/*****************************************************************************/ /* */ /* 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 1.6 */ /* July 28, 2005 */ /* */ /* Copyright 1993, 1995, 1997, 1998, 2002, 2005 */ /* Jonathan Richard Shewchuk */ /* 2360 Woolsey #H */ /* Berkeley, California 94705-1927 */ /* jrs@cs.berkeley.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 */ /* */ /* Disclaimer: Neither I nor Carnegie Mellon warrant this code in any way */ /* whatsoever. This code is provided "as-is". Use at your own risk. */ /* */ /* Some of the references listed below are marked with an asterisk. [*] */ /* These references are available for downloading from the Web page */ /* */ /* http://www.cs.cmu.edu/~quake/triangle.research.html */ /* */ /* Three papers discussing aspects of Triangle are available. A short */ /* overview appears in "Triangle: Engineering a 2D Quality Mesh */ /* Generator and Delaunay Triangulator," in Applied Computational */ /* Geometry: Towards Geometric Engineering, Ming C. Lin and Dinesh */ /* Manocha, editors, Lecture Notes in Computer Science volume 1148, */ /* pages 203-222, Springer-Verlag, Berlin, May 1996 (from the First ACM */ /* Workshop on Applied Computational Geometry). [*] */ /* */ /* The algorithms are discussed in the greatest detail in "Delaunay */ /* Refinement Algorithms for Triangular Mesh Generation," Computational */ /* Geometry: Theory and Applications 22(1-3):21-74, May 2002. [*] */ /* */ /* More detail about the data structures may be found in my dissertation: */ /* "Delaunay Refinement Mesh Generation," Ph.D. thesis, Technical Report */ /* CMU-CS-97-137, School of Computer Science, Carnegie Mellon University, */ /* Pittsburgh, Pennsylvania, 18 May 1997. [*] */ /* */ /* Triangle was created as part of the Quake Project in the School of */ /* Computer Science at Carnegie Mellon University. For further */ /* information, see Hesheng Bao, Jacobo Bielak, Omar Ghattas, Loukas F. */ /* Kallivokas, David R. O'Hallaron, Jonathan R. Shewchuk, and Jifeng Xu, */ /* "Large-scale Simulation of Elastic Wave Propagation in Heterogeneous */ /* Media on Parallel Computers," Computer Methods in Applied Mechanics */ /* and Engineering 152(1-2):85-102, 22 January 1998. */ /* */ /* Triangle's Delaunay refinement algorithm for quality mesh generation is */ /* a hybrid of one due to Jim Ruppert, "A Delaunay Refinement Algorithm */ /* for Quality 2-Dimensional Mesh Generation," Journal of Algorithms */ /* 18(3):548-585, May 1995 [*], and one due to L. Paul Chew, "Guaranteed- */ /* Quality Mesh Generation for Curved Surfaces," Proceedings of the Ninth */ /* Annual Symposium on Computational Geometry (San Diego, California), */ /* pages 274-280, Association for Computing Machinery, May 1993, */ /* http://portal.acm.org/citation.cfm?id=161150 . */ /* */ /* The Delaunay refinement algorithm has been modified so that it meshes */ /* domains with small input angles well, as described in Gary L. Miller, */ /* Steven E. Pav, and Noel J. Walkington, "When and Why Ruppert's */ /* Algorithm Works," Twelfth International Meshing Roundtable, pages */ /* 91-102, Sandia National Laboratories, September 2003. [*] */ /* */ /* 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 the switch 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, http://portal.acm.org/citation.cfm?id=282923 .*/ /* */ /* 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. Triangle's improvement of 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 vertex */ /* insertion (I currently don't bother), their result combined with the */ /* result of Kenneth L. Clarkson and Peter W. Shor, "Applications of */ /* Random Sampling in Computational Geometry II," Discrete & */ /* Computational Geometry 4(1):387-421, 1989, 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, */ /* http://portal.acm.org/citation.cfm?id=3835 . */ /* */ /* 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," Discrete & */ /* Computational Geometry 18(3):305-363, October 1997. (Also available */ /* as Technical Report CMU-CS-96-140, School of Computer Science, */ /* Carnegie Mellon University, Pittsburgh, Pennsylvania, May 1996.) [*] */ /* An abbreviated version appears as Jonathan Richard Shewchuk, "Robust */ /* Adaptive Floating-Point Geometric Predicates," Proceedings of the */ /* Twelfth Annual Symposium on Computational 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, pp. 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 Computa- */ /* tional 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 Triangu- */ /* lations," International Journal of Computational Geometry & Applica- */ /* tions 5(1-2):193-213, March-June 1995. */ /* */ /* The method of inserting new vertices off-center (not precisely at the */ /* circumcenter of every poor-quality triangle) is from Alper Ungor, */ /* "Off-centers: A New Type of Steiner Points for Computing Size-Optimal */ /* Quality-Guaranteed Delaunay Triangulations," Proceedings of LATIN */ /* 2004 (Buenos Aires, Argentina), April 2004. */ /* */ /* 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(t^2) per segment in the worst case and O(t) per segment in the */ /* common case, where t 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. I could improve this to O(d log d) */ /* time, but d is usually quite small, so it's not worth the bother. */ /* (This note does not apply when the -s switch is used, invoking a */ /* different method is used to insert segments.) */ /* */ /* The time for deleting a vertex from a Delaunay triangulation is O(d^2) */ /* in the worst case and O(d) in the common case, where d is the degree */ /* of the vertex being deleted. I could improve this to O(d log d) time, */ /* but d is usually quite small, so it's not worth the bother. */ /* */ /* 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 */ /* quadratic time is required, but these never arise in practice. */ /* */ /* The geometric predicates (circumcenter calculations, segment */ /* intersection formulae, etc.) appear in my "Lecture Notes on Geometric */ /* Robustness" at http://www.cs.berkeley.edu/~jrs/mesh . */ /* */ /* 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. */ /* */ /*****************************************************************************/ #include "triangle_private.h" /* Fast lookup arrays to speed some of the mesh manipulation primitives. */ int plus1mod3[3] = { 1, 2, 0 }; int minus1mod3[3] = { 2, 0, 1 }; /********* User-defined triangle evaluation routine begins here *********/ /** **/ /** **/ /*****************************************************************************/ /* */ /* triunsuitable() Determine if a triangle is unsuitable, and thus must */ /* be further refined. */ /* */ /* You may write your own procedure that decides whether or not a selected */ /* triangle is too big (and needs to be refined). There are two ways to do */ /* this. */ /* */ /* (1) Modify the procedure `triunsuitable' below, then recompile */ /* Triangle. */ /* */ /* (2) Define the symbol EXTERNAL_TEST (either by adding the definition */ /* to this file, or by using the appropriate compiler switch). This way, */ /* you can compile triangle.c separately from your test. Write your own */ /* `triunsuitable' procedure in a separate C file (using the same prototype */ /* as below). Compile it and link the object code with triangle.o. */ /* */ /* This procedure returns 1 if the triangle is too large and should be */ /* refined; 0 otherwise. */ /* */ /*****************************************************************************/ #ifdef EXTERNAL_TEST int triunsuitable(); #else /* not EXTERNAL_TEST */ int triunsuitable(vertex triorg, vertex tridest, vertex triapex, REAL area) { REAL dxoa, dxda, dxod; REAL dyoa, dyda, dyod; REAL oalen, dalen, odlen; REAL maxlen; dxoa = triorg[0] - triapex[0]; dyoa = triorg[1] - triapex[1]; dxda = tridest[0] - triapex[0]; dyda = tridest[1] - triapex[1]; dxod = triorg[0] - tridest[0]; dyod = triorg[1] - tridest[1]; /* Find the squares of the lengths of the triangle's three edges. */ oalen = dxoa * dxoa + dyoa * dyoa; dalen = dxda * dxda + dyda * dyda; odlen = dxod * dxod + dyod * dyod; /* Find the square of the length of the longest edge. */ maxlen = (dalen > oalen) ? dalen : oalen; maxlen = (odlen > maxlen) ? odlen : maxlen; if (maxlen > 0.05 * (triorg[0] * triorg[0] + triorg[1] * triorg[1]) + 0.02) { return 1; } else { return 0; } } #endif /* not EXTERNAL_TEST */ /** **/ /** **/ /********* User-defined triangle evaluation routine ends here *********/ /********* Memory allocation and program exit wrappers begin here *********/ /** **/ /** **/ void triexit(int status) { printf("Exit %d.\n", status); exit(status); } VOID *trimalloc(int size) { VOID *memptr; memptr = (VOID *) malloc((unsigned int) size); if (memptr == (VOID *) NULL) { printf("Error: Out of memory.\n"); triexit(1); } return (memptr); } void trifree(VOID *memptr) { free(memptr); } /** **/ /** **/ /********* Memory allocation and program exit wrappers end here *********/ /********* User interaction routines begin here *********/ /** **/ /** **/ /*****************************************************************************/ /* */ /* internalerror() Ask the user to send me the defective product. Exit. */ /* */ /*****************************************************************************/ int error_set = 0; void internalerror() { error_set = 1; printf("Triangle is going to quit its job now\n"); //printf(" Please report this bug to jrs@cs.berkeley.edu\n"); ///printf(" Include the message above, your input data set, and the exact\n"); //printf(" command line you used to run Triangle.\n"); //triexit(1); } /*****************************************************************************/ /* */ /* parsecommandline() Read the command line, identify switches, and set */ /* up options and file names. */ /* */ /*****************************************************************************/ void parsecommandline(int argc, char **argv, struct behavior *b) { error_set = 0; #define STARTINDEX 0 int i, j, k; char workstring[FILENAMESIZE]; b->poly = b->refine = b->quality = 0; b->vararea = b->fixedarea = b->usertest = 0; b->regionattrib = b->convex = b->weighted = b->jettison = 0; b->firstnumber = 1; b->edgesout = b->voronoi = b->neighbors = b->geomview = 0; b->nobound = b->nopolywritten = b->nonodewritten = b->noelewritten = 0; b->noiterationnum = 0; b->noholes = b->noexact = 0; b->incremental = b->sweepline = 0; b->dwyer = 1; b->splitseg = 0; b->docheck = 0; b->nobisect = 0; b->conformdel = 0; b->steiner = -1; b->order = 1; b->minangle = 0.0; b->maxarea = -1.0; b->quiet = b->verbose = 0; for (i = STARTINDEX; i < argc; i++) { for (j = STARTINDEX; argv[i][j] != '\0'; j++) { if (argv[i][j] == 'p') { b->poly = 1; } #ifndef CDT_ONLY if (argv[i][j] == 'r') { b->refine = 1; } if (argv[i][j] == 'q') { b->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'; b->minangle = (REAL) strtod(workstring, (char **) NULL); } else { b->minangle = 20.0; } } if (argv[i][j] == 'a') { b->quality = 1; if (((argv[i][j + 1] >= '0') && (argv[i][j + 1] <= '9')) || (argv[i][j + 1] == '.')) { b->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'; b->maxarea = (REAL) strtod(workstring, (char **) NULL); if (b->maxarea <= 0.0) { printf("Error: Maximum area must be greater than zero.