that knows something about the structure */ /* of the item. */ /* */ /*****************************************************************************/ VOID *traverse( pool ) struct memorypool *pool; { VOID *newitem; unsigned long alignptr; /* Stop upon exhausting the list of items. */ if ( pool->pathitem == pool->nextitem ) { return (VOID *) NULL; } /* Check whether any untraversed items remain in the current block. */ if ( pool->pathitemsleft == 0 ) { /* Find the next block. */ pool->pathblock = (VOID **) *( pool->pathblock ); /* Find the first item in the block. Increment by the size of (VOID *). */ alignptr = (unsigned long) ( pool->pathblock + 1 ); /* Align with item on an `alignbytes'-byte boundary. */ pool->pathitem = (VOID *) ( alignptr + (unsigned long) pool->alignbytes - ( alignptr % (unsigned long) pool->alignbytes ) ); /* Set the number of items left in the current block. */ pool->pathitemsleft = pool->itemsperblock; } newitem = pool->pathitem; /* Find the next item in the block. */ if ( pool->itemwordtype == POINTER ) { pool->pathitem = (VOID *) ( (VOID **) pool->pathitem + pool->itemwords ); } else { pool->pathitem = (VOID *) ( (REAL *) pool->pathitem + pool->itemwords ); } pool->pathitemsleft--; return newitem; } /*****************************************************************************/ /* */ /* dummyinit() Initialize the triangle that fills "outer space" and the */ /* omnipresent shell edge. */ /* */ /* The triangle that fills "outer space", called `dummytri', is pointed to */ /* by every triangle and shell edge on a boundary (be it outer or inner) of */ /* the triangulation. Also, `dummytri' points to one of the triangles on */ /* the convex hull (until the holes and concavities are carved), making it */ /* possible to find a starting triangle for point location. */ /* */ /* The omnipresent shell edge, `dummysh', is pointed to by every triangle */ /* or shell edge that doesn't have a full complement of real shell edges */ /* to point to. */ /* */ /*****************************************************************************/ void dummyinit( trianglewords, shellewords ) int trianglewords; int shellewords; { unsigned long alignptr; /* `triwords' and `shwords' are used by the mesh manipulation primitives */ /* to extract orientations of triangles and shell edges from pointers. */ triwords = trianglewords; /* Initialize `triwords' once and for all. */ shwords = shellewords; /* Initialize `shwords' once and for all. */ /* Set up `dummytri', the `triangle' that occupies "outer space". */ dummytribase = (triangle *) malloc( triwords * sizeof( triangle ) + triangles.alignbytes ); if ( dummytribase == (triangle *) NULL ) { printf( "Error: Out of memory.\n" ); exit( 1 ); } /* Align `dummytri' on a `triangles.alignbytes'-byte boundary. */ alignptr = (unsigned long) dummytribase; dummytri = (triangle *) ( alignptr + (unsigned long) triangles.alignbytes - ( alignptr % (unsigned long) triangles.alignbytes ) ); /* Initialize the three adjoining triangles to be "outer space". These */ /* will eventually be changed by various bonding operations, but their */ /* values don't really matter, as long as they can legally be */ /* dereferenced. */ dummytri[0] = (triangle) dummytri; dummytri[1] = (triangle) dummytri; dummytri[2] = (triangle) dummytri; /* Three NULL vertex points. */ dummytri[3] = (triangle) NULL; dummytri[4] = (triangle) NULL; dummytri[5] = (triangle) NULL; if ( useshelles ) { /* Set up `dummysh', the omnipresent "shell edge" pointed to by any */ /* triangle side or shell edge end that isn't attached to a real shell */ /* edge. */ dummyshbase = (shelle *) malloc( shwords * sizeof( shelle ) + shelles.alignbytes ); if ( dummyshbase == (shelle *) NULL ) { printf( "Error: Out of memory.\n" ); exit( 1 ); } /* Align `dummysh' on a `shelles.alignbytes'-byte boundary. */ alignptr = (unsigned long) dummyshbase; dummysh = (shelle *) ( alignptr + (unsigned long) shelles.alignbytes - ( alignptr % (unsigned long) shelles.alignbytes ) ); /* Initialize the two adjoining shell edges to be the omnipresent shell */ /* edge. These will eventually be changed by various bonding */ /* operations, but their values don't really matter, as long as they */ /* can legally be dereferenced. */ dummysh[0] = (shelle) dummysh; dummysh[1] = (shelle) dummysh; /* Two NULL vertex points. */ dummysh[2] = (shelle) NULL; dummysh[3] = (shelle) NULL; /* Initialize the two adjoining triangles to be "outer space". */ dummysh[4] = (shelle) dummytri; dummysh[5] = (shelle) dummytri; /* Set the boundary marker to zero. */ *(int *) ( dummysh + 6 ) = 0; /* Initialize the three adjoining shell edges of `dummytri' to be */ /* the omnipresent shell edge. */ dummytri[6] = (triangle) dummysh; dummytri[7] = (triangle) dummysh; dummytri[8] = (triangle) dummysh; } } /*****************************************************************************/ /* */ /* initializepointpool() Calculate the size of the point data structure */ /* and initialize its memory pool. */ /* */ /* This routine also computes the `pointmarkindex' and `point2triindex' */ /* indices used to find values within each point. */ /* */ /*****************************************************************************/ void initializepointpool(){ int pointsize; /* The index within each point at which the boundary marker is found. */ /* Ensure the point marker is aligned to a sizeof(int)-byte address. */ pointmarkindex = ( ( mesh_dim + nextras ) * sizeof( REAL ) + sizeof( int ) - 1 ) / sizeof( int ); pointsize = ( pointmarkindex + 1 ) * sizeof( int ); if ( poly ) { /* The index within each point at which a triangle pointer is found. */ /* Ensure the pointer is aligned to a sizeof(triangle)-byte address. */ point2triindex = ( pointsize + sizeof( triangle ) - 1 ) / sizeof( triangle ); pointsize = ( point2triindex + 1 ) * sizeof( triangle ); } /* Initialize the pool of points. */ poolinit( &points, pointsize, POINTPERBLOCK, ( sizeof( REAL ) >= sizeof( triangle ) ) ? FLOATINGPOINT : POINTER, 0 ); } /*****************************************************************************/ /* */ /* initializetrisegpools() Calculate the sizes of the triangle and shell */ /* edge data structures and initialize their */ /* memory pools. */ /* */ /* This routine also computes the `highorderindex', `elemattribindex', and */ /* `areaboundindex' indices used to find values within each triangle. */ /* */ /*****************************************************************************/ void initializetrisegpools(){ int trisize; /* The index within each triangle at which the extra nodes (above three) */ /* associated with high order elements are found. There are three */ /* pointers to other triangles, three pointers to corners, and possibly */ /* three pointers to shell edges before the extra nodes. */ highorderindex = 6 + ( useshelles * 3 ); /* The number of bytes occupied by a triangle. */ trisize = ( ( order + 1 ) * ( order + 2 ) / 2 + ( highorderindex - 3 ) ) * sizeof( triangle ); /* The index within each triangle at which its attributes are found, */ /* where the index is measured in REALs. */ elemattribindex = ( trisize + sizeof( REAL ) - 1 ) / sizeof( REAL ); /* The index within each triangle at which the maximum area constraint */ /* is found, where the index is measured in REALs. Note that if the */ /* `regionattrib' flag is set, an additional attribute will be added. */ areaboundindex = elemattribindex + eextras + regionattrib; /* If triangle attributes or an area bound are needed, increase the number */ /* of bytes occupied by a triangle. */ if ( vararea ) { trisize = ( areaboundindex + 1 ) * sizeof( REAL ); } else if ( eextras + regionattrib > 0 ) { trisize = areaboundindex * sizeof( REAL ); } /* If a Voronoi diagram or triangle neighbor graph is requested, make */ /* sure there's room to store an integer index in each triangle. This */ /* integer index can occupy the same space as the shell edges or */ /* attributes or area constraint or extra nodes. */ if ( ( voronoi || neighbors ) && ( trisize < 6 * sizeof( triangle ) + sizeof( int ) ) ) { trisize = 6 * sizeof( triangle ) + sizeof( int ); } /* Having determined the memory size of a triangle, initialize the pool. */ poolinit( &triangles, trisize, TRIPERBLOCK, POINTER, 4 ); if ( useshelles ) { /* Initialize the pool of shell edges. */ poolinit( &shelles, 6 * sizeof( triangle ) + sizeof( int ), SHELLEPERBLOCK, POINTER, 4 ); /* Initialize the "outer space" triangle and omnipresent shell edge. */ dummyinit( triangles.itemwords, shelles.itemwords ); } else { /* Initialize the "outer space" triangle. */ dummyinit( triangles.itemwords, 0 ); } } /*****************************************************************************/ /* */ /* triangledealloc() Deallocate space for a triangle, marking it dead. */ /* */ /*****************************************************************************/ void triangledealloc( dyingtriangle ) triangle * dyingtriangle; { /* Set triangle's vertices to NULL. This makes it possible to */ /* detect dead triangles when traversing the list of all triangles. */ dyingtriangle[3] = (triangle) NULL; dyingtriangle[4] = (triangle) NULL; dyingtriangle[5] = (triangle) NULL; pooldealloc( &triangles, (VOID *) dyingtriangle ); } /*****************************************************************************/ /* */ /* triangletraverse() Traverse the triangles, skipping dead ones. */ /* */ /*****************************************************************************/ triangle *triangletraverse(){ triangle *newtriangle; do { newtriangle = (triangle *) traverse( &triangles ); if ( newtriangle == (triangle *) NULL ) { return (triangle *) NULL; } } while ( newtriangle[3] == (triangle) NULL ); /* Skip dead ones. */ return newtriangle; } /*****************************************************************************/ /* */ /* shelledealloc() Deallocate space for a shell edge, marking it dead. */ /* */ /*****************************************************************************/ void shelledealloc( dyingshelle ) shelle * dyingshelle; { /* Set shell edge's vertices to NULL. This makes it possible to */ /* detect dead shells when traversing the list of all shells. */ dyingshelle[2] = (shelle) NULL; dyingshelle[3] = (shelle) NULL; pooldealloc( &shelles, (VOID *) dyingshelle ); } /*****************************************************************************/ /* */ /* shelletraverse() Traverse the shell edges, skipping dead ones. */ /* */ /*****************************************************************************/ shelle *shelletraverse(){ shelle *newshelle; do { newshelle = (shelle *) traverse( &shelles ); if ( newshelle == (shelle *) NULL ) { return (shelle *) NULL; } } while ( newshelle[2] == (shelle) NULL ); /* Skip dead ones. */ return newshelle; } /*****************************************************************************/ /* */ /* pointdealloc() Deallocate space for a point, marking it dead. */ /* */ /*****************************************************************************/ void pointdealloc( dyingpoint ) point dyingpoint; { /* Mark the point as dead. This makes it possible to detect dead points */ /* when traversing the list of all points. */ setpointmark( dyingpoint, DEADPOINT ); pooldealloc( &points, (VOID *) dyingpoint ); } /*****************************************************************************/ /* */ /* pointtraverse() Traverse the points, skipping dead ones. */ /* */ /*****************************************************************************/ point pointtraverse(){ point newpoint; do { newpoint = (point) traverse( &points ); if ( newpoint == (point) NULL ) { return (point) NULL; } } while ( pointmark( newpoint ) == DEADPOINT ); /* Skip dead ones. */ return newpoint; } /*****************************************************************************/ /* */ /* badsegmentdealloc() Deallocate space for a bad segment, marking it */ /* dead. */ /* */ /*****************************************************************************/ #ifndef CDT_ONLY void badsegmentdealloc( dyingseg ) struct edge *dyingseg; { /* Set segment's orientation to -1. This makes it possible to */ /* detect dead segments when traversing the list of all segments. */ dyingseg->shorient = -1; pooldealloc( &badsegments, (VOID *) dyingseg ); } #endif /* not CDT_ONLY */ /*****************************************************************************/ /* */ /* badsegmenttraverse() Traverse the bad segments, skipping dead ones. */ /* */ /*****************************************************************************/ #ifndef CDT_ONLY struct edge *badsegmenttraverse(){ struct edge *newseg; do { newseg = (struct edge *) traverse( &badsegments ); if ( newseg == (struct edge *) NULL ) { return (struct edge *) NULL; } } while ( newseg->shorient == -1 ); /* Skip dead ones. */ return newseg; } #endif /* not CDT_ONLY */ /*****************************************************************************/ /* */ /* getpoint() Get a specific point, by number, from the list. */ /* */ /* The first point is number 'firstnumber'. */ /* */ /* Note that this takes O(n) time (with a small constant, if POINTPERBLOCK */ /* is large). I don't care to take the trouble to make it work in constant */ /* time. */ /* */ /*****************************************************************************/ point getpoint( number ) int number; { VOID **getblock; point foundpoint; unsigned long alignptr; int current; getblock = points.firstblock; current = firstnumber; /* Find the right block. */ while ( current + points.itemsperblock <= number ) { getblock = (VOID **) *getblock; current += points.itemsperblock; } /* Now find the right point. */ alignptr = (unsigned long) ( getblock + 1 ); foundpoint = (point) ( alignptr + (unsigned long) points.alignbytes - ( alignptr % (unsigned long) points.alignbytes ) ); while ( current < number ) { foundpoint += points.itemwords; current++; } return foundpoint; } /*****************************************************************************/ /* */ /* triangledeinit() Free all remaining allocated memory. */ /* */ /*****************************************************************************/ void triangledeinit(){ pooldeinit( &triangles ); free( dummytribase ); if ( useshelles ) { pooldeinit( &shelles ); free( dummyshbase ); } pooldeinit( &points ); #ifndef CDT_ONLY if ( quality ) { pooldeinit( &badsegments ); if ( ( minangle > 0.0 ) || vararea || fixedarea ) { pooldeinit( &badtriangles ); } } #endif /* not CDT_ONLY */ } /** **/ /** **/ /********* Memory management routines end here *********/ /********* Constructors begin here *********/ /** **/ /** **/ /*****************************************************************************/ /* */ /* maketriangle() Create a new triangle with orientation zero. */ /* */ /*****************************************************************************/ void maketriangle( newtriedge ) struct triedge *newtriedge; { int i; newtriedge->tri = (triangle *) poolalloc( &triangles ); /* Initialize the three adjoining triangles to be "outer space". */ newtriedge->tri[0] = (triangle) dummytri; newtriedge->tri[1] = (triangle) dummytri; newtriedge->tri[2] = (triangle) dummytri; /* Three NULL vertex points. */ newtriedge->tri[3] = (triangle) NULL; newtriedge->tri[4] = (triangle) NULL; newtriedge->tri[5] = (triangle) NULL; /* Initialize the three adjoining shell edges to be the omnipresent */ /* shell edge. */ if ( useshelles ) { newtriedge->tri[6] = (triangle) dummysh; newtriedge->tri[7] = (triangle) dummysh; newtriedge->tri[8] = (triangle) dummysh; } for ( i = 0; i < eextras; i++ ) { setelemattribute( *newtriedge, i, 0.0 ); } if ( vararea ) { setareabound( *newtriedge, -1.0 ); } newtriedge->orient = 0; } /*****************************************************************************/ /* */ /* makeshelle() Create a new shell edge with orientation zero. */ /* */ /*****************************************************************************/ void makeshelle( newedge ) struct edge *newedge; { newedge->sh = (shelle *) poolalloc( &shelles ); /* Initialize the two adjoining shell edges to be the omnipresent */ /* shell edge. */ newedge->sh[0] = (shelle) dummysh; newedge->sh[1] = (shelle) dummysh; /* Two NULL vertex points. */ newedge->sh[2] = (shelle) NULL; newedge->sh[3] = (shelle) NULL; /* Initialize the two adjoining triangles to be "outer space". */ newedge->sh[4] = (shelle) dummytri; newedge->sh[5] = (shelle) dummytri; /* Set the boundary marker to zero. */ setmark( *newedge, 0 ); newedge->shorient = 0; } /** **/ /** **/ /********* Constructors end here *********/ /********* Determinant evaluation routines begin here *********/ /** **/ /** **/ /* The adaptive exact arithmetic geometric predicates implemented herein are */ /* described in detail in my Technical Report CMU-CS-96-140. The complete */ /* reference is given in the header. */ /* Which of the following two methods of finding the absolute values is */ /* fastest is compiler-dependent. A few compilers can inline and optimize */ /* the fabs() call; but most will incur the overhead of a function call, */ /* which is disastrously slow. A faster way on IEEE machines might be to */ /* mask the appropriate bit, but that's difficult to do in C. */ #define Absolute( a ) ( ( a ) >= 0.0 ? ( a ) : -( a ) ) /* #define Absolute(a) fabs(a) */ /* Many of the operations are broken up into two pieces, a main part that */ /* performs an approximate operation, and a "tail" that computes the */ /* roundoff error of that operation. */ /* */ /* The operations Fast_Two_Sum(), Fast_Two_Diff(), Two_Sum(), Two_Diff(), */ /* Split(), and Two_Product() are all implemented as described in the */ /* reference. Each of these macros requires certain variables to be */ /* defined in the calling routine. The variables `bvirt', `c', `abig', */ /* `_i', `_j', `_k', `_l', `_m', and `_n' are declared `INEXACT' because */ /* they store the result of an operation that may incur roundoff error. */ /* The input parameter `x' (or the highest numbered `x_' parameter) must */ /* also be declared `INEXACT'. */ #define Fast_Two_Sum_Tail( a, b, x, y ) \ bvirt = x - a; \ y = b - bvirt #define Fast_Two_Sum( a, b, x, y ) \ x = (REAL) ( a + b ); \ Fast_Two_Sum_Tail( a, b, x, y ) #define Two_Sum_Tail( a, b, x, y ) \ bvirt = (REAL) ( x - a ); \ avirt = x - bvirt; \ bround = b - bvirt; \ around = a - avirt; \ y = around + bround #define Two_Sum( a, b, x, y ) \ x = (REAL) ( a + b ); \ Two_Sum_Tail( a, b, x, y ) #define Two_Diff_Tail( a, b, x, y ) \ bvirt = (REAL) ( a - x ); \ avirt = x + bvirt; \ bround = bvirt - b; \ around = a - avirt; \ y = around + bround #define Two_Diff( a, b, x, y ) \ x = (REAL) ( a - b ); \ Two_Diff_Tail( a, b, x, y ) #define Split( a, ahi, alo ) \ c = (REAL) ( splitter * a ); \ abig = (REAL) ( c - a ); \ ahi = (REAL)( c - abig ); \ alo = (REAL)( a - ahi ) #define Two_Product_Tail( a, b, x, y ) \ Split( a, ahi, alo ); \ Split( b, bhi, blo ); \ err1 = x - ( ahi * bhi ); \ err2 = err1 - ( alo * bhi ); \ err3 = err2 - ( ahi * blo ); \ y = ( alo * blo ) - err3 #define Two_Product( a, b, x, y ) \ x = (REAL) ( a * b ); \ Two_Product_Tail( a, b, x, y ) /* Two_Product_Presplit() is Two_Product() where one of the inputs has */ /* already been split. Avoids redundant splitting. */ #define Two_Product_Presplit( a, b, bhi, blo, x, y ) \ x = (REAL) ( a * b ); \ Split( a, ahi, alo ); \ err1 = x - ( ahi * bhi ); \ err2 = err1 - ( alo * bhi ); \ err3 = err2 - ( ahi * blo ); \ y = ( alo * blo ) - err3 /* Square() can be done more quickly than Two_Product(). */ #define Square_Tail( a, x, y ) \ Split( a, ahi, alo ); \ err1 = x - ( ahi * ahi ); \ err3 = err1 - ( ( ahi + ahi ) * alo ); \ y = ( alo * alo ) - err3 #define Square( a, x, y ) \ x = (REAL) ( a * a ); \ Square_Tail( a, x, y ) /* Macros for summing expansions of various fixed lengths. These are all */ /* unrolled versions of Expansion_Sum(). */ #define Two_One_Sum( a1, a0, b, x2, x1, x0 ) \ Two_Sum( a0, b, _i, x0 ); \ Two_Sum( a1, _i, x2, x1 ) #define Two_One_Diff( a1, a0, b, x2, x1, x0 ) \ Two_Diff( a0, b, _i, x0 ); \ Two_Sum( a1, _i, x2, x1 ) #define Two_Two_Sum( a1, a0, b1, b0, x3, x2, x1, x0 ) \ Two_One_Sum( a1, a0, b0, _j, _0, x0 ); \ Two_One_Sum( _j, _0, b1, x3, x2, x1 ) #define Two_Two_Diff( a1, a0, b1, b0, x3, x2, x1, x0 ) \ Two_One_Diff( a1, a0, b0, _j, _0, x0 ); \ Two_One_Diff( _j, _0, b1, x3, x2, x1 ) /*****************************************************************************/ /* */ /* exactinit() Initialize the variables used for exact arithmetic. */ /* */ /* `epsilon' is the largest power of two such that 1.0 + epsilon = 1.0 in */ /* floating-point arithmetic. `epsilon' bounds the relative roundoff */ /* error. It is used for floating-point error analysis. */ /* */ /* `splitter' is used to split floating-point numbers into two half- */ /* length significands for exact multiplication. */ /* */ /* I imagine that a highly optimizing compiler might be too smart for its */ /* own good, and somehow cause this routine to fail, if it pretends that */ /* floating-point arithmetic is too much like real arithmetic. */ /* */ /* Don't change this routine unless you fully understand it. */ /* */ /*****************************************************************************/ void exactinit(){ REAL half; REAL check, lastcheck; int every_other; every_other = 1; half = 0.5; epsilon = 1.0; splitter = 1.0; check = 1.0; /* Repeatedly divide `epsilon' by two until it is too small to add to */ /* one without causing roundoff. (Also check if the sum is equal to */ /* the previous sum, for machines that round up instead of using exact */ /* rounding. Not that these routines will work on such machines anyway. */ do { lastcheck = check; epsilon *= half; if ( every_other ) { splitter *= 2.0; } every_other = !every_other; check = (REAL)( 1.0 + epsilon ); } while ( ( check != 1.0 ) && ( check != lastcheck ) ); splitter += 1.0; if ( verbose > 1 ) { printf( "Floating point roundoff is of magnitude %.17g\n", epsilon ); printf( "Floating point splitter is %.17g\n", splitter ); } /* Error bounds for orientation and incircle tests. */ resulterrbound = (REAL)( ( 3.0 + 8.0 * epsilon ) * epsilon ); ccwerrboundA = (REAL)( ( 3.0 + 16.0 * epsilon ) * epsilon ); ccwerrboundB = (REAL)( ( 2.0 + 12.0 * epsilon ) * epsilon ); ccwerrboundC = (REAL)( ( 9.0 + 64.0 * epsilon ) * epsilon * epsilon ); iccerrboundA = (REAL)( ( 10.0 + 96.0 * epsilon ) * epsilon ); iccerrboundB = (REAL)( ( 4.0 + 48.0 * epsilon ) * epsilon ); iccerrboundC = (REAL)( ( 44.0 + 576.0 * epsilon ) * epsilon * epsilon ); } /*****************************************************************************/ /* */ /* fast_expansion_sum_zeroelim() Sum two expansions, eliminating zero */ /* components from the output expansion. */ /* */ /* Sets h = e + f. See my Robust Predicates paper for details. */ /* */ /* If round-to-even is used (as with IEEE 754), maintains the strongly */ /* nonoverlapping property. (That is, if e is strongly nonoverlapping, h */ /* will be also.) Does NOT maintain the nonoverlapping or nonadjacent */ /* properties. */ /* */ /*****************************************************************************/ int fast_expansion_sum_zeroelim( elen, e, flen, f, h ) /* h cannot be e or f. */ int elen; REAL *e; int flen; REAL *f; REAL *h; { REAL Q; INEXACT REAL Qnew; INEXACT REAL hh; INEXACT REAL bvirt; REAL avirt, bround, around; int eindex, findex, hindex; REAL enow, fnow; enow = e[0]; fnow = f[0]; eindex = findex = 0; if ( ( fnow > enow ) == ( fnow > -enow ) ) { Q = enow; enow = e[++eindex]; } else { Q = fnow; fnow = f[++findex]; } hindex = 0; if ( ( eindex < elen ) && ( findex < flen ) ) { if ( ( fnow > enow ) == ( fnow > -enow ) ) { Fast_Two_Sum( enow, Q, Qnew, hh ); enow = e[++eindex]; } else { Fast_Two_Sum( fnow, Q, Qnew, hh ); fnow = f[++findex]; } Q = Qnew; if ( hh != 0.0 ) { h[hindex++] = hh; } while ( ( eindex < elen ) && ( findex < flen ) ) { if ( ( fnow > enow ) == ( fnow > -enow ) ) { Two_Sum( Q, enow, Qnew, hh ); enow = e[++eindex]; } else { Two_Sum( Q, fnow, Qnew, hh ); fnow = f[++findex]; } Q = Qnew; if ( hh != 0.0 ) { h[hindex++] = hh; } } } while ( eindex < elen ) { Two_Sum( Q, enow, Qnew, hh ); enow = e[++eindex]; Q = Qnew; if ( hh != 0.0 ) { h[hindex++] = hh; } } while ( findex < flen ) { Two_Sum( Q, fnow, Qnew, hh ); fnow = f[++findex]; Q = Qnew; if ( hh != 0.0 ) { h[hindex++] = hh; } } if ( ( Q != 0.0 ) || ( hindex == 0 ) ) { h[hindex++] = Q; } return hindex; } /*****************************************************************************/ /* */ /* scale_expansion_zeroelim() Multiply an expansion by a scalar, */ /* eliminating zero components from the */ /* output expansion. */ /* */ /* Sets h = be. See my Robust Predicates paper for details. */ /* */ /* Maintains the nonoverlapping property. If round-to-even is used (as */ /* with IEEE 754), maintains the strongly nonoverlapping and nonadjacent */ /* properties as well. (That is, if e has one of these properties, so */ /* will h.) */ /* */ /*****************************************************************************/ int scale_expansion_zeroelim( elen, e, b, h ) /* e and h cannot be the same. */ int elen; REAL *e; REAL b; REAL *h; { INEXACT REAL Q, sum; REAL hh; INEXACT REAL product1; REAL product0; int eindex, hindex; REAL enow; INEXACT REAL bvirt; REAL avirt, bround, around; INEXACT REAL c; INEXACT REAL abig; REAL ahi, alo, bhi, blo; REAL err1, err2, err3; Split( b, bhi, blo ); Two_Product_Presplit( e[0], b, bhi, blo, Q, hh ); hindex = 0; if ( hh != 0 ) { h[hindex++] = hh; } for ( eindex = 1; eindex < elen; eindex++ ) { enow = e[eindex]; Two_Product_Presplit( enow, b, bhi, blo, product1, product0 ); Two_Sum( Q, product0, sum, hh ); if ( hh != 0 ) { h[hindex++] = hh; } Fast_Two_Sum( product1, sum, Q, hh ); if ( hh != 0 ) { h[hindex++] = hh; } } if ( ( Q != 0.0 ) || ( hindex == 0 ) ) { h[hindex++] = Q; } return hindex; } /*****************************************************************************/ /* */ /* estimate() Produce a one-word estimate of an expansion's value. */ /* */ /* See my Robust Predicates paper for details. */ /* */ /*****************************************************************************/ REAL estimate( elen, e ) int elen; REAL *e; { REAL Q; int eindex; Q = e[0]; for ( eindex = 1; eindex < elen; eindex++ ) { Q += e[eindex]; } return Q; } /*****************************************************************************/ /* */ /* counterclockwise() Return a positive value if the points pa, pb, and */ /* pc occur in counterclockwise order; a negative */ /* value if they occur in clockwise order; and zero */ /* if they are collinear. The result is also a rough */ /* approximation of twice the signed area of the */ /* triangle defined by the three points. */ /* */ /* Uses exact arithmetic if necessary to ensure a correct answer. The */ /* result returned is the determinant of a matrix. This determinant is */ /* computed adaptively, in the sense that exact arithmetic is used only to */ /* the degree it is needed to ensure that the returned value has the */ /* correct sign. Hence, this function is usually quite fast, but will run */ /* more slowly when the input points are collinear or nearly so. */ /* */ /* See my Robust Predicates paper for details. */ /* */ /*****************************************************************************/ REAL counterclockwiseadapt( pa, pb, pc, detsum ) point pa; point pb; point pc; REAL detsum; { INEXACT REAL acx, acy, bcx, bcy; REAL acxtail, acytail, bcxtail, bcytail; INEXACT REAL detleft, detright; REAL detlefttail, detrighttail; REAL det, errbound; REAL B[4], C1[8], C2[12], D[16]; INEXACT REAL B3; int C1length, C2length, Dlength; REAL u[4]; INEXACT REAL u3; INEXACT REAL s1, t1; REAL s0, t0; INEXACT REAL bvirt; REAL avirt, bround, around; INEXACT REAL c; INEXACT REAL abig; REAL ahi, alo, bhi, blo; REAL err1, err2, err3; INEXACT REAL _i, _j; REAL _0; acx = (REAL) ( pa[0] - pc[0] ); bcx = (REAL) ( pb[0] - pc[0] ); acy = (REAL) ( pa[1] - pc[1] ); bcy = (REAL) ( pb[1] - pc[1] ); Two_Product( acx, bcy, detleft, detlefttail ); Two_Product( acy, bcx, detright, detrighttail ); Two_Two_Diff( detleft, detlefttail, detright, detrighttail, B3, B[2], B[1], B[0] ); B[3] = B3; det = estimate( 4, B ); errbound = (REAL)( ccwerrboundB * detsum ); if ( ( det >= errbound ) || ( -det >= errbound ) ) { return det; } Two_Diff_Tail( pa[0], pc[0], acx, acxtail ); Two_Diff_Tail( pb[0], pc[0], bcx, bcxtail ); Two_Diff_Tail( pa[1], pc[1], acy, acytail ); Two_Diff_Tail( pb[1], pc[1], bcy, bcytail ); if ( ( acxtail == 0.0 ) && ( acytail == 0.0 ) && ( bcxtail == 0.0 ) && ( bcytail == 0.0 ) ) { return det; } errbound = (REAL)( ccwerrboundC * detsum + resulterrbound * Absolute( det ) ); det += ( acx * bcytail + bcy * acxtail ) - ( acy * bcxtail + bcx * acytail ); if ( ( det >= errbound ) || ( -det >= errbound ) ) { return det; } Two_Product( acxtail, bcy, s1, s0 ); Two_Product( acytail, bcx, t1, t0 ); Two_Two_Diff( s1, s0, t1, t0, u3, u[2], u[1], u[0] ); u[3] = u3; C1length = fast_expansion_sum_zeroelim( 4, B, 4, u, C1 ); Two_Product( acx, bcytail, s1, s0 ); Two_Product( acy, bcxtail, t1, t0 ); Two_Two_Diff( s1, s0, t1, t0, u3, u[2], u[1], u[0] ); u[3] = u3; C2length = fast_expansion_sum_zeroelim( C1length, C1, 4, u, C2 ); Two_Product( acxtail, bcytail, s1, s0 ); Two_Product( acytail, bcxtail, t1, t0 ); Two_Two_Diff( s1, s0, t1, t0, u3, u[2], u[1], u[0] ); u[3] = u3; Dlength = fast_expansion_sum_zeroelim( C2length, C2, 4, u, D ); return( D[Dlength - 1] ); } REAL counterclockwise( pa, pb, pc ) point pa; point pb; point pc; { REAL detleft, detright, det; REAL detsum, errbound; counterclockcount++; detleft = ( pa[0] - pc[0] ) * ( pb[1] - pc[1] ); detright = ( pa[1] - pc[1] ) * ( pb[0] - pc[0] ); det = detleft - detright; if ( noexact ) { return det; } if ( detleft > 0.0 ) { if ( detright <= 0.0 ) { return det; } else { detsum = detleft + detright; } } else if ( detleft < 0.0 ) { if ( detright >= 0.0 ) { return det; } else { detsum = -detleft - detright; } } else { return det; } errbound = ccwerrboundA * detsum; if ( ( det >= errbound ) || ( -det >= errbound ) ) { return det; } return counterclockwiseadapt( pa, pb, pc, detsum ); } /*****************************************************************************/ /* */ /* incircle() Return a positive value if the point pd lies inside the */ /* circle passing through pa, pb, and pc; a negative value if */ /* it lies outside; and zero if the four points are cocircular.*/ /* The points pa, pb, and pc must be in counterclockwise */ /* order, or the sign of the result will be reversed. */ /* */ /* Uses exact arithmetic if necessary to ensure a correct answer. The */ /* result returned is the determinant of a matrix. This determinant is */ /* computed adaptively, in the sense that exact arithmetic is used only to */ /* the degree it is needed to ensure that the returned value has the */ /* correct sign. Hence, this function is usually quite fast, but will run */ /* more slowly when the input points are cocircular or nearly so. */ /* */ /* See my Robust Predicates paper for details. */ /* */ /*****************************************************************************/ REAL incircleadapt( pa, pb, pc, pd, permanent ) point pa; point pb; point pc; point pd; REAL permanent; { INEXACT REAL adx, bdx, cdx, ady, bdy, cdy; REAL det, errbound; INEXACT REAL bdxcdy1, cdxbdy1, cdxady1, adxcdy1, adxbdy1, bdxady1; REAL bdxcdy0, cdxbdy0, cdxady0, adxcdy0, adxbdy0, bdxady0; REAL bc[4], ca[4], ab[4]; INEXACT REAL bc3, ca3, ab3; REAL axbc[8], axxbc[16], aybc[8], ayybc[16], adet[32]; int axbclen, axxbclen, aybclen, ayybclen, alen; REAL bxca[8], bxxca[16], byca[8], byyca[16], bdet[32]; int bxcalen, bxxcalen, bycalen, byycalen, blen; REAL cxab[8], cxxab[16], cyab[8], cyyab[16], cdet[32]; int cxablen, cxxablen, cyablen, cyyablen, clen; REAL abdet[64]; int ablen; REAL fin1[1152], fin2[1152]; REAL *finnow, *finother, *finswap; int finlength; REAL adxtail, bdxtail, cdxtail, adytail, bdytail, cdytail; INEXACT REAL adxadx1, adyady1, bdxbdx1, bdybdy1, cdxcdx1, cdycdy1; REAL adxadx0, adyady0, bdxbdx0, bdybdy0, cdxcdx0, cdycdy0; REAL aa[4], bb[4], cc[4]; INEXACT REAL aa3, bb3, cc3; INEXACT REAL ti1, tj1; REAL ti0, tj0; REAL u[4], v[4]; INEXACT REAL u3, v3; REAL temp8[8], temp16a[16], temp16b[16], temp16c[16]; REAL temp32a[32], temp32b[32], temp48[48], temp64[64]; int temp8len, temp16alen, temp16blen, temp16clen; int temp32alen, temp32blen, temp48len, temp64len; REAL axtbb[8], axtcc[8], aytbb[8], aytcc[8]; int axtbblen, axtcclen, aytbblen, aytcclen; REAL bxtaa[8], bxtcc[8], bytaa[8], bytcc[8]; int bxtaalen, bxtcclen, bytaalen, bytcclen; REAL cxtaa[8], cxtbb[8], cytaa[8], cytbb[8]; int cxtaalen, cxtbblen, cytaalen, cytbblen; REAL axtbc[8], aytbc[8], bxtca[8], bytca[8], cxtab[8], cytab[8]; int axtbclen, aytbclen, bxtcalen, bytcalen, cxtablen, cytablen; REAL axtbct[16], aytbct[16], bxtcat[16], bytcat[16], cxtabt[16], cytabt[16]; int axtbctlen, aytbctlen, bxtcatlen, bytcatlen, cxtabtlen, cytabtlen; REAL axtbctt[8], aytbctt[8], bxtcatt[8]; REAL bytcatt[8], cxtabtt[8], cytabtt[8]; int axtbcttlen, aytbcttlen, bxtcattlen, bytcattlen, cxtabttlen, cytabttlen; REAL abt[8], bct[8], cat[8]; int abtlen, bctlen, catlen; REAL abtt[4], bctt[4], catt[4]; int abttlen, bcttlen, cattlen; INEXACT REAL abtt3, bctt3, catt3; REAL negate; INEXACT REAL bvirt; REAL avirt, bround, around; INEXACT REAL c; INEXACT REAL abig; REAL ahi, alo, bhi, blo; REAL err1, err2, err3; INEXACT REAL _i, _j; REAL _0; adx = (REAL) ( pa[0] - pd[0] ); bdx = (REAL) ( pb[0] - pd[0] ); cdx = (REAL) ( pc[0] - pd[0] ); ady = (REAL) ( pa[1] - pd[1] ); bdy = (REAL) ( pb[1] - pd[1] ); cdy = (REAL) ( pc[1] - pd[1] ); Two_Product( bdx, cdy, bdxcdy1, bdxcdy0 ); Two_Product( cdx, bdy, cdxbdy1, cdxbdy0 ); Two_Two_Diff( bdxcdy1, bdxcdy0, cdxbdy1, cdxbdy0, bc3, bc[2], bc[1], bc[0] ); bc[3] = bc3; axbclen = scale_expansion_zeroelim( 4, bc, adx, axbc ); axxbclen = scale_expansion_zeroelim( axbclen, axbc, adx, axxbc ); aybclen = scale_expansion_zeroelim( 4, bc, ady, aybc ); ayybclen = scale_expansion_zeroelim( aybclen, aybc, ady, ayybc ); alen = fast_expansion_sum_zeroelim( axxbclen, axxbc, ayybclen, ayybc, adet ); Two_Product( cdx, ady, cdxady1, cdxady0 ); Two_Product( adx, cdy, adxcdy1, adxcdy0 ); Two_Two_Diff( cdxady1, cdxady0, adxcdy1, adxcdy0, ca3, ca[2], ca[1], ca[0] ); ca[3] = ca3; bxcalen = scale_expansion_zeroelim( 4, ca, bdx, bxca ); bxxcalen = scale_expansion_zeroelim( bxcalen, bxca, bdx, bxxca ); bycalen = scale_expansion_zeroelim( 4, ca, bdy, byca ); byycalen = scale_expansion_zeroelim( bycalen, byca, bdy, byyca ); blen = fast_expansion_sum_zeroelim( bxxcalen, bxxca, byycalen, byyca, bdet ); Two_Product( adx, bdy, adxbdy1, adxbdy0 ); Two_Product( bdx, ady, bdxady1, bdxady0 ); Two_Two_Diff( adxbdy1, adxbdy0, bdxady1, bdxady0, ab3, ab[2], ab[1], ab[0] ); ab[3] = ab3; cxablen = scale_expansion_zeroelim( 4, ab, cdx, cxab ); cxxablen = scale_expansion_zeroelim( cxablen, cxab, cdx, cxxab ); cyablen = scale_expansion_zeroelim( 4, ab, cdy, cyab ); cyyablen = scale_expansion_zeroelim( cyablen, cyab, cdy, cyyab ); clen = fast_expansion_sum_zeroelim( cxxablen, cxxab, cyyablen, cyyab, cdet ); ablen = fast_expansion_sum_zeroelim( alen, adet, blen, bdet, abdet ); finlength = fast_expansion_sum_zeroelim( ablen, abdet, clen, cdet, fin1 ); det = estimate( finlength, fin1 ); errbound = (REAL)( iccerrboundB * permanent ); if ( ( det >= errbound ) || ( -det >= errbound ) ) { return det; } Two_Diff_Tail( pa[0], pd[0], adx, adxtail ); Two_Diff_Tail( pa[1], pd[1], ady, adytail ); Two_Diff_Tail( pb[0], pd[0], bdx, bdxtail ); Two_Diff_Tail( pb[1], pd[1], bdy, bdytail ); Two_Diff_Tail( pc[0], pd[0], cdx, cdxtail ); Two_Diff_Tail( pc[1], pd[1], cdy, cdytail ); if ( ( adxtail == 0.0 ) && ( bdxtail == 0.0 ) && ( cdxtail == 0.0 ) && ( adytail == 0.0 ) && ( bdytail == 0.0 ) && ( cdytail == 0.0 ) ) { return det; } errbound = (REAL)( iccerrboundC * permanent + resulterrbound * Absolute( det ) ); det += (REAL)( ( ( adx * adx + ady * ady ) * ( ( bdx * cdytail + cdy * bdxtail ) - ( bdy * cdxtail + cdx * bdytail ) ) + 2.0 * ( adx * adxtail + ady * adytail ) * ( bdx * cdy - bdy * cdx ) ) + ( ( bdx * bdx + bdy * bdy ) * ( ( cdx * adytail + ady * cdxtail ) - ( cdy * adxtail + adx * cdytail ) ) + 2.0 * ( bdx * bdxtail + bdy * bdytail ) * ( cdx * ady - cdy * adx ) ) + ( ( cdx * cdx + cdy * cdy ) * ( ( adx * bdytail + bdy * adxtail ) - ( ady * bdxtail + bdx * adytail ) ) + 2.0 * ( cdx * cdxtail + cdy * cdytail ) * ( adx * bdy - ady * bdx ) ) ); if ( ( det >= errbound ) || ( -det >= errbound ) ) { return det; } finnow = fin1; finother = fin2; if ( ( bdxtail != 0.0 ) || ( bdytail != 0.0 ) || ( cdxtail != 0.0 ) || ( cdytail != 0.0 ) ) { Square( adx, adxadx1, adxadx0 ); Square( ady, adyady1, adyady0 ); Two_Two_Sum( adxadx1, adxadx0, adyady1, adyady0, aa3, aa[2], aa[1], aa[0] ); aa[3] = aa3; } if ( ( cdxtail != 0.0 ) || ( cdytail != 0.0 ) || ( adxtail != 0.0 ) || ( adytail != 0.0 ) ) { Square( bdx, bdxbdx1, bdxbdx0 ); Square( bdy, bdybdy1, bdybdy0 ); Two_Two_Sum( bdxbdx1, bdxbdx0, bdybdy1, bdybdy0, bb3, bb[2], bb[1], bb[0] ); bb[3] = bb3; } if ( ( adxtail != 0.0 ) || ( adytail != 0.0 ) || ( bdxtail != 0.0 ) || ( bdytail != 0.0 ) ) { Square( cdx, cdxcdx1, cdxcdx0 ); Square( cdy, cdycdy1, cdycdy0 ); Two_Two_Sum( cdxcdx1, cdxcdx0, cdycdy1, cdycdy0, cc3, cc[2], cc[1], cc[0] ); cc[3] = cc3; } if ( adxtail != 0.0 ) { axtbclen = scale_expansion_zeroelim( 4, bc, adxtail, axtbc ); temp16alen = scale_expansion_zeroelim( axtbclen, axtbc, 2.0 * adx, temp16a ); axtcclen = scale_expansion_zeroelim( 4, cc, adxtail, axtcc ); temp16blen = scale_expansion_zeroelim( axtcclen, axtcc, bdy, temp16b ); axtbblen = scale_expansion_zeroelim( 4, bb, adxtail, axtbb ); temp16clen = scale_expansion_zeroelim( axtbblen, axtbb, -cdy, temp16c ); temp32alen = fast_expansion_sum_zeroelim( temp16alen, temp16a, temp16blen, temp16b, temp32a ); temp48len = fast_expansion_sum_zeroelim( temp16clen, temp16c, temp32alen, temp32a, temp48 ); finlength = fast_expansion_sum_zeroelim( finlength, finnow, temp48len, temp48, finother ); finswap = finnow; finnow = finother; finother = finswap; } if ( adytail != 0.0 ) { aytbclen = scale_expansion_zeroelim( 4, bc, adytail, aytbc ); temp16alen = scale_expansion_zeroelim( aytbclen, aytbc, 2.0 * ady, temp16a ); aytbblen = scale_expansion_zeroelim( 4, bb, adytail, aytbb ); temp16blen = scale_expansion_zeroelim( aytbblen, aytbb, cdx, temp16b ); aytcclen = scale_expansion_zeroelim( 