\n"); triexit(1); } } else { b->vararea = 1; } } if (argv[i][j] == 'u') { b->quality = 1; b->usertest = 1; } #endif /* not CDT_ONLY */ if (argv[i][j] == 'A') { b->regionattrib = 1; } if (argv[i][j] == 'c') { b->convex = 1; } if (argv[i][j] == 'w') { b->weighted = 1; } if (argv[i][j] == 'W') { b->weighted = 2; } if (argv[i][j] == 'j') { b->jettison = 1; } if (argv[i][j] == 'z') { b->firstnumber = 0; } if (argv[i][j] == 'e') { b->edgesout = 1; } if (argv[i][j] == 'v') { b->voronoi = 1; } if (argv[i][j] == 'n') { b->neighbors = 1; } if (argv[i][j] == 'g') { b->geomview = 1; } if (argv[i][j] == 'B') { b->nobound = 1; } if (argv[i][j] == 'P') { b->nopolywritten = 1; } if (argv[i][j] == 'N') { b->nonodewritten = 1; } if (argv[i][j] == 'E') { b->noelewritten = 1; } if (argv[i][j] == 'O') { b->noholes = 1; } if (argv[i][j] == 'X') { b->noexact = 1; } if (argv[i][j] == 'o') { if (argv[i][j + 1] == '2') { j++; b->order = 2; } } #ifndef CDT_ONLY if (argv[i][j] == 'Y') { b->nobisect++; } if (argv[i][j] == 'S') { b->steiner = 0; while ((argv[i][j + 1] >= '0') && (argv[i][j + 1] <= '9')) { j++; b->steiner = b->steiner * 10 + (int) (argv[i][j] - '0'); } } #endif /* not CDT_ONLY */ #ifndef REDUCED if (argv[i][j] == 'i') { b->incremental = 1; } if (argv[i][j] == 'F') { b->sweepline = 1; } #endif /* not REDUCED */ if (argv[i][j] == 'l') { b->dwyer = 0; } #ifndef REDUCED #ifndef CDT_ONLY if (argv[i][j] == 's') { b->splitseg = 1; } if ((argv[i][j] == 'D') || (argv[i][j] == 'L')) { b->quality = 1; b->conformdel = 1; } #endif /* not CDT_ONLY */ if (argv[i][j] == 'C') { b->docheck = 1; } #endif /* not REDUCED */ if (argv[i][j] == 'Q') { b->quiet = 1; } if (argv[i][j] == 'V') { b->verbose++; } } } b->usesegments = b->poly || b->refine || b->quality || b->convex; b->goodangle = cos(b->minangle * PI / 180.0); if (b->goodangle == 1.0) { b->offconstant = 0.0; } else { b->offconstant = 0.475 * sqrt((1.0 + b->goodangle) / (1.0 - b->goodangle)); } b->goodangle *= b->goodangle; if (b->refine && b->noiterationnum) { printf( "Error: You cannot use the -I switch when refining a triangulation.\n"); triexit(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 (!b->refine && !b->poly) { b->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 (b->refine || !b->poly) { b->regionattrib = 0; } /* Regular/weighted triangulations are incompatible with PSLGs */ /* and meshing. */ if (b->weighted && (b->poly || b->quality)) { b->weighted = 0; if (!b->quiet) { printf("Warning: weighted triangulations (-w, -W) are incompatible\n"); printf(" with PSLGs (-p) and meshing (-q, -a, -u). Weights ignored.\n"); } } if (b->jettison && b->nonodewritten && !b->quiet) { printf("Warning: -j and -N switches are somewhat incompatible.\n"); printf(" If any vertices are jettisoned, you will need the output\n"); printf(" .node file to reconstruct the new node indices."); } } /** **/ /** **/ /********* User interaction routines begin here *********/ /********* Memory management routines begin here *********/ /** **/ /** **/ /*****************************************************************************/ /* */ /* poolzero() Set all of a pool's fields to zero. */ /* */ /* This procedure should never be called on a pool that has any memory */ /* allocated to it, as that memory would leak. */ /* */ /*****************************************************************************/ void poolzero(struct memorypool *pool) { pool->firstblock = (VOID **) NULL; pool->nowblock = (VOID **) NULL; pool->nextitem = (VOID *) NULL; pool->deaditemstack = (VOID *) NULL; pool->pathblock = (VOID **) NULL; pool->pathitem = (VOID *) NULL; pool->alignbytes = 0; pool->itembytes = 0; pool->itemsperblock = 0; pool->itemsfirstblock = 0; pool->items = 0; pool->maxitems = 0; pool->unallocateditems = 0; pool->pathitemsleft = 0; } /*****************************************************************************/ /* */ /* 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(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->itemsfirstblock; /* The stack of deallocated items is empty. */ pool->deaditemstack = (VOID *) NULL; } /*****************************************************************************/ /* */ /* 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(struct memorypool *pool, int bytecount, int itemcount, int firstitemcount, int alignment) { /* Find the proper alignment, which must be at least as large as: */ /* - The parameter `alignment'. */ /* - sizeof(VOID *), so the stack of dead items can be maintained */ /* without unaligned accesses. */ if (alignment > sizeof(VOID *)) { pool->alignbytes = alignment; } else { pool->alignbytes = sizeof(VOID *); } pool->itembytes = ((bytecount - 1) / pool->alignbytes + 1) * pool->alignbytes; pool->itemsperblock = itemcount; if (firstitemcount == 0) { pool->itemsfirstblock = itemcount; } else { pool->itemsfirstblock = firstitemcount; } /* Allocate a block of items. Space for `itemsfirstblock' 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 **) trimalloc( pool->itemsfirstblock * pool->itembytes + (int) sizeof(VOID *) + pool->alignbytes); /* Set the next block pointer to NULL. */ *(pool->firstblock) = (VOID *) NULL; poolrestart(pool); } /*****************************************************************************/ /* */ /* pooldeinit() Free to the operating system all memory taken by a pool. */ /* */ /*****************************************************************************/ void pooldeinit(struct memorypool *pool) { while (pool->firstblock != (VOID **) NULL) { pool->nowblock = (VOID **) *(pool->firstblock); trifree((VOID *) pool->firstblock); pool->firstblock = pool->nowblock; } } /*****************************************************************************/ /* */ /* poolalloc() Allocate space for an item. */ /* */ /*****************************************************************************/ VOID *poolalloc(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 **) trimalloc( pool->itemsperblock * pool->itembytes + (int) sizeof(VOID *) + pool->alignbytes); *(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. */ pool->nextitem = (VOID *) ((char *) pool->nextitem + pool->itembytes); 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(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(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->itemsfirstblock; } /*****************************************************************************/ /* */ /* 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(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. */ pool->pathitem = (VOID *) ((char *) pool->pathitem + pool->itembytes); pool->pathitemsleft--; return newitem; } /*****************************************************************************/ /* */ /* dummyinit() Initialize the triangle that fills "outer space" and the */ /* omnipresent subsegment. */ /* */ /* The triangle that fills "outer space," called `dummytri', is pointed to */ /* by every triangle and subsegment 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 subsegment, `dummysub', is pointed to by every triangle */ /* or subsegment that doesn't have a full complement of real subsegments */ /* to point to. */ /* */ /* `dummytri' and `dummysub' are generally required to fulfill only a few */ /* invariants: their vertices must remain NULL and `dummytri' must always */ /* be bonded (at offset zero) to some triangle on the convex hull of the */ /* mesh, via a boundary edge. Otherwise, the connections of `dummytri' and */ /* `dummysub' may change willy-nilly. This makes it possible to avoid */ /* writing a good deal of special-case code (in the edge flip, for example) */ /* for dealing with the boundary of the mesh, places where no subsegment is */ /* present, and so forth. Other entities are frequently bonded to */ /* `dummytri' and `dummysub' as if they were real mesh entities, with no */ /* harm done. */ /* */ /*****************************************************************************/ void dummyinit(struct mesh *m, struct behavior *b, int trianglebytes, int subsegbytes) { unsigned long alignptr; /* Set up `dummytri', the `triangle' that occupies "outer space." */ m->dummytribase = (triangle *) trimalloc(trianglebytes + m->triangles.alignbytes); /* Align `dummytri' on a `triangles.alignbytes'-byte boundary. */ alignptr = (unsigned long) m->dummytribase; m->dummytri = (triangle *) (alignptr + (unsigned long) m->triangles.alignbytes - (alignptr % (unsigned long) m->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. */ m->dummytri[0] = (triangle) m->dummytri; m->dummytri[1] = (triangle) m->dummytri; m->dummytri[2] = (triangle) m->dummytri; /* Three NULL vertices. */ m->dummytri[3] = (triangle) NULL; m->dummytri[4] = (triangle) NULL; m->dummytri[5] = (triangle) NULL; if (b->usesegments) { /* Set up `dummysub', the omnipresent subsegment pointed to by any */ /* triangle side or subsegment end that isn't attached to a real */ /* subsegment. */ m->dummysubbase = (subseg *) trimalloc(subsegbytes + m->subsegs.alignbytes); /* Align `dummysub' on a `subsegs.alignbytes'-byte boundary. */ alignptr = (unsigned long) m->dummysubbase; m->dummysub = (subseg *) (alignptr + (unsigned long) m->subsegs.alignbytes - (alignptr % (unsigned long) m->subsegs.alignbytes)); /* Initialize the two adjoining subsegments to be the omnipresent */ /* subsegment. These will eventually be changed by various bonding */ /* operations, but their values don't really matter, as long as they */ /* can legally be dereferenced. */ m->dummysub[0] = (subseg) m->dummysub; m->dummysub[1] = (subseg) m->dummysub; /* Four NULL vertices. */ m->dummysub[2] = (subseg) NULL; m->dummysub[3] = (subseg) NULL; m->dummysub[4] = (subseg) NULL; m->dummysub[5] = (subseg) NULL; /* Initialize the two adjoining triangles to be "outer space." */ m->dummysub[6] = (subseg) m->dummytri; m->dummysub[7] = (subseg) m->dummytri; /* Set the boundary marker to zero. */ *(int *) (m->dummysub + 8) = 0; /* Initialize the three adjoining subsegments of `dummytri' to be */ /* the omnipresent subsegment. */ m->dummytri[6] = (triangle) m->dummysub; m->dummytri[7] = (triangle) m->dummysub; m->dummytri[8] = (triangle) m->dummysub; } } /*****************************************************************************/ /* */ /* initializevertexpool() Calculate the size of the vertex data structure */ /* and initialize its memory pool. */ /* */ /* This routine also computes the `vertexmarkindex' and `vertex2triindex' */ /* indices used to find values within each vertex. */ /* */ /*****************************************************************************/ void initializevertexpool(struct mesh *m, struct behavior *b) { int vertexsize; /* The index within each vertex at which the boundary marker is found, */ /* followed by the vertex type. Ensure the vertex marker is aligned to */ /* a sizeof(int)-byte address. */ m->vertexmarkindex = ((m->mesh_dim + m->nextras) * sizeof(REAL) + sizeof(int) - 1) / sizeof(int); vertexsize = (m->vertexmarkindex + 2) * sizeof(int); if (b->poly) { /* The index within each vertex at which a triangle pointer is found. */ /* Ensure the pointer is aligned to a sizeof(triangle)-byte address. */ m->vertex2triindex = (vertexsize + sizeof(triangle) - 1) / sizeof(triangle); vertexsize = (m->vertex2triindex + 1) * sizeof(triangle); } /* Initialize the pool of vertices. */ poolinit(&m->vertices, vertexsize, VERTEXPERBLOCK, m->invertices > VERTEXPERBLOCK ? m->invertices : VERTEXPERBLOCK, sizeof(REAL)); } /*****************************************************************************/ /* */ /* initializetrisubpools() Calculate the sizes of the triangle and */ /* subsegment 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 initializetrisubpools(struct mesh *m, struct behavior *b) { 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 subsegments before the extra nodes. */ m->highorderindex = 6 + (b->usesegments * 3); /* The number of bytes occupied by a triangle. */ trisize = ((b->order + 1) * (b->order + 2) / 2 + (m->highorderindex - 3)) * sizeof(triangle); /* The index within each triangle at which its attributes are found, */ /* where the index is measured in REALs. */ m->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. */ m->areaboundindex = m->elemattribindex + m->eextras + b->regionattrib; /* If triangle attributes or an area bound are needed, increase the number */ /* of bytes occupied by a triangle. */ if (b->vararea) { trisize = (m->areaboundindex + 1) * sizeof(REAL); } else if (m->eextras + b->regionattrib > 0) { trisize = m->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 subsegment pointers */ /* or attributes or area constraint or extra nodes. */ if ((b->voronoi || b->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(&m->triangles, trisize, TRIPERBLOCK, (2 * m->invertices - 2) > TRIPERBLOCK ? (2 * m->invertices - 2) : TRIPERBLOCK, 4); if (b->usesegments) { /* Initialize the pool of subsegments. Take into account all eight */ /* pointers and one boundary marker. */ poolinit(&m->subsegs, 8 * sizeof(triangle) + sizeof(int), SUBSEGPERBLOCK, SUBSEGPERBLOCK, 4); /* Initialize the "outer space" triangle and omnipresent subsegment. */ dummyinit(m, b, m->triangles.itembytes, m->subsegs.itembytes); } else { /* Initialize the "outer space" triangle. */ dummyinit(m, b, m->triangles.itembytes, 0); } } /*****************************************************************************/ /* */ /* triangledealloc() Deallocate space for a triangle, marking it dead. */ /* */ /*****************************************************************************/ void triangledealloc(struct mesh *m, triangle *dyingtriangle) { /* Mark the triangle as dead. This makes it possible to detect dead */ /* triangles when traversing the list of all triangles. */ killtri(dyingtriangle); pooldealloc(&m->triangles, (VOID *) dyingtriangle); } /*****************************************************************************/ /* */ /* triangletraverse() Traverse the triangles, skipping dead ones. */ /* */ /*****************************************************************************/ triangle *triangletraverse(struct mesh *m) { triangle *newtriangle; do { newtriangle = (triangle *) traverse(&m->triangles); if (newtriangle == (triangle *) NULL) { return (triangle *) NULL; } } while (deadtri(newtriangle)); /* Skip dead ones. */ return newtriangle; } /*****************************************************************************/ /* */ /* subsegdealloc() Deallocate space for a subsegment, marking it dead. */ /* */ /*****************************************************************************/ void subsegdealloc(struct mesh *m, subseg *dyingsubseg) { /* Mark the subsegment as dead. This makes it possible to detect dead */ /* subsegments when traversing the list of all subsegments. */ killsubseg(dyingsubseg); pooldealloc(&m->subsegs, (VOID *) dyingsubseg); } /*****************************************************************************/ /* */ /* subsegtraverse() Traverse the subsegments, skipping dead ones. */ /* */ /*****************************************************************************/ subseg *subsegtraverse(struct mesh *m) { subseg *newsubseg; do { newsubseg = (subseg *) traverse(&m->subsegs); if (newsubseg == (subseg *) NULL) { return (subseg *) NULL; } } while (deadsubseg(newsubseg)); /* Skip dead ones. */ return newsubseg; } /*****************************************************************************/ /* */ /* vertexdealloc() Deallocate space for a vertex, marking it dead. */ /* */ /*****************************************************************************/ void vertexdealloc(struct mesh *m, vertex dyingvertex) { /* Mark the vertex as dead. This makes it possible to detect dead */ /* vertices when traversing the list of all vertices. */ setvertextype(dyingvertex, DEADVERTEX); pooldealloc(&m->vertices, (VOID *) dyingvertex); } /*****************************************************************************/ /* */ /* vertextraverse() Traverse the vertices, skipping dead ones. */ /* */ /*****************************************************************************/ vertex vertextraverse(struct mesh *m) { vertex newvertex; do { newvertex = (vertex) traverse(&m->vertices); if (newvertex == (vertex) NULL) { return (vertex) NULL; } } while (vertextype(newvertex) == DEADVERTEX); /* Skip dead ones. */ return newvertex; } /*****************************************************************************/ /* */ /* getvertex() Get a specific vertex, by number, from the list. */ /* */ /* The first vertex is number 'firstnumber'. */ /* */ /* Note that this takes O(n) time (with a small constant, if VERTEXPERBLOCK */ /* is large). I don't care to take the trouble to make it work in constant */ /* time. */ /* */ /*****************************************************************************/ vertex getvertex(struct mesh *m, struct behavior *b, int number) { VOID **getblock; char *foundvertex; unsigned long alignptr; int current; getblock = m->vertices.firstblock; current = b->firstnumber; /* Find the right block. */ if (current + m->vertices.itemsfirstblock <= number) { getblock = (VOID **) *getblock; current += m->vertices.itemsfirstblock; while (current + m->vertices.itemsperblock <= number) { getblock = (VOID **) *getblock; current += m->vertices.itemsperblock; } } /* Now find the right vertex. */ alignptr = (unsigned long) (getblock + 1); foundvertex = (char *) (alignptr + (unsigned long) m->vertices.alignbytes - (alignptr % (unsigned long) m->vertices.alignbytes)); return (vertex) (foundvertex + m->vertices.itembytes * (number - current)); } /*****************************************************************************/ /* */ /* triangledeinit() Free all remaining allocated memory. */ /* */ /*****************************************************************************/ void triangledeinit(struct mesh *m, struct behavior *b) { pooldeinit(&m->triangles); trifree((VOID *) m->dummytribase); if (b->usesegments) { pooldeinit(&m->subsegs); trifree((VOID *) m->dummysubbase); } pooldeinit(&m->vertices); #ifndef CDT_ONLY if (b->quality) { pooldeinit(&m->badsubsegs); if ((b->minangle > 0.0) || b->vararea || b->fixedarea || b->usertest) { pooldeinit(&m->badtriangles); pooldeinit(&m->flipstackers); } } #endif /* not CDT_ONLY */ } /** **/ /** **/ /********* Memory management routines end here *********/ /********* Constructors begin here *********/ /** **/ /** **/ /*****************************************************************************/ /* */ /* maketriangle() Create a new triangle with orientation zero. */ /* */ /*****************************************************************************/ void maketriangle(struct mesh *m, struct behavior *b, struct otri *newotri) { int i; newotri->tri = (triangle *) poolalloc(&m->triangles); /* Initialize the three adjoining triangles to be "outer space". */ newotri->tri[0] = (triangle) m->dummytri; newotri->tri[1] = (triangle) m->dummytri; newotri->tri[2] = (triangle) m->dummytri; /* Three NULL vertices. */ newotri->tri[3] = (triangle) NULL; newotri->tri[4] = (triangle) NULL; newotri->tri[5] = (triangle) NULL; if (b->usesegments) { /* Initialize the three adjoining subsegments to be the omnipresent */ /* subsegment. */ newotri->tri[6] = (triangle) m->dummysub; newotri->tri[7] = (triangle) m->dummysub; newotri->tri[8] = (triangle) m->dummysub; } for (i = 0; i < m->eextras; i++) { setelemattribute(*newotri, i, 0.0); } if (b->vararea) { setareabound(*newotri, -1.0); } newotri->orient = 0; } /*****************************************************************************/ /* */ /* makesubseg() Create a new subsegment with orientation zero. */ /* */ /*****************************************************************************/ void makesubseg(struct mesh *m, struct osub *newsubseg) { newsubseg->ss = (subseg *) poolalloc(&m->subsegs); /* Initialize the two adjoining subsegments to be the omnipresent */ /* subsegment. */ newsubseg->ss[0] = (subseg) m->dummysub; newsubseg->ss[1] = (subseg) m->dummysub; /* Four NULL vertices. */ newsubseg->ss[2] = (subseg) NULL; newsubseg->ss[3] = (subseg) NULL; newsubseg->ss[4] = (subseg) NULL; newsubseg->ss[5] = (subseg) NULL; /* Initialize the two adjoining triangles to be "outer space." */ newsubseg->ss[6] = (subseg) m->dummytri; newsubseg->ss[7] = (subseg) m->dummytri; /* Set the boundary marker to zero. */ setmark(*newsubseg, 0); newsubseg->ssorient = 0; } /** **/ /** **/ /********* Constructors end here *********/ /********* Geometric primitives begin here *********/ /** **/ /** **/ /* The adaptive exact arithmetic geometric predicates implemented herein are */ /* described in detail in my paper, "Adaptive Precision Floating-Point */ /* Arithmetic and Fast Robust Geometric Predicates." See the header for a */ /* full citation. */ /* 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 without */ /* forcing the value to be stored to memory (rather than be kept in the */ /* register to which the optimizer assigned it). */ #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 `' 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 `'. */ #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 = c - abig; \ alo = 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) /* Macro for multiplying a two-component expansion by a single component. */ #define Two_One_Product(a1, a0, b, x3, x2, x1, x0) \ Split(b, bhi, blo); \ Two_Product_Presplit(a0, b, bhi, blo, _i, x0); \ Two_Product_Presplit(a1, b, bhi, blo, _j, _0); \ Two_Sum(_i, _0, _k, x1); \ Fast_Two_Sum(_j, _k, x3, x2) /*****************************************************************************/ /* */ /* 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; #ifdef LINUX int cword; #endif /* LINUX */ #ifdef CPU86 #ifdef SINGLE _control87(_PC_24, _MCW_PC); /* Set FPU control word for single precision. */ #else /* not SINGLE */ _control87(_PC_53, _MCW_PC); /* Set FPU control word for double precision. */ #endif /* not SINGLE */ #endif /* CPU86 */ #ifdef LINUX #ifdef SINGLE /* cword = 4223; */ cword = 4210; /* set FPU control word for single precision */ #else /* not SINGLE */ /* cword = 4735; */ cword = 4722; /* set FPU control word for double precision */ #endif /* not SINGLE */ _FPU_SETCW(cword); #endif /* LINUX */ 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.) */ do { lastcheck = check; epsilon *= half; if (every_other) { splitter *= 2.0; } every_other = !every_other; check = 1.0 + epsilon; } while ((check != 1.0) && (check != lastcheck)); splitter += 1.0; /* Error bounds for orientation and incircle tests. */ resulterrbound = (3.0 + 8.0 * epsilon) * epsilon; ccwerrboundA = (3.0 + 16.0 * epsilon) * epsilon; ccwerrboundB = (2.0 + 12.0 * epsilon) * epsilon; ccwerrboundC = (9.0 + 64.0 * epsilon) * epsilon * epsilon; iccerrboundA = (10.0 + 96.0 * epsilon) * epsilon; iccerrboundB = (4.0 + 48.0 * epsilon) * epsilon; iccerrboundC = (44.0 + 576.0 * epsilon) * epsilon * epsilon; o3derrboundA = (7.0 + 56.0 * epsilon) * epsilon; o3derrboundB = (3.0 + 28.0 * epsilon) * epsilon; o3derrboundC = (26.0 + 288.0 * 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(int elen, REAL *e, int flen, REAL *f, REAL *h) { REAL Q; REAL Qnew; REAL hh; 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(int elen, REAL *e, REAL b, REAL *h) { REAL Q, sum; REAL hh; REAL product1; REAL product0; int eindex, hindex; REAL enow; REAL bvirt; REAL avirt, bround, around; REAL c; 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(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(vertex pa, vertex pb, vertex pc, REAL detsum) { REAL acx, acy, bcx, bcy; REAL acxtail, acytail, bcxtail, bcytail; REAL detleft, detright; REAL detlefttail, detrighttail; REAL det, errbound; REAL B[4], C1[8], C2[12], D[16]; REAL B3; int C1length, C2length, Dlength; REAL u[4]; REAL u3; REAL s1, t1; REAL s0, t0; REAL bvirt; REAL avirt, bround, around; REAL c; REAL abig; REAL ahi, alo, bhi, blo; REAL err1, err2, err3; 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 = 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 = 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(struct mesh *m, struct behavior *b, vertex pa, vertex pb, vertex pc) { REAL detleft, detright, det; REAL detsum, errbound; m->counterclockcount++; detleft = (pa[0] - pc[0]) * (pb[1] - pc[1]); detright = (pa[1] - pc[1]) * (pb[0] - pc[0]); det = detleft - detright; if (b->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(vertex pa, vertex pb, vertex pc, vertex pd, REAL permanent) { REAL adx, bdx, cdx, ady, bdy, cdy; REAL det, errbound; REAL bdxcdy1, cdxbdy1, cdxady1, adxcdy1, adxbdy1, bdxady1; REAL bdxcdy0, cdxbdy0, cdxady0, adxcdy0, adxbdy0, bdxady0; REAL bc[4], ca[4], ab[4]; 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; REAL adxadx1, adyady1, bdxbdx1, bdybdy1, cdxcdx1, cdycdy1; REAL adxadx0, adyady0, bdxbdx0, bdybdy0, cdxcdx0, cdycdy0; REAL aa[4], bb[4], cc[4]; REAL aa3, bb3, cc3; REAL ti1, tj1; REAL ti0, tj0; REAL u[4], v[4]; 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; REAL abtt3, bctt3, catt3; REAL negate; REAL bvirt; REAL avirt, bround, around; REAL c; REAL abig; REAL ahi, alo, bhi, blo; REAL err1, err2, err3; 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 = 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 = iccerrboundC * permanent + resulterrbound * Absolute(det); det += ((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(struct mesh *m, struct behavior *b, vertex pa, vertex pb, vertex pc, vertex pd) { REAL adx, bdx, cdx, ady, bdy, cdy; REAL bdxcdy, cdxbdy, cdxady, adxcdy, adxbdy, bdxady; REAL alift, blift, clift; REAL det; REAL permanent, errbound; m->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 (b->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); } /*****************************************************************************/ /* */ /* orient3d() Return a positive value if the point pd lies below the */ /* plane passing through pa, pb, and pc; "below" is defined so */ /* that pa, pb, and pc appear in counterclockwise order when */ /* viewed from above the plane. Returns a negative value if */ /* pd lies above the plane. Returns zero if the points are */ /* coplanar. The result is also a rough approximation of six */ /* times the signed volume of the tetrahedron defined by the */ /* four 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 coplanar or nearly so. */ /* */ /* See my Robust Predicates paper for details. */ /* */ /*****************************************************************************/ REAL orient3dadapt(vertex pa, vertex pb, vertex pc, vertex pd, REAL aheight, REAL bheight, REAL cheight, REAL dheight, REAL permanent) { REAL adx, bdx, cdx, ady, bdy, cdy, adheight, bdheight, cdheight; REAL det, errbound; REAL bdxcdy1, cdxbdy1, cdxady1, adxcdy1, adxbdy1, bdxady1; REAL bdxcdy0, cdxbdy0, cdxady0, adxcdy0, adxbdy0, bdxady0; REAL bc[4], ca[4], ab[4]; REAL bc3, ca3, ab3; REAL adet[8], bdet[8], cdet[8]; int alen, blen, clen; REAL abdet[16]; int ablen; REAL *finnow, *finother, *finswap; REAL fin1[192], fin2[192]; int finlength; REAL adxtail, bdxtail, cdxtail; REAL adytail, bdytail, cdytail; REAL adheighttail, bdheighttail, cdheighttail; REAL at_blarge, at_clarge; REAL bt_clarge, bt_alarge; REAL ct_alarge, ct_blarge; REAL at_b[4], at_c[4], bt_c[4], bt_a[4], ct_a[4], ct_b[4]; int at_blen, at_clen, bt_clen, bt_alen, ct_alen, ct_blen; REAL bdxt_cdy1, cdxt_bdy1, cdxt_ady1; REAL adxt_cdy1, adxt_bdy1, bdxt_ady1; REAL bdxt_cdy0, cdxt_bdy0, cdxt_ady0; REAL adxt_cdy0, adxt_bdy0, bdxt_ady0; REAL bdyt_cdx1, cdyt_bdx1, cdyt_adx1; REAL adyt_cdx1, adyt_bdx1, bdyt_adx1; REAL bdyt_cdx0, cdyt_bdx0, cdyt_adx0; REAL adyt_cdx0, adyt_bdx0, bdyt_adx0; REAL bct[8], cat[8], abt[8]; int bctlen, catlen, abtlen; REAL bdxt_cdyt1, cdxt_bdyt1, cdxt_adyt1; REAL adxt_cdyt1, adxt_bdyt1, bdxt_adyt1; REAL bdxt_cdyt0, cdxt_bdyt0, cdxt_adyt0; REAL adxt_cdyt0, adxt_bdyt0, bdxt_adyt0; REAL u[4], v[12], w[16]; REAL u3; int vlength, wlength; REAL negate; REAL bvirt; REAL avirt, bround, around; REAL c; REAL abig; REAL ahi, alo, bhi, blo; REAL err1, err2, err3; REAL _i, _j, _k; 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]); adheight = (REAL) (aheight - dheight); bdheight = (REAL) (bheight - dheight); cdheight = (REAL) (cheight - dheight); 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; alen = scale_expansion_zeroelim(4, bc, adheight, 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; blen = scale_expansion_zeroelim(4, ca, bdheight, 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; clen = scale_expansion_zeroelim(4, ab, cdheight, 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 = o3derrboundB * permanent; if ((det >= errbound) || (-det >= errbound)) { return