4, cc, adytail, aytcc ); temp16clen = scale_expansion_zeroelim( aytcclen, aytcc, -bdx, temp16c ); temp32alen = fast_expansion_sum_zeroelim( temp16alen, temp16a, temp16blen, temp16b, temp32a ); temp48len = fast_expansion_sum_zeroelim( temp16clen, temp16c, temp32alen, temp32a, temp48 ); finlength = fast_expansion_sum_zeroelim( finlength, finnow, temp48len, temp48, finother ); finswap = finnow; finnow = finother; finother = finswap; } if ( bdxtail != 0.0 ) { bxtcalen = scale_expansion_zeroelim( 4, ca, bdxtail, bxtca ); temp16alen = scale_expansion_zeroelim( bxtcalen, bxtca, 2.0 * bdx, temp16a ); bxtaalen = scale_expansion_zeroelim( 4, aa, bdxtail, bxtaa ); temp16blen = scale_expansion_zeroelim( bxtaalen, bxtaa, cdy, temp16b ); bxtcclen = scale_expansion_zeroelim( 4, cc, bdxtail, bxtcc ); temp16clen = scale_expansion_zeroelim( bxtcclen, bxtcc, -ady, temp16c ); temp32alen = fast_expansion_sum_zeroelim( temp16alen, temp16a, temp16blen, temp16b, temp32a ); temp48len = fast_expansion_sum_zeroelim( temp16clen, temp16c, temp32alen, temp32a, temp48 ); finlength = fast_expansion_sum_zeroelim( finlength, finnow, temp48len, temp48, finother ); finswap = finnow; finnow = finother; finother = finswap; } if ( bdytail != 0.0 ) { bytcalen = scale_expansion_zeroelim( 4, ca, bdytail, bytca ); temp16alen = scale_expansion_zeroelim( bytcalen, bytca, 2.0 * bdy, temp16a ); bytcclen = scale_expansion_zeroelim( 4, cc, bdytail, bytcc ); temp16blen = scale_expansion_zeroelim( bytcclen, bytcc, adx, temp16b ); bytaalen = scale_expansion_zeroelim( 4, aa, bdytail, bytaa ); temp16clen = scale_expansion_zeroelim( bytaalen, bytaa, -cdx, temp16c ); temp32alen = fast_expansion_sum_zeroelim( temp16alen, temp16a, temp16blen, temp16b, temp32a ); temp48len = fast_expansion_sum_zeroelim( temp16clen, temp16c, temp32alen, temp32a, temp48 ); finlength = fast_expansion_sum_zeroelim( finlength, finnow, temp48len, temp48, finother ); finswap = finnow; finnow = finother; finother = finswap; } if ( cdxtail != 0.0 ) { cxtablen = scale_expansion_zeroelim( 4, ab, cdxtail, cxtab ); temp16alen = scale_expansion_zeroelim( cxtablen, cxtab, 2.0 * cdx, temp16a ); cxtbblen = scale_expansion_zeroelim( 4, bb, cdxtail, cxtbb ); temp16blen = scale_expansion_zeroelim( cxtbblen, cxtbb, ady, temp16b ); cxtaalen = scale_expansion_zeroelim( 4, aa, cdxtail, cxtaa ); temp16clen = scale_expansion_zeroelim( cxtaalen, cxtaa, -bdy, temp16c ); temp32alen = fast_expansion_sum_zeroelim( temp16alen, temp16a, temp16blen, temp16b, temp32a ); temp48len = fast_expansion_sum_zeroelim( temp16clen, temp16c, temp32alen, temp32a, temp48 ); finlength = fast_expansion_sum_zeroelim( finlength, finnow, temp48len, temp48, finother ); finswap = finnow; finnow = finother; finother = finswap; } if ( cdytail != 0.0 ) { cytablen = scale_expansion_zeroelim( 4, ab, cdytail, cytab ); temp16alen = scale_expansion_zeroelim( cytablen, cytab, 2.0 * cdy, temp16a ); cytaalen = scale_expansion_zeroelim( 4, aa, cdytail, cytaa ); temp16blen = scale_expansion_zeroelim( cytaalen, cytaa, bdx, temp16b ); cytbblen = scale_expansion_zeroelim( 4, bb, cdytail, cytbb ); temp16clen = scale_expansion_zeroelim( cytbblen, cytbb, -adx, temp16c ); temp32alen = fast_expansion_sum_zeroelim( temp16alen, temp16a, temp16blen, temp16b, temp32a ); temp48len = fast_expansion_sum_zeroelim( temp16clen, temp16c, temp32alen, temp32a, temp48 ); finlength = fast_expansion_sum_zeroelim( finlength, finnow, temp48len, temp48, finother ); finswap = finnow; finnow = finother; finother = finswap; } if ( ( adxtail != 0.0 ) || ( adytail != 0.0 ) ) { if ( ( bdxtail != 0.0 ) || ( bdytail != 0.0 ) || ( cdxtail != 0.0 ) || ( cdytail != 0.0 ) ) { Two_Product( bdxtail, cdy, ti1, ti0 ); Two_Product( bdx, cdytail, tj1, tj0 ); Two_Two_Sum( ti1, ti0, tj1, tj0, u3, u[2], u[1], u[0] ); u[3] = u3; negate = -bdy; Two_Product( cdxtail, negate, ti1, ti0 ); negate = -bdytail; Two_Product( cdx, negate, tj1, tj0 ); Two_Two_Sum( ti1, ti0, tj1, tj0, v3, v[2], v[1], v[0] ); v[3] = v3; bctlen = fast_expansion_sum_zeroelim( 4, u, 4, v, bct ); Two_Product( bdxtail, cdytail, ti1, ti0 ); Two_Product( cdxtail, bdytail, tj1, tj0 ); Two_Two_Diff( ti1, ti0, tj1, tj0, bctt3, bctt[2], bctt[1], bctt[0] ); bctt[3] = bctt3; bcttlen = 4; } else { bct[0] = 0.0; bctlen = 1; bctt[0] = 0.0; bcttlen = 1; } if ( adxtail != 0.0 ) { temp16alen = scale_expansion_zeroelim( axtbclen, axtbc, adxtail, temp16a ); axtbctlen = scale_expansion_zeroelim( bctlen, bct, adxtail, axtbct ); temp32alen = scale_expansion_zeroelim( axtbctlen, axtbct, 2.0 * adx, temp32a ); temp48len = fast_expansion_sum_zeroelim( temp16alen, temp16a, temp32alen, temp32a, temp48 ); finlength = fast_expansion_sum_zeroelim( finlength, finnow, temp48len, temp48, finother ); finswap = finnow; finnow = finother; finother = finswap; if ( bdytail != 0.0 ) { temp8len = scale_expansion_zeroelim( 4, cc, adxtail, temp8 ); temp16alen = scale_expansion_zeroelim( temp8len, temp8, bdytail, temp16a ); finlength = fast_expansion_sum_zeroelim( finlength, finnow, temp16alen, temp16a, finother ); finswap = finnow; finnow = finother; finother = finswap; } if ( cdytail != 0.0 ) { temp8len = scale_expansion_zeroelim( 4, bb, -adxtail, temp8 ); temp16alen = scale_expansion_zeroelim( temp8len, temp8, cdytail, temp16a ); finlength = fast_expansion_sum_zeroelim( finlength, finnow, temp16alen, temp16a, finother ); finswap = finnow; finnow = finother; finother = finswap; } temp32alen = scale_expansion_zeroelim( axtbctlen, axtbct, adxtail, temp32a ); axtbcttlen = scale_expansion_zeroelim( bcttlen, bctt, adxtail, axtbctt ); temp16alen = scale_expansion_zeroelim( axtbcttlen, axtbctt, 2.0 * adx, temp16a ); temp16blen = scale_expansion_zeroelim( axtbcttlen, axtbctt, adxtail, temp16b ); temp32blen = fast_expansion_sum_zeroelim( temp16alen, temp16a, temp16blen, temp16b, temp32b ); temp64len = fast_expansion_sum_zeroelim( temp32alen, temp32a, temp32blen, temp32b, temp64 ); finlength = fast_expansion_sum_zeroelim( finlength, finnow, temp64len, temp64, finother ); finswap = finnow; finnow = finother; finother = finswap; } if ( adytail != 0.0 ) { temp16alen = scale_expansion_zeroelim( aytbclen, aytbc, adytail, temp16a ); aytbctlen = scale_expansion_zeroelim( bctlen, bct, adytail, aytbct ); temp32alen = scale_expansion_zeroelim( aytbctlen, aytbct, 2.0 * ady, temp32a ); temp48len = fast_expansion_sum_zeroelim( temp16alen, temp16a, temp32alen, temp32a, temp48 ); finlength = fast_expansion_sum_zeroelim( finlength, finnow, temp48len, temp48, finother ); finswap = finnow; finnow = finother; finother = finswap; temp32alen = scale_expansion_zeroelim( aytbctlen, aytbct, adytail, temp32a ); aytbcttlen = scale_expansion_zeroelim( bcttlen, bctt, adytail, aytbctt ); temp16alen = scale_expansion_zeroelim( aytbcttlen, aytbctt, 2.0 * ady, temp16a ); temp16blen = scale_expansion_zeroelim( aytbcttlen, aytbctt, adytail, temp16b ); temp32blen = fast_expansion_sum_zeroelim( temp16alen, temp16a, temp16blen, temp16b, temp32b ); temp64len = fast_expansion_sum_zeroelim( temp32alen, temp32a, temp32blen, temp32b, temp64 ); finlength = fast_expansion_sum_zeroelim( finlength, finnow, temp64len, temp64, finother ); finswap = finnow; finnow = finother; finother = finswap; } } if ( ( bdxtail != 0.0 ) || ( bdytail != 0.0 ) ) { if ( ( cdxtail != 0.0 ) || ( cdytail != 0.0 ) || ( adxtail != 0.0 ) || ( adytail != 0.0 ) ) { Two_Product( cdxtail, ady, ti1, ti0 ); Two_Product( cdx, adytail, tj1, tj0 ); Two_Two_Sum( ti1, ti0, tj1, tj0, u3, u[2], u[1], u[0] ); u[3] = u3; negate = -cdy; Two_Product( adxtail, negate, ti1, ti0 ); negate = -cdytail; Two_Product( adx, negate, tj1, tj0 ); Two_Two_Sum( ti1, ti0, tj1, tj0, v3, v[2], v[1], v[0] ); v[3] = v3; catlen = fast_expansion_sum_zeroelim( 4, u, 4, v, cat ); Two_Product( cdxtail, adytail, ti1, ti0 ); Two_Product( adxtail, cdytail, tj1, tj0 ); Two_Two_Diff( ti1, ti0, tj1, tj0, catt3, catt[2], catt[1], catt[0] ); catt[3] = catt3; cattlen = 4; } else { cat[0] = 0.0; catlen = 1; catt[0] = 0.0; cattlen = 1; } if ( bdxtail != 0.0 ) { temp16alen = scale_expansion_zeroelim( bxtcalen, bxtca, bdxtail, temp16a ); bxtcatlen = scale_expansion_zeroelim( catlen, cat, bdxtail, bxtcat ); temp32alen = scale_expansion_zeroelim( bxtcatlen, bxtcat, 2.0 * bdx, temp32a ); temp48len = fast_expansion_sum_zeroelim( temp16alen, temp16a, temp32alen, temp32a, temp48 ); finlength = fast_expansion_sum_zeroelim( finlength, finnow, temp48len, temp48, finother ); finswap = finnow; finnow = finother; finother = finswap; if ( cdytail != 0.0 ) { temp8len = scale_expansion_zeroelim( 4, aa, bdxtail, temp8 ); temp16alen = scale_expansion_zeroelim( temp8len, temp8, cdytail, temp16a ); finlength = fast_expansion_sum_zeroelim( finlength, finnow, temp16alen, temp16a, finother ); finswap = finnow; finnow = finother; finother = finswap; } if ( adytail != 0.0 ) { temp8len = scale_expansion_zeroelim( 4, cc, -bdxtail, temp8 ); temp16alen = scale_expansion_zeroelim( temp8len, temp8, adytail, temp16a ); finlength = fast_expansion_sum_zeroelim( finlength, finnow, temp16alen, temp16a, finother ); finswap = finnow; finnow = finother; finother = finswap; } temp32alen = scale_expansion_zeroelim( bxtcatlen, bxtcat, bdxtail, temp32a ); bxtcattlen = scale_expansion_zeroelim( cattlen, catt, bdxtail, bxtcatt ); temp16alen = scale_expansion_zeroelim( bxtcattlen, bxtcatt, 2.0 * bdx, temp16a ); temp16blen = scale_expansion_zeroelim( bxtcattlen, bxtcatt, bdxtail, temp16b ); temp32blen = fast_expansion_sum_zeroelim( temp16alen, temp16a, temp16blen, temp16b, temp32b ); temp64len = fast_expansion_sum_zeroelim( temp32alen, temp32a, temp32blen, temp32b, temp64 ); finlength = fast_expansion_sum_zeroelim( finlength, finnow, temp64len, temp64, finother ); finswap = finnow; finnow = finother; finother = finswap; } if ( bdytail != 0.0 ) { temp16alen = scale_expansion_zeroelim( bytcalen, bytca, bdytail, temp16a ); bytcatlen = scale_expansion_zeroelim( catlen, cat, bdytail, bytcat ); temp32alen = scale_expansion_zeroelim( bytcatlen, bytcat, 2.0 * bdy, temp32a ); temp48len = fast_expansion_sum_zeroelim( temp16alen, temp16a, temp32alen, temp32a, temp48 ); finlength = fast_expansion_sum_zeroelim( finlength, finnow, temp48len, temp48, finother ); finswap = finnow; finnow = finother; finother = finswap; temp32alen = scale_expansion_zeroelim( bytcatlen, bytcat, bdytail, temp32a ); bytcattlen = scale_expansion_zeroelim( cattlen, catt, bdytail, bytcatt ); temp16alen = scale_expansion_zeroelim( bytcattlen, bytcatt, 2.0 * bdy, temp16a ); temp16blen = scale_expansion_zeroelim( bytcattlen, bytcatt, bdytail, temp16b ); temp32blen = fast_expansion_sum_zeroelim( temp16alen, temp16a, temp16blen, temp16b, temp32b ); temp64len = fast_expansion_sum_zeroelim( temp32alen, temp32a, temp32blen, temp32b, temp64 ); finlength = fast_expansion_sum_zeroelim( finlength, finnow, temp64len, temp64, finother ); finswap = finnow; finnow = finother; finother = finswap; } } if ( ( cdxtail != 0.0 ) || ( cdytail != 0.0 ) ) { if ( ( adxtail != 0.0 ) || ( adytail != 0.0 ) || ( bdxtail != 0.0 ) || ( bdytail != 0.0 ) ) { Two_Product( adxtail, bdy, ti1, ti0 ); Two_Product( adx, bdytail, tj1, tj0 ); Two_Two_Sum( ti1, ti0, tj1, tj0, u3, u[2], u[1], u[0] ); u[3] = u3; negate = -ady; Two_Product( bdxtail, negate, ti1, ti0 ); negate = -adytail; Two_Product( bdx, negate, tj1, tj0 ); Two_Two_Sum( ti1, ti0, tj1, tj0, v3, v[2], v[1], v[0] ); v[3] = v3; abtlen = fast_expansion_sum_zeroelim( 4, u, 4, v, abt ); Two_Product( adxtail, bdytail, ti1, ti0 ); Two_Product( bdxtail, adytail, tj1, tj0 ); Two_Two_Diff( ti1, ti0, tj1, tj0, abtt3, abtt[2], abtt[1], abtt[0] ); abtt[3] = abtt3; abttlen = 4; } else { abt[0] = 0.0; abtlen = 1; abtt[0] = 0.0; abttlen = 1; } if ( cdxtail != 0.0 ) { temp16alen = scale_expansion_zeroelim( cxtablen, cxtab, cdxtail, temp16a ); cxtabtlen = scale_expansion_zeroelim( abtlen, abt, cdxtail, cxtabt ); temp32alen = scale_expansion_zeroelim( cxtabtlen, cxtabt, 2.0 * cdx, temp32a ); temp48len = fast_expansion_sum_zeroelim( temp16alen, temp16a, temp32alen, temp32a, temp48 ); finlength = fast_expansion_sum_zeroelim( finlength, finnow, temp48len, temp48, finother ); finswap = finnow; finnow = finother; finother = finswap; if ( adytail != 0.0 ) { temp8len = scale_expansion_zeroelim( 4, bb, cdxtail, temp8 ); temp16alen = scale_expansion_zeroelim( temp8len, temp8, adytail, temp16a ); finlength = fast_expansion_sum_zeroelim( finlength, finnow, temp16alen, temp16a, finother ); finswap = finnow; finnow = finother; finother = finswap; } if ( bdytail != 0.0 ) { temp8len = scale_expansion_zeroelim( 4, aa, -cdxtail, temp8 ); temp16alen = scale_expansion_zeroelim( temp8len, temp8, bdytail, temp16a ); finlength = fast_expansion_sum_zeroelim( finlength, finnow, temp16alen, temp16a, finother ); finswap = finnow; finnow = finother; finother = finswap; } temp32alen = scale_expansion_zeroelim( cxtabtlen, cxtabt, cdxtail, temp32a ); cxtabttlen = scale_expansion_zeroelim( abttlen, abtt, cdxtail, cxtabtt ); temp16alen = scale_expansion_zeroelim( cxtabttlen, cxtabtt, 2.0 * cdx, temp16a ); temp16blen = scale_expansion_zeroelim( cxtabttlen, cxtabtt, cdxtail, temp16b ); temp32blen = fast_expansion_sum_zeroelim( temp16alen, temp16a, temp16blen, temp16b, temp32b ); temp64len = fast_expansion_sum_zeroelim( temp32alen, temp32a, temp32blen, temp32b, temp64 ); finlength = fast_expansion_sum_zeroelim( finlength, finnow, temp64len, temp64, finother ); finswap = finnow; finnow = finother; finother = finswap; } if ( cdytail != 0.0 ) { temp16alen = scale_expansion_zeroelim( cytablen, cytab, cdytail, temp16a ); cytabtlen = scale_expansion_zeroelim( abtlen, abt, cdytail, cytabt ); temp32alen = scale_expansion_zeroelim( cytabtlen, cytabt, 2.0 * cdy, temp32a ); temp48len = fast_expansion_sum_zeroelim( temp16alen, temp16a, temp32alen, temp32a, temp48 ); finlength = fast_expansion_sum_zeroelim( finlength, finnow, temp48len, temp48, finother ); finswap = finnow; finnow = finother; finother = finswap; temp32alen = scale_expansion_zeroelim( cytabtlen, cytabt, cdytail, temp32a ); cytabttlen = scale_expansion_zeroelim( abttlen, abtt, cdytail, cytabtt ); temp16alen = scale_expansion_zeroelim( cytabttlen, cytabtt, 2.0 * cdy, temp16a ); temp16blen = scale_expansion_zeroelim( cytabttlen, cytabtt, cdytail, temp16b ); temp32blen = fast_expansion_sum_zeroelim( temp16alen, temp16a, temp16blen, temp16b, temp32b ); temp64len = fast_expansion_sum_zeroelim( temp32alen, temp32a, temp32blen, temp32b, temp64 ); finlength = fast_expansion_sum_zeroelim( finlength, finnow, temp64len, temp64, finother ); finswap = finnow; finnow = finother; finother = finswap; } } return finnow[finlength - 1]; } REAL incircle( pa, pb, pc, pd ) point pa; point pb; point pc; point pd; { REAL adx, bdx, cdx, ady, bdy, cdy; REAL bdxcdy, cdxbdy, cdxady, adxcdy, adxbdy, bdxady; REAL alift, blift, clift; REAL det; REAL permanent, errbound; incirclecount++; adx = pa[0] - pd[0]; bdx = pb[0] - pd[0]; cdx = pc[0] - pd[0]; ady = pa[1] - pd[1]; bdy = pb[1] - pd[1]; cdy = pc[1] - pd[1]; bdxcdy = bdx * cdy; cdxbdy = cdx * bdy; alift = adx * adx + ady * ady; cdxady = cdx * ady; adxcdy = adx * cdy; blift = bdx * bdx + bdy * bdy; adxbdy = adx * bdy; bdxady = bdx * ady; clift = cdx * cdx + cdy * cdy; det = alift * ( bdxcdy - cdxbdy ) + blift * ( cdxady - adxcdy ) + clift * ( adxbdy - bdxady ); if ( noexact ) { return det; } permanent = ( Absolute( bdxcdy ) + Absolute( cdxbdy ) ) * alift + ( Absolute( cdxady ) + Absolute( adxcdy ) ) * blift + ( Absolute( adxbdy ) + Absolute( bdxady ) ) * clift; errbound = iccerrboundA * permanent; if ( ( det > errbound ) || ( -det > errbound ) ) { return det; } return incircleadapt( pa, pb, pc, pd, permanent ); } /** **/ /** **/ /********* Determinant evaluation routines end here *********/ /*****************************************************************************/ /* */ /* triangleinit() Initialize some variables. */ /* */ /*****************************************************************************/ void triangleinit(){ points.maxitems = triangles.maxitems = shelles.maxitems = viri.maxitems = badsegments.maxitems = badtriangles.maxitems = splaynodes.maxitems = 0l; points.itembytes = triangles.itembytes = shelles.itembytes = viri.itembytes = badsegments.itembytes = badtriangles.itembytes = splaynodes.itembytes = 0; recenttri.tri = (triangle *) NULL; /* No triangle has been visited yet. */ samples = 1; /* Point location should take at least one sample. */ checksegments = 0; /* There are no segments in the triangulation yet. */ incirclecount = counterclockcount = hyperbolacount = 0; circumcentercount = circletopcount = 0; randomseed = 1; exactinit(); /* Initialize exact arithmetic constants. */ } /*****************************************************************************/ /* */ /* randomnation() Generate a random number between 0 and `choices' - 1. */ /* */ /* This is a simple linear congruential random number generator. Hence, it */ /* is a bad random number generator, but good enough for most randomized */ /* geometric algorithms. */ /* */ /*****************************************************************************/ unsigned long randomnation( choices ) unsigned int choices; { randomseed = ( randomseed * 1366l + 150889l ) % 714025l; return randomseed / ( 714025l / choices + 1 ); } /********* Mesh quality testing routines begin here *********/ /** **/ /** **/ /*****************************************************************************/ /* */ /* checkmesh() Test the mesh for topological consistency. */ /* */ /*****************************************************************************/ #ifndef REDUCED void checkmesh(){ struct triedge triangleloop; struct triedge oppotri, oppooppotri; point triorg, tridest, triapex; point oppoorg, oppodest; int horrors; int saveexact; triangle ptr; /* Temporary variable used by sym(). */ /* Temporarily turn on exact arithmetic if it's off. */ saveexact = noexact; noexact = 0; if ( !quiet ) { printf( " Checking consistency of mesh...\n" ); } horrors = 0; /* Run through the list of triangles, checking each one. */ traversalinit( &triangles ); triangleloop.tri = triangletraverse(); while ( triangleloop.tri != (triangle *) NULL ) { /* Check all three edges of the triangle. */ for ( triangleloop.orient = 0; triangleloop.orient < 3; triangleloop.orient++ ) { org( triangleloop, triorg ); dest( triangleloop, tridest ); if ( triangleloop.orient == 0 ) { /* Only test for inversion once. */ /* Test if the triangle is flat or inverted. */ apex( triangleloop, triapex ); if ( counterclockwise( triorg, tridest, triapex ) <= 0.0 ) { printf( " !! !! Inverted " ); printtriangle( &triangleloop ); horrors++; } } /* Find the neighboring triangle on this edge. */ sym( triangleloop, oppotri ); if ( oppotri.tri != dummytri ) { /* Check that the triangle's neighbor knows it's a neighbor. */ sym( oppotri, oppooppotri ); if ( ( triangleloop.tri != oppooppotri.tri ) || ( triangleloop.orient != oppooppotri.orient ) ) { printf( " !! !! Asymmetric triangle-triangle bond:\n" ); if ( triangleloop.tri == oppooppotri.tri ) { printf( " (Right triangle, wrong orientation)\n" ); } printf( " First " ); printtriangle( &triangleloop ); printf( " Second (nonreciprocating) " ); printtriangle( &oppotri ); horrors++; } /* Check that both triangles agree on the identities */ /* of their shared vertices. */ org( oppotri, oppoorg ); dest( oppotri, oppodest ); if ( ( triorg != oppodest ) || ( tridest != oppoorg ) ) { printf( " !! !! Mismatched edge coordinates between two triangles:\n" ); printf( " First mismatched " ); printtriangle( &triangleloop ); printf( " Second mismatched " ); printtriangle( &oppotri ); horrors++; } } } triangleloop.tri = triangletraverse(); } if ( horrors == 0 ) { if ( !quiet ) { printf( " In my studied opinion, the mesh appears to be consistent.\n" ); } } else if ( horrors == 1 ) { printf( " !! !! !! !! Precisely one festering wound discovered.\n" ); } else { printf( " !! !! !! !! %d abominations witnessed.\n", horrors ); } /* Restore the status of exact arithmetic. */ noexact = saveexact; } #endif /* not REDUCED */ /*****************************************************************************/ /* */ /* checkdelaunay() Ensure that the mesh is (constrained) Delaunay. */ /* */ /*****************************************************************************/ #ifndef REDUCED void checkdelaunay(){ struct triedge triangleloop; struct triedge oppotri; struct edge opposhelle; point triorg, tridest, triapex; point oppoapex; int shouldbedelaunay; int horrors; int saveexact; triangle ptr; /* Temporary variable used by sym(). */ shelle sptr; /* Temporary variable used by tspivot(). */ /* Temporarily turn on exact arithmetic if it's off. */ saveexact = noexact; noexact = 0; if ( !quiet ) { printf( " Checking Delaunay property of mesh...\n" ); } horrors = 0; /* Run through the list of triangles, checking each one. */ traversalinit( &triangles ); triangleloop.tri = triangletraverse(); while ( triangleloop.tri != (triangle *) NULL ) { /* Check all three edges of the triangle. */ for ( triangleloop.orient = 0; triangleloop.orient < 3; triangleloop.orient++ ) { org( triangleloop, triorg ); dest( triangleloop, tridest ); apex( triangleloop, triapex ); sym( triangleloop, oppotri ); apex( oppotri, oppoapex ); /* Only test that the edge is locally Delaunay if there is an */ /* adjoining triangle whose pointer is larger (to ensure that */ /* each pair isn't tested twice). */ shouldbedelaunay = ( oppotri.tri != dummytri ) && ( triapex != (point) NULL ) && ( oppoapex != (point) NULL ) && ( triangleloop.tri < oppotri.tri ); if ( checksegments && shouldbedelaunay ) { /* If a shell edge separates the triangles, then the edge is */ /* constrained, so no local Delaunay test should be done. */ tspivot( triangleloop, opposhelle ); if ( opposhelle.sh != dummysh ) { shouldbedelaunay = 0; } } if ( shouldbedelaunay ) { if ( incircle( triorg, tridest, triapex, oppoapex ) > 0.0 ) { printf( " !! !! Non-Delaunay pair of triangles:\n" ); printf( " First non-Delaunay " ); printtriangle( &triangleloop ); printf( " Second non-Delaunay " ); printtriangle( &oppotri ); horrors++; } } } triangleloop.tri = triangletraverse(); } if ( horrors == 0 ) { if ( !quiet ) { printf( " By virtue of my perceptive intelligence, I declare the mesh Delaunay.\n" ); } } else if ( horrors == 1 ) { printf( " !! !! !! !! Precisely one terrifying transgression identified.\n" ); } else { printf( " !! !! !! !! %d obscenities viewed with horror.\n", horrors ); } /* Restore the status of exact arithmetic. */ noexact = saveexact; } #endif /* not REDUCED */ /*****************************************************************************/ /* */ /* enqueuebadtri() Add a bad triangle to the end of a queue. */ /* */ /* The queue is actually a set of 64 queues. I use multiple queues to give */ /* priority to smaller angles. I originally implemented a heap, but the */ /* queues are (to my surprise) much faster. */ /* */ /*****************************************************************************/ #ifndef CDT_ONLY void enqueuebadtri( instri, angle, insapex, insorg, insdest ) struct triedge *instri; REAL angle; point insapex; point insorg; point insdest; { struct badface *newface; int queuenumber; if ( verbose > 2 ) { printf( " Queueing bad triangle:\n" ); printf( " (%.12g, %.12g) (%.12g, %.12g) (%.12g, %.12g)\n", insorg[0], insorg[1], insdest[0], insdest[1], insapex[0], insapex[1] ); } /* Allocate space for the bad triangle. */ newface = (struct badface *) poolalloc( &badtriangles ); triedgecopy( *instri, newface->badfacetri ); newface->key = angle; newface->faceapex = insapex; newface->faceorg = insorg; newface->facedest = insdest; newface->nextface = (struct badface *) NULL; /* Determine the appropriate queue to put the bad triangle into. */ if ( angle > 0.6 ) { queuenumber = (int) ( 160.0 * ( angle - 0.6 ) ); if ( queuenumber > 63 ) { queuenumber = 63; } } else { /* It's not a bad angle; put the triangle in the lowest-priority queue. */ queuenumber = 0; } /* Add the triangle to the end of a queue. */ *queuetail[queuenumber] = newface; /* Maintain a pointer to the NULL pointer at the end of the queue. */ queuetail[queuenumber] = &newface->nextface; } #endif /* not CDT_ONLY */ /*****************************************************************************/ /* */ /* dequeuebadtri() Remove a triangle from the front of the queue. */ /* */ /*****************************************************************************/ #ifndef CDT_ONLY struct badface *dequeuebadtri(){ struct badface *result; int queuenumber; /* Look for a nonempty queue. */ for ( queuenumber = 63; queuenumber >= 0; queuenumber-- ) { result = queuefront[queuenumber]; if ( result != (struct badface *) NULL ) { /* Remove the triangle from the queue. */ queuefront[queuenumber] = result->nextface; /* Maintain a pointer to the NULL pointer at the end of the queue. */ if ( queuefront[queuenumber] == (struct badface *) NULL ) { queuetail[queuenumber] = &queuefront[queuenumber]; } return result; } } return (struct badface *) NULL; } #endif /* not CDT_ONLY */ /*****************************************************************************/ /* */ /* checkedge4encroach() Check a segment to see if it is encroached; add */ /* it to the list if it is. */ /* */ /* An encroached segment is an unflippable edge that has a point in its */ /* diametral circle (that is, it faces an angle greater than 90 degrees). */ /* This definition is due to Ruppert. */ /* */ /* Returns a nonzero value if the edge is encroached. */ /* */ /*****************************************************************************/ #ifndef CDT_ONLY int checkedge4encroach( testedge ) struct edge *testedge; { struct triedge neighbortri; struct edge testsym; struct edge *badedge; int addtolist; int sides; point eorg, edest, eapex; triangle ptr; /* Temporary variable used by stpivot(). */ addtolist = 0; sides = 0; sorg( *testedge, eorg ); sdest( *testedge, edest ); /* Check one neighbor of the shell edge. */ stpivot( *testedge, neighbortri ); /* Does the neighbor exist, or is this a boundary edge? */ if ( neighbortri.tri != dummytri ) { sides++; /* Find a vertex opposite this edge. */ apex( neighbortri, eapex ); /* Check whether the vertex is inside the diametral circle of the */ /* shell edge. Pythagoras' Theorem is used to check whether the */ /* angle at the vertex is greater than 90 degrees. */ if ( eapex[0] * ( eorg[0] + edest[0] ) + eapex[1] * ( eorg[1] + edest[1] ) > eapex[0] * eapex[0] + eorg[0] * edest[0] + eapex[1] * eapex[1] + eorg[1] * edest[1] ) { addtolist = 1; } } /* Check the other neighbor of the shell edge. */ ssym( *testedge, testsym ); stpivot( testsym, neighbortri ); /* Does the neighbor exist, or is this a boundary edge? */ if ( neighbortri.tri != dummytri ) { sides++; /* Find the other vertex opposite this edge. */ apex( neighbortri, eapex ); /* Check whether the vertex is inside the diametral circle of the */ /* shell edge. Pythagoras' Theorem is used to check whether the */ /* angle at the vertex is greater than 90 degrees. */ if ( eapex[0] * ( eorg[0] + edest[0] ) + eapex[1] * ( eorg[1] + edest[1] ) > eapex[0] * eapex[0] + eorg[0] * edest[0] + eapex[1] * eapex[1] + eorg[1] * edest[1] ) { addtolist += 2; } } if ( addtolist && ( !nobisect || ( ( nobisect == 1 ) && ( sides == 2 ) ) ) ) { if ( verbose > 2 ) { printf( " Queueing encroached segment (%.12g, %.12g) (%.12g, %.12g).\n", eorg[0], eorg[1], edest[0], edest[1] ); } /* Add the shell edge to the list of encroached segments. */ /* Be sure to get the orientation right. */ badedge = (struct edge *) poolalloc( &badsegments ); if ( addtolist == 1 ) { shellecopy( *testedge, *badedge ); } else { shellecopy( testsym, *badedge ); } } return addtolist; } #endif /* not CDT_ONLY */ /*****************************************************************************/ /* */ /* testtriangle() Test a face for quality measures. */ /* */ /* Tests a triangle to see if it satisfies the minimum angle condition and */ /* the maximum area condition. Triangles that aren't up to spec are added */ /* to the bad triangle queue. */ /* */ /*****************************************************************************/ #ifndef CDT_ONLY void testtriangle( testtri ) struct triedge *testtri; { struct triedge sametesttri; struct edge edge1, edge2; point torg, tdest, tapex; point anglevertex; REAL dxod, dyod, dxda, dyda, dxao, dyao; REAL dxod2, dyod2, dxda2, dyda2, dxao2, dyao2; REAL apexlen, orglen, destlen; REAL angle; REAL area; shelle sptr; /* Temporary variable used by tspivot(). */ org( *testtri, torg ); dest( *testtri, tdest ); apex( *testtri, tapex ); dxod = torg[0] - tdest[0]; dyod = torg[1] - tdest[1]; dxda = tdest[0] - tapex[0]; dyda = tdest[1] - tapex[1]; dxao = tapex[0] - torg[0]; dyao = tapex[1] - torg[1]; dxod2 = dxod * dxod; dyod2 = dyod * dyod; dxda2 = dxda * dxda; dyda2 = dyda * dyda; dxao2 = dxao * dxao; dyao2 = dyao * dyao; /* Find the lengths of the triangle's three edges. */ apexlen = dxod2 + dyod2; orglen = dxda2 + dyda2; destlen = dxao2 + dyao2; if ( ( apexlen < orglen ) && ( apexlen < destlen ) ) { /* The edge opposite the apex is shortest. */ /* Find the square of the cosine of the angle at the apex. */ angle = dxda * dxao + dyda * dyao; angle = angle * angle / ( orglen * destlen ); anglevertex = tapex; lnext( *testtri, sametesttri ); tspivot( sametesttri, edge1 ); lnextself( sametesttri ); tspivot( sametesttri, edge2 ); } else if ( orglen < destlen ) { /* The edge opposite the origin is shortest. */ /* Find the square of the cosine of the angle at the origin. */ angle = dxod * dxao + dyod * dyao; angle = angle * angle / ( apexlen * destlen ); anglevertex = torg; tspivot( *testtri, edge1 ); lprev( *testtri, sametesttri ); tspivot( sametesttri, edge2 ); } else { /* The edge opposite the destination is shortest. */ /* Find the square of the cosine of the angle at the destination. */ angle = dxod * dxda + dyod * dyda; angle = angle * angle / ( apexlen * orglen ); anglevertex = tdest; tspivot( *testtri, edge1 ); lnext( *testtri, sametesttri ); tspivot( sametesttri, edge2 ); } /* Check if both edges that form the angle are segments. */ if ( ( edge1.sh != dummysh ) && ( edge2.sh != dummysh ) ) { /* The angle is a segment intersection. */ if ( ( angle > 0.9924 ) && !quiet ) { /* Roughly 5 degrees. */ if ( angle > 1.0 ) { /* Beware of a floating exception in acos(). */ angle = 1.0; } /* Find the actual angle in degrees, for printing. */ angle = acos( sqrt( angle ) ) * ( 180.0 / PI ); printf( "Warning: Small angle (%.4g degrees) between segments at point\n", angle ); printf( " (%.12g, %.12g)\n", anglevertex[0], anglevertex[1] ); } /* Don't add this bad triangle to the list; there's nothing that */ /* can be done about a small angle between two segments. */ angle = 0.0; } /* Check whether the angle is smaller than permitted. */ if ( angle > goodangle ) { /* Add this triangle to the list of bad triangles. */ enqueuebadtri( testtri, angle, tapex, torg, tdest ); return; } if ( vararea || fixedarea ) { /* Check whether the area is larger than permitted. */ area = 0.5 * ( dxod * dyda - dyod * dxda ); if ( fixedarea && ( area > maxarea ) ) { /* Add this triangle to the list of bad triangles. */ enqueuebadtri( testtri, angle, tapex, torg, tdest ); } else if ( vararea ) { /* Nonpositive area constraints are treated as unconstrained. */ if ( ( area > areabound( *testtri ) ) && ( areabound( *testtri ) > 0.0 ) ) { /* Add this triangle to the list of bad triangles. */ enqueuebadtri( testtri, angle, tapex, torg, tdest ); } } } } #endif /* not CDT_ONLY */ /** **/ /** **/ /********* Mesh quality testing routines end here *********/ /********* Point location routines begin here *********/ /** **/ /** **/ /*****************************************************************************/ /* */ /* makepointmap() Construct a mapping from points to triangles to improve */ /* the speed of point location for segment insertion. */ /* */ /* Traverses all the triangles, and provides each corner of each triangle */ /* with a pointer to that triangle. Of course, pointers will be */ /* overwritten by other pointers because (almost) each point is a corner */ /* of several triangles, but in the end every point will point to some */ /* triangle that contains it. */ /* */ /*****************************************************************************/ void makepointmap(){ struct triedge triangleloop; point triorg; if ( verbose ) { printf( " Constructing mapping from points to triangles.\n" ); } traversalinit( &triangles ); triangleloop.tri = triangletraverse(); while ( triangleloop.tri != (triangle *) NULL ) { /* Check all three points of the triangle. */ for ( triangleloop.orient = 0; triangleloop.orient < 3; triangleloop.orient++ ) { org( triangleloop, triorg ); setpoint2tri( triorg, encode( triangleloop ) ); } triangleloop.tri = triangletraverse(); } } /*****************************************************************************/ /* */ /* preciselocate() Find a triangle or edge containing a given point. */ /* */ /* Begins its search from `searchtri'. It is important that `searchtri' */ /* be a handle with the property that `searchpoint' is strictly to the left */ /* of the edge denoted by `searchtri', or is collinear with that edge and */ /* does not intersect that edge. (In particular, `searchpoint' should not */ /* be the origin or destination of that edge.) */ /* */ /* These conditions are imposed because preciselocate() is normally used in */ /* one of two situations: */ /* */ /* (1) To try to find the location to insert a new point. Normally, we */ /* know an edge that the point is strictly to the left of. In the */ /* incremental Delaunay algorithm, that edge is a bounding box edge. */ /* In Ruppert's Delaunay refinement algorithm for quality meshing, */ /* that edge is the shortest edge of the triangle whose circumcenter */ /* is being inserted. */ /* */ /* (2) To try to find an existing point. In this case, any edge on the */ /* convex hull is a good starting edge. The possibility that the */ /* vertex one seeks is an endpoint of the starting edge must be */ /* screened out before preciselocate() is called. */ /* */ /* On completion, `searchtri' is a triangle that contains `searchpoint'. */ /* */ /* This implementation differs from that given by Guibas and Stolfi. It */ /* walks from triangle to triangle, crossing an edge only if `searchpoint' */ /* is on the other side of the line containing that edge. After entering */ /* a triangle, there are two edges by which one can leave that triangle. */ /* If both edges are valid (`searchpoint' is on the other side of both */ /* edges), one of the two is chosen by drawing a line perpendicular to */ /* the entry edge (whose endpoints are `forg' and `fdest') passing through */ /* `fapex'. Depending on which side of this perpendicular `searchpoint' */ /* falls on, an exit edge is chosen. */ /* */ /* This implementation is empirically faster than the Guibas and Stolfi */ /* point location routine (which I originally used), which tends to spiral */ /* in toward its target. */ /* */ /* Returns ONVERTEX if the point lies on an existing vertex. `searchtri' */ /* is a handle whose origin is the existing vertex. */ /* */ /* Returns ONEDGE if the point lies on a mesh edge. `searchtri' is a */ /* handle whose primary edge is the edge on which the point lies. */ /* */ /* Returns INTRIANGLE if the point lies strictly within a triangle. */ /* `searchtri' is a handle on the triangle that contains the point. */ /* */ /* Returns OUTSIDE if the point lies outside the mesh. `searchtri' is a */ /* handle whose primary edge the point is to the right of. This might */ /* occur when the circumcenter of a triangle falls just slightly outside */ /* the mesh due to floating-point roundoff error. It also occurs when */ /* seeking a hole or region point that a foolish user has placed outside */ /* the mesh. */ /* */ /* WARNING: This routine is designed for convex triangulations, and will */ /* not generally work after the holes and concavities have been carved. */ /* However, it can still be used to find the circumcenter of a triangle, as */ /* long as the search is begun from the triangle in question. */ /* */ /*****************************************************************************/ enum locateresult preciselocate( searchpoint, searchtri ) point searchpoint; struct triedge *searchtri; { struct triedge backtracktri; point forg, fdest, fapex; point swappoint; REAL orgorient, destorient; int moveleft; triangle ptr; /* Temporary variable used by sym(). */ if ( verbose > 2 ) { printf( " Searching for point (%.12g, %.12g).\n", searchpoint[0], searchpoint[1] ); } /* Where are we? */ org( *searchtri, forg ); dest( *searchtri, fdest ); apex( *searchtri, fapex ); while ( 1 ) { if ( verbose > 2 ) { printf( " At (%.12g, %.12g) (%.12g, %.12g) (%.12g, %.12g)\n", forg[0], forg[1], fdest[0], fdest[1], fapex[0], fapex[1] ); } /* Check whether the apex is the point we seek. */ if ( ( fapex[0] == searchpoint[0] ) && ( fapex[1] == searchpoint[1] ) ) { lprevself( *searchtri ); return ONVERTEX; } /* Does the point lie on the other side of the line defined by the */ /* triangle edge opposite the triangle's destination? */ destorient = counterclockwise( forg, fapex, searchpoint ); /* Does the point lie on the other side of the line defined by the */ /* triangle edge opposite the triangle's origin? */ orgorient = counterclockwise( fapex, fdest, searchpoint ); if ( destorient > 0.0 ) { if ( orgorient > 0.0 ) { /* Move left if the inner product of (fapex - searchpoint) and */ /* (fdest - forg) is positive. This is equivalent to drawing */ /* a line perpendicular to the line (forg, fdest) passing */ /* through `fapex', and determining which side of this line */ /* `searchpoint' falls on. */ moveleft = ( fapex[0] - searchpoint[0] ) * ( fdest[0] - forg[0] ) + ( fapex[1] - searchpoint[1] ) * ( fdest[1] - forg[1] ) > 0.0; } else { moveleft = 1; } } else { if ( orgorient > 0.0 ) { moveleft = 0; } else { /* The point we seek must be on the boundary of or inside this */ /* triangle. */ if ( destorient == 0.0 ) { lprevself( *searchtri ); return ONEDGE; } if ( orgorient == 0.0 ) { lnextself( *searchtri ); return ONEDGE; } return INTRIANGLE; } } /* Move to another triangle. Leave a trace `backtracktri' in case */ /* floating-point roundoff or some such bogey causes us to walk */ /* off a boundary of the triangulation. We can just bounce off */ /* the boundary as if it were an elastic band. */ if ( moveleft ) { lprev( *searchtri, backtracktri ); fdest = fapex; } else { lnext( *searchtri, backtracktri ); forg = fapex; } sym( backtracktri, *searchtri ); /* Check for walking off the edge. */ if ( searchtri->tri == dummytri ) { /* Turn around. */ triedgecopy( backtracktri, *searchtri ); swappoint = forg; forg = fdest; fdest = swappoint; apex( *searchtri, fapex ); /* Check if the point really is beyond the triangulation boundary. */ destorient = counterclockwise( forg, fapex, searchpoint ); orgorient = counterclockwise( fapex, fdest, searchpoint ); if ( ( orgorient < 0.0 ) && ( destorient < 0.0 ) ) { return OUTSIDE; } } else { apex( *searchtri, fapex ); } } } /*****************************************************************************/ /* */ /* locate() Find a triangle or edge containing a given point. */ /* */ /* Searching begins from one of: the input `searchtri', a recently */ /* encountered triangle `recenttri', or from a triangle chosen from a */ /* random sample. The choice is made by determining which triangle's */ /* origin is closest to the point we are searcing for. Normally, */ /* `searchtri' should be a handle on the convex hull of the triangulation. */ /* */ /* Details on the random sampling method can be found in the Mucke, Saias, */ /* and Zhu paper cited in the header of this code. */ /* */ /* On completion, `searchtri' is a triangle that contains `searchpoint'. */ /* */ /* Returns ONVERTEX if the point lies on an existing vertex. `searchtri' */ /* is a handle whose origin is the existing vertex. */ /* */ /* Returns ONEDGE if the point lies on a mesh edge. `searchtri' is a */ /* handle whose primary edge is the edge on which the point lies. */ /* */ /* Returns INTRIANGLE if the point lies strictly within a triangle. */ /* `searchtri' is a handle on the triangle that contains the point. */ /* */ /* Returns OUTSIDE if the point lies outside the mesh. `searchtri' is a */ /* handle whose primary edge the point is to the right of. This might */ /* occur when the circumcenter of a triangle falls just slightly outside */ /* the mesh due to floating-point roundoff error. It also occurs when */ /* seeking a hole or region point that a foolish user has placed outside */ /* the mesh. */ /* */ /* WARNING: This routine is designed for convex triangulations, and will */ /* not generally work after the holes and concavities have been carved. */ /* */ /*****************************************************************************/ enum locateresult locate( searchpoint, searchtri ) point searchpoint; struct triedge *searchtri; { VOID **sampleblock; triangle *firsttri; struct triedge sampletri; point torg, tdest; unsigned long alignptr; REAL searchdist, dist; REAL ahead; long sampleblocks, samplesperblock, samplenum; long triblocks; long i, j; triangle ptr; /* Temporary variable used by sym(). */ if ( verbose > 2 ) { printf( " Randomly sampling for a triangle near point (%.12g, %.12g).\n", searchpoint[0], searchpoint[1] ); } /* Record the distance from the suggested starting triangle to the */ /* point we seek. */ org( *searchtri, torg ); searchdist = ( searchpoint[0] - torg[0] ) * ( searchpoint[0] - torg[0] ) + ( searchpoint[1] - torg[1] ) * ( searchpoint[1] - torg[1] ); if ( verbose > 2 ) { printf( " Boundary triangle has origin (%.12g, %.12g).\n", torg[0], torg[1] ); } /* If a recently encountered triangle has been recorded and has not been */ /* deallocated, test it as a good starting point. */ if ( recenttri.tri != (triangle *) NULL ) { if ( recenttri.tri[3] != (triangle) NULL ) { org( recenttri, torg ); if ( ( torg[0] == searchpoint[0] ) && ( torg[1] == searchpoint[1] ) ) { triedgecopy( recenttri, *searchtri ); return ONVERTEX; } dist = ( searchpoint[0] - torg[0] ) * ( searchpoint[0] - torg[0] ) + ( searchpoint[1] - torg[1] ) * ( searchpoint[1] - torg[1] ); if ( dist < searchdist ) { triedgecopy( recenttri, *searchtri ); searchdist = dist; if ( verbose > 2 ) { printf( " Choosing recent triangle with origin (%.12g, %.12g).\n", torg[0], torg[1] ); } } } } /* The number of random samples taken is proportional to the cube root of */ /* the number of triangles in the mesh. The next bit of code assumes */ /* that the number of triangles increases monotonically. */ while ( SAMPLEFACTOR * samples * samples * samples < triangles.items ) { samples++; } triblocks = ( triangles.maxitems + TRIPERBLOCK - 1 ) / TRIPERBLOCK; samplesperblock = 1 + ( samples / triblocks ); sampleblocks = samples / samplesperblock; sampleblock = triangles.firstblock; sampletri.orient = 0; for ( i = 0; i < sampleblocks; i++ ) { alignptr = (unsigned long) ( sampleblock + 1 ); firsttri = (triangle *) ( alignptr + (unsigned long) triangles.alignbytes - ( alignptr % (unsigned long) triangles.alignbytes ) ); for ( j = 0; j < samplesperblock; j++ ) { if ( i == triblocks - 1 ) { samplenum = randomnation( (int) ( triangles.maxitems - ( i * TRIPERBLOCK ) ) ); } else { samplenum = randomnation( TRIPERBLOCK ); } sampletri.tri = (triangle *) ( firsttri + ( samplenum * triangles.itemwords ) ); if ( sampletri.tri[3] != (triangle) NULL ) { org( sampletri, torg ); dist = ( searchpoint[0] - torg[0] ) * ( searchpoint[0] - torg[0] ) + ( searchpoint[1] - torg[1] ) * ( searchpoint[1] - torg[1] ); if ( dist < searchdist ) { triedgecopy( sampletri, *searchtri ); searchdist = dist; if ( verbose > 2 ) { printf( " Choosing triangle with origin (%.12g, %.12g).\n", torg[0], torg[1] ); } } } } sampleblock = (VOID **) *sampleblock; } /* Where are we? */ org( *searchtri, torg ); dest( *searchtri, tdest ); /* Check the starting triangle's vertices. */ if ( ( torg[0] == searchpoint[0] ) && ( torg[1] == searchpoint[1] ) ) { return ONVERTEX; } if ( ( tdest[0] == searchpoint[0] ) && ( tdest[1] == searchpoint[1] ) ) { lnextself( *searchtri ); return ONVERTEX; } /* Orient `searchtri' to fit the preconditions of calling preciselocate(). */ ahead = counterclockwise( torg, tdest, searchpoint ); if ( ahead < 0.0 ) { /* Turn around so that `searchpoint' is to the left of the */ /* edge specified by `searchtri'. */ symself( *searchtri ); } else if ( ahead == 0.0 ) { /* Check if `searchpoint' is between `torg' and `tdest'. */ if ( ( ( torg[0] < searchpoint[0] ) == ( searchpoint[0] < tdest[0] ) ) && ( ( torg[1] < searchpoint[1] ) == ( searchpoint[1] < tdest[1] ) ) ) { return ONEDGE; } } return preciselocate( searchpoint, searchtri ); } /** **/ /** **/ /********* Point location routines end here *********/ /********* Mesh transformation routines begin here *********/ /** **/ /** **/ /*****************************************************************************/ /* */ /* insertshelle() Create a new shell edge and insert it between two */ /* triangles. */ /* */ /* The new shell edge is inserted at the edge described by the handle */ /* `tri'. Its vertices are properly initialized. The marker `shellemark' */ /* is applied to the shell edge and, if appropriate, its vertices. */ /* */ /*****************************************************************************/ void insertshelle( tri, shellemark ) struct triedge *tri; /* Edge at which to insert the new shell edge. */ int shellemark; /* Marker for the new shell edge. */ { struct triedge oppotri; struct edge newshelle; point triorg, tridest; triangle ptr; /* Temporary variable used by sym(). */ shelle sptr; /* Temporary variable used by tspivot(). */ /* Mark points if possible. */ org( *tri, triorg ); dest( *tri, tridest ); if ( pointmark( triorg ) == 0 ) { setpointmark( triorg, shellemark ); } if ( pointmark( tridest ) == 0 ) { setpointmark( tridest, shellemark ); } /* Check if there's already a shell edge here. */ tspivot( *tri, newshelle ); if ( newshelle.sh == dummysh ) { /* Make new shell edge and initialize its vertices. */ makeshelle( &newshelle ); setsorg( newshelle, tridest ); setsdest( newshelle, triorg ); /* Bond new shell edge to the two triangles it is sandwiched between. */ /* Note that the facing triangle `oppotri' might be equal to */ /* `dummytri' (outer space), but the new shell edge is bonded to it */ /* all the same. */ tsbond( *tri, newshelle ); sym( *tri, oppotri ); ssymself( newshelle ); tsbond( oppotri, newshelle ); setmark( newshelle, shellemark ); if ( verbose > 2 ) { printf( " Inserting new " ); printshelle( &newshelle ); } } else { if ( mark( newshelle ) == 0 ) { setmark( newshelle, shellemark ); } } } /*****************************************************************************/ /* */ /* Terminology */ /* */ /* A "local transformation" replaces a small set of triangles with another */ /* set of triangles. This may or may not involve inserting or deleting a */ /* point. */ /* */ /* The term "casing" is used to describe the set of triangles that are */ /* attached to the triangles being transformed, but are not transformed */ /* themselves. Think of the casing as a fixed hollow structure inside */ /* which all the action happens. A "casing" is only defined relative to */ /* a single transformation; each occurrence of a transformation will */ /* involve a different casing. */ /* */ /* A "shell" is similar to a "casing". The term "shell" describes the set */ /* of shell edges (if any) that are attached to the triangles being */ /* transformed. However, I sometimes use "shell" to refer to a single */ /* shell edge, so don't get confused. */ /* */ /*****************************************************************************/ /*****************************************************************************/ /* */ /* flip() Transform two triangles to two different triangles by flipping */ /* an edge within a quadrilateral. */ /* */ /* Imagine the original triangles, abc and bad, oriented so that the */ /* shared edge ab lies in a horizontal plane, with the point b on the left */ /* and the point a on the right. The point c lies below the edge, and the */ /* point d lies above the edge. The `flipedge' handle holds the edge ab */ /* of triangle abc, and is directed left, from vertex a to vertex b. */ /* */ /* The triangles abc and bad are deleted and replaced by the triangles cdb */ /* and dca. The triangles that represent abc and bad are NOT deallocated; */ /* they are reused for dca and cdb, respectively. Hence, any handles that */ /* may have held the original triangles are still valid, although not */ /* directed as they were before. */ /* */ /* Upon completion of this routine, the `flipedge' handle holds the edge */ /* dc of triangle dca, and is directed down, from vertex d to vertex c. */ /* (Hence, the two triangles have rotated counterclockwise.) */ /* */ /* WARNING: This transformation is geometrically valid only if the */ /* quadrilateral adbc is convex. Furthermore, this transformation is */ /* valid only if there is not a shell edge between the triangles abc and */ /* bad. This routine does not check either of these preconditions, and */ /* it is the responsibility of the calling routine to ensure that they are */ /* met. If they are not, the streets shall be filled with wailing and */ /* gnashing of teeth. */ /* */ /*****************************************************************************/ void flip( flipedge ) struct triedge *flipedge; /* Handle for the triangle abc. */ { struct triedge botleft, botright; struct triedge topleft, topright; struct triedge top; struct triedge botlcasing, botrcasing; struct triedge toplcasing, toprcasing; struct edge botlshelle, botrshelle; struct edge toplshelle, toprshelle; point leftpoint, rightpoint, botpoint; point farpoint; triangle ptr; /* Temporary variable used by sym(). */ shelle sptr; /* Temporary variable used by tspivot(). */ /* Identify the vertices of the quadrilateral. */ org( *flipedge, rightpoint ); dest( *flipedge, leftpoint ); apex( *flipedge, botpoint ); sym( *flipedge, top ); #ifdef SELF_CHECK if ( top.tri == dummytri ) { printf( "Internal error in flip(): Attempt to flip on boundary.\n" ); lnextself( *flipedge ); return; } if ( checksegments ) { tspivot( *flipedge, toplshelle ); if ( toplshelle.sh != dummysh ) { printf( "Internal error in flip(): Attempt to flip a segment.\n" ); lnextself( *flipedge ); return; } } #endif /* SELF_CHECK */ apex( top, farpoint ); /* Identify the casing of the quadrilateral. */ lprev( top, topleft ); sym( topleft, toplcasing ); lnext( top, topright ); sym( topright, toprcasing ); lnext( *flipedge, botleft ); sym( botleft, botlcasing ); lprev( *flipedge, botright ); sym( botright, botrcasing ); /* Rotate the quadrilateral one-quarter turn counterclockwise. */ bond( topleft, botlcasing ); bond( botleft, botrcasing ); bond( botright, toprcasing ); bond( topright, toplcasing ); if ( checksegments ) { /* Check for shell edges and rebond them to the quadrilateral. */ tspivot( topleft, toplshelle ); tspivot( botleft, botlshelle ); tspivot( botright, botrshelle ); tspivot( topright, toprshelle ); if ( toplshelle.sh == dummysh ) { tsdissolve( topright ); } else { tsbond( topright, toplshelle ); } if ( botlshelle.sh == dummysh ) { tsdissolve( topleft ); } else { tsbond( topleft, botlshelle ); } if ( botrshelle.sh == dummysh ) { tsdissolve( botleft ); } else { tsbond( botleft, botrshelle ); } if ( toprshelle.sh == dummysh ) { tsdissolve( botright ); } else { tsbond( botright, toprshelle ); } } /* New point assignments for the rotated quadrilateral. */ setorg( *flipedge, farpoint ); setdest( *flipedge, botpoint ); setapex( *flipedge, rightpoint ); setorg( top, botpoint ); setdest( top, farpoint ); setapex( top, leftpoint ); if ( verbose > 2 ) { printf( " Edge flip results in left " ); lnextself( topleft ); printtriangle( &topleft ); printf( " and right " ); printtriangle( flipedge ); } } /*****************************************************************************/ /* */ /* insertsite() Insert a vertex into a Delaunay triangulation, */ /* performing flips as necessary to maintain the Delaunay */ /* property. */ /* */ /* The point `insertpoint' is located. If `searchtri.tri' is not NULL, */ /* the search for the containing triangle begins from `searchtri'. If */ /* `searchtri.tri' is NULL, a full point location procedure is called. */ /* If `insertpoint' is found inside a triangle, the triangle is split into */ /* three; if `insertpoint' lies on an edge, the edge is split in two, */ /* thereby splitting the two adjacent triangles into four. Edge flips are */ /* used to restore the Delaunay property. If `insertpoint' lies on an */ /* existing vertex, no action is taken, and the value DUPLICATEPOINT is */ /* returned. On return, `searchtri' is set to a handle whose origin is the */ /* existing vertex. */ /* */ /* Normally, the parameter `splitedge' is set to NULL, implying that no */ /* segment should be split. In this case, if `insertpoint' is found to */ /* lie on a segment, no action is taken, and the value VIOLATINGPOINT is */ /* returned. On return, `searchtri' is set to a handle whose primary edge */ /* is the violated segment. */ /* */ /* If the calling routine wishes to split a segment by inserting a point in */ /* it, the parameter `splitedge' should be that segment. In this case, */ /* `searchtri' MUST be the triangle handle reached by pivoting from that */ /* segment; no point location is done. */ /* */ /* `segmentflaws' and `triflaws' are flags that indicate whether or not */ /* there should be checks for the creation of encroached segments or bad */ /* quality faces. If a newly inserted point encroaches upon segments, */ /* these segments are added to the list of segments to be split if */ /* `segmentflaws' is set. If bad triangles are created, these are added */ /* to the queue if `triflaws' is set. */ /* */ /* If a duplicate point or violated segment does not prevent the point */ /* from being inserted, the return value will be ENCROACHINGPOINT if the */ /* point encroaches upon a segment (and checking is enabled), or */ /* SUCCESSFULPOINT otherwise. In either case, `searchtri' is set to a */ /* handle whose origin is the newly inserted vertex. */ /* */ /* insertsite() does not use flip() for reasons of speed; some */ /* information can be reused from edge flip to edge flip, like the */ /* locations of shell edges. */ /* */ /*****************************************************************************/ enum insertsiteresult insertsite( insertpoint, searchtri, splitedge, segmentflaws, triflaws ) point insertpoint; struct triedge *searchtri; struct edge *splitedge; int segmentflaws; int triflaws; { struct triedge horiz; struct triedge top; struct triedge botleft, botright; struct triedge topleft, topright; struct triedge newbotleft, newbotright; struct triedge newtopright; struct triedge botlcasing, botrcasing; struct triedge toplcasing, toprcasing; struct triedge testtri; struct edge botlshelle, botrshelle; struct edge toplshelle, toprshelle; struct edge brokenshelle; struct edge checkshelle; struct edge rightedge; struct edge newedge; struct edge *encroached; point first; point leftpoint, rightpoint, botpoint, toppoint, farpoint; REAL attrib; REAL area; enum insertsiteresult success; enum locateresult intersect; int doflip; int mirrorflag; int i; triangle ptr; /* Temporary variable used by sym(). */ shelle sptr; /* Temporary variable used by spivot() and tspivot(). */ if ( verbose > 1 ) { printf( " Inserting (%.12g, %.12g).\n", insertpoint[0], insertpoint[1] ); } if ( splitedge == (struct edge *) NULL ) { /* Find the location of the point to be inserted. Check if a good */ /* starting triangle has already been provided by the caller. */ if ( searchtri->tri == (triangle *) NULL ) { /* Find a boundary triangle. */ horiz.tri = dummytri; horiz.orient = 0; symself( horiz ); /* Search for a triangle containing `insertpoint'. */ intersect = locate( insertpoint, &horiz ); } else { /* Start searching from the triangle provided by the caller. */ triedgecopy( *searchtri, horiz ); intersect = preciselocate( insertpoint, &horiz ); } } else { /* The calling routine provides the edge in which the point is inserted. */ triedgecopy( *searchtri, horiz ); intersect = ONEDGE; } if ( intersect == ONVERTEX ) { /* There's already a vertex there. Return in `searchtri' a triangle */ /* whose origin is the existing vertex. */ triedgecopy( horiz, *searchtri ); triedgecopy( horiz, recenttri ); return DUPLICATEPOINT; } if ( ( intersect == ONEDGE ) || ( intersect == OUTSIDE ) ) { /* The vertex falls on an edge or boundary. */ if ( checksegments && ( splitedge == (struct edge *) NULL ) ) { /* Check whether the vertex falls on a shell edge. */ tspivot( horiz, brokenshelle ); if ( brokenshelle.sh != dummysh ) { /* The vertex falls on a shell edge. */ if ( segmentflaws ) { if ( nobisect == 0 ) { /* Add the shell edge to the list of encroached segments. */ encroached = (struct edge *) poolalloc( &badsegments ); shellecopy( brokenshelle, *encroached ); } else if ( ( nobisect == 1 ) && ( intersect == ONEDGE ) ) { /* This segment may be split only if it is an internal boundary. */ sym( horiz, testtri ); if ( testtri.tri != dummytri ) { /* Add the shell edge to the list of encroached segments. */ encroached = (struct edge *) poolalloc( &badsegments ); shellecopy( brokenshelle, *encroached ); } } } /* Return a handle whose primary edge contains the point, */ /* which has not been inserted. */ triedgecopy( horiz, *searchtri ); triedgecopy( horiz, recenttri ); return VIOLATINGPOINT; } } /* Insert the point on an edge, dividing one triangle into two (if */ /* the edge lies on a boundary) or two triangles into four. */ lprev( horiz, botright ); sym( botright, botrcasing ); sym( horiz, topright ); /* Is there a second triangle? (Or does this edge lie on a boundary?) */ mirrorflag = topright.tri != dummytri; if ( mirrorflag ) { lnextself( topright ); sym( topright, toprcasing ); maketriangle( &newtopright ); } else { /* Splitting the boundary edge increases the number of boundary edges. */ hullsize++; } maketriangle( &newbotright ); /* Set the vertices of changed and new triangles. */ org( horiz, rightpoint ); dest( horiz, leftpoint ); apex( horiz, botpoint ); setorg( newbotright, botpoint ); setdest( newbotright, rightpoint ); setapex( newbotright, insertpoint ); setorg( horiz, insertpoint ); for ( i = 0; i < eextras; i++ ) { /* Set the element attributes of a new triangle. */ setelemattribute( newbotright, i, elemattribute( botright, i ) ); } if ( vararea ) { /* Set the area constraint of a new triangle. */ setareabound( newbotright, areabound( botright ) ); } if ( mirrorflag ) { dest( topright, toppoint ); setorg( newtopright, rightpoint ); setdest( newtopright, toppoint ); setapex( newtopright, insertpoint ); setorg( topright, insertpoint ); for ( i = 0; i < eextras; i++ ) { /* Set the element attributes of another new triangle. */ setelemattribute( newtopright, i, elemattribute( topright, i ) ); } if ( vararea ) { /* Set the area constraint of another new triangle. */ setareabound( newtopright, areabound( topright ) ); } } /* There may be shell edges that need to be bonded */ /* to the new triangle(s). */ if ( checksegments ) { tspivot( botright, botrshelle ); if ( botrshelle.sh != dummysh ) { tsdissolve( botright ); tsbond( newbotright, botrshelle ); } if ( mirrorflag ) { tspivot( topright, toprshelle ); if ( toprshelle.sh != dummysh ) { tsdissolve( topright ); tsbond( newtopright, toprshelle ); } } } /* Bond the new triangle(s) to the surrounding triangles. */ bond( newbotright, botrcasing ); lprevself( newbotright ); bond( newbotright, botright ); lprevself( newbotright ); if ( mirrorflag ) { bond( newtopright, toprcasing ); lnextself( newtopright ); bond( newtopright, topright ); lnextself( newtopright ); bond( newtopright, newbotright ); } if ( splitedge != (struct edge *) NULL ) { /* Split the shell edge into two. */ setsdest( *splitedge, insertpoint ); ssymself( *splitedge ); spivot( *splitedge, rightedge ); insertshelle( &newbotright, mark( *splitedge ) ); tspivot( newbotright, newedge ); sbond( *splitedge, newedge ); ssymself( newedge ); sbond( newedge, rightedge ); ssymself( *splitedge ); } #ifdef SELF_CHECK if ( counterclockwise( rightpoint, leftpoint, botpoint ) < 0.0 ) { printf( "Internal error in insertsite():\n" ); printf( " Clockwise triangle prior to edge point insertion (bottom).\n" ); } if ( mirrorflag ) { if ( counterclockwise( leftpoint, rightpoint, toppoint ) < 0.0 ) { printf( "Internal error in insertsite():\n" ); printf( " Clockwise triangle prior to edge point insertion (top).\n" ); } if ( counterclockwise( rightpoint, toppoint, insertpoint ) < 0.0 ) { printf( "Internal error in insertsite():\n" ); printf( " Clockwise triangle after edge point insertion (top right).\n" ); } if ( counterclockwise( toppoint, leftpoint, insertpoint ) < 0.0 ) { printf( "Internal error in insertsite():\n" ); printf( " Clockwise triangle after edge point insertion (top left).\n" ); } } if ( counterclockwise( leftpoint, botpoint, insertpoint ) < 0.0 ) { printf( "Internal error in insertsite():\n" ); printf( " Clockwise triangle after edge point insertion (bottom left).\n" ); } if ( counterclockwise( botpoint, rightpoint, insertpoint ) < 0.0 ) { printf( "Internal error in insertsite():\n" ); printf( " Clockwise triangle after edge point insertion (bottom right).\n" ); } #endif /* SELF_CHECK */ if ( verbose > 2 ) { printf( " Updating bottom left " ); printtriangle( &botright ); if ( mirrorflag ) { printf( " Updating top left " ); printtriangle( &topright ); printf( " Creating top right " ); printtriangle( &newtopright ); } printf( " Creating bottom right " ); printtriangle( &newbotright ); } /* Position `horiz' on the first edge to check for */ /* the Delaunay property. */ lnextself( horiz ); } else { /* Insert the point in a triangle, splitting it into three. */ lnext( horiz, botleft ); lprev( horiz, botright ); sym( botleft, botlcasing ); sym( botright, botrcasing ); maketriangle( &newbotleft ); maketriangle( &newbotright ); /* Set the vertices of changed and new triangles. */ org( horiz, rightpoint ); dest( horiz, leftpoint ); apex( horiz, botpoint ); setorg( newbotleft, leftpoint ); setdest( newbotleft, botpoint ); setapex( newbotleft, insertpoint ); setorg( newbotright, botpoint ); setdest( newbotright, rightpoint ); setapex( newbotright, insertpoint ); setapex( horiz, insertpoint ); for ( i = 0; i < eextras; i++ ) { /* Set the element attributes of the new triangles. */ attrib = elemattribute( horiz, i ); setelemattribute( newbotleft, i, attrib ); setelemattribute( newbotright, i, attrib ); } if ( vararea ) { /* Set the area constraint of the new triangles. */ area = areabound( horiz ); setareabound( newbotleft, area ); setareabound( newbotright, area ); } /* There may be shell edges that need to be bonded */ /* to the new triangles. */ if ( checksegments ) { tspivot( botleft, botlshelle ); if ( botlshelle.sh != dummysh ) { tsdissolve( botleft ); tsbond( newbotleft, botlshelle ); } tspivot( botright, botrshelle ); if ( botrshelle.sh != dummysh ) { tsdissolve( botright ); tsbond( newbotright, botrshelle ); } } /* Bond the new triangles to the surrounding triangles. */ bond( newbotleft, botlcasing ); bond( newbotright, botrcasing ); lnextself( newbotleft ); lprevself( newbotright ); bond( newbotleft, newbotright ); lnextself( newbotleft ); bond( botleft, newbotleft ); lprevself( newbotright ); bond( botright, newbotright ); #ifdef SELF_CHECK if ( counterclockwise( rightpoint, leftpoint, botpoint ) < 0.0 ) { printf( "Internal error in insertsite():\n" ); printf( " Clockwise triangle prior to point insertion.\n" ); } if ( counterclockwise( rightpoint, leftpoint, insertpoint ) < 0.0 ) { printf( "Internal error in insertsite():\n" ); printf( " Clockwise triangle after point insertion (top).\n" ); } if ( counterclockwise( leftpoint, botpoint, insertpoint ) < 0.0 ) { printf( "Internal error in insertsite():\n" ); printf( " Clockwise triangle after point insertion (left).\n" ); } if ( counterclockwise( botpoint, rightpoint, insertpoint ) < 0.0 ) { printf( "Internal error in insertsite():\n" ); printf( " Clockwise triangle after point insertion (right).\n" ); } #endif /* SELF_CHECK */ if ( verbose > 2 ) { printf( " Updating top " ); printtriangle( &horiz ); printf( " Creating left " ); printtriangle( &newbotleft ); printf( " Creating right " ); printtriangle( &newbotright ); } } /* The insertion is successful by default, unless an encroached */ /* edge is found. */ success = SUCCESSFULPOINT; /* Circle around the newly inserted vertex, checking each edge opposite */ /* it for the Delaunay property. Non-Delaunay edges are flipped. */ /* `horiz' is always the edge being checked. `first' marks where to */ /* stop circling. */ org( horiz, first ); rightpoint = first; dest( horiz, leftpoint ); /* Circle until finished. */ while ( 1 ) { /* By default, the edge will be flipped. */ doflip = 1; if ( checksegments ) { /* Check for a segment, which cannot be flipped. */ tspivot( horiz, checkshelle ); if ( checkshelle.sh != dummysh ) { /* The edge is a segment and cannot be flipped. */ doflip = 0; #ifndef CDT_ONLY if ( segmentflaws ) { /* Does the new point encroach upon this segment? */ if ( checkedge4encroach( &checkshelle ) ) { success = ENCROACHINGPOINT; } } #endif /* not CDT_ONLY */ } } if ( doflip ) { /* Check if the edge is a boundary edge. */ sym( horiz, top ); if ( top.tri == dummytri ) { /* The edge is a boundary edge and cannot be flipped. */ doflip = 0; } else { /* Find the point on the other side of the edge. */ apex( top, farpoint ); /* In the incremental Delaunay triangulation algorithm, any of */ /* `leftpoint', `rightpoint', and `farpoint' could be vertices */ /* of the triangular bounding box. These vertices must be */ /* treated as if they are infinitely distant, even though their */ /* "coordinates" are not. */ if ( ( leftpoint == infpoint1 ) || ( leftpoint == infpoint2 ) || ( leftpoint == infpoint3 ) ) { /* `leftpoint' is infinitely distant. Check the convexity of */ /* the boundary of the triangulation. 'farpoint' might be */ /* infinite as well, but trust me, this same condition */ /* should be applied. */ doflip = counterclockwise( insertpoint, rightpoint, farpoint ) > 0.0; } else if ( ( rightpoint == infpoint1 ) || ( rightpoint == infpoint2 ) || ( rightpoint == infpoint3 ) ) { /* `rightpoint' is infinitely distant. Check the convexity of */ /* the boundary of the triangulation. 'farpoint' might be */ /* infinite as well, but trust me, this same condition */ /* should be applied. */ doflip = counterclockwise( farpoint, leftpoint, insertpoint ) > 0.0; } else if ( ( farpoint == infpoint1 ) || ( farpoint == infpoint2 ) || ( farpoint == infpoint3 ) ) { /* `farpoint' is infinitely distant and cannot be inside */ /* the circumcircle of the triangle `horiz'. */ doflip = 0; } else { /* Test whether the edge is locally Delaunay. */ doflip = incircle( leftpoint, insertpoint, rightpoint, farpoint ) > 0.0; } if ( doflip ) { /* We made it! Flip the edge `horiz' by rotating its containing */ /* quadrilateral (the two triangles adjacent to `horiz'). */ /* Identify the casing of the quadrilateral. */ lprev( top, topleft ); sym( topleft, toplcasing ); lnext( top, topright ); sym( topright, toprcasing ); lnext( horiz, botleft ); sym( botleft, botlcasing ); lprev( horiz, botright ); sym( botright, botrcasing ); /* Rotate the quadrilateral one-quarter turn counterclockwise. */ bond( topleft, botlcasing ); bond( botleft, botrcasing ); bond( botright, toprcasing ); bond( topright, toplcasing ); if ( checksegments ) { /* Check for shell edges and rebond them to the quadrilateral. */ tspivot( topleft, toplshelle ); tspivot( botleft, botlshelle ); tspivot( botright, botrshelle ); tspivot( topright, toprshelle ); if ( toplshelle.sh == dummysh ) { tsdissolve( topright ); } else { tsbond( topright, toplshelle ); } if ( botlshelle.sh == dummysh ) { tsdissolve( topleft ); } else { tsbond( topleft, botlshelle ); } if ( botrshelle.sh == dummysh ) { tsdissolve( botleft ); } else { tsbond( botleft, botrshelle ); } if ( toprshelle.sh == dummysh ) { tsdissolve( botright ); } else { tsbond( botright, toprshelle ); } } /* New point assignments for the rotated quadrilateral. */ setorg( horiz, farpoint ); setdest( horiz, insertpoint ); setapex( horiz, rightpoint ); setorg( top, insertpoint ); setdest( top, farpoint ); setapex( top, leftpoint ); for ( i = 0; i < eextras; i++ ) { /* Take the average of the two triangles' attributes. */ attrib = (REAL)( 0.5 * ( elemattribute( top, i ) + elemattribute( horiz, i ) ) ); setelemattribute( top, i, attrib ); setelemattribute( horiz, i, attrib ); } if ( vararea ) { if ( ( areabound( top ) <= 0.0 ) || ( areabound( horiz ) <= 0.0 ) ) { area = -1.0; } else { /* Take the average of the two triangles' area constraints. */ /* This prevents small area constraints from migrating a */ /* long, long way from their original location due to flips. */ area = (REAL)( 0.5 * ( areabound( top ) + areabound( horiz ) ) ); } setareabound( top, area ); setareabound( horiz, area ); } #ifdef SELF_CHECK if ( insertpoint != (point) NULL ) { if ( counterclockwise( leftpoint, insertpoint, rightpoint ) < 0.0 ) { printf( "Internal error in insertsite():\n" ); printf( " Clockwise triangle prior to edge flip (bottom).\n" ); } /* The following test has been removed because constrainededge() */ /* sometimes generates inverted triangles that insertsite() */ /* removes. */ /* if (counterclockwise(rightpoint, farpoint, leftpoint) < 0.0) { printf("Internal error in insertsite():\n"); printf(" Clockwise triangle prior to edge flip (top).\n"); } */ if ( counterclockwise( farpoint, leftpoint, insertpoint ) < 0.0 ) { printf( "Internal error in insertsite():\n" ); printf( " Clockwise triangle after edge flip (left).\n" ); } if ( counterclockwise( insertpoint, rightpoint, farpoint ) < 0.0 ) { printf( "Internal error in insertsite():\n" ); printf( " Clockwise triangle after edge flip (right).\n" ); } } #endif /* SELF_CHECK */ if ( verbose > 2 ) { printf( " Edge flip results in left " ); lnextself( topleft ); printtriangle( &topleft ); printf( " and right " ); printtriangle( &horiz ); } /* On the next iterations, consider the two edges that were */ /* exposed (this is, are now visible to the newly inserted */ /* point) by the edge flip. */ lprevself( horiz ); leftpoint = farpoint; } } } if ( !doflip ) { /* The handle `horiz' is accepted as locally Delaunay. */ #ifndef CDT_ONLY if ( triflaws ) { /* Check the triangle `horiz' for quality. */ testtriangle( &horiz ); } #endif /* not CDT_ONLY */ /* Look for the next edge around the newly inserted point. */ lnextself( horiz ); sym( horiz, testtri ); /* Check for finishing a complete revolution about the new point, or */ /* falling off the edge of the triangulation. The latter will */ /* happen when a point is inserted at a boundary. */ if ( ( leftpoint == first ) || ( testtri.tri == dummytri ) ) { /* We're done. Return a triangle whose origin is the new point. */ lnext( horiz, *searchtri ); lnext( horiz, recenttri ); return success; } /* Finish finding the next edge around the newly inserted point. */ lnext( testtri, horiz ); rightpoint = leftpoint; dest( horiz, leftpoint ); } } } /*****************************************************************************/ /* */ /* triangulatepolygon() Find the Delaunay triangulation of a polygon that */ /* has a certain "nice" shape. This includes the */ /* polygons that result from deletion of a point or */ /* insertion of a segment. */ /* */ /* This is a conceptually difficult routine. The starting assumption is */ /* that we have a polygon with n sides. n - 1 of these sides are currently */ /* represented as edges in the mesh. One side, called the "base", need not */ /* be. */ /* */ /* Inside the polygon is a structure I call a "fan", consisting of n - 1 */ /* triangles that share a common origin. For each of these triangles, the */ /* edge opposite the origin is one of the sides of the polygon. The */ /* primary edge of each triangle is the edge directed from the origin to */ /* the destination; note that this is not the same edge that is a side of */ /* the polygon. `firstedge' is the primary edge of the first triangle. */ /* From there, the triangles follow in counterclockwise order about the */ /* polygon, until `lastedge', the primary edge of the last triangle. */ /* `firstedge' and `lastedge' are probably connected to other triangles */ /* beyond the extremes of the fan, but their identity is not important, as */ /* long as the fan remains connected to them. */ /* */ /* Imagine the polygon oriented so that its base is at the bottom. This */ /* puts `firstedge' on the far right, and `lastedge' on the far left. */ /* The right vertex of the base is the destination of `firstedge', and the */ /* left vertex of the base is the apex of `lastedge'. */ /* */ /* The challenge now is to find the right sequence of edge flips to */ /* transform the fan into a Delaunay triangulation of the polygon. Each */ /* edge flip effectively removes one triangle from the fan, committing it */ /* to the polygon. The resulting polygon has one fewer edge. If `doflip' */ /* is set, the final flip will be performed, resulting in a fan of one */ /* (useless?) triangle. If `doflip' is not set, the final flip is not */ /* performed, resulting in a fan of two triangles, and an unfinished */ /* triangular polygon that is not yet filled out with a single triangle. */ /* On completion of the routine, `lastedge' is the last remaining triangle, */ /* or the leftmost of the last two. */ /* */ /* Although the flips are performed in the order described above, the */ /* decisions about what flips to perform are made in precisely the reverse */ /* order. The recursive triangulatepolygon() procedure makes a decision, */ /* uses up to two recursive calls to triangulate the "subproblems" */ /* (polygons with fewer edges), and then performs an edge flip. */ /* */ /* The "decision" it makes is which vertex of the polygon should be */ /* connected to the base. This decision is made by testing every possible */ /* vertex. Once the best vertex is found, the two edges that connect this */ /* vertex to the base become the bases for two smaller polygons. These */ /* are triangulated recursively. Unfortunately, this approach can take */ /* O(n^2) time not only in the worst case, but in many common cases. It's */ /* rarely a big deal for point deletion, where n is rarely larger than ten, */ /* but it could be a big deal for segment insertion, especially if there's */ /* a lot of long segments that each cut many triangles. I ought to code */ /* a faster algorithm some time. */ /* */ /* The `edgecount' parameter is the number of sides of the polygon, */ /* including its base. `triflaws' is a flag that determines whether the */ /* new triangles should be tested for quality, and enqueued if they are */ /* bad. */ /* */ /*****************************************************************************/ void triangulatepolygon( firstedge, lastedge, edgecount, doflip, triflaws ) struct triedge *firstedge; struct triedge *lastedge; int edgecount; int doflip; int triflaws; { struct triedge testtri; struct triedge besttri; struct triedge tempedge; point leftbasepoint, rightbasepoint; point testpoint; point bestpoint; int bestnumber; int i; triangle ptr; /* Temporary variable used by sym(), onext(), and oprev(). */ /* Identify the base vertices. */ apex( *lastedge, leftbasepoint ); dest( *firstedge, rightbasepoint ); if ( verbose > 2 ) { printf( " Triangulating interior polygon at edge\n" ); printf( " (%.12g, %.12g) (%.12g, %.12g)\n", leftbasepoint[0], leftbasepoint[1], rightbasepoint[0], rightbasepoint[1] ); } /* Find the best vertex to connect the base to. */ onext( *firstedge, besttri ); dest( besttri, bestpoint ); triedgecopy( besttri, testtri ); bestnumber = 1; for ( i = 2; i <= edgecount - 2; i++ ) { onextself( testtri ); dest( testtri, testpoint ); /* Is this a better vertex? */ if ( incircle( leftbasepoint, rightbasepoint, bestpoint, testpoint ) > 0.0 ) { triedgecopy( testtri, besttri ); bestpoint = testpoint; bestnumber = i; } } if ( verbose > 2 ) { printf( " Connecting edge to (%.12g, %.12g)\n", bestpoint[0], bestpoint[1] ); } if ( bestnumber > 1 ) { /* Recursively triangulate the smaller polygon on the right. */ oprev( besttri, tempedge ); triangulatepolygon( firstedge, &tempedge, bestnumber + 1, 1, triflaws ); } if ( bestnumber < edgecount - 2 ) { /* Recursively triangulate the smaller polygon on the left. */ sym( besttri, tempedge ); triangulatepolygon( &besttri, lastedge, edgecount - bestnumber, 1, triflaws ); /* Find `besttri' again; it may have been lost to edge flips. */ sym( tempedge, besttri ); } if ( doflip ) { /* Do one final edge flip. */ flip( &besttri ); #ifndef CDT_ONLY if ( triflaws ) { /* Check the quality of the newly committed triangle. */ sym( besttri, testtri ); testtriangle( &testtri ); } #endif /* not CDT_ONLY */ } /* Return the base triangle. */ triedgecopy( besttri, *lastedge ); } /*****************************************************************************/ /* */ /* deletesite() Delete a vertex from a Delaunay triangulation, ensuring */ /* that the triangulation remains Delaunay. */ /* */ /* The origin of `deltri' is deleted. The union of the triangles adjacent */ /* to this point is a polygon, for which the Delaunay triangulation is */ /* found. Two triangles are removed from the mesh. */ /* */ /* Only interior points that do not lie on segments (shell edges) or */ /* boundaries may be deleted. */ /* */ /*****************************************************************************/ #ifndef CDT_ONLY void deletesite( deltri ) struct triedge *deltri; { struct triedge countingtri; struct triedge firstedge, lastedge; struct triedge deltriright; struct triedge lefttri, righttri; struct triedge leftcasing, rightcasing; struct edge leftshelle, rightshelle; point delpoint; point neworg; int edgecount; triangle ptr; /* Temporary variable used by sym(), onext(), and oprev(). */ shelle sptr; /* Temporary variable used by tspivot(). */ org( *deltri, delpoint ); if ( verbose > 1 ) { printf( " Deleting (%.12g, %.12g).\n", delpoint[0], delpoint[1] ); } pointdealloc( delpoint ); /* Count the degree of the point being deleted. */ onext( *deltri, countingtri ); edgecount = 1; while ( !triedgeequal( *deltri, countingtri ) ) { #ifdef SELF_CHECK if ( countingtri.tri == dummytri ) { printf( "Internal error in deletesite():\n" ); printf( " Attempt to delete boundary point.\n" ); internalerror(); } #endif /* SELF_CHECK */ edgecount++; onextself( countingtri ); } #ifdef SELF_CHECK if ( edgecount < 3 ) { printf( "Internal error in deletesite():\n Point has degree %d.\n", edgecount ); internalerror(); } #endif /* SELF_CHECK */ if ( edgecount > 3 ) { /* Triangulate the polygon defined by the union of all triangles */ /* adjacent to the point being deleted. Check the quality of */ /* the resulting triangles. */ onext( *deltri, firstedge ); oprev( *deltri, lastedge ); triangulatepolygon( &firstedge, &lastedge, edgecount, 0, !nobisect ); } /* Splice out two triangles. */ lprev( *deltri, deltriright ); dnext( *deltri, lefttri ); sym( lefttri, leftcasing ); oprev( deltriright, righttri ); sym( righttri, rightcasing ); bond( *deltri, leftcasing ); bond( deltriright, rightcasing ); tspivot( lefttri, leftshelle ); if ( leftshelle.sh != dummysh ) { tsbond( *deltri, leftshelle ); } tspivot( righttri, rightshelle ); if ( rightshelle.sh != dummysh ) { tsbond( deltriright, rightshelle ); } /* Set the new origin of `deltri' and check its quality. */ org( lefttri, neworg ); setorg( *deltri, neworg ); if ( !nobisect ) { testtriangle( deltri ); } /* Delete the two spliced-out triangles. */ triangledealloc( lefttri.tri ); triangledealloc( righttri.tri ); } #endif /* not CDT_ONLY */ /** **/ /** **/ /********* Mesh transformation routines end here *********/ /********* Divide-and-conquer Delaunay triangulation begins here *********/ /** **/ /** **/ /*****************************************************************************/ /* */ /* The divide-and-conquer bounding box */ /* */ /* I originally implemented the divide-and-conquer and incremental Delaunay */ /* triangulations using the edge-based data structure presented by Guibas */ /* and Stolfi. Switching to a triangle-based data structure doubled the */ /* speed. However, I had to think of a few extra tricks to maintain the */ /* elegance of the original algorithms. */ /* */ /* The "bounding box" used by my variant of the divide-and-conquer */ /* algorithm uses one triangle for each edge of the convex hull of the */ /* triangulation. These bounding triangles all share a common apical */ /* vertex, which is represented by NULL and which represents nothing. */ /* The bounding triangles are linked in a circular fan about this NULL */ /* vertex, and the edges on the convex hull of the triangulation appear */ /* opposite the NULL vertex. You might find it easiest to imagine that */ /* the NULL vertex is a point in 3D space behind the center of the */ /* triangulation, and that the bounding triangles form a sort of cone. */ /* */ /* This bounding box makes it easy to represent degenerate cases. For */ /* instance, the triangulation of two vertices is a single edge. This edge */ /* is represented by two bounding box triangles, one on each "side" of the */ /* edge. These triangles are also linked together in a fan about the NULL */ /* vertex. */ /* */ /* The bounding box also makes it easy to traverse the convex hull, as the */ /* divide-and-conquer algorithm needs to do. */ /* */ /*****************************************************************************/ /*****************************************************************************/ /* */ /* pointsort() Sort an array of points by x-coordinate, using the */ /* y-coordinate as a secondary key. */ /* */ /* Uses quicksort. Randomized O(n log n) time. No, I did not make any of */ /* the usual quicksort mistakes. */ /* */ /*****************************************************************************/ void pointsort( sortarray, arraysize ) point * sortarray; int arraysize; { int left, right; int pivot; REAL pivotx, pivoty; point temp; if ( arraysize == 2 ) { /* Recursive base case. */ if ( ( sortarray[0][0] > sortarray[1][0] ) || ( ( sortarray[0][0] == sortarray[1][0] ) && ( sortarray[0][1] > sortarray[1][1] ) ) ) { temp = sortarray[1]; sortarray[1] = sortarray[0]; sortarray[0] = temp; } return; } /* Choose a random pivot to split the array. */ pivot = (int) randomnation( arraysize ); pivotx = sortarray[pivot][0]; pivoty = sortarray[pivot][1]; /* Split the array. */ left = -1; right = arraysize; while ( left < right ) { /* Search for a point whose x-coordinate is too large for the left. */ do { left++; } while ( ( left <= right ) && ( ( sortarray[left][0] < pivotx ) || ( ( sortarray[left][0] == pivotx ) && ( sortarray[left][1] < pivoty ) ) ) ); /* Search for a point whose x-coordinate is too small for the right. */ do { right--; } while ( ( left <= right ) && ( ( sortarray[right][0] > pivotx ) || ( ( sortarray[right][0] == pivotx ) && ( sortarray[right][1] > pivoty ) ) ) ); if ( left < right ) { /* Swap the left and right points. */ temp = sortarray[left]; sortarray[left] = sortarray[right]; sortarray[right] = temp; } } if ( left > 1 ) { /* Recursively sort the left subset. */ pointsort( sortarray, left ); } if ( right < arraysize - 2 ) { /* Recursively sort the right subset. */ pointsort( &sortarray[right + 1], arraysize - right - 1 ); } } /*****************************************************************************/ /* */ /* pointmedian() An order statistic algorithm, almost. Shuffles an array */ /* of points so that the first `median' points occur */ /* lexicographically before the remaining points. */ /* */ /* Uses the x-coordinate as the primary key if axis == 0; the y-coordinate */ /* if axis == 1. Very similar to the pointsort() procedure, but runs in */ /* randomized linear time. */ /* */ /*****************************************************************************/ void pointmedian( sortarray, arraysize, median, axis ) point * sortarray; int arraysize; int median; int axis; { int left, right; int pivot; REAL pivot1, pivot2; point temp; if ( arraysize == 2 ) { /* Recursive base case. */ if ( ( sortarray[0][axis] > sortarray[1][axis] ) || ( ( sortarray[0][axis] == sortarray[1][axis] ) && ( sortarray[0][1 - axis] > sortarray[1][1 - axis] ) ) ) { temp = sortarray[1]; sortarray[1] = sortarray[0]; sortarray[0] = temp; } return; } /* Choose a random pivot to split the array. */ pivot = (int) randomnation( arraysize ); pivot1 = sortarray[pivot][axis]; pivot2 = sortarray[pivot][1 - axis]; /* Split the array. */ left = -1; right = arraysize; while ( left < right ) { /* Search for a point whose x-coordinate is too large for the left. */ do { left++; } while ( ( left <= right ) && ( ( sortarray[left][axis] < pivot1 ) || ( ( sortarray[left][axis] == pivot1 ) && ( sortarray[left][1 - axis] < pivot2 ) ) ) ); /* Search for a point whose x-coordinate is too small for the right. */ do { right--; } while ( ( left <= right ) && ( ( sortarray[right][axis] > pivot1 ) || ( ( sortarray[right][axis] == pivot1 ) && ( sortarray[right][1 - axis] > pivot2 ) ) ) ); if ( left < right ) { /* Swap the left and right points. */ temp = sortarray[left]; sortarray[left] = sortarray[right]; sortarray[right] = temp; } } /* Unlike in pointsort(), at most one of the following */ /* conditionals is true. */ if ( left > median ) { /* Recursively shuffle the left subset. */ pointmedian( sortarray, left, median, axis ); } if ( right < median - 1 ) { /* Recursively shuffle the right subset. */ pointmedian( &sortarray[right + 1], arraysize - right - 1, median - right - 1, axis ); } } /*****************************************************************************/ /* */ /* alternateaxes() Sorts the points as appropriate for the divide-and- */ /* conquer algorithm with alternating cuts. */ /* */ /* Partitions by x-coordinate if axis == 0; by y-coordinate if axis == 1. */ /* For the base case, subsets containing only two or three points are */ /* always sorted by x-coordinate. */ /* */ /*****************************************************************************/ void alternateaxes( sortarray, arraysize, axis ) point * sortarray; int arraysize; int axis; { int divider; divider = arraysize >> 1; if ( arraysize <= 3 ) { /* Recursive base case: subsets of two or three points will be */ /* handled specially, and should always be sorted by x-coordinate. */ axis = 0; } /* Partition with a horizontal or vertical cut. */ pointmedian( sortarray, arraysize, divider, axis ); /* Recursively partition the subsets with a cross cut. */ if ( arraysize - divider >= 2 ) { if ( divider >= 2 ) { alternateaxes( sortarray, divider, 1 - axis ); } alternateaxes( &sortarray[divider], arraysize - divider, 1 - axis ); } } /*****************************************************************************/ /* */ /* mergehulls() Merge two adjacent Delaunay triangulations into a */ /* single Delaunay triangulation. */ /* */ /* This is similar to the algorithm given by Guibas and Stolfi, but uses */ /* a triangle-based, rather than edge-based, data structure. */ /* */ /* The algorithm walks up the gap between the two triangulations, knitting */ /* them together. As they are merged, some of their bounding triangles */ /* are converted into real triangles of the triangulation. The procedure */ /* pulls each hull's bounding triangles apart, then knits them together */ /* like the teeth of two gears. The Delaunay property determines, at each */ /* step, whether the next "tooth" is a bounding triangle of the left hull */ /* or the right. When a bounding triangle becomes real, its apex is */ /* changed from NULL to a real point. */ /* */ /* Only two new triangles need to be allocated. These become new bounding */ /* triangles at the top and bottom of the seam. They are used to connect */ /* the remaining bounding triangles (those that have not been converted */ /* into real triangles) into a single fan. */ /* */ /* On entry, `farleft' and `innerleft' are bounding triangles of the left */ /* triangulation. The origin of `farleft' is the leftmost vertex, and */ /* the destination of `innerleft' is the rightmost vertex of the */ /* triangulation. Similarly, `innerright' and `farright' are bounding */ /* triangles of the right triangulation. The origin of `innerright' and */ /* destination of `farright' are the leftmost and rightmost vertices. */ /* */ /* On completion, the origin of `farleft' is the leftmost vertex of the */ /* merged triangulation, and the destination of `farright' is the rightmost */ /* vertex. */ /* */ /*****************************************************************************/ void mergehulls( farleft, innerleft, innerright, farright, axis ) struct triedge *farleft; struct triedge *innerleft; struct triedge *innerright; struct triedge *farright; int axis; { struct triedge leftcand, rightcand; struct triedge baseedge; struct triedge nextedge; struct triedge sidecasing, topcasing, outercasing; struct triedge checkedge; point innerleftdest; point innerrightorg; point innerleftapex, innerrightapex; point farleftpt, farrightpt; point farleftapex, farrightapex; point lowerleft, lowerright; point upperleft, upperright; point nextapex; point checkvertex; int changemade; int badedge; int leftfinished, rightfinished; triangle ptr; /* Temporary variable used by sym(). */ dest( *innerleft, innerleftdest ); apex( *innerleft, innerleftapex ); org( *innerright, innerrightorg ); apex( *innerright, innerrightapex ); /* Special treatment for horizontal cuts. */ if ( dwyer && ( axis == 1 ) ) { org( *farleft, farleftpt ); apex( *farleft, farleftapex ); dest( *farright, farrightpt ); apex( *farright, farrightapex ); /* The pointers to the extremal points are shifted to point to the */ /* topmost and bottommost point of each hull, rather than the */ /* leftmost and rightmost points. */ while ( farleftapex[1] < farleftpt[1] ) { lnextself( *farleft ); symself( *farleft ); farleftpt = farleftapex; apex( *farleft, farleftapex ); } sym( *innerleft, checkedge ); apex( checkedge, checkvertex ); while ( checkvertex[1] > innerleftdest[1] ) { lnext( checkedge, *innerleft ); innerleftapex = innerleftdest; innerleftdest = checkvertex; sym( *innerleft, checkedge ); apex( checkedge, checkvertex ); } while ( innerrightapex[1] < innerrightorg[1] ) { lnextself( *innerright ); symself( *innerright ); innerrightorg = innerrightapex; apex( *innerright, innerrightapex ); } sym( *farright, checkedge ); apex( checkedge, checkvertex ); while ( checkvertex[1] > farrightpt[1] ) { lnext( checkedge, *farright ); farrightapex = farrightpt; farrightpt = checkvertex; sym( *farright, checkedge ); apex( checkedge, checkvertex ); } } /* Find a line tangent to and below both hulls. */ do { changemade = 0; /* Make innerleftdest the "bottommost" point of the left hull. */ if ( counterclockwise( innerleftdest, innerleftapex, innerrightorg ) > 0.0 ) { lprevself( *innerleft ); symself( *innerleft ); innerleftdest = innerleftapex; apex( *innerleft, innerleftapex ); changemade = 1; } /* Make innerrightorg the "bottommost" point of the right hull. */ if ( counterclockwise( innerrightapex, innerrightorg, innerleftdest ) > 0.0 ) { lnextself( *innerright ); symself( *innerright ); innerrightorg = innerrightapex; apex( *innerright, innerrightapex ); changemade = 1; } } while ( changemade ); /* Find the two candidates to be the next "gear tooth". */ sym( *innerleft, leftcand ); sym( *innerright, rightcand ); /* Create the bottom new bounding triangle. */ maketriangle( &baseedge ); /* Connect it to the bounding boxes of the left and right triangulations. */ bond( baseedge, *innerleft ); lnextself( baseedge ); bond( baseedge, *innerright ); lnextself( baseedge ); setorg( baseedge, innerrightorg ); setdest( baseedge, innerleftdest ); /* Apex is intentionally left NULL. */ if ( verbose > 2 ) { printf( " Creating base bounding " ); printtriangle( &baseedge ); } /* Fix the extreme triangles if necessary. */ org( *farleft, farleftpt ); if ( innerleftdest == farleftpt ) { lnext( baseedge, *farleft ); } dest( *farright, farrightpt ); if ( innerrightorg == farrightpt ) { lprev( baseedge, *farright ); } /* The vertices of the current knitting edge. */ lowerleft = innerleftdest; lowerright = innerrightorg; /* The candidate vertices for knitting. */ apex( leftcand, upperleft ); apex( rightcand, upperright ); /* Walk up the gap between the two triangulations, knitting them together. */ while ( 1 ) { /* Have we reached the top? (This isn't quite the right question, */ /* because even though the left triangulation might seem finished now, */ /* moving up on the right triangulation might reveal a new point of */ /* the left triangulation. And vice-versa.) */ leftfinished = counterclockwise( upperleft, lowerleft, lowerright ) <= 0.0; rightfinished = counterclockwise( upperright, lowerleft, lowerright ) <= 0.0; if ( leftfinished && rightfinished ) { /* Create the top new bounding triangle. */ maketriangle( &nextedge ); setorg( nextedge, lowerleft ); setdest( nextedge, lowerright ); /* Apex is intentionally left NULL. */ /* Connect it to the bounding boxes of the two triangulations. */ bond( nextedge, baseedge ); lnextself( nextedge ); bond( nextedge, rightcand ); lnextself( nextedge ); bond( nextedge, leftcand ); if ( verbose > 2 ) { printf( " Creating top bounding " ); printtriangle( &baseedge ); } /* Special treatment for horizontal cuts. */ if ( dwyer && ( axis == 1 ) ) { org( *farleft, farleftpt ); apex( *farleft, farleftapex ); dest( *farright, farrightpt ); apex( *farright, farrightapex ); sym( *farleft, checkedge ); apex( checkedge, checkvertex ); /* The pointers to the extremal points are restored to the leftmost */ /* and rightmost points (rather than topmost and bottommost). */ while ( checkvertex[0] < farleftpt[0] ) { lprev( checkedge, *farleft ); farleftapex = farleftpt; farleftpt = checkvertex; sym( *farleft, checkedge ); apex( checkedge, checkvertex ); } while ( farrightapex[0] > farrightpt[0] ) { lprevself( *farright ); symself( *farright ); farrightpt = farrightapex; apex( *farright, farrightapex ); } } return; } /* Consider eliminating edges from the left triangulation. */ if ( !leftfinished ) { /* What vertex would be exposed if an edge were deleted? */ lprev( leftcand, nextedge ); symself( nextedge ); apex( nextedge, nextapex ); /* If nextapex is NULL, then no vertex would be exposed; the */ /* triangulation would have been eaten right through. */ if ( nextapex != (point) NULL ) { /* Check whether the edge is Delaunay. */ badedge = incircle( lowerleft, lowerright, upperleft, nextapex ) > 0.0; while ( badedge ) { /* Eliminate the edge with an edge flip. As a result, the */ /* left triangulation will have one more boundary triangle. */ lnextself( nextedge ); sym( nextedge, topcasing ); lnextself( nextedge ); sym( nextedge, sidecasing ); bond( nextedge, topcasing ); bond( leftcand, sidecasing ); lnextself( leftcand ); sym( leftcand, outercasing ); lprevself( nextedge ); bond( nextedge, outercasing ); /* Correct the vertices to reflect the edge flip. */ setorg( leftcand, lowerleft ); setdest( leftcand, NULL ); setapex( leftcand, nextapex ); setorg( nextedge, NULL ); setdest( nextedge, upperleft ); setapex( nextedge, nextapex ); /* Consider the newly exposed vertex. */ upperleft = nextapex; /* What vertex would be exposed if another edge were deleted? */ triedgecopy( sidecasing, nextedge ); apex( nextedge, nextapex ); if ( nextapex != (point) NULL ) { /* Check whether the edge is Delaunay. */ badedge = incircle( lowerleft, lowerright, upperleft, nextapex ) > 0.0; } else { /* Avoid eating right through the triangulation. */ badedge = 0; } } } } /* Consider eliminating edges from the right triangulation. */ if ( !rightfinished ) { /* What vertex would be exposed if an edge were deleted? */ lnext( rightcand, nextedge ); symself( nextedge ); apex( nextedge, nextapex ); /* If nextapex is NULL, then no vertex would be exposed; the */ /* triangulation would have been eaten right through. */ if ( nextapex != (point) NULL ) { /* Check whether the edge is Delaunay. */ badedge = incircle( lowerleft, lowerright, upperright, nextapex ) > 0.0; while ( badedge ) { /* Eliminate the edge with an edge flip. As a result, the */ /* right triangulation will have one more boundary triangle. */ lprevself( nextedge ); sym( nextedge, topcasing ); lprevself( nextedge ); sym( nextedge, sidecasing ); bond( nextedge, topcasing ); bond( rightcand, sidecasing ); lprevself( rightcand ); sym( rightcand, outercasing ); lnextself( nextedge ); bond( nextedge, outercasing ); /* Correct the vertices to reflect the edge flip. */ setorg( rightcand, NULL ); setdest( rightcand, lowerright ); setapex( rightcand, nextapex ); setorg( nextedge, upperright ); setdest( nextedge, NULL ); setapex( nextedge, nextapex ); /* Consider the newly exposed vertex. */ upperright = nextapex; /* What vertex would be exposed if another edge were deleted? */ triedgecopy( sidecasing, nextedge ); apex( nextedge, nextapex ); if ( nextapex != (point) NULL ) { /* Check whether the edge is Delaunay. */ badedge = incircle( lowerleft, lowerright, upperright, nextapex ) > 0.0; } else { /* Avoid eating right through the triangulation. */ badedge = 0; } } } } if ( leftfinished || ( !rightfinished && ( incircle( upperleft, lowerleft, lowerright, upperright ) > 0.0 ) ) ) { /* Knit the triangulations, adding an edge from `lowerleft' */ /* to `upperright'. */ bond( baseedge, rightcand ); lprev( rightcand, baseedge ); setdest( baseedge, lowerleft ); lowerright = upperright; sym( baseedge, rightcand ); apex( rightcand, upperright ); } else { /* Knit the triangulations, adding an edge from `upperleft' */ /* to `lowerright'. */ bond( baseedge, leftcand ); lnext( leftcand, baseedge ); setorg( baseedge, lowerright ); lowerleft = upperleft; sym( baseedge, leftcand ); apex( leftcand, upperleft ); } if ( verbose > 2 ) { printf( " Connecting " ); printtriangle( &baseedge ); } } } /*****************************************************************************/ /* */ /* divconqrecurse() Recursively form a Delaunay triangulation by the */ /* divide-and-conquer method. */ /* */ /* Recursively breaks down the problem into smaller pieces, which are */ /* knitted together by mergehulls(). The base cases (problems of two or */ /* three points) are handled specially here. */ /* */ /* On completion, `farleft' and `farright' are bounding triangles such that */ /* the origin of `farleft' is the leftmost vertex (breaking ties by */ /* choosing the highest leftmost vertex), and the destination of */ /* `farright' is the rightmost vertex (breaking ties by choosing the */ /* lowest rightmost vertex). */ /* */ /*****************************************************************************/ void divconqrecurse( sortarray, vertices, axis, farleft, farright ) point * sortarray; int vertices; int axis; struct triedge *farleft; struct triedge *farright; { struct triedge midtri, tri1, tri2, tri3; struct triedge innerleft, innerright; REAL area; int divider; if ( verbose > 2 ) { printf( " Triangulating %d points.\n", vertices ); } if ( vertices == 2 ) { /* The triangulation of two vertices is an edge. An edge is */ /* represented by two bounding triangles. */ maketriangle( farleft ); setorg( *farleft, sortarray[0] ); setdest( *farleft, sortarray[1] ); /* The apex is intentionally left NULL. */ maketriangle( farright ); setorg( *farright, sortarray[1] ); setdest( *farright, sortarray[0] ); /* The apex is intentionally left NULL. */ bond( *farleft, *farright ); lprevself( *farleft ); lnextself( *farright ); bond( *farleft, *farright ); lprevself( *farleft ); lnextself( *farright ); bond( *farleft, *farright ); if ( verbose > 2 ) { printf( " Creating " ); printtriangle( farleft ); printf( " Creating " ); printtriangle( farright ); } /* Ensure that the origin of `farleft' is sortarray[0]. */ lprev( *farright, *farleft ); return; } else if ( vertices == 3 ) { /* The triangulation of three vertices is either a triangle (with */ /* three bounding triangles) or two edges (with four bounding */ /* triangles). In either case, four triangles are created. */ maketriangle( &midtri ); maketriangle( &tri1 ); maketriangle( &tri2 ); maketriangle( &tri3 ); area = counterclockwise( sortarray[0], sortarray[1], sortarray[2] ); if ( area == 0.0 ) { /* Three collinear points; the triangulation is two edges. */ setorg( midtri, sortarray[0] ); setdest( midtri, sortarray[1] ); setorg( tri1, sortarray[1] ); setdest( tri1, sortarray[0] ); setorg( tri2, sortarray[2] ); setdest( tri2, sortarray[1] ); setorg( tri3, sortarray[1] ); setdest( tri3, sortarray[2] ); /* All apices are intentionally left NULL. */ bond( midtri, tri1 ); bond( tri2, tri3 ); lnextself( midtri ); lprevself( tri1 ); lnextself( tri2 ); lprevself( tri3 ); bond( midtri, tri3 ); bond( tri1, tri2 ); lnextself( midtri ); lprevself( tri1 ); lnextself( tri2 ); lprevself( tri3 ); bond( midtri, tri1 ); bond( tri2, tri3 ); /* Ensure that the origin of `farleft' is sortarray[0]. */ triedgecopy( tri1, *farleft ); /* Ensure that the destination of `farright' is sortarray[2]. */ triedgecopy( tri2, *farright ); } else { /* The three points are not collinear; the triangulation is one */ /* triangle, namely `midtri'. */ setorg( midtri, sortarray[0] ); setdest( tri1, sortarray[0] ); setorg( tri3, sortarray[0] ); /* Apices of tri1, tri2, and tri3 are left NULL. */ if ( area > 0.0 ) { /* The vertices are in counterclockwise order. */ setdest( midtri, sortarray[1] ); setorg( tri1, sortarray[1] ); setdest( tri2, sortarray[1] ); setapex( midtri, sortarray[2] ); setorg( tri2, sortarray[2] ); setdest( tri3, sortarray[2] ); } else { /* The vertices are in clockwise order. */ setdest( midtri, sortarray[2] ); setorg( tri1, sortarray[2] ); setdest( tri2, sortarray[2] ); setapex( midtri, sortarray[1] ); setorg( tri2, sortarray[1] ); setdest( tri3, sortarray[1] ); } /* The topology does not depend on how the vertices are ordered. */ bond( midtri, tri1 ); lnextself( midtri ); bond( midtri, tri2 ); lnextself( midtri ); bond( midtri, tri3 ); lprevself( tri1 ); lnextself( tri2 ); bond( tri1, tri2 ); lprevself( tri1 ); lprevself( tri3 ); bond( tri1, tri3 ); lnextself( tri2 ); lprevself( tri3 ); bond( tri2, tri3 ); /* Ensure that the origin of `farleft' is sortarray[0]. */ triedgecopy( tri1, *farleft ); /* Ensure that the destination of `farright' is sortarray[2]. */ if ( area > 0.0 ) { triedgecopy( tri2, *farright ); } else { lnext( *farleft, *farright ); } } if ( verbose > 2 ) { printf( " Creating " ); printtriangle( &midtri ); printf( " Creating " ); printtriangle( &tri1 ); printf( " Creating " ); printtriangle( &tri2 ); printf( " Creating " ); printtriangle( &tri3 ); } return; } else { /* Split the vertices in half. */ divider = vertices >> 1; /* Recursively triangulate each half. */ divconqrecurse( sortarray, divider, 1 - axis, farleft, &innerleft ); divconqrecurse( &sortarray[divider], vertices - divider, 1 - axis, &innerright, farright ); if ( verbose > 1 ) { printf( " Joining triangulations with %d and %d vertices.\n", divider, vertices - divider ); } /* Merge the two triangulations into one. */ mergehulls( farleft, &innerleft, &innerright, farright, axis ); } } long removeghosts( startghost ) struct triedge *startghost; { struct triedge searchedge; struct triedge dissolveedge; struct triedge deadtri; point markorg; long hullsize; triangle ptr; /* Temporary variable used by sym(). */ if ( verbose ) { printf( " Removing ghost triangles.\n" ); } /* Find an edge on the convex hull to start point location from. */ lprev( *startghost, searchedge ); symself( searchedge ); dummytri[0] = encode( searchedge ); /* Remove the bounding box and count the convex hull edges. */ triedgecopy( *startghost, dissolveedge ); hullsize = 0; do { hullsize++; lnext( dissolveedge, deadtri ); lprevself( dissolveedge ); symself( dissolveedge ); /* If no PSLG is involved, set the boundary markers of all the points */ /* on the convex hull. If a PSLG is used, this step is done later. */ if ( !poly ) { /* Watch out for the case where all the input points are collinear. */ if ( dissolveedge.tri != dummytri ) { org( dissolveedge, markorg ); if ( pointmark( markorg ) == 0 ) { setpointmark( markorg, 1 ); } } } /* Remove a bounding triangle from a convex hull triangle. */ dissolve( dissolveedge ); /* Find the next bounding triangle. */ sym( deadtri, dissolveedge ); /* Delete the bounding triangle. */ triangledealloc( deadtri.tri ); } while ( !triedgeequal( dissolveedge, *startghost ) ); return hullsize; } /*****************************************************************************/ /* */ /* divconqdelaunay() Form a Delaunay triangulation by the divide-and- */ /* conquer method. */ /* */ /* Sorts the points, calls a recursive procedure to triangulate them, and */ /* removes the bounding box, setting boundary markers as appropriate. */ /* */ /*****************************************************************************/ long divconqdelaunay(){ point *sortarray; struct triedge hullleft, hullright; int divider; int i, j; /* Allocate an array of pointers to points for sorting. */ sortarray = (point *) malloc( inpoints * sizeof( point ) ); if ( sortarray == (point *) NULL ) { printf( "Error: Out of memory.\n" ); exit( 1 ); } traversalinit( &points ); for ( i = 0; i < inpoints; i++ ) { sortarray[i] = pointtraverse(); } if ( verbose ) { printf( " Sorting points.\n" ); } /* Sort the points. */ pointsort( sortarray, inpoints ); /* Discard duplicate points, which can really mess up the algorithm. */ i = 0; for ( j = 1; j < inpoints; j++ ) { if ( ( sortarray[i][0] == sortarray[j][0] ) && ( sortarray[i][1] == sortarray[j][1] ) ) { if ( !quiet ) { printf( "Warning: A duplicate point at (%.12g, %.12g) appeared and was ignored.\n", sortarray[j][0], sortarray[j][1] ); } /* Commented out - would eliminate point from output .node file, but causes a failure if some segment has this point as an endpoint. setpointmark(sortarray[j], DEADPOINT); */ } else { i++; sortarray[i] = sortarray[j]; } } i++; if ( dwyer ) { /* Re-sort the array of points to accommodate alternating cuts. */ divider = i >> 1; if ( i - divider >= 2 ) { if ( divider >= 2 ) { alternateaxes( sortarray, divider, 1 ); } alternateaxes( &sortarray[divider], i - divider, 1 ); } } if ( verbose ) { printf( " Forming triangulation.\n" ); } /* Form the Delaunay triangulation. */ divconqrecurse( sortarray, i, 0, &hullleft, &hullright ); free( sortarray ); return removeghosts( &hullleft ); } /** **/ /** **/ /********* Divide-and-conquer Delaunay triangulation ends here *********/ /********* Incremental Delaunay triangulation begins here *********/ /** **/ /** **/ /*****************************************************************************/ /* */ /* boundingbox() Form an "infinite" bounding triangle to insert points */ /* into. */ /* */ /* The points at "infinity" are assigned finite coordinates, which are used */ /* by the point location routines, but (mostly) ignored by the Delaunay */ /* edge flip routines. */ /* */ /*****************************************************************************/ #ifndef REDUCED void boundingbox(){ struct triedge inftri; /* Handle for the triangular bounding box. */ REAL width; if ( verbose ) { printf( " Creating triangular bounding box.\n" ); } /* Find the width (or height, whichever is larger) of the triangulation. */ width = xmax - xmin; if ( ymax - ymin > width ) { width = ymax - ymin; } if ( width == 0.0 ) { width = 1.0; } /* Create the vertices of the bounding box. */ infpoint1 = (point) malloc( points.itembytes ); infpoint2 = (point) malloc( points.itembytes ); infpoint3 = (point) malloc( points.itembytes ); if ( ( infpoint1 == (point) NULL ) || ( infpoint2 == (point) NULL ) || ( infpoint3 == (point) NULL ) ) { printf( "Error: Out of memory.\n" ); exit( 1 ); } infpoint1[0] = xmin - 50.0 * width; infpoint1[1] = ymin - 40.0 * width; infpoint2[0] = xmax + 50.0 * width; infpoint2[1] = ymin - 40.0 * width; infpoint3[0] = 0.5 * ( xmin + xmax ); infpoint3[1] = ymax + 60.0 * width; /* Create the bounding box. */ maketriangle( &inftri ); setorg( inftri, infpoint1 ); setdest( inftri, infpoint2 ); setapex( inftri, infpoint3 ); /* Link dummytri to the bounding box so we can always find an */ /* edge to begin searching (point location) from. */ dummytri[0] = (triangle) inftri.tri; if ( verbose > 2 ) { printf( " Creating " ); printtriangle( &inftri ); } } #endif /* not REDUCED */ /*****************************************************************************/ /* */ /* removebox() Remove the "infinite" bounding triangle, setting boundary */ /* markers as appropriate. */ /* */ /* The triangular bounding box has three boundary triangles (one for each */ /* side of the bounding box), and a bunch of triangles fanning out from */ /* the three bounding box vertices (one triangle for each edge of the */ /* convex hull of the inner mesh). This routine removes these triangles. */ /* */ /*****************************************************************************/ #ifndef REDUCED long removebox(){ struct triedge deadtri; struct triedge searchedge; struct triedge checkedge; struct triedge nextedge, finaledge, dissolveedge; point markorg; long hullsize; triangle ptr; /* Temporary variable used by sym(). */ if ( verbose ) { printf( " Removing triangular bounding box.\n" ); } /* Find a boundary triangle. */ nextedge.tri = dummytri; nextedge.orient = 0; symself( nextedge ); /* Mark a place to stop. */ lprev( nextedge, finaledge ); lnextself( nextedge ); symself( nextedge ); /* Find a triangle (on the boundary of the point set) that isn't */ /* a bounding box triangle. */ lprev( nextedge, searchedge ); symself( searchedge ); /* Check whether nextedge is another boundary triangle */ /* adjacent to the first one. */ lnext( nextedge, checkedge ); symself( checkedge ); if ( checkedge.tri == dummytri ) { /* Go on to the next triangle. There are only three boundary */ /* triangles, and this next triangle cannot be the third one, */ /* so it's safe to stop here. */ lprevself( searchedge ); symself( searchedge ); } /* Find a new boundary edge to search from, as the current search */ /* edge lies on a bounding box triangle and will be deleted. */ dummytri[0] = encode( searchedge ); hullsize = -2l; while ( !triedgeequal( nextedge, finaledge ) ) { hullsize++; lprev( nextedge, dissolveedge ); symself( dissolveedge ); /* If not using a PSLG, the vertices should be marked now. */ /* (If using a PSLG, markhull() will do the job.) */ if ( !poly ) { /* Be careful! One must check for the case where all the input */ /* points are collinear, and thus all the triangles are part of */ /* the bounding box. Otherwise, the setpointmark() call below */ /* will cause a bad pointer reference. */ if ( dissolveedge.tri != dummytri ) { org( dissolveedge, markorg ); if ( pointmark( markorg ) == 0 ) { setpointmark( markorg, 1 ); } } } /* Disconnect the bounding box triangle from the mesh triangle. */ dissolve( dissolveedge ); lnext( nextedge, deadtri ); sym( deadtri, nextedge ); /* Get rid of the bounding box triangle. */ triangledealloc( deadtri.tri ); /* Do we need to turn the corner? */ if ( nextedge.tri == dummytri ) { /* Turn the corner. */ triedgecopy( dissolveedge, nextedge ); } } triangledealloc( finaledge.tri ); free( infpoint1 ); /* Deallocate the bounding box vertices. */ free( infpoint2 ); free( infpoint3 ); return hullsize; } #endif /* not REDUCED */ /*****************************************************************************/ /* */ /* incrementaldelaunay() Form a Delaunay triangulation by incrementally */ /* adding vertices. */ /* */ /*****************************************************************************/ #ifndef REDUCED long incrementaldelaunay(){ struct triedge starttri; point pointloop; int i; /* Create a triangular bounding box. */ boundingbox(); if ( verbose ) { printf( " Incrementally inserting points.\n" ); } traversalinit( &points ); pointloop = pointtraverse(); i = 1; while ( pointloop != (point) NULL ) { /* Find a boundary triangle to search from. */ starttri.tri = (triangle *) NULL; if ( insertsite( pointloop, &starttri, (struct edge *) NULL, 0, 0 ) == DUPLICATEPOINT ) { if ( !quiet ) { printf( "Warning: A duplicate point at (%.12g, %.12g) appeared and was ignored.\n", pointloop[0], pointloop[1] ); } /* Commented out - would eliminate point from output .node file. setpointmark(pointloop, DEADPOINT); */ } pointloop = pointtraverse(); i++; } /* Remove the bounding box. */ return removebox(); } #endif /* not REDUCED */ /** **/ /** **/ /********* Incremental Delaunay triangulation ends here *********/ /********* Sweepline Delaunay triangulation begins here *********/ /** **/ /** **/ #ifndef REDUCED void eventheapinsert( heap, heapsize, newevent ) struct event **heap; int heapsize; struct event *newevent; { REAL eventx, eventy; int eventnum; int parent; int notdone; eventx = newevent->xkey; eventy = newevent->ykey; eventnum = heapsize; notdone = eventnum > 0; while ( notdone ) { parent = ( eventnum - 1 ) >> 1; if ( ( heap[parent]->ykey < eventy ) || ( ( heap[parent]->ykey == eventy ) && ( heap[parent]->xkey <= eventx ) ) ) { notdone = 0; } else { heap[eventnum] = heap[parent]; heap[eventnum]->heapposition = eventnum; eventnum = parent; notdone = eventnum > 0; } } heap[eventnum] = newevent; newevent->heapposition = eventnum; } #endif /* not REDUCED */ #ifndef REDUCED void eventheapify( heap, heapsize, eventnum ) struct event **heap; int heapsize; int eventnum; { struct event *thisevent; REAL eventx, eventy; int leftchild, rightchild; int smallest; int notdone; thisevent = heap[eventnum]; eventx = thisevent->xkey; eventy = thisevent->ykey; leftchild = 2 * eventnum + 1; notdone = leftchild < heapsize; while ( notdone ) { if ( ( heap[leftchild]->ykey < eventy ) || ( ( heap[leftchild]->ykey == eventy ) && ( heap[leftchild]->xkey < eventx ) ) ) { smallest = leftchild; } else { smallest = eventnum; } rightchild = leftchild + 1; if ( rightchild < heapsize ) { if ( ( heap[rightchild]->ykey < heap[smallest]->ykey ) || ( ( heap[rightchild]->ykey == heap[smallest]->ykey ) && ( heap[rightchild]->xkey < heap[smallest]->xkey ) ) ) { smallest = rightchild; } } if ( smallest == eventnum ) { notdone = 0; } else { heap[eventnum] = heap[smallest]; heap[eventnum]->heapposition = eventnum; heap[smallest] = thisevent; thisevent->heapposition = smallest; eventnum = smallest; leftchild = 2 * eventnum + 1; notdone = leftchild < heapsize; } } } #endif /* not REDUCED */ #ifndef REDUCED void eventheapdelete( heap, heapsize, eventnum ) struct event **heap; int heapsize; int eventnum; { struct event *moveevent; REAL eventx, eventy; int parent; int notdone; moveevent = heap[heapsize - 1]; if ( eventnum > 0 ) { eventx = moveevent->xkey; eventy = moveevent->ykey; do { parent = ( eventnum - 1 ) >> 1; if ( ( heap[parent]->ykey < eventy ) || ( ( heap[parent]->ykey == eventy ) && ( heap[parent]->xkey <= eventx ) ) ) { notdone = 0; } else { heap[eventnum] = heap[parent]; heap[eventnum]->heapposition = eventnum; eventnum = parent; notdone = eventnum > 0; } } while ( notdone ); } heap[eventnum] = moveevent; moveevent->heapposition = eventnum; eventheapify( heap, heapsize - 1, eventnum ); } #endif /* not REDUCED */ #ifndef REDUCED void createeventheap( eventheap, events, freeevents ) struct event ***eventheap; struct event **events; struct event **freeevents; { point thispoint; int maxevents; int i; maxevents = ( 3 * inpoints ) / 2; *eventheap = (struct event **) malloc( maxevents * sizeof( struct event * ) ); if ( *eventheap == (struct event **) NULL ) { printf( "Error: Out of memory.\n" ); exit( 1 ); } *events = (struct event *) malloc( maxevents * sizeof( struct event ) ); if ( *events == (struct event *) NULL ) { printf( "Error: Out of memory.\n" ); exit( 1 ); } traversalinit( &points ); for ( i = 0; i < inpoints; i++ ) { thispoint = pointtraverse(); ( *events )[i].eventptr = (VOID *) thispoint; ( *events )[i].xkey = thispoint[0]; ( *events )[i].ykey = thispoint[1]; eventheapinsert( *eventheap, i, *events + i ); } *freeevents = (struct event *) NULL; for ( i = maxevents - 1; i >= inpoints; i-- ) { ( *events )[i].eventptr = (VOID *) *freeevents; *freeevents = *events + i; } } #endif /* not REDUCED */ #ifndef REDUCED int rightofhyperbola( fronttri, newsite ) struct triedge *fronttri; point newsite; { point leftpoint, rightpoint; REAL dxa, dya, dxb, dyb; hyperbolacount++; dest( *fronttri, leftpoint ); apex( *fronttri, rightpoint ); if ( ( leftpoint[1] < rightpoint[1] ) || ( ( leftpoint[1] == rightpoint[1] ) && ( leftpoint[0] < rightpoint[0] ) ) ) { if ( newsite[0] >= rightpoint[0] ) { return 1; } } else { if ( newsite[0] <= leftpoint[0] ) { return 0; } } dxa = leftpoint[0] - newsite[0]; dya = leftpoint[1] - newsite[1]; dxb = rightpoint[0] - newsite[0]; dyb = rightpoint[1] - newsite[1]; return dya * ( dxb * dxb + dyb * dyb ) > dyb * ( dxa * dxa + dya * dya ); } #endif /* not REDUCED */ #ifndef REDUCED REAL circletop( pa, pb, pc, ccwabc ) point pa; point pb; point pc; REAL ccwabc; { REAL xac, yac, xbc, ybc, xab, yab; REAL aclen2, bclen2, ablen2; circletopcount++; xac = pa[0] - pc[0]; yac = pa[1] - pc[1]; xbc = pb[0] - pc[0]; ybc = pb[1] - pc[1]; xab = pa[0] - pb[0]; yab = pa[1] - pb[1]; aclen2 = xac * xac + yac * yac; bclen2 = xbc * xbc + ybc * ybc; ablen2 = xab * xab + yab * yab; return pc[1] + ( xac * bclen2 - xbc * aclen2 + sqrt( aclen2 * bclen2 * ablen2 ) ) / ( 2.0 * ccwabc ); } #endif /* not REDUCED */ #ifndef REDUCED void check4deadevent( checktri, freeevents, eventheap, heapsize ) struct triedge *checktri; struct event **freeevents; struct event **eventheap; int *heapsize; { struct event *deadevent; point eventpoint; int eventnum; org( *checktri, eventpoint ); if ( eventpoint != (point) NULL ) { deadevent = (struct event *) eventpoint; eventnum = deadevent->heapposition; deadevent->eventptr = (VOID *) *freeevents; *freeevents = deadevent; eventheapdelete( eventheap, *heapsize, eventnum ); ( *heapsize )--; setorg( *checktri, NULL ); } } #endif /* not REDUCED */ #ifndef REDUCED struct splaynode *splay( splaytree, searchpoint, searchtri ) struct splaynode *splaytree; point searchpoint; struct triedge *searchtri; { struct splaynode *child, *grandchild; struct splaynode *lefttree, *righttree; struct splaynode *leftright; point checkpoint; int rightofroot, rightofchild; if ( splaytree == (struct splaynode *) NULL ) { return (struct splaynode *) NULL; } dest( splaytree->keyedge, checkpoint ); if ( checkpoint == splaytree->keydest ) { rightofroot = rightofhyperbola( &splaytree->keyedge, searchpoint ); if ( rightofroot ) { triedgecopy( splaytree->keyedge, *searchtri ); child = splaytree->rchild; } else { child = splaytree->lchild; } if ( child == (struct splaynode *) NULL ) { return splaytree; } dest( child->keyedge, checkpoint ); if ( checkpoint != child->keydest ) { child = splay( child, searchpoint, searchtri ); if ( child == (struct splaynode *) NULL ) { if ( rightofroot ) { splaytree->rchild = (struct splaynode *) NULL; } else { splaytree->lchild = (struct splaynode *) NULL; } return splaytree; } } rightofchild = rightofhyperbola( &child->keyedge, searchpoint ); if ( rightofchild ) { triedgecopy( child->keyedge, *searchtri ); grandchild = splay( child->rchild, searchpoint, searchtri ); child->rchild = grandchild; } else { grandchild = splay( child->lchild, searchpoint, searchtri ); child->lchild = grandchild; } if ( grandchild == (struct splaynode *) NULL ) { if ( rightofroot ) { splaytree->rchild = child->lchild; child->lchild = splaytree; } else { splaytree->lchild = child->rchild; child->rchild = splaytree; } return child; } if ( rightofchild ) { if ( rightofroot ) { splaytree->rchild = child->lchild; child->lchild = splaytree; } else { splaytree->lchild = grandchild->rchild; grandchild->rchild = splaytree; } child->rchild = grandchild->lchild; grandchild->lchild = child; } else { if ( rightofroot ) { splaytree->rchild = grandchild->lchild; grandchild->lchild = splaytree; } else { splaytree->lchild = child->rchild; child->rchild = splaytree; } child->lchild = grandchild->rchild; grandchild->rchild = child; } return grandchild; } else { lefttree = splay( splaytree->lchild, searchpoint, searchtri ); righttree = splay( splaytree->rchild, searchpoint, searchtri ); pooldealloc( &splaynodes, (VOID *) splaytree ); if ( lefttree == (struct splaynode *) NULL ) { return righttree; } else if ( righttree == (struct splaynode *) NULL ) { return lefttree; } else if ( lefttree->rchild == (struct splaynode *) NULL ) { lefttree->rchild = righttree->lchild; righttree->lchild = lefttree; return righttree; } else if ( righttree->lchild == (struct splaynode *) NULL ) { righttree->lchild = lefttree->rchild; lefttree->rchild = righttree; return lefttree; } else { /* printf("Holy Toledo!!!\n"); */ leftright = lefttree->rchild; while ( leftright->rchild != (struct splaynode *) NULL ) { leftright = leftright->rchild; } leftright->rchild = righttree; return lefttree; } } } #endif /* not REDUCED */ #ifndef REDUCED struct splaynode *splayinsert( splayroot, newkey, searchpoint ) struct splaynode *splayroot; struct triedge *newkey; point searchpoint; { struct splaynode *newsplaynode; newsplaynode = (struct splaynode *) poolalloc( &splaynodes ); triedgecopy( *newkey, newsplaynode->keyedge ); dest( *newkey, newsplaynode->keydest ); if ( splayroot == (struct splaynode *) NULL ) { newsplaynode->lchild = (struct splaynode *) NULL; newsplaynode->rchild = (struct splaynode *) NULL; } else if ( rightofhyperbola( &splayroot->keyedge, searchpoint ) ) { newsplaynode->lchild = splayroot; newsplaynode->rchild = splayroot->rchild; splayroot->rchild = (struct splaynode *) NULL; } else { newsplaynode->lchild = splayroot->lchild; newsplaynode->rchild = splayroot; splayroot->lchild = (struct splaynode *) NULL; } return newsplaynode; } #endif /* not REDUCED */ #ifndef REDUCED struct splaynode *circletopinsert( splayroot, newkey, pa, pb, pc, topy ) struct splaynode *splayroot; struct triedge *newkey; point pa; point pb; point pc; REAL topy; { REAL ccwabc; REAL xac, yac, xbc, ybc; REAL aclen2, bclen2; REAL searchpoint[2]; struct triedge dummytri; ccwabc = counterclockwise( pa, pb, pc ); xac = pa[0] - pc[0]; yac = pa[1] - pc[1]; xbc = pb[0] - pc[0]; ybc = pb[1] - pc[1]; aclen2 = xac * xac + yac * yac; bclen2 = xbc * xbc + ybc * ybc; searchpoint[0] = pc[0] - ( yac * bclen2 - ybc * aclen2 ) / ( 2.0 * ccwabc ); searchpoint[1] = topy; return splayinsert( splay( splayroot, (point) searchpoint, &dummytri ), newkey, (point) searchpoint ); } #endif /* not REDUCED */ #ifndef REDUCED struct splaynode *frontlocate( splayroot, bottommost, searchpoint, searchtri, farright ) struct splaynode *splayroot; struct triedge *bottommost; point searchpoint; struct triedge *searchtri; int *farright; { int farrightflag; triangle ptr; /* Temporary variable used by onext(). */ triedgecopy( *bottommost, *searchtri ); splayroot = splay( splayroot, searchpoint, searchtri ); farrightflag = 0; while ( !farrightflag && rightofhyperbola( searchtri, searchpoint ) ) { onextself( *searchtri ); farrightflag = triedgeequal( *searchtri, *bottommost ); } *farright = farrightflag; return splayroot; } #endif /* not REDUCED */ #ifndef REDUCED long sweeplinedelaunay(){ struct event **eventheap; struct event *events; struct event *freeevents; struct event *nextevent; struct event *newevent; struct splaynode *splayroot; struct triedge bottommost; struct triedge searchtri; struct triedge fliptri; struct triedge lefttri, righttri, farlefttri, farrighttri; struct triedge inserttri; point firstpoint, secondpoint; point nextpoint, lastpoint; point connectpoint; point leftpoint, midpoint, rightpoint; REAL lefttest, righttest; int heapsize; int check4events, farrightflag; triangle ptr; /* Temporary variable used by sym(), onext(), and oprev(). */ poolinit( &splaynodes, sizeof( struct splaynode ), SPLAYNODEPERBLOCK, POINTER, 0 ); splayroot = (struct splaynode *) NULL; if ( verbose ) { printf( " Placing points in event heap.\n" ); } createeventheap( &eventheap, &events, &freeevents ); heapsize = inpoints; if ( verbose ) { printf( " Forming triangulation.\n" ); } maketriangle( &lefttri ); maketriangle( &righttri ); bond( lefttri, righttri ); lnextself( lefttri ); lprevself( righttri ); bond( lefttri, righttri ); lnextself( lefttri ); lprevself( righttri ); bond( lefttri, righttri ); firstpoint = (point) eventheap[0]->eventptr; eventheap[0]->eventptr = (VOID *) freeevents; freeevents = eventheap[0]; eventheapdelete( eventheap, heapsize, 0 ); heapsize--; do { if ( heapsize == 0 ) { printf( "Error: Input points are all identical.\n" ); exit( 1 ); } secondpoint = (point) eventheap[0]->eventptr; eventheap[0]->eventptr = (VOID *) freeevents; freeevents = eventheap[0]; eventheapdelete( eventheap, heapsize, 0 ); heapsize--; if ( ( firstpoint[0] == secondpoint[0] ) && ( firstpoint[1] == secondpoint[1] ) ) { printf( "Warning: A duplicate point at (%.12g, %.12g) appeared and was ignored.\n", secondpoint[0], secondpoint[1] ); /* Commented out - would eliminate point from output .node file. setpointmark(secondpoint, DEADPOINT); */ } } while ( ( firstpoint[0] == secondpoint[0] ) && ( firstpoint[1] == secondpoint[1] ) ); setorg( lefttri, firstpoint ); setdest( lefttri, secondpoint ); setorg( righttri, secondpoint ); setdest( righttri, firstpoint ); lprev( lefttri, bottommost ); lastpoint = secondpoint; while ( heapsize > 0 ) { nextevent = eventheap[0]; eventheapdelete( eventheap, heapsize, 0 ); heapsize--; check4events = 1; if ( nextevent->xkey < xmin ) { decode( nextevent->eventptr, fliptri ); oprev( fliptri, farlefttri ); check4deadevent( &farlefttri, &freeevents, eventheap, &heapsize ); onext( fliptri, farrighttri ); check4deadevent( &farrighttri, &freeevents, eventheap, &heapsize ); if ( triedgeequal( farlefttri, bottommost ) ) { lprev( fliptri, bottommost ); } flip( &fliptri ); setapex( fliptri, NULL ); lprev( fliptri, lefttri ); lnext( fliptri, righttri ); sym( lefttri, farlefttri ); if ( randomnation( SAMPLERATE ) == 0 ) { symself( fliptri ); dest( fliptri, leftpoint ); apex( fliptri, midpoint ); org( fliptri, rightpoint ); splayroot = circletopinsert( splayroot, &lefttri, leftpoint, midpoint, rightpoint, nextevent->ykey ); } } else { nextpoint = (point) nextevent->eventptr; if ( ( nextpoint[0] == lastpoint[0] ) && ( nextpoint[1] == lastpoint[1] ) ) { printf( "Warning: A duplicate point at (%.12g, %.12g) appeared and was ignored.\n", nextpoint[0], nextpoint[1] ); /* Commented out - would eliminate point from output .node file. setpointmark(nextpoint, DEADPOINT); */ check4events = 0; } else { lastpoint = nextpoint; splayroot = frontlocate( splayroot, &bottommost, nextpoint, &searchtri, &farrightflag ); /* triedgecopy(bottommost, searchtri); farrightflag = 0; while (!farrightflag && rightofhyperbola(&searchtri, nextpoint)) { onextself(searchtri); farrightflag = triedgeequal(searchtri, bottommost); } */ check4deadevent( &searchtri, &freeevents, eventheap, &heapsize ); triedgecopy( searchtri, farrighttri ); sym( searchtri, farlefttri ); maketriangle( &lefttri ); maketriangle( &righttri ); dest( farrighttri, connectpoint ); setorg( lefttri, connectpoint ); setdest( lefttri, nextpoint ); setorg( righttri, nextpoint ); setdest( righttri, connectpoint ); bond( lefttri, righttri ); lnextself( lefttri ); lprevself( righttri ); bond( lefttri, righttri ); lnextself( lefttri ); lprevself( righttri ); bond( lefttri, farlefttri ); bond( righttri, farrighttri ); if ( !farrightflag && triedgeequal( farrighttri, bottommost ) ) { triedgecopy( lefttri, bottommost ); } if ( randomnation( SAMPLERATE ) == 0 ) { splayroot = splayinsert( splayroot, &lefttri, nextpoint ); } else if ( randomnation( SAMPLERATE ) == 0 ) { lnext( righttri, inserttri ); splayroot = splayinsert( splayroot, &inserttri, nextpoint ); } } } nextevent->eventptr = (VOID *) freeevents; freeevents = nextevent; if ( check4events ) { apex( farlefttri, leftpoint ); dest( lefttri, midpoint ); apex( lefttri, rightpoint ); lefttest = counterclockwise( leftpoint, midpoint, rightpoint ); if ( lefttest > 0.0 ) { newevent = freeevents; freeevents = (struct event *) freeevents->eventptr; newevent->xkey = xminextreme; newevent->ykey = circletop( leftpoint, midpoint, rightpoint, lefttest ); newevent->eventptr = (VOID *) encode( lefttri ); eventheapinsert( eventheap, heapsize, newevent ); heapsize++; setorg( lefttri, newevent ); } apex( righttri, leftpoint ); org( righttri, midpoint ); apex( farrighttri, rightpoint ); righttest = counterclockwise( leftpoint, midpoint, rightpoint ); if ( righttest > 0.0 ) { newevent = freeevents; freeevents = (struct event *) freeevents->eventptr; newevent->xkey = xminextreme; newevent->ykey = circletop( leftpoint, midpoint, rightpoint, righttest ); newevent->eventptr = (VOID *) encode( farrighttri ); eventheapinsert( eventheap, heapsize, newevent ); heapsize++; setorg( farrighttri, newevent ); } } } pooldeinit( &splaynodes ); lprevself( bottommost ); return removeghosts( &bottommost ); } #endif /* not REDUCED */ /** **/ /** **/ /********* Sweepline Delaunay triangulation ends here *********/ /********* General mesh construction routines begin here *********/ /** **/ /** **/ /*****************************************************************************/ /* */ /* delaunay() Form a Delaunay triangulation. */ /* */ /*****************************************************************************/ long delaunay(){ eextras = 0; initializetrisegpools(); #ifdef REDUCED if ( !quiet ) { printf( "Constructing Delaunay triangulation by divide-and-conquer method.\n" ); } return divconqdelaunay(); #else /* not REDUCED */ if ( !quiet ) { printf( "Constructing Delaunay triangulation " ); if ( incremental ) { printf( "by incremental method.\n" ); } else if ( sweepline ) { printf( "by sweepline method.\n" ); } else { printf( "by divide-and-conquer method.\n" ); } } if ( incremental ) { return incrementaldelaunay(); } else if ( sweepline ) { return sweeplinedelaunay(); } else { return divconqdelaunay(); } #endif /* not REDUCED */ } /*****************************************************************************/ /* */ /* reconstruct() Reconstruct a triangulation from its .ele (and possibly */ /* .poly) file. Used when the -r switch is used. */ /* */ /* Reads an .ele file and reconstructs the original mesh. If the -p switch */ /* is used, this procedure will also read a .poly file and reconstruct the */ /* shell edges of the original mesh. If the -a switch is used, this */ /* procedure will also read an .area file and set a maximum area constraint */ /* on each triangle. */ /* */ /* Points that are not corners of triangles, such as nodes on edges of */ /* subparametric elements, are discarded. */ /* */ /* This routine finds the adjacencies between triangles (and shell edges) */ /* by forming one stack of triangles for each vertex. Each triangle is on */ /* three different stacks simultaneously. Each triangle's shell edge */ /* pointers are used to link the items in each stack. This memory-saving */ /* feature makes the code harder to read. The most important thing to keep */ /* in mind is that each triangle is removed from a stack precisely when */ /* the corresponding pointer is adjusted to refer to a shell edge rather */ /* than the next triangle of the stack. */ /* */ /*****************************************************************************/ #ifndef CDT_ONLY #ifdef TRILIBRARY int reconstruct( trianglelist, triangleattriblist, trianglearealist, elements, corners, attribs, segmentlist, segmentmarkerlist, numberofsegments ) int *trianglelist; REAL *triangleattriblist; REAL *trianglearealist; int elements; int corners; int attribs; int *segmentlist; int *segmentmarkerlist; int numberofsegments; #else /* not TRILIBRARY */ long reconstruct( elefilename, areafilename, polyfilename, polyfile ) char *elefilename; char *areafilename; char *polyfilename; FILE *polyfile; #endif /* not TRILIBRARY */ { #ifdef TRILIBRARY int pointindex; int attribindex; #else /* not TRILIBRARY */ FILE *elefile; FILE *areafile; char inputline[INPUTLINESIZE]; char *stringptr; int areaelements; #endif /* not TRILIBRARY */ struct triedge triangleloop; struct triedge triangleleft; struct triedge checktri; struct triedge checkleft; struct triedge checkneighbor; struct edge shelleloop; triangle *vertexarray; triangle *prevlink; triangle nexttri; point tdest, tapex; point checkdest, checkapex; point shorg; point killpoint; REAL area; int corner[3]; int end[2]; int killpointindex; int incorners; int segmentmarkers; int boundmarker; int aroundpoint; long hullsize; int notfound; int elementnumber, segmentnumber; int i, j; triangle ptr; /* Temporary variable used by sym(). */ #ifdef TRILIBRARY inelements = elements; incorners = corners; if ( incorners < 3 ) { printf( "Error: Triangles must have at least 3 points.\n" ); exit( 1 ); } eextras = attribs; #else /* not TRILIBRARY */ /* Read the triangles from an .ele file. */ if ( !quiet ) { printf( "Opening %s.\n", elefilename ); } elefile = fopen( elefilename, "r" ); if ( elefile == (FILE *) NULL ) { printf( " Error: Cannot access file %s.\n", elefilename ); exit( 1 ); } /* Read number of triangles, number of points per triangle, and */ /* number of triangle attributes from .ele file. */ stringptr = readline( inputline, elefile, elefilename ); inelements = (int) strtol( stringptr, &stringptr, 0 ); stringptr = findfield( stringptr ); if ( *stringptr == '\0' ) { incorners = 3; } else { incorners = (int) strtol( stringptr, &stringptr, 0 ); if ( incorners < 3 ) { printf( "Error: Triangles in %s must have at least 3 points.\n", elefilename ); exit( 1 ); } } stringptr = findfield( stringptr ); if ( *stringptr == '\0' ) { eextras = 0; } else { eextras = (int) strtol( stringptr, &stringptr, 0 ); } #endif /* not TRILIBRARY */ initializetrisegpools(); /* Create the triangles. */ for ( elementnumber = 1; elementnumber <= inelements; elementnumber++ ) { maketriangle( &triangleloop ); /* Mark the triangle as living. */ triangleloop.tri[3] = (triangle) triangleloop.tri; } if ( poly ) { #ifdef TRILIBRARY insegments = numberofsegments; segmentmarkers = segmentmarkerlist != (int *) NULL; #else /* not TRILIBRARY */ /* Read number of segments and number of segment */ /* boundary markers from .poly file. */ stringptr = readline( inputline, polyfile, inpolyfilename ); insegments = (int) strtol( stringptr, &stringptr, 0 ); stringptr = findfield( stringptr ); if ( *stringptr == '\0' ) { segmentmarkers = 0; } else { segmentmarkers = (int) strtol( stringptr, &stringptr, 0 ); } #endif /* not TRILIBRARY */ /* Create the shell edges. */ for ( segmentnumber = 1; segmentnumber <= insegments; segmentnumber++ ) { makeshelle( &shelleloop ); /* Mark the shell edge as living. */ shelleloop.sh[2] = (shelle) shelleloop.sh; } } #ifdef TRILIBRARY pointindex = 0; attribindex = 0; #else /* not TRILIBRARY */ if ( vararea ) { /* Open an .area file, check for consistency with the .ele file. */ if ( !quiet ) { printf( "Opening %s.\n", areafilename ); } areafile = fopen( areafilename, "r" ); if ( areafile == (FILE *) NULL ) { printf( " Error: Cannot access file %s.\n", areafilename ); exit( 1 ); } stringptr = readline( inputline, areafile, areafilename ); areaelements = (int) strtol( stringptr, &stringptr, 0 ); if ( areaelements != inelements ) { printf( "Error: %s and %s disagree on number of triangles.\n", elefilename, areafilename ); exit( 1 ); } } #endif /* not TRILIBRARY */ if ( !quiet ) { printf( "Reconstructing mesh.\n" ); } /* Allocate a temporary array that maps each point to some adjacent */ /* triangle. I took care to allocate all the permanent memory for */ /* triangles and shell edges first. */ vertexarray = (triangle *) malloc( points.items * sizeof( triangle ) ); if ( vertexarray == (triangle *) NULL ) { printf( "Error: Out of memory.\n" ); exit( 1 ); } /* Each point is initially unrepresented. */ for ( i = 0; i < points.items; i++ ) { vertexarray[i] = (triangle) dummytri; } if ( verbose ) { printf( " Assembling triangles.\n" ); } /* Read the triangles from the .ele file, and link */ /* together those that share an edge. */ traversalinit( &triangles ); triangleloop.tri = triangletraverse(); elementnumber = firstnumber; while ( triangleloop.tri != (triangle *) NULL ) { #ifdef TRILIBRARY /* Copy the triangle's three corners. */ for ( j = 0; j < 3; j++ ) { corner[j] = trianglelist[pointindex++]; if ( ( corner[j] < firstnumber ) || ( corner[j] >= firstnumber + inpoints ) ) { printf( "Error: Triangle %d has an invalid vertex index.\n", elementnumber ); exit( 1 ); } } #else /* not TRILIBRARY */ /* Read triangle number and the triangle's three corners. */ stringptr = readline( inputline, elefile, elefilename ); for ( j = 0; j < 3; j++ ) { stringptr = findfield( stringptr ); if ( *stringptr == '\0' ) { printf( "Error: Triangle %d is missing point %d in %s.\n", elementnumber, j + 1, elefilename ); exit( 1 ); } else { corner[j] = (int) strtol( stringptr, &stringptr, 0 ); if ( ( corner[j] < firstnumber ) || ( corner[j] >= firstnumber + inpoints ) ) { printf( "Error: Triangle %d has an invalid vertex index.\n", elementnumber ); exit( 1 ); } } } #endif /* not TRILIBRARY */ /* Find out about (and throw away) extra nodes. */ for ( j = 3; j < incorners; j++ ) { #ifdef TRILIBRARY killpointindex = trianglelist[pointindex++]; #else /* not TRILIBRARY */ stringptr = findfield( stringptr ); if ( *stringptr != '\0' ) { killpointindex = (int) strtol( stringptr, &stringptr, 0 ); #endif /* not TRILIBRARY */ if ( ( killpointindex >= firstnumber ) && ( killpointindex < firstnumber + inpoints ) ) { /* Delete the non-corner point if it's not already deleted. */ killpoint = getpoint( killpointindex ); if ( pointmark( killpoint ) != DEADPOINT ) { pointdealloc( killpoint ); } } #ifndef TRILIBRARY } #endif /* not TRILIBRARY */ } /* Read the triangle's attributes. */ for ( j = 0; j < eextras; j++ ) { #ifdef TRILIBRARY setelemattribute( triangleloop, j, triangleattriblist[attribindex++] ); #else /* not TRILIBRARY */ stringptr = findfield( stringptr ); if ( *stringptr == '\0' ) { setelemattribute( triangleloop, j, 0 ); } else { setelemattribute( triangleloop, j, (REAL) strtod( stringptr, &stringptr ) ); } #endif /* not TRILIBRARY */ } if ( vararea ) { #ifdef TRILIBRARY area = trianglearealist[elementnumber - firstnumber]; #else /* not TRILIBRARY */ /* Read an area constraint from the .area file. */ stringptr = readline( inputline, areafile, areafilename ); stringptr = findfield( stringptr ); if ( *stringptr == '\0' ) { area = -1.0; /* No constraint on this triangle. */ } else { area = (REAL) strtod( stringptr, &stringptr ); } #endif /* not TRILIBRARY */ setareabound( triangleloop, area ); } /* Set the triangle's vertices. */ triangleloop.orient = 0; setorg( triangleloop, getpoint( corner[0] ) ); setdest( triangleloop, getpoint( corner[1] ) ); setapex( triangleloop, getpoint( corner[2] ) ); /* Try linking the triangle to others that share these vertices. */ for ( triangleloop.orient = 0; triangleloop.orient < 3; triangleloop.orient++ ) { /* Take the number for the origin of triangleloop. */ aroundpoint = corner[triangleloop.orient]; /* Look for other triangles having this vertex. */ nexttri = vertexarray[aroundpoint - firstnumber]; /* Link the current triangle to the next one in the stack. */ triangleloop.tri[6 + triangleloop.orient] = nexttri; /* Push the current triangle onto the stack. */ vertexarray[aroundpoint - firstnumber] = encode( triangleloop ); decode( nexttri, checktri ); if ( checktri.tri != dummytri ) { dest( triangleloop, tdest ); apex( triangleloop, tapex ); /* Look for other triangles that share an edge. */ do { dest( checktri, checkdest ); apex( checktri, checkapex ); if ( tapex == checkdest ) { /* The two triangles share an edge; bond them together. */ lprev( triangleloop, triangleleft ); bond( triangleleft, checktri ); } if ( tdest == checkapex ) { /* The two triangles share an edge; bond them together. */ lprev( checktri, checkleft ); bond( triangleloop, checkleft ); } /* Find the next triangle in the stack. */ nexttri = checktri.tri[6 + checktri.orient]; decode( nexttri, checktri ); } while ( checktri.tri != dummytri ); } } triangleloop.tri = triangletraverse(); elementnumber++; } #ifdef TRILIBRARY pointindex = 0; #else /* not TRILIBRARY */ fclose( elefile ); if ( vararea ) { fclose( areafile ); } #endif /* not TRILIBRARY */ hullsize = 0; /* Prepare to count the boundary edges. */ if ( poly ) { if ( verbose ) { printf( " Marking segments in triangulation.\n" ); } /* Read the segments from the .poly file, and link them */ /* to their neighboring triangles. */ boundmarker = 0; traversalinit( &shelles ); shelleloop.sh = shelletraverse(); segmentnumber = firstnumber; while ( shelleloop.sh != (shelle *) NULL ) { #ifdef TRILIBRARY end[0] = segmentlist[pointindex++]; end[1] = segmentlist[pointindex++]; if ( segmentmarkers ) { boundmarker = segmentmarkerlist[segmentnumber - firstnumber]; } #else /* not TRILIBRARY */ /* Read the endpoints of each segment, and possibly a boundary marker. */ stringptr = readline( inputline, polyfile, inpolyfilename ); /* Skip the first (segment number) field. */ stringptr = findfield( stringptr ); if ( *stringptr == '\0' ) { printf( "Error: Segment %d has no endpoints in %s.\n", segmentnumber, polyfilename ); exit( 1 ); } else { end[0] = (int) strtol( stringptr, &stringptr, 0 ); } stringptr = findfield( stringptr ); if ( *stringptr == '\0' ) { printf( "Error: Segment %d is missing its second endpoint in %s.\n", segmentnumber, polyfilename ); exit( 1 ); } else { end[1] = (int) strtol( stringptr, &stringptr, 0 ); } if ( segmentmarkers ) { stringptr = findfield( stringptr ); if ( *stringptr == '\0' ) { boundmarker = 0; } else { boundmarker = (int) strtol( stringptr, &stringptr, 0 ); } } #endif /* not TRILIBRARY */ for ( j = 0; j < 2; j++ ) { if ( ( end[j] < firstnumber ) || ( end[j] >= firstnumber + inpoints ) ) { printf( "Error: Segment %d has an invalid vertex index.\n", segmentnumber ); exit( 1 ); } } /* set the shell edge's vertices. */ shelleloop.shorient = 0; setsorg( shelleloop, getpoint( end[0] ) ); setsdest( shelleloop, getpoint( end[1] ) ); setmark( shelleloop, boundmarker ); /* Try linking the shell edge to triangles that share these vertices. */ for ( shelleloop.shorient = 0; shelleloop.shorient < 2; shelleloop.shorient++ ) { /* Take the number for the destination of shelleloop. */ aroundpoint = end[1 - shelleloop.shorient]; /* Look for triangles having this vertex. */ prevlink = &vertexarray[aroundpoint - firstnumber]; nexttri = vertexarray[aroundpoint - firstnumber]; decode( nexttri, checktri ); sorg( shelleloop, shorg ); notfound = 1; /* Look for triangles having this edge. Note that I'm only */ /* comparing each triangle's destination with the shell edge; */ /* each triangle's apex is handled through a different vertex. */ /* Because each triangle appears on three vertices' lists, each */ /* occurrence of a triangle on a list can (and does) represent */ /* an edge. In this way, most edges are represented twice, and */ /* every triangle-segment bond is represented once. */ while ( notfound && ( checktri.tri != dummytri ) ) { dest( checktri, checkdest ); if ( shorg == checkdest ) { /* We have a match. Remove this triangle from the list. */ *prevlink = checktri.tri[6 + checktri.orient]; /* Bond the shell edge to the triangle. */ tsbond( checktri, shelleloop ); /* Check if this is a boundary edge. */ sym( checktri, checkneighbor ); if ( checkneighbor.tri == dummytri ) { /* The next line doesn't insert a shell edge (because there's */ /* already one there), but it sets the boundary markers of */ /* the existing shell edge and its vertices. */ insertshelle( &checktri, 1 ); hullsize++; } notfound = 0; } /* Find the next triangle in the stack. */ prevlink = &checktri.tri[6 + checktri.orient]; nexttri = checktri.tri[6 + checktri.orient]; decode( nexttri, checktri ); } } shelleloop.sh = shelletraverse(); segmentnumber++; } } /* Mark the remaining edges as not being attached to any shell edge. */ /* Also, count the (yet uncounted) boundary edges. */ for ( i = 0; i < points.items; i++ ) { /* Search the stack of triangles adjacent to a point. */ nexttri = vertexarray[i]; decode( nexttri, checktri ); while ( checktri.tri != dummytri ) { /* Find the next triangle in the stack before this */ /* information gets overwritten. */ nexttri = checktri.tri[6 + checktri.orient]; /* No adjacent shell edge. (This overwrites the stack info.) */ tsdissolve( checktri ); sym( checktri, checkneighbor ); if ( checkneighbor.tri == dummytri ) { insertshelle( &checktri, 1 ); hullsize++; } decode( nexttri, checktri ); } } free( vertexarray ); return hullsize; } #endif /* not CDT_ONLY */ /** **/ /** **/ /********* General mesh construction routines end here *********/ /********* Segment (shell edge) insertion begins here *********/ /** **/ /** **/ /*****************************************************************************/ /* */ /* finddirection() Find the first triangle on the path from one point */ /* to another. */ /* */ /* Finds the triangle that intersects a line segment drawn from the */ /* origin of `searchtri' to the point `endpoint', and returns the result */ /* in `searchtri'. The origin of `searchtri' does not change, even though */ /* the triangle returned may differ from the one passed in. This routine */ /* is used to find the direction to move in to get from one point to */ /* another. */ /* */ /* The return value notes whether the destination or apex of the found */ /* triangle is collinear with the two points in question. */ /* */ /*****************************************************************************/ enum finddirectionresult finddirection( searchtri, endpoint ) struct triedge *searchtri; point endpoint; { struct triedge checktri; point startpoint; point leftpoint, rightpoint; REAL leftccw, rightccw; int leftflag, rightflag; triangle ptr; /* Temporary variable used by onext() and oprev(). */ org( *searchtri, startpoint ); dest( *searchtri, rightpoint ); apex( *searchtri, leftpoint ); /* Is `endpoint' to the left? */ leftccw = counterclockwise( endpoint, startpoint, leftpoint ); leftflag = leftccw > 0.0; /* Is `endpoint' to the right? */ rightccw = counterclockwise( startpoint, endpoint, rightpoint ); rightflag = rightccw > 0.0; if ( leftflag && rightflag ) { /* `searchtri' faces directly away from `endpoint'. We could go */ /* left or right. Ask whether it's a triangle or a boundary */ /* on the left. */ onext( *searchtri, checktri ); if ( checktri.tri == dummytri ) { leftflag = 0; } else { rightflag = 0; } } while ( leftflag ) { /* Turn left until satisfied. */ onextself( *searchtri ); if ( searchtri->tri == dummytri ) { printf( "Internal error in finddirection(): Unable to find a\n" ); printf( " triangle leading from (%.12g, %.12g) to", startpoint[0], startpoint[1] ); printf( " (%.12g, %.12g).\n", endpoint[0], endpoint[1] ); internalerror(); } apex( *searchtri, leftpoint ); rightccw = leftccw; leftccw = counterclockwise( endpoint, startpoint, leftpoint ); leftflag = leftccw > 0.0; } while ( rightflag ) { /* Turn right until satisfied. */ oprevself( *searchtri ); if ( searchtri->tri == dummytri ) { printf( "Internal error in finddirection(): Unable to find a\n" ); printf( " triangle leading from (%.12g, %.12g) to", startpoint[0], startpoint[1] ); printf( " (%.12g, %.12g).\n", endpoint[0], endpoint[1] ); internalerror(); } dest( *searchtri, rightpoint ); leftccw = rightccw; rightccw = counterclockwise( startpoint, endpoint, rightpoint ); rightflag = rightccw > 0.0; } if ( leftccw == 0.0 ) { return LEFTCOLLINEAR; } else if ( rightccw == 0.0 ) { return RIGHTCOLLINEAR; } else { return WITHIN; } } /*****************************************************************************/ /* */ /* segmentintersection() Find the intersection of an existing segment */ /* and a segment that is being inserted. Insert */ /* a point at the intersection, splitting an */ /* existing shell edge. */ /* */ /* The segment being inserted connects the apex of splittri to endpoint2. */ /* splitshelle is the shell edge being split, and MUST be opposite */ /* splittri. Hence, the edge being split connects the origin and */ /* destination of splittri. */ /* */ /* On completion, splittri is a handle having the newly inserted */ /* intersection point as its origin, and endpoint1 as its destination. */ /* */ /*****************************************************************************/ void segmentintersection( splittri, splitshelle, endpoint2 ) struct triedge *splittri; struct edge *splitshelle; point endpoint2; { point endpoint1; point torg, tdest; point leftpoint, rightpoint; point newpoint; enum insertsiteresult success; enum finddirectionresult collinear; REAL ex, ey; REAL tx, ty; REAL etx, ety; REAL split, denom; int i; triangle ptr; /* Temporary variable used by onext(). */ /* Find the other three segment endpoints. */ apex( *splittri, endpoint1 ); org( *splittri, torg ); dest( *splittri, tdest ); /* Segment intersection formulae; see the Antonio reference. */ tx = tdest[0] - torg[0]; ty = tdest[1] - torg[1]; ex = endpoint2[0] - endpoint1[0]; ey = endpoint2[1] - endpoint1[1]; etx = torg[0] - endpoint2[0]; ety = torg[1] - endpoint2[1]; denom = ty * ex - tx * ey; if ( denom == 0.0 ) { printf( "Internal error in segmentintersection():" ); printf( " Attempt to find intersection of parallel segments.\n" ); internalerror(); } split = ( ey * etx - ex * ety ) / denom; /* Create the new point. */ newpoint = (point) poolalloc( &points ); /* Interpolate its coordinate and attributes. */ for ( i = 0; i < 2 + nextras; i++ ) { newpoint[i] = torg[i] + split * ( tdest[i] - torg[i] ); } setpointmark( newpoint, mark( *splitshelle ) ); if ( verbose > 1 ) { printf( " Splitting edge (%.12g, %.12g) (%.12g, %.12g) at (%.12g, %.12g).\n", torg[0], torg[1], tdest[0], tdest[1], newpoint[0], newpoint[1] ); } /* Insert the intersection point. This should always succeed. */ success = insertsite( newpoint, splittri, splitshelle, 0, 0 ); if ( success != SUCCESSFULPOINT ) { printf( "Internal error in segmentintersection():\n" ); printf( " Failure to split a segment.\n" ); internalerror(); } if ( steinerleft > 0 ) { steinerleft--; } /* Inserting the point may have caused edge flips. We wish to rediscover */ /* the edge connecting endpoint1 to the new intersection point. */ collinear = finddirection( splittri, endpoint1 ); dest( *splittri, rightpoint ); apex( *splittri, leftpoint ); if ( ( leftpoint[0] == endpoint1[0] ) && ( leftpoint[1] == endpoint1[1] ) ) { onextself( *splittri ); } else if ( ( rightpoint[0] != endpoint1[0] ) || ( rightpoint[1] != endpoint1[1] ) ) { printf( "Internal error in segmentintersection():\n" ); printf( " Topological inconsistency after splitting a segment.\n" ); internalerror(); } /* `splittri' should have destination endpoint1. */ } /*****************************************************************************/ /* */ /* scoutsegment() Scout the first triangle on the path from one endpoint */ /* to another, and check for completion (reaching the */ /* second endpoint), a collinear point, and the */ /* intersection of two segments. */ /* */ /* Returns one if the entire segment is successfully inserted, and zero if */ /* the job must be finished by conformingedge() or constrainededge(). */ /* */ /* If the first triangle on the path has the second endpoint as its */ /* destination or apex, a shell edge is inserted and the job is done. */ /* */ /* If the first triangle on the path has a destination or apex that lies on */ /* the segment, a shell edge is inserted connecting the first endpoint to */ /* the collinear point, and the search is continued from the collinear */ /* point. */ /* */ /* If the first triangle on the path has a shell edge opposite its origin, */ /* then there is a segment that intersects the segment being inserted. */ /* Their intersection point is inserted, splitting the shell edge. */ /* */ /* Otherwise, return zero. */ /* */ /*****************************************************************************/ int scoutsegment( searchtri, endpoint2, newmark ) struct triedge *searchtri; point endpoint2; int newmark; { struct triedge crosstri; struct edge crossedge; point leftpoint, rightpoint; point endpoint1; enum finddirectionresult collinear; shelle sptr; /* Temporary variable used by tspivot(). */ collinear = finddirection( searchtri, endpoint2 ); dest( *searchtri, rightpoint ); apex( *searchtri, leftpoint ); if ( ( ( leftpoint[0] == endpoint2[0] ) && ( leftpoint[1] == endpoint2[1] ) ) || ( ( rightpoint[0] == endpoint2[0] ) && ( rightpoint[1] == endpoint2[1] ) ) ) { /* The segment is already an edge in the mesh. */ if ( ( leftpoint[0] == endpoint2[0] ) && ( leftpoint[1] == endpoint2[1] ) ) { lprevself( *searchtri ); } /* Insert a shell edge, if there isn't already one there. */ insertshelle( searchtri, newmark ); return 1; } else if ( collinear == LEFTCOLLINEAR ) { /* We've collided with a point between the segment's endpoints. */ /* Make the collinear point be the triangle's origin. */ lprevself( *searchtri ); insertshelle( searchtri, newmark ); /* Insert the remainder of the segment. */ return scoutsegment( searchtri, endpoint2, newmark ); } else if ( collinear == RIGHTCOLLINEAR ) { /* We've collided with a point between the segment's endpoints. */ insertshelle( searchtri, newmark ); /* Make the collinear point be the triangle's origin. */ lnextself( *searchtri ); /* Insert the remainder of the segment. */ return scoutsegment( searchtri, endpoint2, newmark ); } else { lnext( *searchtri, crosstri ); tspivot( crosstri, crossedge ); /* Check for a crossing segment. */ if ( crossedge.sh == dummysh ) { return 0; } else { org( *searchtri, endpoint1 ); /* Insert a point at the intersection. */ segmentintersection( &crosstri, &crossedge, endpoint2 ); triedgecopy( crosstri, *searchtri ); insertshelle( searchtri, newmark ); /* Insert the remainder of the segment. */ return scoutsegment( searchtri, endpoint2, newmark ); } } } /*****************************************************************************/ /* */ /* conformingedge() Force a segment into a conforming Delaunay */ /* triangulation by inserting a point at its midpoint, */ /* and recursively forcing in the two half-segments if */ /* necessary. */ /* */ /* Generates a sequence of edges connecting `endpoint1' to `endpoint2'. */ /* `newmark' is the boundary marker of the segment, assigned to each new */ /* splitting point and shell edge. */ /* */ /* Note that conformingedge() does not always maintain the conforming */ /* Delaunay property. Once inserted, segments are locked into place; */ /* points inserted later (to force other segments in) may render these */ /* fixed segments non-Delaunay. The conforming Delaunay property will be */ /* restored by enforcequality() by splitting encroached segments. */ /* */ /*****************************************************************************/ #ifndef REDUCED #ifndef CDT_ONLY void conformingedge( endpoint1, endpoint2, newmark ) point endpoint1; point endpoint2; int newmark; { struct triedge searchtri1, searchtri2; struct edge brokenshelle; point newpoint; point midpoint1, midpoint2; enum insertsiteresult success; int result1, result2; int i; shelle sptr; /* Temporary variable used by tspivot(). */ if ( verbose > 2 ) { printf( "Forcing segment into triangulation by recursive splitting:\n" ); printf( " (%.12g, %.12g) (%.12g, %.12g)\n", endpoint1[0], endpoint1[1], endpoint2[0], endpoint2[1] ); } /* Create a new point to insert in the middle of the segment. */ newpoint = (point) poolalloc( &points ); /* Interpolate coordinates and attributes. */ for ( i = 0; i < 2 + nextras; i++ ) { newpoint[i] = 0.5 * ( endpoint1[i] + endpoint2[i] ); } setpointmark( newpoint, newmark ); /* Find a boundary triangle to search from. */ searchtri1.tri = (triangle *) NULL; /* Attempt to insert the new point. */ success = insertsite( newpoint, &searchtri1, (struct edge *) NULL, 0, 0 ); if ( success == DUPLICATEPOINT ) { if ( verbose > 2 ) { printf( " Segment intersects existing point (%.12g, %.12g).\n", newpoint[0], newpoint[1] ); } /* Use the point that's already there. */ pointdealloc( newpoint ); org( searchtri1, newpoint ); } else { if ( success == VIOLATINGPOINT ) { if ( verbose > 2 ) { printf( " Two segments intersect at (%.12g, %.12g).\n", newpoint[0], newpoint[1] ); } /* By fluke, we've landed right on another segment. Split it. */ tspivot( searchtri1, brokenshelle ); success = insertsite( newpoint, &searchtri1, &brokenshelle, 0, 0 ); if ( success != SUCCESSFULPOINT ) { printf( "Internal error in conformingedge():\n" ); printf( " Failure to split a segment.\n" ); internalerror(); } } /* The point has been inserted successfully. */ if ( steinerleft > 0 ) { steinerleft--; } } triedgecopy( searchtri1, searchtri2 ); result1 = scoutsegment( &searchtri1, endpoint1, newmark ); result2 = scoutsegment( &searchtri2, endpoint2, newmark ); if ( !result1 ) { /* The origin of searchtri1 may have changed if a collision with an */ /* intervening vertex on the segment occurred. */ org( searchtri1, midpoint1 ); conformingedge( midpoint1, endpoint1, newmark ); } if ( !result2 ) { /* The origin of searchtri2 may have changed if a collision with an */ /* intervening vertex on the segment occurred. */ org( searchtri2, midpoint2 ); conformingedge( midpoint2, endpoint2, newmark ); } } #endif /* not CDT_ONLY */ #endif /* not REDUCED */ /*****************************************************************************/ /* */ /* delaunayfixup() Enforce the Delaunay condition at an edge, fanning out */ /* recursively from an existing point. Pay special */ /* attention to stacking inverted triangles. */ /* */ /* This is a support routine for inserting segments into a constrained */ /* Delaunay triangulation. */ /* */ /* The origin of fixuptri is treated as if it has just been inserted, and */ /* the local Delaunay condition needs to be enforced. It is only enforced */ /* in one sector, however, that being the angular range defined by */ /* fixuptri. */ /* */ /* This routine also needs to make decisions regarding the "stacking" of */ /* triangles. (Read the description of constrainededge() below before */ /* reading on here, so you understand the algorithm.) If the position of */ /* the new point (the origin of fixuptri) indicates that the vertex before */ /* it on the polygon is a reflex vertex, then "stack" the triangle by */ /* doing nothing. (fixuptri is an inverted triangle, which is how stacked */ /* triangles are identified.) */ /* */ /* Otherwise, check whether the vertex before that was a reflex vertex. */ /* If so, perform an edge flip, thereby eliminating an inverted triangle */ /* (popping it off the stack). The edge flip may result in the creation */ /* of a new inverted triangle, depending on whether or not the new vertex */ /* is visible to the vertex three edges behind on the polygon. */ /* */ /* If neither of the two vertices behind the new vertex are reflex */ /* vertices, fixuptri and fartri, the triangle opposite it, are not */ /* inverted; hence, ensure that the edge between them is locally Delaunay. */ /* */ /* `leftside' indicates whether or not fixuptri is to the left of the */ /* segment being inserted. (Imagine that the segment is pointing up from */ /* endpoint1 to endpoint2.) */ /* */ /*****************************************************************************/ void delaunayfixup( fixuptri, leftside ) struct triedge *fixuptri; int leftside; { struct triedge neartri; struct triedge fartri; struct edge faredge; point nearpoint, leftpoint, rightpoint, farpoint; triangle ptr; /* Temporary variable used by sym(). */ shelle sptr; /* Temporary variable used by tspivot(). */ lnext( *fixuptri, neartri ); sym( neartri, fartri ); /* Check if the edge opposite the origin of fixuptri can be flipped. */ if ( fartri.tri == dummytri ) { return; } tspivot( neartri, faredge ); if ( faredge.sh != dummysh ) { return; } /* Find all the relevant vertices. */ apex( neartri, nearpoint ); org( neartri, leftpoint ); dest( neartri, rightpoint ); apex( fartri, farpoint ); /* Check whether the previous polygon vertex is a reflex vertex. */ if ( leftside ) { if ( counterclockwise( nearpoint, leftpoint, farpoint ) <= 0.0 ) { /* leftpoint is a reflex vertex too. Nothing can */ /* be done until a convex section is found. */ return; } } else { if ( counterclockwise( farpoint, rightpoint, nearpoint ) <= 0.0 ) { /* rightpoint is a reflex vertex too. Nothing can */ /* be done until a convex section is found. */ return; } } if ( counterclockwise( rightpoint, leftpoint, farpoint ) > 0.0 ) { /* fartri is not an inverted triangle, and farpoint is not a reflex */ /* vertex. As there are no reflex vertices, fixuptri isn't an */ /* inverted triangle, either. Hence, test the edge between the */ /* triangles to ensure it is locally Delaunay. */ if ( incircle( leftpoint, farpoint, rightpoint, nearpoint ) <= 0.0 ) { return; } /* Not locally Delaunay; go on to an edge flip. */ } /* else fartri is inverted; remove it from the stack by flipping. */ flip( &neartri ); lprevself( *fixuptri ); /* Restore the origin of fixuptri after the flip. */ /* Recursively process the two triangles that result from the flip. */ delaunayfixup( fixuptri, leftside ); delaunayfixup( &fartri, leftside ); } /*****************************************************************************/ /* */ /* constrainededge() Force a segment into a constrained Delaunay */ /* triangulation by deleting the triangles it */ /* intersects, and triangulating the polygons that */ /* form on each side of it. */ /* */ /* Generates a single edge connecting `endpoint1' to `endpoint2'. The */ /* triangle `starttri' has `endpoint1' as its origin. `newmark' is the */ /* boundary marker of the segment. */ /* */ /* To insert a segment, every triangle whose interior intersects the */ /* segment is deleted. The union of these deleted triangles is a polygon */ /* (which is not necessarily monotone, but is close enough), which is */ /* divided into two polygons by the new segment. This routine's task is */ /* to generate the Delaunay triangulation of these two polygons. */ /* */ /* You might think of this routine's behavior as a two-step process. The */ /* first step is to walk from endpoint1 to endpoint2, flipping each edge */ /* encountered. This step creates a fan of edges connected to endpoint1, */ /* including the desired edge to endpoint2. The second step enforces the */ /* Delaunay condition on each side of the segment in an incremental manner: */ /* proceeding along the polygon from endpoint1 to endpoint2 (this is done */ /* independently on each side of the segment), each vertex is "enforced" */ /* as if it had just been inserted, but affecting only the previous */ /* vertices. The result is the same as if the vertices had been inserted */ /* in the order they appear on the polygon, so the result is Delaunay. */ /* */ /* In truth, constrainededge() interleaves these two steps. The procedure */ /* walks from endpoint1 to endpoint2, and each time an edge is encountered */ /* and flipped, the newly exposed vertex (at the far end of the flipped */ /* edge) is "enforced" upon the previously flipped edges, usually affecting */ /* only one side of the polygon (depending upon which side of the segment */ /* the vertex falls on). */ /* */ /* The algorithm is complicated by the need to handle polygons that are not */ /* convex. Although the polygon is not necessarily monotone, it can be */ /* triangulated in a manner similar to the stack-based algorithms for */ /* monotone polygons. For each reflex vertex (local concavity) of the */ /* polygon, there will be an inverted triangle formed by one of the edge */ /* flips. (An inverted triangle is one with negative area - that is, its */ /* vertices are arranged in clockwise order - and is best thought of as a */ /* wrinkle in the fabric of the mesh.) Each inverted triangle can be */ /* thought of as a reflex vertex pushed on the stack, waiting to be fixed */ /* later. */ /* */ /* A reflex vertex is popped from the stack when a vertex is inserted that */ /* is visible to the reflex vertex. (However, if the vertex behind the */ /* reflex vertex is not visible to the reflex vertex, a new inverted */ /* triangle will take its place on the stack.) These details are handled */ /* by the delaunayfixup() routine above. */ /* */ /*****************************************************************************/ void constrainededge( starttri, endpoint2, newmark ) struct triedge *starttri; point endpoint2; int newmark; { struct triedge fixuptri, fixuptri2; struct edge fixupedge; point endpoint1; point farpoint; REAL area; int collision; int done; triangle ptr; /* Temporary variable used by sym() and oprev(). */ shelle sptr; /* Temporary variable used by tspivot(). */ org( *starttri, endpoint1 ); lnext( *starttri, fixuptri ); flip( &fixuptri ); /* `collision' indicates whether we have found a point directly */ /* between endpoint1 and endpoint2. */ collision = 0; done = 0; do { org( fixuptri, farpoint ); /* `farpoint' is the extreme point of the polygon we are "digging" */ /* to get from endpoint1 to endpoint2. */ if ( ( farpoint[0] == endpoint2[0] ) && ( farpoint[1] == endpoint2[1] ) ) { oprev( fixuptri, fixuptri2 ); /* Enforce the Delaunay condition around endpoint2. */ delaunayfixup( &fixuptri, 0 ); delaunayfixup( &fixuptri2, 1 ); done = 1; } else { /* Check whether farpoint is to the left or right of the segment */ /* being inserted, to decide which edge of fixuptri to dig */ /* through next. */ area = counterclockwise( endpoint1, endpoint2, farpoint ); if ( area == 0.0 ) { /* We've collided with a point between endpoint1 and endpoint2. */ collision = 1; oprev( fixuptri, fixuptri2 ); /* Enforce the Delaunay condition around farpoint. */ delaunayfixup( &fixuptri, 0 ); delaunayfixup( &fixuptri2, 1 ); done = 1; } else { if ( area > 0.0 ) { /* farpoint is to the left of the segment. */ oprev( fixuptri, fixuptri2 ); /* Enforce the Delaunay condition around farpoint, on the */ /* left side of the segment only. */ delaunayfixup( &fixuptri2, 1 ); /* Flip the edge that crosses the segment. After the edge is */ /* flipped, one of its endpoints is the fan vertex, and the */ /* destination of fixuptri is the fan vertex. */ lprevself( fixuptri ); } else { /* farpoint is to the right of the segment. */ delaunayfixup( &fixuptri, 0 ); /* Flip the edge that crosses the segment. After the edge is */ /* flipped, one of its endpoints is the fan vertex, and the */ /* destination of fixuptri is the fan vertex. */ oprevself( fixuptri ); } /* Check for two intersecting segments. */ tspivot( fixuptri, fixupedge ); if ( fixupedge.sh == dummysh ) { flip( &fixuptri ); /* May create an inverted triangle on the left. */ } else { /* We've collided with a segment between endpoint1 and endpoint2. */ collision = 1; /* Insert a point at the intersection. */ segmentintersection( &fixuptri, &fixupedge, endpoint2 ); done = 1; } } } } while ( !done ); /* Insert a shell edge to make the segment permanent. */ insertshelle( &fixuptri, newmark ); /* If there was a collision with an interceding vertex, install another */ /* segment connecting that vertex with endpoint2. */ if ( collision ) { /* Insert the remainder of the segment. */ if ( !scoutsegment( &fixuptri, endpoint2, newmark ) ) { constrainededge( &fixuptri, endpoint2, newmark ); } } } /*****************************************************************************/ /* */ /* insertsegment() Insert a PSLG segment into a triangulation. */ /* */ /*****************************************************************************/ void insertsegment( endpoint1, endpoint2, newmark ) point endpoint1; point endpoint2; int newmark; { struct triedge searchtri1, searchtri2; triangle encodedtri; point checkpoint; triangle ptr; /* Temporary variable used by sym(). */ if ( verbose > 1 ) { printf( " Connecting (%.12g, %.12g) to (%.12g, %.12g).\n", endpoint1[0], endpoint1[1], endpoint2[0], endpoint2[1] ); } /* Find a triangle whose origin is the segment's first endpoint. */ checkpoint = (point) NULL; encodedtri = point2tri( endpoint1 ); if ( encodedtri != (triangle) NULL ) { decode( encodedtri, searchtri1 ); org( searchtri1, checkpoint ); } if ( checkpoint != endpoint1 ) { /* Find a boundary triangle to search from. */ searchtri1.tri = dummytri; searchtri1.orient = 0; symself( searchtri1 ); /* Search for the segment's first endpoint by point location. */ if ( locate( endpoint1, &searchtri1 ) != ONVERTEX ) { printf( "Internal error in insertsegment(): Unable to locate PSLG point\n" ); printf( " (%.12g, %.12g) in triangulation.\n", endpoint1[0], endpoint1[1] ); internalerror(); } } /* Remember this triangle to improve subsequent point location. */ triedgecopy( searchtri1, recenttri ); /* Scout the beginnings of a path from the first endpoint */ /* toward the second. */ if ( scoutsegment( &searchtri1, endpoint2, newmark ) ) { /* The segment was easily inserted. */ return; } /* The first endpoint may have changed if a collision with an intervening */ /* vertex on the segment occurred. */ org( searchtri1, endpoint1 ); /* Find a triangle whose origin is the segment's second endpoint. */ checkpoint = (point) NULL; encodedtri = point2tri( endpoint2 ); if ( encodedtri != (triangle) NULL ) { decode( encodedtri, searchtri2 ); org( searchtri2, checkpoint ); } if ( checkpoint != endpoint2 ) { /* Find a boundary triangle to search from. */ searchtri2.tri = dummytri; searchtri2.orient = 0; symself( searchtri2 ); /* Search for the segment's second endpoint by point location. */ if ( locate( endpoint2, &searchtri2 ) != ONVERTEX ) { printf( "Internal error in insertsegment(): Unable to locate PSLG point\n" ); printf( " (%.12g, %.12g) in triangulation.\n", endpoint2[0], endpoint2[1] ); internalerror(); } } /* Remember this triangle to improve subsequent point location. */ triedgecopy( searchtri2, recenttri ); /* Scout the beginnings of a path from the second endpoint */ /* toward the first. */ if ( scoutsegment( &searchtri2, endpoint1, newmark ) ) { /* The segment was easily inserted. */ return; } /* The second endpoint may have changed if a collision with an intervening */ /* vertex on the segment occurred. */ org( searchtri2, endpoint2 ); #ifndef REDUCED #ifndef CDT_ONLY if ( splitseg ) { /* Insert vertices to force the segment into the triangulation. */ conformingedge( endpoint1, endpoint2, newmark ); } else { #endif /* not CDT_ONLY */ #endif /* not REDUCED */ /* Insert the segment directly into the triangulation. */ constrainededge( &searchtri1, endpoint2, newmark ); #ifndef REDUCED #ifndef CDT_ONLY } #endif /* not CDT_ONLY */ #endif /* not REDUCED */ } /*****************************************************************************/ /* */ /* markhull() Cover the convex hull of a triangulation with shell edges. */ /* */ /*****************************************************************************/ void markhull(){ struct triedge hulltri; struct triedge nexttri; struct triedge starttri; triangle ptr; /* Temporary variable used by sym() and oprev(). */ /* Find a triangle handle on the hull. */ hulltri.tri = dummytri; hulltri.orient = 0; symself( hulltri ); /* Remember where we started so we know when to stop. */ triedgecopy( hulltri, starttri ); /* Go once counterclockwise around the convex hull. */ do { /* Create a shell edge if there isn't already one here. */ insertshelle( &hulltri, 1 ); /* To find the next hull edge, go clockwise around the next vertex. */ lnextself( hulltri ); oprev( hulltri, nexttri ); while ( nexttri.tri != dummytri ) { triedgecopy( nexttri, hulltri ); oprev( hulltri, nexttri ); } } while ( !triedgeequal( hulltri, starttri ) ); } /*****************************************************************************/ /* */ /* formskeleton() Create the shell edges of a triangulation, including */ /* PSLG edges and edges on the convex hull. */ /* */ /* The PSLG edges are read from a .poly file. The return value is the */ /* number of segments in the file. */ /* */ /*****************************************************************************/ #ifdef TRILIBRARY int formskeleton( segmentlist, segmentmarkerlist, numberofsegments ) int *segmentlist; int *segmentmarkerlist; int numberofsegments; #else /* not TRILIBRARY */ int formskeleton( polyfile, polyfilename ) FILE * polyfile; char *polyfilename; #endif /* not TRILIBRARY */ { #ifdef TRILIBRARY char polyfilename[6]; int index; #else /* not TRILIBRARY */ char inputline[INPUTLINESIZE]; char *stringptr; #endif /* not TRILIBRARY */ point endpoint1, endpoint2; int segments; int segmentmarkers; int end1, end2; int boundmarker; int i; if ( poly ) { if ( !quiet ) { printf( "Inserting segments into Delaunay triangulation.\n" ); } #ifdef TRILIBRARY strcpy( polyfilename, "input" ); segments = numberofsegments; segmentmarkers = segmentmarkerlist != (int *) NULL; index = 0; #else /* not TRILIBRARY */ /* Read the segments from a .poly file. */ /* Read number of segments and number of boundary markers. */ stringptr = readline( inputline, polyfile, polyfilename ); segments = (int) strtol( stringptr, &stringptr, 0 ); stringptr = findfield( stringptr ); if ( *stringptr == '\0' ) { segmentmarkers = 0; } else { segmentmarkers = (int) strtol( stringptr, &stringptr, 0 ); } #endif /* not TRILIBRARY */ /* If segments are to be inserted, compute a mapping */ /* from points to triangles. */ if ( segments > 0 ) { if ( verbose ) { printf( " Inserting PSLG segments.\n" ); } makepointmap(); } boundmarker = 0; /* Read and insert the segments. */ for ( i = 1; i <= segments; i++ ) { #ifdef TRILIBRARY end1 = segmentlist[index++]; end2 = segmentlist[index++]; if ( segmentmarkers ) { boundmarker = segmentmarkerlist[i - 1]; } #else /* not TRILIBRARY */ stringptr = readline( inputline, polyfile, inpolyfilename ); stringptr = findfield( stringptr ); if ( *stringptr == '\0' ) { printf( "Error: Segment %d has no endpoints in %s.\n", i, polyfilename ); exit( 1 ); } else { end1 = (int) strtol( stringptr, &stringptr, 0 ); } stringptr = findfield( stringptr ); if ( *stringptr == '\0' ) { printf( "Error: Segment %d is missing its second endpoint in %s.\n", i, polyfilename ); exit( 1 ); } else { end2 = (int) strtol( stringptr, &stringptr, 0 ); } if ( segmentmarkers ) { stringptr = findfield( stringptr ); if ( *stringptr == '\0' ) { boundmarker = 0; } else { boundmarker = (int) strtol( stringptr, &stringptr, 0 ); } } #endif /* not TRILIBRARY */ if ( ( end1 < firstnumber ) || ( end1 >= firstnumber + inpoints ) ) { if ( !quiet ) { printf( "Warning: Invalid first endpoint of segment %d in %s.\n", i, polyfilename ); } } else if ( ( end2 < firstnumber ) || ( end2 >= firstnumber + inpoints ) ) { if ( !quiet ) { printf( "Warning: Invalid second endpoint of segment %d in %s.\n", i, polyfilename ); } } else { endpoint1 = getpoint( end1 ); endpoint2 = getpoint( end2 ); if ( ( endpoint1[0] == endpoint2[0] ) && ( endpoint1[1] == endpoint2[1] ) ) { if ( !quiet ) { printf( "Warning: Endpoints of segment %d are coincident in %s.\n", i, polyfilename ); } } else { insertsegment( endpoint1, endpoint2, boundmarker ); } } } } else { segments = 0; } if ( convex || !poly ) { /* Enclose the convex hull with shell edges. */ if ( verbose ) { printf( " Enclosing convex hull with segments.\n" ); } markhull(); } return segments; } /** **/ /** **/ /********* Segment (shell edge) insertion ends here *********/ /********* Carving out holes and concavities begins here *********/ /** **/ /** **/ /*****************************************************************************/ /* */ /* infecthull() Virally infect all of the triangles of the convex hull */ /* that are not protected by shell edges. Where there are */ /* shell edges, set boundary markers as appropriate. */ /* */ /*****************************************************************************/ void infecthull(){ struct triedge hulltri; struct triedge nexttri; struct triedge starttri; struct edge hulledge; triangle **deadtri; point horg, hdest; triangle ptr; /* Temporary variable used by sym(). */ shelle sptr; /* Temporary variable used by tspivot(). */ if ( verbose ) { printf( " Marking concavities (external triangles) for elimination.\n" ); } /* Find a triangle handle on the hull. */ hulltri.tri = dummytri; hulltri.orient = 0; symself( hulltri ); /* Remember where we started so we know when to stop. */ triedgecopy( hulltri, starttri ); /* Go once counterclockwise around the convex hull. */ do { /* Ignore triangles that are already infected. */ if ( !infected( hulltri ) ) { /* Is the triangle protected by a shell edge? */ tspivot( hulltri, hulledge ); if ( hulledge.sh == dummysh ) { /* The triangle is not protected; infect it. */ infect( hulltri ); deadtri = (triangle **) poolalloc( &viri ); *deadtri = hulltri.tri; } else { /* The triangle is protected; set boundary markers if appropriate. */ if ( mark( hulledge ) == 0 ) { setmark( hulledge, 1 ); org( hulltri, horg ); dest( hulltri, hdest ); if ( pointmark( horg ) == 0 ) { setpointmark( horg, 1 ); } if ( pointmark( hdest ) == 0 ) { setpointmark( hdest, 1 ); } } } } /* To find the next hull edge, go clockwise around the next vertex. */ lnextself( hulltri ); oprev( hulltri, nexttri ); while ( nexttri.tri != dummytri ) { triedgecopy( nexttri, hulltri ); oprev( hulltri, nexttri ); } } while ( !triedgeequal( hulltri, starttri ) ); } /*****************************************************************************/ /* */ /* plague() Spread the virus from all infected triangles to any neighbors */ /* not protected by shell edges. Delete all infected triangles. */ /* */ /* This is the procedure that actually creates holes and concavities. */ /* */ /* This procedure operates in two phases. The first phase identifies all */ /* the triangles that will die, and marks them as infected. They are */ /* marked to ensure that each triangle is added to the virus pool only */ /* once, so the procedure will terminate. */ /* */ /* The second phase actually eliminates the infected triangles. It also */ /* eliminates orphaned points. */ /* */ /*****************************************************************************/ void plague(){ struct triedge testtri; struct triedge neighbor; triangle **virusloop; triangle **deadtri; struct edge neighborshelle; point testpoint; point norg, ndest; point deadorg, deaddest, deadapex; int killorg; triangle ptr; /* Temporary variable used by sym() and onext(). */ shelle sptr; /* Temporary variable used by tspivot(). */ if ( verbose ) { printf( " Marking neighbors of marked triangles.\n" ); } /* Loop through all the infected triangles, spreading the virus to */ /* their neighbors, then to their neighbors' neighbors. */ traversalinit( &viri ); virusloop = (triangle **) traverse( &viri ); while ( virusloop != (triangle **) NULL ) { testtri.tri = *virusloop; /* A triangle is marked as infected by messing with one of its shell */ /* edges, setting it to an illegal value. Hence, we have to */ /* temporarily uninfect this triangle so that we can examine its */ /* adjacent shell edges. */ uninfect( testtri ); if ( verbose > 2 ) { /* Assign the triangle an orientation for convenience in */ /* checking its points. */ testtri.orient = 0; org( testtri, deadorg ); dest( testtri, deaddest ); apex( testtri, deadapex ); printf( " Checking (%.12g, %.12g) (%.12g, %.12g) (%.12g, %.12g)\n", deadorg[0], deadorg[1], deaddest[0], deaddest[1], deadapex[0], deadapex[1] ); } /* Check each of the triangle's three neighbors. */ for ( testtri.orient = 0; testtri.orient < 3; testtri.orient++ ) { /* Find the neighbor. */ sym( testtri, neighbor ); /* Check for a shell between the triangle and its neighbor. */ tspivot( testtri, neighborshelle ); /* Check if the neighbor is nonexistent or already infected. */ if ( ( neighbor.tri == dummytri ) || infected( neighbor ) ) { if ( neighborshelle.sh != dummysh ) { /* There is a shell edge separating the triangle from its */ /* neighbor, but both triangles are dying, so the shell */ /* edge dies too. */ shelledealloc( neighborshelle.sh ); if ( neighbor.tri != dummytri ) { /* Make sure the shell edge doesn't get deallocated again */ /* later when the infected neighbor is visited. */ uninfect( neighbor ); tsdissolve( neighbor ); infect( neighbor ); } } } else { /* The neighbor exists and is not infected. */ if ( neighborshelle.sh == dummysh ) { /* There is no shell edge protecting the neighbor, so */ /* the neighbor becomes infected. */ if ( verbose > 2 ) { org( neighbor, deadorg ); dest( neighbor, deaddest ); apex( neighbor, deadapex ); printf( " Marking (%.12g, %.12g) (%.12g, %.12g) (%.12g, %.12g)\n", deadorg[0], deadorg[1], deaddest[0], deaddest[1], deadapex[0], deadapex[1] ); } infect( neighbor ); /* Ensure that the neighbor's neighbors will be infected. */ deadtri = (triangle **) poolalloc( &viri ); *deadtri = neighbor.tri; } else { /* The neighbor is protected by a shell edge. */ /* Remove this triangle from the shell edge. */ stdissolve( neighborshelle ); /* The shell edge becomes a boundary. Set markers accordingly. */ if ( mark( neighborshelle ) == 0 ) { setmark( neighborshelle, 1 ); } org( neighbor, norg ); dest( neighbor, ndest ); if ( pointmark( norg ) == 0 ) { setpointmark( norg, 1 ); } if ( pointmark( ndest ) == 0 ) { setpointmark( ndest, 1 ); } } } } /* Remark the triangle as infected, so it doesn't get added to the */ /* virus pool again. */ infect( testtri ); virusloop = (triangle **) traverse( &viri ); } if ( verbose ) { printf( " Deleting marked triangles.\n" ); } traversalinit( &viri ); virusloop = (triangle **) traverse( &viri ); while ( virusloop != (triangle **) NULL ) { testtri.tri = *virusloop; /* Check each of the three corners of the triangle for elimination. */ /* This is done by walking around each point, checking if it is */ /* still connected to at least one live triangle. */ for ( testtri.orient = 0; testtri.orient < 3; testtri.orient++ ) { org( testtri, testpoint ); /* Check if the point has already been tested. */ if ( testpoint != (point) NULL ) { killorg = 1; /* Mark the corner of the triangle as having been tested. */ setorg( testtri, NULL ); /* Walk counterclockwise about the point. */ onext( testtri, neighbor ); /* Stop upon reaching a boundary or the starting triangle. */ while ( ( neighbor.tri != dummytri ) && ( !triedgeequal( neighbor, testtri ) ) ) { if ( infected( neighbor ) ) { /* Mark the corner of this triangle as having been tested. */ setorg( neighbor, NULL ); } else { /* A live triangle. The point survives. */ killorg = 0; } /* Walk counterclockwise about the point. */ onextself( neighbor ); } /* If we reached a boundary, we must walk clockwise as well. */ if ( neighbor.tri == dummytri ) { /* Walk clockwise about the point. */ oprev( testtri, neighbor ); /* Stop upon reaching a boundary. */ while ( neighbor.tri != dummytri ) { if ( infected( neighbor ) ) { /* Mark the corner of this triangle as having been tested. */ setorg( neighbor, NULL ); } else { /* A live triangle. The point survives. */ killorg = 0; } /* Walk clockwise about the point. */ oprevself( neighbor ); } } if ( killorg ) { if ( verbose > 1 ) { printf( " Deleting point (%.12g, %.12g)\n", testpoint[0], testpoint[1] ); } pointdealloc( testpoint ); } } } /* Record changes in the number of boundary edges, and disconnect */ /* dead triangles from their neighbors. */ for ( testtri.orient = 0; testtri.orient < 3; testtri.orient++ ) { sym( testtri, neighbor ); if ( neighbor.tri == dummytri ) { /* There is no neighboring triangle on this edge, so this edge */ /* is a boundary edge. This triangle is being deleted, so this */ /* boundary edge is deleted. */ hullsize--; } else { /* Disconnect the triangle from its neighbor. */ dissolve( neighbor ); /* There is a neighboring triangle on this edge, so this edge */ /* becomes a boundary edge when this triangle is deleted. */ hullsize++; } } /* Return the dead triangle to the pool of triangles. */ triangledealloc( testtri.tri ); virusloop = (triangle **) traverse( &viri ); } /* Empty the virus pool. */ poolrestart( &viri ); } /*****************************************************************************/ /* */ /* regionplague() Spread regional attributes and/or area constraints */ /* (from a .poly file) throughout the mesh. */ /* */ /* This procedure operates in two phases. The first phase spreads an */ /* attribute and/or an area constraint through a (segment-bounded) region. */ /* The triangles are marked to ensure that each triangle is added to the */ /* virus pool only once, so the procedure will terminate. */ /* */ /* The second phase uninfects all infected triangles, returning them to */ /* normal. */ /* */ /*****************************************************************************/ void regionplague( attribute, area ) REAL attribute; REAL area; { struct triedge testtri; struct triedge neighbor; triangle **virusloop; triangle **regiontri; struct edge neighborshelle; point regionorg, regiondest, regionapex; triangle ptr; /* Temporary variable used by sym() and onext(). */ shelle sptr; /* Temporary variable used by tspivot(). */ if ( verbose > 1 ) { printf( " Marking neighbors of marked triangles.\n" ); } /* Loop through all the infected triangles, spreading the attribute */ /* and/or area constraint to their neighbors, then to their neighbors' */ /* neighbors. */ traversalinit( &viri ); virusloop = (triangle **) traverse( &viri ); while ( virusloop != (triangle **) NULL ) { testtri.tri = *virusloop; /* A triangle is marked as infected by messing with one of its shell */ /* edges, setting it to an illegal value. Hence, we have to */ /* temporarily uninfect this triangle so that we can examine its */ /* adjacent shell edges. */ uninfect( testtri ); if ( regionattrib ) { /* Set an attribute. */ setelemattribute( testtri, eextras, attribute ); } if ( vararea ) { /* Set an area constraint. */ setareabound( testtri, area ); } if ( verbose > 2 ) { /* Assign the triangle an orientation for convenience in */ /* checking its points. */ testtri.orient = 0; org( testtri, regionorg ); dest( testtri, regiondest ); apex( testtri, regionapex ); printf( " Checking (%.12g, %.12g) (%.12g, %.12g) (%.12g, %.12g)\n", regionorg[0], regionorg[1], regiondest[0], regiondest[1], regionapex[0], regionapex[1] ); } /* Check each of the triangle's three neighbors. */ for ( testtri.orient = 0; testtri.orient < 3; testtri.orient++ ) { /* Find the neighbor. */ sym( testtri, neighbor ); /* Check for a shell between the triangle and its neighbor. */ tspivot( testtri, neighborshelle ); /* Make sure the neighbor exists, is not already infected, and */ /* isn't protected by a shell edge. */ if ( ( neighbor.tri != dummytri ) && !infected( neighbor ) && ( neighborshelle.sh == dummysh ) ) { if ( verbose > 2 ) { org( neighbor, regionorg ); dest( neighbor, regiondest ); apex( neighbor, regionapex ); printf( " Marking (%.12g, %.12g) (%.12g, %.12g) (%.12g, %.12g)\n", regionorg[0], regionorg[1], regiondest[0], regiondest[1], regionapex[0], regionapex[1] ); } /* Infect the neighbor. */ infect( neighbor ); /* Ensure that the neighbor's neighbors will be infected. */ regiontri = (triangle **) poolalloc( &viri ); *regiontri = neighbor.tri; } } /* Remark the triangle as infected, so it doesn't get added to the */ /* virus pool again. */ infect( testtri ); virusloop = (triangle **) traverse( &viri ); } /* Uninfect all triangles. */ if ( verbose > 1 ) { printf( " Unmarking marked triangles.\n" ); } traversalinit( &viri ); virusloop = (triangle **) traverse( &viri ); while ( virusloop != (triangle **) NULL ) { testtri.tri = *virusloop; uninfect( testtri ); virusloop = (triangle **) traverse( &viri ); } /* Empty the virus pool. */ poolrestart( &viri ); } /*****************************************************************************/ /* */ /* carveholes() Find the holes and infect them. Find the area */ /* constraints and infect them. Infect the convex hull. */ /* Spread the infection and kill triangles. Spread the */ /* area constraints. */ /* */ /* This routine mainly calls other routines to carry out all these */ /* functions. */ /* */ /*****************************************************************************/ void carveholes( holelist, holes, regionlist, regions ) REAL * holelist; int holes; REAL *regionlist; int regions; { struct triedge searchtri; struct triedge triangleloop; struct triedge *regiontris; triangle **holetri; triangle **regiontri; point searchorg, searchdest; enum locateresult intersect; int i; triangle ptr; /* Temporary variable used by sym(). */ if ( !( quiet || ( noholes && convex ) ) ) { printf( "Removing unwanted triangles.\n" ); if ( verbose && ( holes > 0 ) ) { printf( " Marking holes for elimination.\n" ); } } if ( regions > 0 ) { /* Allocate storage for the triangles in which region points fall. */ regiontris = (struct triedge *) malloc( regions * sizeof( struct triedge ) ); if ( regiontris == (struct triedge *) NULL ) { printf( "Error: Out of memory.\n" ); exit( 1 ); } } if ( ( ( holes > 0 ) && !noholes ) || !convex || ( regions > 0 ) ) { /* Initialize a pool of viri to be used for holes, concavities, */ /* regional attributes, and/or regional area constraints. */ poolinit( &viri, sizeof( triangle * ), VIRUSPERBLOCK, POINTER, 0 ); } if ( !convex ) { /* Mark as infected any unprotected triangles on the boundary. */ /* This is one way by which concavities are created. */ infecthull(); } if ( ( holes > 0 ) && !noholes ) { /* Infect each triangle in which a hole lies. */ for ( i = 0; i < 2 * holes; i += 2 ) { /* Ignore holes that aren't within the bounds of the mesh. */ if ( ( holelist[i] >= xmin ) && ( holelist[i] <= xmax ) && ( holelist[i + 1] >= ymin ) && ( holelist[i + 1] <= ymax ) ) { /* Start searching from some triangle on the outer boundary. */ searchtri.tri = dummytri; searchtri.orient = 0; symself( searchtri ); /* Ensure that the hole is to the left of this boundary edge; */ /* otherwise, locate() will falsely report that the hole */ /* falls within the starting triangle. */ org( searchtri, searchorg ); dest( searchtri, searchdest ); if ( counterclockwise( searchorg, searchdest, &holelist[i] ) > 0.0 ) { /* Find a triangle that contains the hole. */ intersect = locate( &holelist[i], &searchtri ); if ( ( intersect != OUTSIDE ) && ( !infected( searchtri ) ) ) { /* Infect the triangle. This is done by marking the triangle */ /* as infect and including the triangle in the virus pool. */ infect( searchtri ); holetri = (triangle **) poolalloc( &viri ); *holetri = searchtri.tri; } } } } } /* Now, we have to find all the regions BEFORE we carve the holes, because */ /* locate() won't work when the triangulation is no longer convex. */ /* (Incidentally, this is the reason why regional attributes and area */ /* constraints can't be used when refining a preexisting mesh, which */ /* might not be convex; they can only be used with a freshly */ /* triangulated PSLG.) */ if ( regions > 0 ) { /* Find the starting triangle for each region. */ for ( i = 0; i < regions; i++ ) { regiontris[i].tri = dummytri; /* Ignore region points that aren't within the bounds of the mesh. */ if ( ( regionlist[4 * i] >= xmin ) && ( regionlist[4 * i] <= xmax ) && ( regionlist[4 * i + 1] >= ymin ) && ( regionlist[4 * i + 1] <= ymax ) ) { /* Start searching from some triangle on the outer boundary. */ searchtri.tri = dummytri; searchtri.orient = 0; symself( searchtri ); /* Ensure that the region point is to the left of this boundary */ /* edge; otherwise, locate() will falsely report that the */ /* region point falls within the starting triangle. */ org( searchtri, searchorg ); dest( searchtri, searchdest ); if ( counterclockwise( searchorg, searchdest, ®ionlist[4 * i] ) > 0.0 ) { /* Find a triangle that contains the region point. */ intersect = locate( ®ionlist[4 * i], &searchtri ); if ( ( intersect != OUTSIDE ) && ( !infected( searchtri ) ) ) { /* Record the triangle for processing after the */ /* holes have been carved. */ triedgecopy( searchtri, regiontris[i] ); } } } } } if ( viri.items > 0 ) { /* Carve the holes and concavities. */ plague(); } /* The virus pool should be empty now. */ if ( regions > 0 ) { if ( !quiet ) { if ( regionattrib ) { if ( vararea ) { printf( "Spreading regional attributes and area constraints.\n" ); } else { printf( "Spreading regional attributes.\n" ); } } else { printf( "Spreading regional area constraints.\n" ); } } if ( regionattrib && !refine ) { /* Assign every triangle a regional attribute of zero. */ traversalinit( &triangles ); triangleloop.orient = 0; triangleloop.tri = triangletraverse(); while ( triangleloop.tri != (triangle *) NULL ) { setelemattribute( triangleloop, eextras, 0.0 ); triangleloop.tri = triangletraverse(); } } for ( i = 0; i < regions; i++ ) { if ( regiontris[i].tri != dummytri ) { /* Make sure the triangle under consideration still exists. */ /* It may have been eaten by the virus. */ if ( regiontris[i].tri[3] != (triangle) NULL ) { /* Put one triangle in the virus pool. */ infect( regiontris[i] ); regiontri = (triangle **) poolalloc( &viri ); *regiontri = regiontris[i].tri; /* Apply one region's attribute and/or area constraint. */ regionplague( regionlist[4 * i + 2], regionlist[4 * i + 3] ); /* The virus pool should be empty now. */ } } } if ( regionattrib && !refine ) { /* Note the fact that each triangle has an additional attribute. */ eextras++; } } /* Free up memory. */ if ( ( ( holes > 0 ) && !noholes ) || !convex || ( regions > 0 ) ) { pooldeinit( &viri ); } if ( regions > 0 ) { free( regiontris ); } } /** **/ /** **/ /********* Carving out holes and concavities ends here *********/ /********* Mesh quality maintenance begins here *********/ /** **/ /** **/ /*****************************************************************************/ /* */ /* tallyencs() Traverse the entire list of shell edges, check each edge */ /* to see if it is encroached. If so, add it to the list. */ /* */ /*****************************************************************************/ #ifndef CDT_ONLY void tallyencs(){ struct edge edgeloop; int dummy; traversalinit( &shelles ); edgeloop.shorient = 0; edgeloop.sh = shelletraverse(); while ( edgeloop.sh != (shelle *) NULL ) { /* If the segment is encroached, add it to the list. */ dummy = checkedge4encroach( &edgeloop ); edgeloop.sh = shelletraverse(); } } #endif /* not CDT_ONLY */ /*****************************************************************************/ /* */ /* precisionerror() Print an error message for precision problems. */ /* */ /*****************************************************************************/ #ifndef CDT_ONLY void precisionerror(){ printf( "Try increasing the area criterion and/or reducing the minimum\n" ); printf( " allowable angle so that tiny triangles are not created.\n" ); #ifdef SINGLE printf( "Alternatively, try recompiling me with double precision\n" ); printf( " arithmetic (by removing \"#define SINGLE\" from the\n" ); printf( " source file or \"-DSINGLE\" from the makefile).\n" ); #endif /* SINGLE */ } #endif /* not CDT_ONLY */ /*****************************************************************************/ /* */ /* repairencs() Find and repair all the encroached segments. */ /* */ /* Encroached segments are repaired by splitting them by inserting a point */ /* at or near their centers. */ /* */ /* `flaws' is a flag that specifies whether one should take note of new */ /* encroached segments and bad triangles that result from inserting points */ /* to repair existing encroached segments. */ /* */ /* When a segment is split, the two resulting subsegments are always */ /* tested to see if they are encroached upon, regardless of the value */ /* of `flaws'. */ /* */ /*****************************************************************************/ #ifndef CDT_ONLY void repairencs( flaws ) int flaws; { struct triedge enctri; struct triedge testtri; struct edge *encloop; struct edge testsh; point eorg, edest; point newpoint; enum insertsiteresult success; REAL segmentlength, nearestpoweroftwo; REAL split; int acuteorg, acutedest; int dummy; int i; triangle ptr; /* Temporary variable used by stpivot(). */ shelle sptr; /* Temporary variable used by snext(). */ while ( ( badsegments.items > 0 ) && ( steinerleft != 0 ) ) { traversalinit( &badsegments ); encloop = badsegmenttraverse(); while ( ( encloop != (struct edge *) NULL ) && ( steinerleft != 0 ) ) { /* To decide where to split a segment, we need to know if the */ /* segment shares an endpoint with an adjacent segment. */ /* The concern is that, if we simply split every encroached */ /* segment in its center, two adjacent segments with a small */ /* angle between them might lead to an infinite loop; each */ /* point added to split one segment will encroach upon the */ /* other segment, which must then be split with a point that */ /* will encroach upon the first segment, and so on forever. */ /* To avoid this, imagine a set of concentric circles, whose */ /* radii are powers of two, about each segment endpoint. */ /* These concentric circles determine where the segment is */ /* split. (If both endpoints are shared with adjacent */ /* segments, split the segment in the middle, and apply the */ /* concentric shells for later splittings.) */ /* Is the origin shared with another segment? */ stpivot( *encloop, enctri ); lnext( enctri, testtri ); tspivot( testtri, testsh ); acuteorg = testsh.sh != dummysh; /* Is the destination shared with another segment? */ lnextself( testtri ); tspivot( testtri, testsh ); acutedest = testsh.sh != dummysh; /* Now, check the other side of the segment, if there's a triangle */ /* there. */ sym( enctri, testtri ); if ( testtri.tri != dummytri ) { /* Is the destination shared with another segment? */ lnextself( testtri ); tspivot( testtri, testsh ); acutedest = acutedest || ( testsh.sh != dummysh ); /* Is the origin shared with another segment? */ lnextself( testtri ); tspivot( testtri, testsh ); acuteorg = acuteorg || ( testsh.sh != dummysh ); } sorg( *encloop, eorg ); sdest( *encloop, edest ); /* Use the concentric circles if exactly one endpoint is shared */ /* with another adjacent segment. */ if ( acuteorg ^ acutedest ) { segmentlength = sqrt( ( edest[0] - eorg[0] ) * ( edest[0] - eorg[0] ) + ( edest[1] - eorg[1] ) * ( edest[1] - eorg[1] ) ); /* Find the power of two nearest the segment's length. */ nearestpoweroftwo = 1.0; while ( segmentlength > SQUAREROOTTWO * nearestpoweroftwo ) { nearestpoweroftwo *= 2.0; } while ( segmentlength < ( 0.5 * SQUAREROOTTWO ) * nearestpoweroftwo ) { nearestpoweroftwo *= 0.5; } /* Where do we split the segment? */ split = 0.5 * nearestpoweroftwo / segmentlength; if ( acutedest ) { split = 1.0 - split; } } else { /* If we're not worried about adjacent segments, split */ /* this segment in the middle. */ split = 0.5; } /* Create the new point. */ newpoint = (point) poolalloc( &points ); /* Interpolate its coordinate and attributes. */ for ( i = 0; i < 2 + nextras; i++ ) { newpoint[i] = ( 1.0 - split ) * eorg[i] + split * edest[i]; } setpointmark( newpoint, mark( *encloop ) ); if ( verbose > 1 ) { printf( " Splitting edge (%.12g, %.12g) (%.12g, %.12g) at (%.12g, %.12g).\n", eorg[0], eorg[1], edest[0], edest[1], newpoint[0], newpoint[1] ); } /* Check whether the new point lies on an endpoint. */ if ( ( ( newpoint[0] == eorg[0] ) && ( newpoint[1] == eorg[1] ) ) || ( ( newpoint[0] == edest[0] ) && ( newpoint[1] == edest[1] ) ) ) { printf( "Error: Ran out of precision at (%.12g, %.12g).\n", newpoint[0], newpoint[1] ); printf( "I attempted to split a segment to a smaller size than can\n" ); printf( " be accommodated by the finite precision of floating point\n" ); printf( " arithmetic.\n" ); precisionerror(); exit( 1 ); } /* Insert the splitting point. This should always succeed. */ success = insertsite( newpoint, &enctri, encloop, flaws, flaws ); if ( ( success != SUCCESSFULPOINT ) && ( success != ENCROACHINGPOINT ) ) { printf( "Internal error in repairencs():\n" ); printf( " Failure to split a segment.\n" ); internalerror(); } if ( steinerleft > 0 ) { steinerleft--; } /* Check the two new subsegments to see if they're encroached. */ dummy = checkedge4encroach( encloop ); snextself( *encloop ); dummy = checkedge4encroach( encloop ); badsegmentdealloc( encloop ); encloop = badsegmenttraverse(); } } } #endif /* not CDT_ONLY */ /*****************************************************************************/ /* */ /* tallyfaces() Test every triangle in the mesh for quality measures. */ /* */ /*****************************************************************************/ #ifndef CDT_ONLY void tallyfaces(){ struct triedge triangleloop; if ( verbose ) { printf( " Making a list of bad triangles.\n" ); } traversalinit( &triangles ); triangleloop.orient = 0; triangleloop.tri = triangletraverse(); while ( triangleloop.tri != (triangle *) NULL ) { /* If the triangle is bad, enqueue it. */ testtriangle( &triangleloop ); triangleloop.tri = triangletraverse(); } } #endif /* not CDT_ONLY */ /*****************************************************************************/ /* */ /* findcircumcenter() Find the circumcenter of a triangle. */ /* */ /* The result is returned both in terms of x-y coordinates and xi-eta */ /* coordinates. The xi-eta coordinate system is defined in terms of the */ /* triangle: the origin of the triangle is the origin of the coordinate */ /* system; the destination of the triangle is one unit along the xi axis; */ /* and the apex of the triangle is one unit along the eta axis. */ /* */ /* The return value indicates which edge of the triangle is shortest. */ /* */ /*****************************************************************************/ enum circumcenterresult findcircumcenter( torg, tdest, tapex, circumcenter, xi, eta ) point torg; point tdest; point tapex; point circumcenter; REAL *xi; REAL *eta; { REAL xdo, ydo, xao, yao, xad, yad; REAL dodist, aodist, addist; REAL denominator; REAL dx, dy; circumcentercount++; /* Compute the circumcenter of the triangle. */ xdo = tdest[0] - torg[0]; ydo = tdest[1] - torg[1]; xao = tapex[0] - torg[0]; yao = tapex[1] - torg[1]; dodist = xdo * xdo + ydo * ydo; aodist = xao * xao + yao * yao; if ( noexact ) { denominator = (REAL)( 0.5 / ( xdo * yao - xao * ydo ) ); } else { /* Use the counterclockwise() routine to ensure a positive (and */ /* reasonably accurate) result, avoiding any possibility of */ /* division by zero. */ denominator = (REAL)( 0.5 / counterclockwise( tdest, tapex, torg ) ); /* Don't count the above as an orientation test. */ counterclockcount--; } circumcenter[0] = torg[0] - ( ydo * aodist - yao * dodist ) * denominator; circumcenter[1] = torg[1] + ( xdo * aodist - xao * dodist ) * denominator; /* To interpolate point attributes for the new point inserted at */ /* the circumcenter, define a coordinate system with a xi-axis, */ /* directed from the triangle's origin to its destination, and */ /* an eta-axis, directed from its origin to its apex. */ /* Calculate the xi and eta coordinates of the circumcenter. */ dx = circumcenter[0] - torg[0]; dy = circumcenter[1] - torg[1]; *xi = (REAL)( ( dx * yao - xao * dy ) * ( 2.0 * denominator ) ); *eta = (REAL)( ( xdo * dy - dx * ydo ) * ( 2.0 * denominator ) ); xad = tapex[0] - tdest[0]; yad = tapex[1] - tdest[1]; addist = xad * xad + yad * yad; if ( ( addist < dodist ) && ( addist < aodist ) ) { return OPPOSITEORG; } else if ( dodist < aodist ) { return OPPOSITEAPEX; } else { return OPPOSITEDEST; } } /*****************************************************************************/ /* */ /* splittriangle() Inserts a point at the circumcenter of a triangle. */ /* Deletes the newly inserted point if it encroaches upon */ /* a segment. */ /* */ /*****************************************************************************/ #ifndef CDT_ONLY void splittriangle( badtri ) struct badface *badtri; { point borg, bdest, bapex; point newpoint; REAL xi, eta; enum insertsiteresult success; enum circumcenterresult shortedge; int errorflag; int i; org( badtri->badfacetri, borg ); dest( badtri->badfacetri, bdest ); apex( badtri->badfacetri, bapex ); /* Make sure that this triangle is still the same triangle it was */ /* when it was tested and determined to be of bad quality. */ /* Subsequent transformations may have made it a different triangle. */ if ( ( borg == badtri->faceorg ) && ( bdest == badtri->facedest ) && ( bapex == badtri->faceapex ) ) { if ( verbose > 1 ) { printf( " Splitting this triangle at its circumcenter:\n" ); printf( " (%.12g, %.12g) (%.12g, %.12g) (%.12g, %.12g)\n", borg[0], borg[1], bdest[0], bdest[1], bapex[0], bapex[1] ); } errorflag = 0; /* Create a new point at the triangle's circumcenter. */ newpoint = (point) poolalloc( &points ); shortedge = findcircumcenter( borg, bdest, bapex, newpoint, &xi, &eta ); /* Check whether the new point lies on a triangle vertex. */ if ( ( ( newpoint[0] == borg[0] ) && ( newpoint[1] == borg[1] ) ) || ( ( newpoint[0] == bdest[0] ) && ( newpoint[1] == bdest[1] ) ) || ( ( newpoint[0] == bapex[0] ) && ( newpoint[1] == bapex[1] ) ) ) { if ( !quiet ) { printf( "Warning: New point (%.12g, %.12g) falls on existing vertex.\n" , newpoint[0], newpoint[1] ); errorflag = 1; } pointdealloc( newpoint ); } else { for ( i = 2; i < 2 + nextras; i++ ) { /* Interpolate the point attributes at the circumcenter. */ newpoint[i] = borg[i] + xi * ( bdest[i] - borg[i] ) + eta * ( bapex[i] - borg[i] ); } /* The new point must be in the interior, and have a marker of zero. */ setpointmark( newpoint, 0 ); /* Ensure that the handle `badtri->badfacetri' represents the shortest */ /* edge of the triangle. This ensures that the circumcenter must */ /* fall to the left of this edge, so point location will work. */ if ( shortedge == OPPOSITEORG ) { lnextself( badtri->badfacetri ); } else if ( shortedge == OPPOSITEDEST ) { lprevself( badtri->badfacetri ); } /* Insert the circumcenter, searching from the edge of the triangle, */ /* and maintain the Delaunay property of the triangulation. */ success = insertsite( newpoint, &( badtri->badfacetri ), (struct edge *) NULL, 1, 1 ); if ( success == SUCCESSFULPOINT ) { if ( steinerleft > 0 ) { steinerleft--; } } else if ( success == ENCROACHINGPOINT ) { /* If the newly inserted point encroaches upon a segment, delete it. */ deletesite( &( badtri->badfacetri ) ); } else if ( success == VIOLATINGPOINT ) { /* Failed to insert the new point, but some segment was */ /* marked as being encroached. */ pointdealloc( newpoint ); } else { /* success == DUPLICATEPOINT */ /* Failed to insert the new point because a vertex is already there. */ if ( !quiet ) { printf( "Warning: New point (%.12g, %.12g) falls on existing vertex.\n" , newpoint[0], newpoint[1] ); errorflag = 1; } pointdealloc( newpoint ); } } if ( errorflag ) { if ( verbose ) { printf( " The new point is at the circumcenter of triangle\n" ); printf( " (%.12g, %.12g) (%.12g, %.12g) (%.12g, %.12g)\n", borg[0], borg[1], bdest[0], bdest[1], bapex[0], bapex[1] ); } printf( "This probably means that I am trying to refine triangles\n" ); printf( " to a smaller size than can be accommodated by the finite\n" ); printf( " precision of floating point arithmetic. (You can be\n" ); printf( " sure of this if I fail to terminate.)\n" ); precisionerror(); } } /* Return the bad triangle to the pool. */ pooldealloc( &badtriangles, (VOID *) badtri ); } #endif /* not CDT_ONLY */ /*****************************************************************************/ /* */ /* enforcequality() Remove all the encroached edges and bad triangles */ /* from the triangulation. */ /* */ /*****************************************************************************/ #ifndef CDT_ONLY void enforcequality(){ int i; if ( !quiet ) { printf( "Adding Steiner points to enforce quality.\n" ); } /* Initialize the pool of encroached segments. */ poolinit( &badsegments, sizeof( struct edge ), BADSEGMENTPERBLOCK, POINTER, 0 ); if ( verbose ) { printf( " Looking for encroached segments.\n" ); } /* Test all segments to see if they're encroached. */ tallyencs(); if ( verbose && ( badsegments.items > 0 ) ) { printf( " Splitting encroached segments.\n" ); } /* Note that steinerleft == -1 if an unlimited number */ /* of Steiner points is allowed. */ while ( ( badsegments.items > 0 ) && ( steinerleft != 0 ) ) { /* Fix the segments without noting newly encroached segments or */ /* bad triangles. The reason we don't want to note newly */ /* encroached segments is because some encroached segments are */ /* likely to be noted multiple times, and would then be blindly */ /* split multiple times. I should fix that some time. */ repairencs( 0 ); /* Now, find all the segments that became encroached while adding */ /* points to split encroached segments. */ tallyencs(); } /* At this point, if we haven't run out of Steiner points, the */ /* triangulation should be (conforming) Delaunay. */ /* Next, we worry about enforcing triangle quality. */ if ( ( minangle > 0.0 ) || vararea || fixedarea ) { /* Initialize the pool of bad triangles. */ poolinit( &badtriangles, sizeof( struct badface ), BADTRIPERBLOCK, POINTER, 0 ); /* Initialize the queues of bad triangles. */ for ( i = 0; i < 64; i++ ) { queuefront[i] = (struct badface *) NULL; queuetail[i] = &queuefront[i]; } /* Test all triangles to see if they're bad. */ tallyfaces(); if ( verbose ) { printf( " Splitting bad triangles.\n" ); } while ( ( badtriangles.items > 0 ) && ( steinerleft != 0 ) ) { /* Fix one bad triangle by inserting a point at its circumcenter. */ splittriangle( dequeuebadtri() ); /* Fix any encroached segments that may have resulted. Record */ /* any new bad triangles or encroached segments that result. */ if ( badsegments.items > 0 ) { repairencs( 1 ); } } } /* At this point, if we haven't run out of Steiner points, the */ /* triangulation should be (conforming) Delaunay and have no */ /* low-quality triangles. */ /* Might we have run out of Steiner points too soon? */ if ( !quiet && ( badsegments.items > 0 ) && ( steinerleft == 0 ) ) { printf( "\nWarning: I ran out of Steiner points, but the mesh has\n" ); if ( badsegments.items == 1 ) { printf( " an encroached segment, and therefore might not be truly\n" ); } else { printf( " %ld encroached segments, and therefore might not be truly\n", badsegments.items ); } printf( " Delaunay. If the Delaunay property is important to you,\n" ); printf( " try increasing the number of Steiner points (controlled by\n" ); printf( " the -S switch) slightly and try again.\n\n" ); } } #endif /* not CDT_ONLY */ /** **/ /** **/ /********* Mesh quality maintenance ends here *********/ /*****************************************************************************/ /* */ /* highorder() Create extra nodes for quadratic subparametric elements. */ /* */ /*****************************************************************************/ void highorder(){ struct triedge triangleloop, trisym; struct edge checkmark; point newpoint; point torg, tdest; int i; triangle ptr; /* Temporary variable used by sym(). */ shelle sptr; /* Temporary variable used by tspivot(). */ if ( !quiet ) { printf( "Adding vertices for second-order triangles.\n" ); } /* The following line ensures that dead items in the pool of nodes */ /* cannot be allocated for the extra nodes associated with high */ /* order elements. This ensures that the primary nodes (at the */ /* corners of elements) will occur earlier in the output files, and */ /* have lower indices, than the extra nodes. */ points.deaditemstack = (VOID *) NULL; traversalinit( &triangles ); triangleloop.tri = triangletraverse(); /* To loop over the set of edges, loop over all triangles, and look at */ /* the three edges of each triangle. If there isn't another triangle */ /* adjacent to the edge, operate on the edge. If there is another */ /* adjacent triangle, operate on the edge only if the current triangle */ /* has a smaller pointer than its neighbor. This way, each edge is */ /* considered only once. */ while ( triangleloop.tri != (triangle *) NULL ) { for ( triangleloop.orient = 0; triangleloop.orient < 3; triangleloop.orient++ ) { sym( triangleloop, trisym ); if ( ( triangleloop.tri < trisym.tri ) || ( trisym.tri == dummytri ) ) { org( triangleloop, torg ); dest( triangleloop, tdest ); /* Create a new node in the middle of the edge. Interpolate */ /* its attributes. */ newpoint = (point) poolalloc( &points ); for ( i = 0; i < 2 + nextras; i++ ) { newpoint[i] = (REAL)( 0.5 * ( torg[i] + tdest[i] ) ); } /* Set the new node's marker to zero or one, depending on */ /* whether it lies on a boundary. */ setpointmark( newpoint, trisym.tri == dummytri ); if ( useshelles ) { tspivot( triangleloop, checkmark ); /* If this edge is a segment, transfer the marker to the new node. */ if ( checkmark.sh != dummysh ) { setpointmark( newpoint, mark( checkmark ) ); } } if ( verbose > 1 ) { printf( " Creating (%.12g, %.12g).\n", newpoint[0], newpoint[1] ); } /* Record the new node in the (one or two) adjacent elements. */ triangleloop.tri[highorderindex + triangleloop.orient] = (triangle) newpoint; if ( trisym.tri != dummytri ) { trisym.tri[highorderindex + trisym.orient] = (triangle) newpoint; } } } triangleloop.tri = triangletraverse(); } } /********* File I/O routines begin here *********/ /** **/ /** **/ /*****************************************************************************/ /* */ /* readline() Read a nonempty line from a file. */ /* */ /* A line is considered "nonempty" if it contains something that looks like */ /* a number. */ /* */ /*****************************************************************************/ #ifndef TRILIBRARY char *readline( string, infile, infilename ) char *string; FILE *infile; char *infilename; { char *result; /* Search for something that looks like a number. */ do { result = fgets( string, INPUTLINESIZE, infile ); if ( result == (char *) NULL ) { printf( " Error: Unexpected end of file in %s.\n", infilename ); exit( 1 ); } /* Skip anything that doesn't look like a number, a comment, */ /* or the end of a line. */ while ( ( *result != '\0' ) && ( *result != '#' ) && ( *result != '.' ) && ( *result != '+' ) && ( *result != '-' ) && ( ( *result < '0' ) || ( *result > '9' ) ) ) { result++; } /* If it's a comment or end of line, read another line and try again. */ } while ( ( *result == '#' ) || ( *result == '\0' ) ); return result; } #endif /* not TRILIBRARY */ /*****************************************************************************/ /* */ /* findfield() Find the next field of a string. */ /* */ /* Jumps past the current field by searching for whitespace, then jumps */ /* past the whitespace to find the next field. */ /* */ /*****************************************************************************/ #ifndef TRILIBRARY char *findfield( string ) char *string; { char *result; result = string; /* Skip the current field. Stop upon reaching whitespace. */ while ( ( *result != '\0' ) && ( *result != '#' ) && ( *result != ' ' ) && ( *result != '\t' ) ) { result++; } /* Now skip the whitespace and anything else that doesn't look like a */ /* number, a comment, or the end of a line. */ while ( ( *result != '\0' ) && ( *result != '#' ) && ( *result != '.' ) && ( *result != '+' ) && ( *result != '-' ) && ( ( *result < '0' ) || ( *result > '9' ) ) ) { result++; } /* Check for a comment (prefixed with `#'). */ if ( *result == '#' ) { *result = '\0'; } return result; } #endif /* not TRILIBRARY */ /*****************************************************************************/ /* */ /* readnodes() Read the points from a file, which may be a .node or .poly */ /* file. */ /* */ /*****************************************************************************/ #ifndef TRILIBRARY void readnodes( nodefilename, polyfilename, polyfile ) char *nodefilename; char *polyfilename; FILE **polyfile; { FILE *infile; point pointloop; char inputline[INPUTLINESIZE]; char *stringptr; char *infilename; REAL x, y; int firstnode; int nodemarkers; int currentmarker; int i, j; if ( poly ) { /* Read the points from a .poly file. */ if ( !quiet ) { printf( "Opening %s.\n", polyfilename ); } *polyfile = fopen( polyfilename, "r" ); if ( *polyfile == (FILE *) NULL ) { printf( " Error: Cannot access file %s.\n", polyfilename ); exit( 1 ); } /* Read number of points, number of dimensions, number of point */ /* attributes, and number of boundary markers. */ stringptr = readline( inputline, *polyfile, polyfilename ); inpoints = (int) strtol( stringptr, &stringptr, 0 ); stringptr = findfield( stringptr ); if ( *stringptr == '\0' ) { mesh_dim = 2; } else { mesh_dim = (int) strtol( stringptr, &stringptr, 0 ); } stringptr = findfield( stringptr ); if ( *stringptr == '\0' ) { nextras = 0; } else { nextras = (int) strtol( stringptr, &stringptr, 0 ); } stringptr = findfield( stringptr ); if ( *stringptr == '\0' ) { nodemarkers = 0; } else { nodemarkers = (int) strtol( stringptr, &stringptr, 0 ); } if ( inpoints > 0 ) { infile = *polyfile; infilename = polyfilename; readnodefile = 0; } else { /* If the .poly file claims there are zero points, that means that */ /* the points should be read from a separate .node file. */ readnodefile = 1; infilename = innodefilename; } } else { readnodefile = 1; infilename = innodefilename; *polyfile = (FILE *) NULL; } if ( readnodefile ) { /* Read the points from a .node file. */ if ( !quiet ) { printf( "Opening %s.\n", innodefilename ); } infile = fopen( innodefilename, "r" ); if ( infile == (FILE *) NULL ) { printf( " Error: Cannot access file %s.\n", innodefilename ); exit( 1 ); } /* Read number of points, number of dimensions, number of point */ /* attributes, and number of boundary markers. */ stringptr = readline( inputline, infile, innodefilename ); inpoints = (int) strtol( stringptr, &stringptr, 0 ); stringptr = findfield( stringptr ); if ( *stringptr == '\0' ) { mesh_dim = 2; } else { mesh_dim = (int) strtol( stringptr, &stringptr, 0 ); } stringptr = findfield( stringptr ); if ( *stringptr == '\0' ) { nextras = 0; } else { nextras = (int) strtol( stringptr, &stringptr, 0 ); } stringptr = findfield( stringptr ); if ( *stringptr == '\0' ) { nodemarkers = 0; } else { nodemarkers = (int) strtol( stringptr, &stringptr, 0 ); } } if ( inpoints < 3 ) { printf( "Error: Input must have at least three input points.\n" ); exit( 1 ); } if ( mesh_dim != 2 ) { printf( "Error: Triangle only works with two-dimensional meshes.\n" ); exit( 1 ); } initializepointpool(); /* Read the points. */ for ( i = 0; i < inpoints; i++ ) { pointloop = (point) poolalloc( &points ); stringptr = readline( inputline, infile, infilename ); if ( i == 0 ) { firstnode = (int) strtol( stringptr, &stringptr, 0 ); if ( ( firstnode == 0 ) || ( firstnode == 1 ) ) { firstnumber = firstnode; } } stringptr = findfield( stringptr ); if ( *stringptr == '\0' ) { printf( "Error: Point %d has no x coordinate.\n", firstnumber + i ); exit( 1 ); } x = (REAL) strtod( stringptr, &stringptr ); stringptr = findfield( stringptr ); if ( *stringptr == '\0' ) { printf( "Error: Point %d has no y coordinate.\n", firstnumber + i ); exit( 1 ); } y = (REAL) strtod( stringptr, &stringptr ); pointloop[0] = x; pointloop[1] = y; /* Read the point attributes. */ for ( j = 2; j < 2 + nextras; j++ ) { stringptr = findfield( stringptr ); if ( *stringptr == '\0' ) { pointloop[j] = 0.0; } else { pointloop[j] = (REAL) strtod( stringptr, &stringptr ); } } if ( nodemarkers ) { /* Read a point marker. */ stringptr = findfield( stringptr ); if ( *stringptr == '\0' ) { setpointmark( pointloop, 0 ); } else { currentmarker = (int) strtol( stringptr, &stringptr, 0 ); setpointmark( pointloop, currentmarker ); } } else { /* If no markers are specified in the file, they default to zero. */ setpointmark( pointloop, 0 ); } /* Determine the smallest and largest x and y coordinates. */ if ( i == 0 ) { xmin = xmax = x; ymin = ymax = y; } else { xmin = ( x < xmin ) ? x : xmin; xmax = ( x > xmax ) ? x : xmax; ymin = ( y < ymin ) ? y : ymin; ymax = ( y > ymax ) ? y : ymax; } } if ( readnodefile ) { fclose( infile ); } /* Nonexistent x value used as a flag to mark circle events in sweepline */ /* Delaunay algorithm. */ xminextreme = 10 * xmin - 9 * xmax; } #endif /* not TRILIBRARY */ /*****************************************************************************/ /* */ /* transfernodes() Read the points from memory. */ /* */ /*****************************************************************************/ #ifdef TRILIBRARY void transfernodes( pointlist, pointattriblist, pointmarkerlist, numberofpoints, numberofpointattribs ) REAL * pointlist; REAL *pointattriblist; int *pointmarkerlist; int numberofpoints; int numberofpointattribs; { point pointloop; REAL x, y; int i, j; int coordindex; int attribindex; inpoints = numberofpoints; mesh_dim = 2; nextras = numberofpointattribs; readnodefile = 0; if ( inpoints < 3 ) { printf( "Error: Input must have at least three input points.\n" ); exit( 1 ); } initializepointpool(); /* Read the points. */ coordindex = 0; attribindex = 0; for ( i = 0; i < inpoints; i++ ) { pointloop = (point) poolalloc( &points ); /* Read the point coordinates. */ x = pointloop[0] = pointlist[coordindex++]; y = pointloop[1] = pointlist[coordindex++]; /* Read the point attributes. */ for ( j = 0; j < numberofpointattribs; j++ ) { pointloop[2 + j] = pointattriblist[attribindex++]; } if ( pointmarkerlist != (int *) NULL ) { /* Read a point marker. */ setpointmark( pointloop, pointmarkerlist[i] ); } else { /* If no markers are specified, they default to zero. */ setpointmark( pointloop, 0 ); } x = pointloop[0]; y = pointloop[1]; /* Determine the smallest and largest x and y coordinates. */ if ( i == 0 ) { xmin = xmax = x; ymin = ymax = y; } else { xmin = ( x < xmin ) ? x : xmin; xmax = ( x > xmax ) ? x : xmax; ymin = ( y < ymin ) ? y : ymin; ymax = ( y > ymax ) ? y : ymax; } } /* Nonexistent x value used as a flag to mark circle events in sweepline */ /* Delaunay algorithm. */ xminextreme = 10 * xmin - 9 * xmax; } #endif /* TRILIBRARY */ /*****************************************************************************/ /* */ /* readholes() Read the holes, and possibly regional attributes and area */ /* constraints, from a .poly file. */ /* */ /*****************************************************************************/ #ifndef TRILIBRARY void readholes( polyfile, polyfilename, hlist, holes, rlist, regions ) FILE * polyfile; char *polyfilename; REAL **hlist; int *holes; REAL **rlist; int *regions; { REAL *holelist; REAL *regionlist; char inputline[INPUTLINESIZE]; char *stringptr; int index; int i; /* Read the holes. */ stringptr = readline( inputline, polyfile, polyfilename ); *holes = (int) strtol( stringptr, &stringptr, 0 ); if ( *holes > 0 ) { holelist = (REAL *) malloc( 2 * *holes * sizeof( REAL ) ); *hlist = holelist; if ( holelist == (REAL *) NULL ) { printf( "Error: Out of memory.\n" ); exit( 1 ); } for ( i = 0; i < 2 * *holes; i += 2 ) { stringptr = readline( inputline, polyfile, polyfilename ); stringptr = findfield( stringptr ); if ( *stringptr == '\0' ) { printf( "Error: Hole %d has no x coordinate.\n", firstnumber + ( i >> 1 ) ); exit( 1 ); } else { holelist[i] = (REAL) strtod( stringptr, &stringptr ); } stringptr = findfield( stringptr ); if ( *stringptr == '\0' ) { printf( "Error: Hole %d has no y coordinate.\n", firstnumber + ( i >> 1 ) ); exit( 1 ); } else { holelist[i + 1] = (REAL) strtod( stringptr, &stringptr ); } } } else { *hlist = (REAL *) NULL; } #ifndef CDT_ONLY if ( ( regionattrib || vararea ) && !refine ) { /* Read the area constraints. */ stringptr = readline( inputline, polyfile, polyfilename ); *regions = (int) strtol( stringptr, &stringptr, 0 ); if ( *regions > 0 ) { regionlist = (REAL *) malloc( 4 * *regions * sizeof( REAL ) ); *rlist = regionlist; if ( regionlist == (REAL *) NULL ) { printf( "Error: Out of memory.\n" ); exit( 1 ); } index = 0; for ( i = 0; i < *regions; i++ ) { stringptr = readline( inputline, polyfile, polyfilename ); stringptr = findfield( stringptr ); if ( *stringptr == '\0' ) { printf( "Error: Region %d has no x coordinate.\n", firstnumber + i ); exit( 1 ); } else { regionlist[index++] = (REAL) strtod( stringptr, &stringptr ); } stringptr = findfield( stringptr ); if ( *stringptr == '\0' ) { printf( "Error: Region %d has no y coordinate.\n", firstnumber + i ); exit( 1 ); } else { regionlist[index++] = (REAL) strtod( stringptr, &stringptr ); } stringptr = findfield( stringptr ); if ( *stringptr == '\0' ) { printf( "Error: Region %d has no region attribute or area constraint.\n", firstnumber + i ); exit( 1 ); } else { regionlist[index++] = (REAL) strtod( stringptr, &stringptr ); } stringptr = findfield( stringptr ); if ( *stringptr == '\0' ) { regionlist[index] = regionlist[index - 1]; } else { regionlist[index] = (REAL) strtod( stringptr, &stringptr ); } index++; } } } else { /* Set `*regions' to zero to avoid an accidental free() later. */ *regions = 0; *rlist = (REAL *) NULL; } #endif /* not CDT_ONLY */ fclose( polyfile ); } #endif /* not TRILIBRARY */ /*****************************************************************************/ /* */ /* finishfile() Write the command line to the output file so the user */ /* can remember how the file was generated. Close the file. */ /* */ /*****************************************************************************/ #ifndef TRILIBRARY void finishfile( outfile, argc, argv ) FILE * outfile; int argc; char **argv; { int i; fprintf( outfile, "# Generated by" ); for ( i = 0; i < argc; i++ ) { fprintf( outfile, " " ); fputs( argv[i], outfile ); } fprintf( outfile, "\n" ); fclose( outfile ); } #endif /* not TRILIBRARY */ /*****************************************************************************/ /* */ /* writenodes() Number the points and write them to a .node file. */ /* */ /* To save memory, the point numbers are written over the shell markers */ /* after the points are written to a file. */ /* */ /*****************************************************************************/ #ifdef TRILIBRARY void writenodes( pointlist, pointattriblist, pointmarkerlist ) REAL * *pointlist; REAL **pointattriblist; int **pointmarkerlist; #else /* not TRILIBRARY */ void writenodes( nodefilename, argc, argv ) char *nodefilename; int argc; char **argv; #endif /* not TRILIBRARY */ { #ifdef TRILIBRARY REAL *plist; REAL *palist; int *pmlist; int coordindex; int attribindex; #else /* not TRILIBRARY */ FILE *outfile; #endif /* not TRILIBRARY */ point pointloop; int pointnumber; int i; #ifdef TRILIBRARY if ( !quiet ) { printf( "Writing points.\n" ); } /* Allocate memory for output points if necessary. */ if ( *pointlist == (REAL *) NULL ) { *pointlist = (REAL *) malloc( points.items * 2 * sizeof( REAL ) ); if ( *pointlist == (REAL *) NULL ) { printf( "Error: Out of memory.\n" ); exit( 1 ); } } /* Allocate memory for output point attributes if necessary. */ if ( ( nextras > 0 ) && ( *pointattriblist == (REAL *) NULL ) ) { *pointattriblist = (REAL *) malloc( points.items * nextras * sizeof( REAL ) ); if ( *pointattriblist == (REAL *) NULL ) { printf( "Error: Out of memory.\n" ); exit( 1 ); } } /* Allocate memory for output point markers if necessary. */ if ( !nobound && ( *pointmarkerlist == (int *) NULL ) ) { *pointmarkerlist = (int *) malloc( points.items * sizeof( int ) ); if ( *pointmarkerlist == (int *) NULL ) { printf( "Error: Out of memory.\n" ); exit( 1 ); } } plist = *pointlist; palist = *pointattriblist; pmlist = *pointmarkerlist; coordindex = 0; attribindex = 0; #else /* not TRILIBRARY */ if ( !quiet ) { printf( "Writing %s.\n", nodefilename ); } outfile = fopen( nodefilename, "w" ); if ( outfile == (FILE *) NULL ) { printf( " Error: Cannot create file %s.\n", nodefilename ); exit( 1 ); } /* Number of points, number of dimensions, number of point attributes, */ /* and number of boundary markers (zero or one). */ fprintf( outfile, "%ld %d %d %d\n", points.items, mesh_dim, nextras, 1 - nobound ); #endif /* not TRILIBRARY */ traversalinit( &points ); pointloop = pointtraverse(); pointnumber = firstnumber; while ( pointloop != (point) NULL ) { #ifdef TRILIBRARY /* X and y coordinates. */ plist[coordindex++] = pointloop[0]; plist[coordindex++] = pointloop[1]; /* Point attributes. */ for ( i = 0; i < nextras; i++ ) { palist[attribindex++] = pointloop[2 + i]; } if ( !nobound ) { /* Copy the boundary marker. */ pmlist[pointnumber - firstnumber] = pointmark( pointloop ); } #else /* not TRILIBRARY */ /* Point number, x and y coordinates. */ fprintf( outfile, "%4d %.17g %.17g", pointnumber, pointloop[0], pointloop[1] ); for ( i = 0; i < nextras; i++ ) { /* Write an attribute. */ fprintf( outfile, " %.17g", pointloop[i + 2] ); } if ( nobound ) { fprintf( outfile, "\n" ); } else { /* Write the boundary marker. */ fprintf( outfile, " %d\n", pointmark( pointloop ) ); } #endif /* not TRILIBRARY */ setpointmark( pointloop, pointnumber ); pointloop = pointtraverse(); pointnumber++; } #ifndef TRILIBRARY finishfile( outfile, argc, argv ); #endif /* not TRILIBRARY */ } /*****************************************************************************/ /* */ /* numbernodes() Number the points. */ /* */ /* Each point is assigned a marker equal to its number. */ /* */ /* Used when writenodes() is not called because no .node file is written. */ /* */ /*****************************************************************************/ void numbernodes(){ point pointloop; int pointnumber; traversalinit( &points ); pointloop = pointtraverse(); pointnumber = firstnumber; while ( pointloop != (point) NULL ) { setpointmark( pointloop, pointnumber ); pointloop = pointtraverse(); pointnumber++; } } /*****************************************************************************/ /* */ /* writeelements() Write the triangles to an .ele file. */ /* */ /*****************************************************************************/ #ifdef TRILIBRARY void writeelements( trianglelist, triangleattriblist ) int **trianglelist; REAL **triangleattriblist; #else /* not TRILIBRARY */ void writeelements( elefilename, argc, argv ) char *elefilename; int argc; char **argv; #endif /* not TRILIBRARY */ { #ifdef TRILIBRARY int *tlist; REAL *talist; int pointindex; int attribindex; #else /* not TRILIBRARY */ FILE *outfile; #endif /* not TRILIBRARY */ struct triedge triangleloop; point p1, p2, p3; point mid1, mid2, mid3; int elementnumber; int i; #ifdef TRILIBRARY if ( !quiet ) { printf( "Writing triangles.\n" ); } /* Allocate memory for output triangles if necessary. */ if ( *trianglelist == (int *) NULL ) { *trianglelist = (int *) malloc( triangles.items * ( ( order + 1 ) * ( order + 2 ) / 2 ) * sizeof( int ) ); if ( *trianglelist == (int *) NULL ) { printf( "Error: Out of memory.\n" ); exit( 1 ); } } /* Allocate memory for output triangle attributes if necessary. */ if ( ( eextras > 0 ) && ( *triangleattriblist == (REAL *) NULL ) ) { *triangleattriblist = (REAL *) malloc( triangles.items * eextras * sizeof( REAL ) ); if ( *triangleattriblist == (REAL *) NULL ) { printf( "Error: Out of memory.\n" ); exit( 1 ); } } tlist = *trianglelist; talist = *triangleattriblist; pointindex = 0; attribindex = 0; #else /* not TRILIBRARY */ if ( !quiet ) { printf( "Writing %s.\n", elefilename ); } outfile = fopen( elefilename, "w" ); if ( outfile == (FILE *) NULL ) { printf( " Error: Cannot create file %s.\n", elefilename ); exit( 1 ); } /* Number of triangles, points per triangle, attributes per triangle. */ fprintf( outfile, "%ld %d %d\n", triangles.items, ( order + 1 ) * ( order + 2 ) / 2, eextras ); #endif /* not TRILIBRARY */ traversalinit( &triangles ); triangleloop.tri = triangletraverse(); triangleloop.orient = 0; elementnumber = firstnumber; while ( triangleloop.tri != (triangle *) NULL ) { org( triangleloop, p1 ); dest( triangleloop, p2 ); apex( triangleloop, p3 ); if ( order == 1 ) { #ifdef TRILIBRARY tlist[pointindex++] = pointmark( p1 ); tlist[pointindex++] = pointmark( p2 ); tlist[pointindex++] = pointmark( p3 ); #else /* not TRILIBRARY */ /* Triangle number, indices for three points. */ fprintf( outfile, "%4d %4d %4d %4d", elementnumber, pointmark( p1 ), pointmark( p2 ), pointmark( p3 ) ); #endif /* not TRILIBRARY */ } else { mid1 = (point) triangleloop.tri[highorderindex + 1]; mid2 = (point) triangleloop.tri[highorderindex + 2]; mid3 = (point) triangleloop.tri[highorderindex]; #ifdef TRILIBRARY tlist[pointindex++] = pointmark( p1 ); tlist[pointindex++] = pointmark( p2 ); tlist[pointindex++] = pointmark( p3 ); tlist[pointindex++] = pointmark( mid1 ); tlist[pointindex++] = pointmark( mid2 ); tlist[pointindex++] = pointmark( mid3 ); #else /* not TRILIBRARY */ /* Triangle number, indices for six points. */ fprintf( outfile, "%4d %4d %4d %4d %4d %4d %4d", elementnumber, pointmark( p1 ), pointmark( p2 ), pointmark( p3 ), pointmark( mid1 ), pointmark( mid2 ), pointmark( mid3 ) ); #endif /* not TRILIBRARY */ } #ifdef TRILIBRARY for ( i = 0; i < eextras; i++ ) { talist[attribindex++] = elemattribute( triangleloop, i ); } #else /* not TRILIBRARY */ for ( i = 0; i < eextras; i++ ) { fprintf( outfile, " %.17g", elemattribute( triangleloop, i ) ); } fprintf( outfile, "\n" ); #endif /* not TRILIBRARY */ triangleloop.tri = triangletraverse(); elementnumber++; } #ifndef TRILIBRARY finishfile( outfile, argc, argv ); #endif /* not TRILIBRARY */ } /*****************************************************************************/ /* */ /* writepoly() Write the segments and holes to a .poly file. */ /* */ /*****************************************************************************/ #ifdef TRILIBRARY void writepoly( segmentlist, segmentmarkerlist ) int **segmentlist; int **segmentmarkerlist; #else /* not TRILIBRARY */ void writepoly( polyfilename, holelist, holes, regionlist, regions, argc, argv ) char *polyfilename; REAL *holelist; int holes; REAL *regionlist; int regions; int argc; char **argv; #endif /* not TRILIBRARY */ { #ifdef TRILIBRARY int *slist; int *smlist; int index; #else /* not TRILIBRARY */ FILE *outfile; int i; #endif /* not TRILIBRARY */ struct edge shelleloop; point endpoint1, endpoint2; int shellenumber; #ifdef TRILIBRARY if ( !quiet ) { printf( "Writing segments.\n" ); } /* Allocate memory for output segments if necessary. */ if ( *segmentlist == (int *) NULL ) { *segmentlist = (int *) malloc( shelles.items * 2 * sizeof( int ) ); if ( *segmentlist == (int *) NULL ) { printf( "Error: Out of memory.\n" ); exit( 1 ); } } /* Allocate memory for output segment markers if necessary. */ if ( !nobound && ( *segmentmarkerlist == (int *) NULL ) ) { *segmentmarkerlist = (int *) malloc( shelles.items * sizeof( int ) ); if ( *segmentmarkerlist == (int *) NULL ) { printf( "Error: Out of memory.\n" ); exit( 1 ); } } slist = *segmentlist; smlist = *segmentmarkerlist; index = 0; #else /* not TRILIBRARY */ if ( !quiet ) { printf( "Writing %s.\n", polyfilename ); } outfile = fopen( polyfilename, "w" ); if ( outfile == (FILE *) NULL ) { printf( " Error: Cannot create file %s.\n", polyfilename ); exit( 1 ); } /* The zero indicates that the points are in a separate .node file. */ /* Followed by number of dimensions, number of point attributes, */ /* and number of boundary markers (zero or one). */ fprintf( outfile, "%d %d %d %d\n", 0, mesh_dim, nextras, 1 - nobound ); /* Number of segments, number of boundary markers (zero or one). */ fprintf( outfile, "%ld %d\n", shelles.items, 1 - nobound ); #endif /* not TRILIBRARY */ traversalinit( &shelles ); shelleloop.sh = shelletraverse(); shelleloop.shorient = 0; shellenumber = firstnumber; while ( shelleloop.sh != (shelle *) NULL ) { sorg( shelleloop, endpoint1 ); sdest( shelleloop, endpoint2 ); #ifdef TRILIBRARY /* Copy indices of the segment's two endpoints. */ slist[index++] = pointmark( endpoint1 ); slist[index++] = pointmark( endpoint2 ); if ( !nobound ) { /* Copy the boundary marker. */ smlist[shellenumber - firstnumber] = mark( shelleloop ); } #else /* not TRILIBRARY */ /* Segment number, indices of its two endpoints, and possibly a marker. */ if ( nobound ) { fprintf( outfile, "%4d %4d %4d\n", shellenumber, pointmark( endpoint1 ), pointmark( endpoint2 ) ); } else { fprintf( outfile, "%4d %4d %4d %4d\n", shellenumber, pointmark( endpoint1 ), pointmark( endpoint2 ), mark( shelleloop ) ); } #endif /* not TRILIBRARY */ shelleloop.sh = shelletraverse(); shellenumber++; } #ifndef TRILIBRARY #ifndef CDT_ONLY fprintf( outfile, "%d\n", holes ); if ( holes > 0 ) { for ( i = 0; i < holes; i++ ) { /* Hole number, x and y coordinates. */ fprintf( outfile, "%4d %.17g %.17g\n", firstnumber + i, holelist[2 * i], holelist[2 * i + 1] ); } } if ( regions > 0 ) { fprintf( outfile, "%d\n", regions ); for ( i = 0; i < regions; i++ ) { /* Region number, x and y coordinates, attribute, maximum area. */ fprintf( outfile, "%4d %.17g %.17g %.17g %.17g\n", firstnumber + i, regionlist[4 * i], regionlist[4 * i + 1], regionlist[4 * i + 2], regionlist[4 * i + 3] ); } } #endif /* not CDT_ONLY */ finishfile( outfile, argc, argv ); #endif /* not TRILIBRARY */ } /*****************************************************************************/ /* */ /* writeedges() Write the edges to a .edge file. */ /* */ /*****************************************************************************/ #ifdef TRILIBRARY void writeedges( edgelist, edgemarkerlist ) int **edgelist; int **edgemarkerlist; #else /* not TRILIBRARY */ void writeedges( edgefilename, argc, argv ) char *edgefilename; int argc; char **argv; #endif /* not TRILIBRARY */ { #ifdef TRILIBRARY int *elist; int *emlist; int index; #else /* not TRILIBRARY */ FILE *outfile; #endif /* not TRILIBRARY */ struct triedge triangleloop, trisym; struct edge checkmark; point p1, p2; int edgenumber; triangle ptr; /* Temporary variable used by sym(). */ shelle sptr; /* Temporary variable used by tspivot(). */ #ifdef TRILIBRARY if ( !quiet ) { printf( "Writing edges.\n" ); } /* Allocate memory for edges if necessary. */ if ( *edgelist == (int *) NULL ) { *edgelist = (int *) malloc( edges * 2 * sizeof( int ) ); if ( *edgelist == (int *) NULL ) { printf( "Error: Out of memory.\n" ); exit( 1 ); } } /* Allocate memory for edge markers if necessary. */ if ( !nobound && ( *edgemarkerlist == (int *) NULL ) ) { *edgemarkerlist = (int *) malloc( edges * sizeof( int ) ); if ( *edgemarkerlist == (int *) NULL ) { printf( "Error: Out of memory.\n" ); exit( 1 ); } } elist = *edgelist; emlist = *edgemarkerlist; index = 0; #else /* not TRILIBRARY */ if ( !quiet ) { printf( "Writing %s.\n", edgefilename ); } outfile = fopen( edgefilename, "w" ); if ( outfile == (FILE *) NULL ) { printf( " Error: Cannot create file %s.\n", edgefilename ); exit( 1 ); } /* Number of edges, number of boundary markers (zero or one). */ fprintf( outfile, "%ld %d\n", edges, 1 - nobound ); #endif /* not TRILIBRARY */ traversalinit( &triangles ); triangleloop.tri = triangletraverse(); edgenumber = firstnumber; /* To loop over the set of edges, loop over all triangles, and look at */ /* the three edges of each triangle. If there isn't another triangle */ /* adjacent to the edge, operate on the edge. If there is another */ /* adjacent triangle, operate on the edge only if the current triangle */ /* has a smaller pointer than its neighbor. This way, each edge is */ /* considered only once. */ while ( triangleloop.tri != (triangle *) NULL ) { for ( triangleloop.orient = 0; triangleloop.orient < 3; triangleloop.orient++ ) { sym( triangleloop, trisym ); if ( ( triangleloop.tri < trisym.tri ) || ( trisym.tri == dummytri ) ) { org( triangleloop, p1 ); dest( triangleloop, p2 ); #ifdef TRILIBRARY elist[index++] = pointmark( p1 ); elist[index++] = pointmark( p2 ); #endif /* TRILIBRARY */ if ( nobound ) { #ifndef TRILIBRARY /* Edge number, indices of two endpoints. */ fprintf( outfile, "%4d %d %d\n", edgenumber, pointmark( p1 ), pointmark( p2 ) ); #endif /* not TRILIBRARY */ } else { /* Edge number, indices of two endpoints, and a boundary marker. */ /* If there's no shell edge, the boundary marker is zero. */ if ( useshelles ) { tspivot( triangleloop, checkmark ); if ( checkmark.sh == dummysh ) { #ifdef TRILIBRARY emlist[edgenumber - firstnumber] = 0; #else /* not TRILIBRARY */ fprintf( outfile, "%4d %d %d %d\n", edgenumber, pointmark( p1 ), pointmark( p2 ), 0 ); #endif /* not TRILIBRARY */ } else { #ifdef TRILIBRARY emlist[edgenumber - firstnumber] = mark( checkmark ); #else /* not TRILIBRARY */ fprintf( outfile, "%4d %d %d %d\n", edgenumber, pointmark( p1 ), pointmark( p2 ), mark( checkmark ) ); #endif /* not TRILIBRARY */ } } else { #ifdef TRILIBRARY emlist[edgenumber - firstnumber] = trisym.tri == dummytri; #else /* not TRILIBRARY */ fprintf( outfile, "%4d %d %d %d\n", edgenumber, pointmark( p1 ), pointmark( p2 ), trisym.tri == dummytri ); #endif /* not TRILIBRARY */ } } edgenumber++; } } triangleloop.tri = triangletraverse(); } #ifndef TRILIBRARY finishfile( outfile, argc, argv ); #endif /* not TRILIBRARY */ } /*****************************************************************************/ /* */ /* writevoronoi() Write the Voronoi diagram to a .v.node and .v.edge */ /* file. */ /* */ /* The Voronoi diagram is the geometric dual of the Delaunay triangulation. */ /* Hence, the Voronoi vertices are listed by traversing the Delaunay */ /* triangles, and the Voronoi edges are listed by traversing the Delaunay */ /* edges. */ /* */ /* WARNING: In order to assign numbers to the Voronoi vertices, this */ /* procedure messes up the shell edges or the extra nodes of every */ /* element. Hence, you should call this procedure last. */ /* */ /*****************************************************************************/ #ifdef TRILIBRARY void writevoronoi( vpointlist, vpointattriblist, vpointmarkerlist, vedgelist, vedgemarkerlist, vnormlist ) REAL * *vpointlist; REAL **vpointattriblist; int **vpointmarkerlist; int **vedgelist; int **vedgemarkerlist; REAL **vnormlist; #else /* not TRILIBRARY */ void writevoronoi( vnodefilename, vedgefilename, argc, argv ) char *vnodefilename; char *vedgefilename; int argc; char **argv; #endif /* not TRILIBRARY */ { #ifdef TRILIBRARY REAL *plist; REAL *palist; int *elist; REAL *normlist; int coordindex; int attribindex; #else /* not TRILIBRARY */ FILE *outfile; #endif /* not TRILIBRARY */ struct triedge triangleloop, trisym; point torg, tdest, tapex; REAL circumcenter[2]; REAL xi, eta; int vnodenumber, vedgenumber; int p1, p2; int i; triangle ptr; /* Temporary variable used by sym(). */ #ifdef TRILIBRARY if ( !quiet ) { printf( "Writing Voronoi vertices.\n" ); } /* Allocate memory for Voronoi vertices if necessary. */ if ( *vpointlist == (REAL *) NULL ) { *vpointlist = (REAL *) malloc( triangles.items * 2 * sizeof( REAL ) ); if ( *vpointlist == (REAL *) NULL ) { printf( "Error: Out of memory.\n" ); exit( 1 ); } } /* Allocate memory for Voronoi vertex attributes if necessary. */ if ( *vpointattriblist == (REAL *) NULL ) { *vpointattriblist = (REAL *) malloc( triangles.items * nextras * sizeof( REAL ) ); if ( *vpointattriblist == (REAL *) NULL ) { printf( "Error: Out of memory.\n" ); exit( 1 ); } } *vpointmarkerlist = (int *) NULL; plist = *vpointlist; palist = *vpointattriblist; coordindex = 0; attribindex = 0; #else /* not TRILIBRARY */ if ( !quiet ) { printf( "Writing %s.\n", vnodefilename ); } outfile = fopen( vnodefilename, "w" ); if ( outfile == (FILE *) NULL ) { printf( " Error: Cannot create file %s.\n", vnodefilename ); exit( 1 ); } /* Number of triangles, two dimensions, number of point attributes, */ /* zero markers. */ fprintf( outfile, "%ld %d %d %d\n", triangles.items, 2, nextras, 0 ); #endif /* not TRILIBRARY */ traversalinit( &triangles ); triangleloop.tri = triangletraverse(); triangleloop.orient = 0; vnodenumber = firstnumber; while ( triangleloop.tri != (triangle *) NULL ) { org( triangleloop, torg ); dest( triangleloop, tdest ); apex( triangleloop, tapex ); findcircumcenter( torg, tdest, tapex, circumcenter, &xi, &eta ); #ifdef TRILIBRARY /* X and y coordinates. */ plist[coordindex++] = circumcenter[0]; plist[coordindex++] = circumcenter[1]; for ( i = 2; i < 2 + nextras; i++ ) { /* Interpolate the point attributes at the circumcenter. */ palist[attribindex++] = torg[i] + xi * ( tdest[i] - torg[i] ) + eta * ( tapex[i] - torg[i] ); } #else /* not TRILIBRARY */ /* Voronoi vertex number, x and y coordinates. */ fprintf( outfile, "%4d %.17g %.17g", vnodenumber, circumcenter[0], circumcenter[1] ); for ( i = 2; i < 2 + nextras; i++ ) { /* Interpolate the point attributes at the circumcenter. */ fprintf( outfile, " %.17g", torg[i] + xi * ( tdest[i] - torg[i] ) + eta * ( tapex[i] - torg[i] ) ); } fprintf( outfile, "\n" ); #endif /* not TRILIBRARY */ *(int *) ( triangleloop.tri + 6 ) = vnodenumber; triangleloop.tri = triangletraverse(); vnodenumber++; } #ifndef TRILIBRARY finishfile( outfile, argc, argv ); #endif /* not TRILIBRARY */ #ifdef TRILIBRARY if ( !quiet ) { printf( "Writing Voronoi edges.\n" ); } /* Allocate memory for output Voronoi edges if necessary. */ if ( *vedgelist == (int *) NULL ) { *vedgelist = (int *) malloc( edges * 2 * sizeof( int ) ); if ( *vedgelist == (int *) NULL ) { printf( "Error: Out of memory.\n" ); exit( 1 ); } } *vedgemarkerlist = (int *) NULL; /* Allocate memory for output Voronoi norms if necessary. */ if ( *vnormlist == (REAL *) NULL ) { *vnormlist = (REAL *) malloc( edges * 2 * sizeof( REAL ) ); if ( *vnormlist == (REAL *) NULL ) { printf( "Error: Out of memory.\n" ); exit( 1 ); } } elist = *vedgelist; normlist = *vnormlist; coordindex = 0; #else /* not TRILIBRARY */ if ( !quiet ) { printf( "Writing %s.\n", vedgefilename ); } outfile = fopen( vedgefilename, "w" ); if ( outfile == (FILE *) NULL ) { printf( " Error: Cannot create file %s.\n", vedgefilename ); exit( 1 ); } /* Number of edges, zero boundary markers. */ fprintf( outfile, "%ld %d\n", edges, 0 ); #endif /* not TRILIBRARY */ traversalinit( &triangles ); triangleloop.tri = triangletraverse(); vedgenumber = firstnumber; /* To loop over the set of edges, loop over all triangles, and look at */ /* the three edges of each triangle. If there isn't another triangle */ /* adjacent to the edge, operate on the edge. If there is another */ /* adjacent triangle, operate on the edge only if the current triangle */ /* has a smaller pointer than its neighbor. This way, each edge is */ /* considered only once. */ while ( triangleloop.tri != (triangle *) NULL ) { for ( triangleloop.orient = 0; triangleloop.orient < 3; triangleloop.orient++ ) { sym( triangleloop, trisym ); if ( ( triangleloop.tri < trisym.tri ) || ( trisym.tri == dummytri ) ) { /* Find the number of this triangle (and Voronoi vertex). */ p1 = *(int *) ( triangleloop.tri + 6 ); if ( trisym.tri == dummytri ) { org( triangleloop, torg ); dest( triangleloop, tdest ); #ifdef TRILIBRARY /* Copy an infinite ray. Index of one endpoint, and -1. */ elist[coordindex] = p1; normlist[coordindex++] = tdest[1] - torg[1]; elist[coordindex] = -1; normlist[coordindex++] = torg[0] - tdest[0]; #else /* not TRILIBRARY */ /* Write an infinite ray. Edge number, index of one endpoint, -1, */ /* and x and y coordinates of a vector representing the */ /* direction of the ray. */ fprintf( outfile, "%4d %d %d %.17g %.17g\n", vedgenumber, p1, -1, tdest[1] - torg[1], torg[0] - tdest[0] ); #endif /* not TRILIBRARY */ } else { /* Find the number of the adjacent triangle (and Voronoi vertex). */ p2 = *(int *) ( trisym.tri + 6 ); /* Finite edge. Write indices of two endpoints. */ #ifdef TRILIBRARY elist[coordindex] = p1; normlist[coordindex++] = 0.0; elist[coordindex] = p2; normlist[coordindex++] = 0.0; #else /* not TRILIBRARY */ fprintf( outfile, "%4d %d %d\n", vedgenumber, p1, p2 ); #endif /* not TRILIBRARY */ } vedgenumber++; } } triangleloop.tri = triangletraverse(); } #ifndef TRILIBRARY finishfile( outfile, argc, argv ); #endif /* not TRILIBRARY */ } #ifdef TRILIBRARY void writeneighbors( neighborlist ) int **neighborlist; #else /* not TRILIBRARY */ void writeneighbors( neighborfilename, argc, argv ) char *neighborfilename; int argc; char **argv; #endif /* not TRILIBRARY */ { #ifdef TRILIBRARY int *nlist; int index; #else /* not TRILIBRARY */ FILE *outfile; #endif /* not TRILIBRARY */ struct triedge triangleloop, trisym; int elementnumber; int neighbor1, neighbor2, neighbor3; triangle ptr; /* Temporary variable used by sym(). */ #ifdef TRILIBRARY if ( !quiet ) { printf( "Writing neighbors.\n" ); } /* Allocate memory for neighbors if necessary. */ if ( *neighborlist == (int *) NULL ) { *neighborlist = (int *) malloc( triangles.items * 3 * sizeof( int ) ); if ( *neighborlist == (int *) NULL ) { printf( "Error: Out of memory.\n" ); exit( 1 ); } } nlist = *neighborlist; index = 0; #else /* not TRILIBRARY */ if ( !quiet ) { printf( "Writing %s.\n", neighborfilename ); } outfile = fopen( neighborfilename, "w" ); if ( outfile == (FILE *) NULL ) { printf( " Error: Cannot create file %s.\n", neighborfilename ); exit( 1 ); } /* Number of triangles, three edges per triangle. */ fprintf( outfile, "%ld %d\n", triangles.items, 3 ); #endif /* not TRILIBRARY */ traversalinit( &triangles ); triangleloop.tri = triangletraverse(); triangleloop.orient = 0; elementnumber = firstnumber; while ( triangleloop.tri != (triangle *) NULL ) { *(int *) ( triangleloop.tri + 6 ) = elementnumber; triangleloop.tri = triangletraverse(); elementnumber++; } *(int *) ( dummytri + 6 ) = -1; traversalinit( &triangles ); triangleloop.tri = triangletraverse(); elementnumber = firstnumber; while ( triangleloop.tri != (triangle *) NULL ) { triangleloop.orient = 1; sym( triangleloop, trisym ); neighbor1 = *(int *) ( trisym.tri + 6 ); triangleloop.orient = 2; sym( triangleloop, trisym ); neighbor2 = *(int *) ( trisym.tri + 6 ); triangleloop.orient = 0; sym( triangleloop, trisym ); neighbor3 = *(int *) ( trisym.tri + 6 ); #ifdef TRILIBRARY nlist[index++] = neighbor1; nlist[index++] = neighbor2; nlist[index++] = neighbor3; #else /* not TRILIBRARY */ /* Triangle number, neighboring triangle numbers. */ fprintf( outfile, "%4d %d %d %d\n", elementnumber, neighbor1, neighbor2, neighbor3 ); #endif /* not TRILIBRARY */ triangleloop.tri = triangletraverse(); elementnumber++; } #ifndef TRILIBRARY finishfile( outfile, argc, argv ); #endif /* TRILIBRARY */ } /*****************************************************************************/ /* */ /* writeoff() Write the triangulation to an .off file. */ /* */ /* OFF stands for the Object File Format, a format used by the Geometry */ /* Center's Geomview package. */ /* */ /*****************************************************************************/ #ifndef TRILIBRARY void writeoff( offfilename, argc, argv ) char *offfilename; int argc; char **argv; { FILE *outfile; struct triedge triangleloop; point pointloop; point p1, p2, p3; if ( !quiet ) { printf( "Writing %s.\n", offfilename ); } outfile = fopen( offfilename, "w" ); if ( outfile == (FILE *) NULL ) { printf( " Error: Cannot create file %s.\n", offfilename ); exit( 1 ); } /* Number of points, triangles, and edges. */ fprintf( outfile, "OFF\n%ld %ld %ld\n", points.items, triangles.items, edges ); /* Write the points. */ traversalinit( &points ); pointloop = pointtraverse(); while ( pointloop != (point) NULL ) { /* The "0.0" is here because the OFF format uses 3D coordinates. */ fprintf( outfile, " %.17g %.17g %.17g\n", pointloop[0], pointloop[1], 0.0 ); pointloop = pointtraverse(); } /* Write the triangles. */ traversalinit( &triangles ); triangleloop.tri = triangletraverse(); triangleloop.orient = 0; while ( triangleloop.tri != (triangle *) NULL ) { org( triangleloop, p1 ); dest( triangleloop, p2 ); apex( triangleloop, p3 ); /* The "3" means a three-vertex polygon. */ fprintf( outfile, " 3 %4d %4d %4d\n", pointmark( p1 ) - 1, pointmark( p2 ) - 1, pointmark( p3 ) - 1 ); triangleloop.tri = triangletraverse(); } finishfile( outfile, argc, argv ); } #endif /* not TRILIBRARY */ /** **/ /** **/ /********* File I/O routines end here *********/ /*****************************************************************************/ /* */ /* quality_statistics() Print statistics about the quality of the mesh. */ /* */ /*****************************************************************************/ void quality_statistics(){ struct triedge triangleloop; point p[3]; REAL cossquaretable[8]; REAL ratiotable[16]; REAL dx[3], dy[3]; REAL edgelength[3]; REAL dotproduct; REAL cossquare; REAL triarea; REAL shortest, longest; REAL trilongest2; REAL smallestarea, biggestarea; REAL triminaltitude2; REAL minaltitude; REAL triaspect2; REAL worstaspect; REAL smallestangle, biggestangle; REAL radconst, degconst; int angletable[18]; int aspecttable[16]; int aspectindex; int tendegree; int acutebiggest; int i, ii, j, k; printf( "Mesh quality statistics:\n\n" ); radconst = (REAL)( PI / 18.0 ); degconst = (REAL)( 180.0 / PI ); for ( i = 0; i < 8; i++ ) { cossquaretable[i] = (REAL)( cos( radconst * (REAL) ( i + 1 ) ) ); cossquaretable[i] = cossquaretable[i] * cossquaretable[i]; } for ( i = 0; i < 18; i++ ) { angletable[i] = 0; } ratiotable[0] = 1.5; ratiotable[1] = 2.0; ratiotable[2] = 2.5; ratiotable[3] = 3.0; ratiotable[4] = 4.0; ratiotable[5] = 6.0; ratiotable[6] = 10.0; ratiotable[7] = 15.0; ratiotable[8] = 25.0; ratiotable[9] = 50.0; ratiotable[10] = 100.0; ratiotable[11] = 300.0; ratiotable[12] = 1000.0; ratiotable[13] = 10000.0; ratiotable[14] = 100000.0; ratiotable[15] = 0.0; for ( i = 0; i < 16; i++ ) { aspecttable[i] = 0; } worstaspect = 0.0; minaltitude = xmax - xmin + ymax - ymin; minaltitude = minaltitude * minaltitude; shortest = minaltitude; longest = 0.0; smallestarea = minaltitude; biggestarea = 0.0; worstaspect = 0.0; smallestangle = 0.0; biggestangle = 2.0; acutebiggest = 1; traversalinit( &triangles ); triangleloop.tri = triangletraverse(); triangleloop.orient = 0; while ( triangleloop.tri != (triangle *) NULL ) { org( triangleloop, p[0] ); dest( triangleloop, p[1] ); apex( triangleloop, p[2] ); trilongest2 = 0.0; for ( i = 0; i < 3; i++ ) { j = plus1mod3[i]; k = minus1mod3[i]; dx[i] = p[j][0] - p[k][0]; dy[i] = p[j][1] - p[k][1]; edgelength[i] = dx[i] * dx[i] + dy[i] * dy[i]; if ( edgelength[i] > trilongest2 ) { trilongest2 = edgelength[i]; } if ( edgelength[i] > longest ) { longest = edgelength[i]; } if ( edgelength[i] < shortest ) { shortest = edgelength[i]; } } triarea = counterclockwise( p[0], p[1], p[2] ); if ( triarea < smallestarea ) { smallestarea = triarea; } if ( triarea > biggestarea ) { biggestarea = triarea; } triminaltitude2 = triarea * triarea / trilongest2; if ( triminaltitude2 < minaltitude ) { minaltitude = triminaltitude2; } triaspect2 = trilongest2 / triminaltitude2; if ( triaspect2 > worstaspect ) { worstaspect = triaspect2; } aspectindex = 0; while ( ( triaspect2 > ratiotable[aspectindex] * ratiotable[aspectindex] ) && ( aspectindex < 15 ) ) { aspectindex++; } aspecttable[aspectindex]++; for ( i = 0; i < 3; i++ ) { j = plus1mod3[i]; k = minus1mod3[i]; dotproduct = dx[j] * dx[k] + dy[j] * dy[k]; cossquare = dotproduct * dotproduct / ( edgelength[j] * edgelength[k] ); tendegree = 8; for ( ii = 7; ii >= 0; ii-- ) { if ( cossquare > cossquaretable[ii] ) { tendegree = ii; } } if ( dotproduct <= 0.0 ) { angletable[tendegree]++; if ( cossquare > smallestangle ) { smallestangle = cossquare; } if ( acutebiggest && ( cossquare < biggestangle ) ) { biggestangle = cossquare; } } else { angletable[17 - tendegree]++; if ( acutebiggest || ( cossquare > biggestangle ) ) { biggestangle = cossquare; acutebiggest = 0; } } } triangleloop.tri = triangletraverse(); } shortest = (REAL)sqrt( shortest ); longest = (REAL)sqrt( longest ); minaltitude = (REAL)sqrt( minaltitude ); worstaspect = (REAL)sqrt( worstaspect ); smallestarea *= 2.0; biggestarea *= 2.0; if ( smallestangle >= 1.0 ) { smallestangle = 0.0; } else { smallestangle = (REAL)( degconst * acos( sqrt( smallestangle ) ) ); } if ( biggestangle >= 1.0 ) { biggestangle = 180.0; } else { if ( acutebiggest ) { biggestangle = (REAL)( degconst * acos( sqrt( biggestangle ) ) ); } else { biggestangle = (REAL)( 180.0 - degconst * acos( sqrt( biggestangle ) ) ); } } printf( " Smallest area: %16.5g | Largest area: %16.5g\n", smallestarea, biggestarea ); printf( " Shortest edge: %16.5g | Longest edge: %16.5g\n", shortest, longest ); printf( " Shortest altitude: %12.5g | Largest aspect ratio: %8.5g\n\n", minaltitude, worstaspect ); printf( " Aspect ratio histogram:\n" ); printf( " 1.1547 - %-6.6g : %8d | %6.6g - %-6.6g : %8d\n", ratiotable[0], aspecttable[0], ratiotable[7], ratiotable[8], aspecttable[8] ); for ( i = 1; i < 7; i++ ) { printf( " %6.6g - %-6.6g : %8d | %6.6g - %-6.6g : %8d\n", ratiotable[i - 1], ratiotable[i], aspecttable[i], ratiotable[i + 7], ratiotable[i + 8], aspecttable[i + 8] ); } printf( " %6.6g - %-6.6g : %8d | %6.6g - : %8d\n", ratiotable[6], ratiotable[7], aspecttable[7], ratiotable[14], aspecttable[15] ); printf( " (Triangle aspect ratio is longest edge divided by shortest altitude)\n\n" ); printf( " Smallest angle: %15.5g | Largest angle: %15.5g\n\n", smallestangle, biggestangle ); printf( " Angle histogram:\n" ); for ( i = 0; i < 9; i++ ) { printf( " %3d - %3d degrees: %8d | %3d - %3d degrees: %8d\n", i * 10, i * 10 + 10, angletable[i], i * 10 + 90, i * 10 + 100, angletable[i + 9] ); } printf( "\n" ); } /*****************************************************************************/ /* */ /* statistics() Print all sorts of cool facts. */ /* */ /*****************************************************************************/ void statistics(){ printf( "\nStatistics:\n\n" ); printf( " Input points: %d\n", inpoints ); if ( refine ) { printf( " Input triangles: %d\n", inelements ); } if ( poly ) { printf( " Input segments: %d\n", insegments ); if ( !refine ) { printf( " Input holes: %d\n", holes ); } } printf( "\n Mesh points: %ld\n", points.items ); printf( " Mesh triangles: %ld\n", triangles.items ); printf( " Mesh edges: %ld\n", edges ); if ( poly || refine ) { printf( " Mesh boundary edges: %ld\n", hullsize ); printf( " Mesh segments: %ld\n\n", shelles.items ); } else { printf( " Mesh convex hull edges: %ld\n\n", hullsize ); } if ( verbose ) { quality_statistics(); printf( "Memory allocation statistics:\n\n" ); printf( " Maximum number of points: %ld\n", points.maxitems ); printf( " Maximum number of triangles: %ld\n", triangles.maxitems ); if ( shelles.maxitems > 0 ) { printf( " Maximum number of segments: %ld\n", shelles.maxitems ); } if ( viri.maxitems > 0 ) { printf( " Maximum number of viri: %ld\n", viri.maxitems ); } if ( badsegments.maxitems > 0 ) { printf( " Maximum number of encroached segments: %ld\n", badsegments.maxitems ); } if ( badtriangles.maxitems > 0 ) { printf( " Maximum number of bad triangles: %ld\n", badtriangles.maxitems ); } if ( splaynodes.maxitems > 0 ) { printf( " Maximum number of splay tree nodes: %ld\n", splaynodes.maxitems ); } printf( " Approximate heap memory use (bytes): %ld\n\n", points.maxitems * points.itembytes + triangles.maxitems * triangles.itembytes + shelles.maxitems * shelles.itembytes + viri.maxitems * viri.itembytes + badsegments.maxitems * badsegments.itembytes + badtriangles.maxitems * badtriangles.itembytes + splaynodes.maxitems * splaynodes.itembytes ); printf( "Algorithmic statistics:\n\n" ); printf( " Number of incircle tests: %ld\n", incirclecount ); printf( " Number of orientation tests: %ld\n", counterclockcount ); if ( hyperbolacount > 0 ) { printf( " Number of right-of-hyperbola tests: %ld\n", hyperbolacount ); } if ( circumcentercount > 0 ) { printf( " Number of circumcenter computations: %ld\n", circumcentercount ); } if ( circletopcount > 0 ) { printf( " Number of circle top computations: %ld\n", circletopcount ); } printf( "\n" ); } } /*****************************************************************************/ /* */ /* main() or triangulate() Gosh, do everything. */ /* */ /* The sequence is roughly as follows. Many of these steps can be skipped, */ /* depending on the command line switches. */ /* */ /* - Initialize constants and parse the command line. */ /* - Read the points from a file and either */ /* - triangulate them (no -r), or */ /* - read an old mesh from files and reconstruct it (-r). */ /* - Insert the PSLG segments (-p), and possibly segments on the convex */ /* hull (-c). */ /* - Read the holes (-p), regional attributes (-pA), and regional area */ /* constraints (-pa). Carve the holes and concavities, and spread the */ /* regional attributes and area constraints. */ /* - Enforce the constraints on minimum angle (-q) and maximum area (-a). */ /* Also enforce the conforming Delaunay property (-q and -a). */ /* - Compute the number of edges in the resulting mesh. */ /* - Promote the mesh's linear triangles to higher order elements (-o). */ /* - Write the output files and print the statistics. */ /* - Check the consistency and Delaunay property of the mesh (-C). */ /* */ /*****************************************************************************/ #ifdef TRILIBRARY void triangulate( triswitches, in, out, vorout ) char *triswitches; struct triangulateio *in; struct triangulateio *out; struct triangulateio *vorout; #else /* not TRILIBRARY */ int main( argc, argv ) int argc; char **argv; #endif /* not TRILIBRARY */ { REAL *holearray; /* Array of holes. */ REAL *regionarray; /* Array of regional attributes and area constraints. */ #ifndef TRILIBRARY FILE *polyfile; #endif /* not TRILIBRARY */ #ifndef NO_TIMER /* Variables for timing the performance of Triangle. The types are */ /* defined in sys/time.h. */ struct timeval tv0, tv1, tv2, tv3, tv4, tv5, tv6; struct timezone tz; #endif /* NO_TIMER */ #ifndef NO_TIMER gettimeofday( &tv0, &tz ); #endif /* NO_TIMER */ triangleinit(); #ifdef TRILIBRARY parsecommandline( 1, &triswitches ); #else /* not TRILIBRARY */ parsecommandline( argc, argv ); #endif /* not TRILIBRARY */ #ifdef TRILIBRARY transfernodes( in->pointlist, in->pointattributelist, in->pointmarkerlist, in->numberofpoints, in->numberofpointattributes ); #else /* not TRILIBRARY */ readnodes( innodefilename, inpolyfilename, &polyfile ); #endif /* not TRILIBRARY */ #ifndef NO_TIMER if ( !quiet ) { gettimeofday( &tv1, &tz ); } #endif /* NO_TIMER */ #ifdef CDT_ONLY hullsize = delaunay(); /* Triangulate the points. */ #else /* not CDT_ONLY */ if ( refine ) { /* Read and reconstruct a mesh. */ #ifdef TRILIBRARY hullsize = reconstruct( in->trianglelist, in->triangleattributelist, in->trianglearealist, in->numberoftriangles, in->numberofcorners, in->numberoftriangleattributes, in->segmentlist, in->segmentmarkerlist, in->numberofsegments ); #else /* not TRILIBRARY */ hullsize = reconstruct( inelefilename, areafilename, inpolyfilename, polyfile ); #endif /* not TRILIBRARY */ } else { hullsize = delaunay(); /* Triangulate the points. */ } #endif /* not CDT_ONLY */ #ifndef NO_TIMER if ( !quiet ) { gettimeofday( &tv2, &tz ); if ( refine ) { printf( "Mesh reconstruction" ); } else { printf( "Delaunay" ); } printf( " milliseconds: %ld\n", 1000l * ( tv2.tv_sec - tv1.tv_sec ) + ( tv2.tv_usec - tv1.tv_usec ) / 1000l ); } #endif /* NO_TIMER */ /* Ensure that no point can be mistaken for a triangular bounding */ /* box point in insertsite(). */ infpoint1 = (point) NULL; infpoint2 = (point) NULL; infpoint3 = (point) NULL; if ( useshelles ) { checksegments = 1; /* Segments will be introduced next. */ if ( !refine ) { /* Insert PSLG segments and/or convex hull segments. */ #ifdef TRILIBRARY insegments = formskeleton( in->segmentlist, in->segmentmarkerlist, in->numberofsegments ); #else /* not TRILIBRARY */ insegments = formskeleton( polyfile, inpolyfilename ); #endif /* not TRILIBRARY */ } } #ifndef NO_TIMER if ( !quiet ) { gettimeofday( &tv3, &tz ); if ( useshelles && !refine ) { printf( "Segment milliseconds: %ld\n", 1000l * ( tv3.tv_sec - tv2.tv_sec ) + ( tv3.tv_usec - tv2.tv_usec ) / 1000l ); } } #endif /* NO_TIMER */ if ( poly ) { #ifdef TRILIBRARY holearray = in->holelist; holes = in->numberofholes; regionarray = in->regionlist; regions = in->numberofregions; #else /* not TRILIBRARY */ readholes( polyfile, inpolyfilename, &holearray, &holes, ®ionarray, ®ions ); #endif /* not TRILIBRARY */ if ( !refine ) { /* Carve out holes and concavities. */ carveholes( holearray, holes, regionarray, regions ); } } else { /* Without a PSLG, there can be no holes or regional attributes */ /* or area constraints. The following are set to zero to avoid */ /* an accidental free() later. */ holes = 0; regions = 0; } #ifndef NO_TIMER if ( !quiet ) { gettimeofday( &tv4, &tz ); if ( poly && !refine ) { printf( "Hole milliseconds: %ld\n", 1000l * ( tv4.tv_sec - tv3.tv_sec ) + ( tv4.tv_usec - tv3.tv_usec ) / 1000l ); } } #endif /* NO_TIMER */ #ifndef CDT_ONLY if ( quality ) { enforcequality(); /* Enforce angle and area constraints. */ } #endif /* not CDT_ONLY */ #ifndef NO_TIMER if ( !quiet ) { gettimeofday( &tv5, &tz ); #ifndef CDT_ONLY if ( quality ) { printf( "Quality milliseconds: %ld\n", 1000l * ( tv5.tv_sec - tv4.tv_sec ) + ( tv5.tv_usec - tv4.tv_usec ) / 1000l ); } #endif /* not CDT_ONLY */ } #endif /* NO_TIMER */ /* Compute the number of edges. */ edges = ( 3l * triangles.items + hullsize ) / 2l; if ( order > 1 ) { highorder(); /* Promote elements to higher polynomial order. */ } if ( !quiet ) { printf( "\n" ); } #ifdef TRILIBRARY out->numberofpoints = points.items; out->numberofpointattributes = nextras; out->numberoftriangles = triangles.items; out->numberofcorners = ( order + 1 ) * ( order + 2 ) / 2; out->numberoftriangleattributes = eextras; out->numberofedges = edges; if ( useshelles ) { out->numberofsegments = shelles.items; } else { out->numberofsegments = hullsize; } if ( vorout != (struct triangulateio *) NULL ) { vorout->numberofpoints = triangles.items; vorout->numberofpointattributes = nextras; vorout->numberofedges = edges; } #endif /* TRILIBRARY */ /* If not using iteration numbers, don't write a .node file if one was */ /* read, because the original one would be overwritten! */ if ( nonodewritten || ( noiterationnum && readnodefile ) ) { if ( !quiet ) { #ifdef TRILIBRARY printf( "NOT writing points.\n" ); #else /* not TRILIBRARY */ printf( "NOT writing a .node file.\n" ); #endif /* not TRILIBRARY */ } numbernodes(); /* We must remember to number the points. */ } else { #ifdef TRILIBRARY writenodes( &out->pointlist, &out->pointattributelist, &out->pointmarkerlist ); #else /* not TRILIBRARY */ writenodes( outnodefilename, argc, argv ); /* Numbers the points too. */ #endif /* TRILIBRARY */ } if ( noelewritten ) { if ( !quiet ) { #ifdef TRILIBRARY printf( "NOT writing triangles.\n" ); #else /* not TRILIBRARY */ printf( "NOT writing an .ele file.\n" ); #endif /* not TRILIBRARY */ } } else { #ifdef TRILIBRARY writeelements( &out->trianglelist, &out->triangleattributelist ); #else /* not TRILIBRARY */ writeelements( outelefilename, argc, argv ); #endif /* not TRILIBRARY */ } /* The -c switch (convex switch) causes a PSLG to be written */ /* even if none was read. */ if ( poly || convex ) { /* If not using iteration numbers, don't overwrite the .poly file. */ if ( nopolywritten || noiterationnum ) { if ( !quiet ) { #ifdef TRILIBRARY printf( "NOT writing segments.\n" ); #else /* not TRILIBRARY */ printf( "NOT writing a .poly file.\n" ); #endif /* not TRILIBRARY */ } } else { #ifdef TRILIBRARY writepoly( &out->segmentlist, &out->segmentmarkerlist ); out->numberofholes = holes; out->numberofregions = regions; if ( poly ) { out->holelist = in->holelist; out->regionlist = in->regionlist; } else { out->holelist = (REAL *) NULL; out->regionlist = (REAL *) NULL; } #else /* not TRILIBRARY */ writepoly( outpolyfilename, holearray, holes, regionarray, regions, argc, argv ); #endif /* not TRILIBRARY */ } } #ifndef TRILIBRARY #ifndef CDT_ONLY if ( regions > 0 ) { free( regionarray ); } #endif /* not CDT_ONLY */ if ( holes > 0 ) { free( holearray ); } if ( geomview ) { writeoff( offfilename, argc, argv ); } #endif /* not TRILIBRARY */ if ( edgesout ) { #ifdef TRILIBRARY writeedges( &out->edgelist, &out->edgemarkerlist ); #else /* not TRILIBRARY */ writeedges( edgefilename, argc, argv ); #endif /* not TRILIBRARY */ } if ( voronoi ) { #ifdef TRILIBRARY writevoronoi( &vorout->pointlist, &vorout->pointattributelist, &vorout->pointmarkerlist, &vorout->edgelist, &vorout->edgemarkerlist, &vorout->normlist ); #else /* not TRILIBRARY */ writevoronoi( vnodefilename, vedgefilename, argc, argv ); #endif /* not TRILIBRARY */ } if ( neighbors ) { #ifdef TRILIBRARY writeneighbors( &out->neighborlist ); #else /* not TRILIBRARY */ writeneighbors( neighborfilename, argc, argv ); #endif /* not TRILIBRARY */ } if ( !quiet ) { #ifndef NO_TIMER gettimeofday( &tv6, &tz ); printf( "\nOutput milliseconds: %ld\n", 1000l * ( tv6.tv_sec - tv5.tv_sec ) + ( tv6.tv_usec - tv5.tv_usec ) / 1000l ); printf( "Total running milliseconds: %ld\n", 1000l * ( tv6.tv_sec - tv0.tv_sec ) + ( tv6.tv_usec - tv0.tv_usec ) / 1000l ); #endif /* NO_TIMER */ statistics(); } #ifndef REDUCED if ( docheck ) { checkmesh(); checkdelaunay(); } #endif /* not REDUCED */ triangledeinit(); #ifndef TRILIBRARY return 0; #endif /* not TRILIBRARY */ }