det; } Two_Diff_Tail(pa[0], pd[0], adx, adxtail); Two_Diff_Tail(pb[0], pd[0], bdx, bdxtail); Two_Diff_Tail(pc[0], pd[0], cdx, cdxtail); Two_Diff_Tail(pa[1], pd[1], ady, adytail); Two_Diff_Tail(pb[1], pd[1], bdy, bdytail); Two_Diff_Tail(pc[1], pd[1], cdy, cdytail); Two_Diff_Tail(aheight, dheight, adheight, adheighttail); Two_Diff_Tail(bheight, dheight, bdheight, bdheighttail); Two_Diff_Tail(cheight, dheight, cdheight, cdheighttail); if ((adxtail == 0.0) && (bdxtail == 0.0) && (cdxtail == 0.0) && (adytail == 0.0) && (bdytail == 0.0) && (cdytail == 0.0) && (adheighttail == 0.0) && (bdheighttail == 0.0) && (cdheighttail == 0.0)) { return det; } errbound = o3derrboundC * permanent + resulterrbound * Absolute(det); det += (adheight * ((bdx * cdytail + cdy * bdxtail) - (bdy * cdxtail + cdx * bdytail)) + adheighttail * (bdx * cdy - bdy * cdx)) + (bdheight * ((cdx * adytail + ady * cdxtail) - (cdy * adxtail + adx * cdytail)) + bdheighttail * (cdx * ady - cdy * adx)) + (cdheight * ((adx * bdytail + bdy * adxtail) - (ady * bdxtail + bdx * adytail)) + cdheighttail * (adx * bdy - ady * bdx)); if ((det >= errbound) || (-det >= errbound)) { return det; } finnow = fin1; finother = fin2; if (adxtail == 0.0) { if (adytail == 0.0) { at_b[0] = 0.0; at_blen = 1; at_c[0] = 0.0; at_clen = 1; } else { negate = -adytail; Two_Product(negate, bdx, at_blarge, at_b[0]); at_b[1] = at_blarge; at_blen = 2; Two_Product(adytail, cdx, at_clarge, at_c[0]); at_c[1] = at_clarge; at_clen = 2; } } else { if (adytail == 0.0) { Two_Product(adxtail, bdy, at_blarge, at_b[0]); at_b[1] = at_blarge; at_blen = 2; negate = -adxtail; Two_Product(negate, cdy, at_clarge, at_c[0]); at_c[1] = at_clarge; at_clen = 2; } else { Two_Product(adxtail, bdy, adxt_bdy1, adxt_bdy0); Two_Product(adytail, bdx, adyt_bdx1, adyt_bdx0); Two_Two_Diff(adxt_bdy1, adxt_bdy0, adyt_bdx1, adyt_bdx0, at_blarge, at_b[2], at_b[1], at_b[0]); at_b[3] = at_blarge; at_blen = 4; Two_Product(adytail, cdx, adyt_cdx1, adyt_cdx0); Two_Product(adxtail, cdy, adxt_cdy1, adxt_cdy0); Two_Two_Diff(adyt_cdx1, adyt_cdx0, adxt_cdy1, adxt_cdy0, at_clarge, at_c[2], at_c[1], at_c[0]); at_c[3] = at_clarge; at_clen = 4; } } if (bdxtail == 0.0) { if (bdytail == 0.0) { bt_c[0] = 0.0; bt_clen = 1; bt_a[0] = 0.0; bt_alen = 1; } else { negate = -bdytail; Two_Product(negate, cdx, bt_clarge, bt_c[0]); bt_c[1] = bt_clarge; bt_clen = 2; Two_Product(bdytail, adx, bt_alarge, bt_a[0]); bt_a[1] = bt_alarge; bt_alen = 2; } } else { if (bdytail == 0.0) { Two_Product(bdxtail, cdy, bt_clarge, bt_c[0]); bt_c[1] = bt_clarge; bt_clen = 2; negate = -bdxtail; Two_Product(negate, ady, bt_alarge, bt_a[0]); bt_a[1] = bt_alarge; bt_alen = 2; } else { Two_Product(bdxtail, cdy, bdxt_cdy1, bdxt_cdy0); Two_Product(bdytail, cdx, bdyt_cdx1, bdyt_cdx0); Two_Two_Diff(bdxt_cdy1, bdxt_cdy0, bdyt_cdx1, bdyt_cdx0, bt_clarge, bt_c[2], bt_c[1], bt_c[0]); bt_c[3] = bt_clarge; bt_clen = 4; Two_Product(bdytail, adx, bdyt_adx1, bdyt_adx0); Two_Product(bdxtail, ady, bdxt_ady1, bdxt_ady0); Two_Two_Diff(bdyt_adx1, bdyt_adx0, bdxt_ady1, bdxt_ady0, bt_alarge, bt_a[2], bt_a[1], bt_a[0]); bt_a[3] = bt_alarge; bt_alen = 4; } } if (cdxtail == 0.0) { if (cdytail == 0.0) { ct_a[0] = 0.0; ct_alen = 1; ct_b[0] = 0.0; ct_blen = 1; } else { negate = -cdytail; Two_Product(negate, adx, ct_alarge, ct_a[0]); ct_a[1] = ct_alarge; ct_alen = 2; Two_Product(cdytail, bdx, ct_blarge, ct_b[0]); ct_b[1] = ct_blarge; ct_blen = 2; } } else { if (cdytail == 0.0) { Two_Product(cdxtail, ady, ct_alarge, ct_a[0]); ct_a[1] = ct_alarge; ct_alen = 2; negate = -cdxtail; Two_Product(negate, bdy, ct_blarge, ct_b[0]); ct_b[1] = ct_blarge; ct_blen = 2; } else { Two_Product(cdxtail, ady, cdxt_ady1, cdxt_ady0); Two_Product(cdytail, adx, cdyt_adx1, cdyt_adx0); Two_Two_Diff(cdxt_ady1, cdxt_ady0, cdyt_adx1, cdyt_adx0, ct_alarge, ct_a[2], ct_a[1], ct_a[0]); ct_a[3] = ct_alarge; ct_alen = 4; Two_Product(cdytail, bdx, cdyt_bdx1, cdyt_bdx0); Two_Product(cdxtail, bdy, cdxt_bdy1, cdxt_bdy0); Two_Two_Diff(cdyt_bdx1, cdyt_bdx0, cdxt_bdy1, cdxt_bdy0, ct_blarge, ct_b[2], ct_b[1], ct_b[0]); ct_b[3] = ct_blarge; ct_blen = 4; } } bctlen = fast_expansion_sum_zeroelim(bt_clen, bt_c, ct_blen, ct_b, bct); wlength = scale_expansion_zeroelim(bctlen, bct, adheight, w); finlength = fast_expansion_sum_zeroelim(finlength, finnow, wlength, w, finother); finswap = finnow; finnow = finother; finother = finswap; catlen = fast_expansion_sum_zeroelim(ct_alen, ct_a, at_clen, at_c, cat); wlength = scale_expansion_zeroelim(catlen, cat, bdheight, w); finlength = fast_expansion_sum_zeroelim(finlength, finnow, wlength, w, finother); finswap = finnow; finnow = finother; finother = finswap; abtlen = fast_expansion_sum_zeroelim(at_blen, at_b, bt_alen, bt_a, abt); wlength = scale_expansion_zeroelim(abtlen, abt, cdheight, w); finlength = fast_expansion_sum_zeroelim(finlength, finnow, wlength, w, finother); finswap = finnow; finnow = finother; finother = finswap; if (adheighttail != 0.0) { vlength = scale_expansion_zeroelim(4, bc, adheighttail, v); finlength = fast_expansion_sum_zeroelim(finlength, finnow, vlength, v, finother); finswap = finnow; finnow = finother; finother = finswap; } if (bdheighttail != 0.0) { vlength = scale_expansion_zeroelim(4, ca, bdheighttail, v); finlength = fast_expansion_sum_zeroelim(finlength, finnow, vlength, v, finother); finswap = finnow; finnow = finother; finother = finswap; } if (cdheighttail != 0.0) { vlength = scale_expansion_zeroelim(4, ab, cdheighttail, v); finlength = fast_expansion_sum_zeroelim(finlength, finnow, vlength, v, finother); finswap = finnow; finnow = finother; finother = finswap; } if (adxtail != 0.0) { if (bdytail != 0.0) { Two_Product(adxtail, bdytail, adxt_bdyt1, adxt_bdyt0); Two_One_Product(adxt_bdyt1, adxt_bdyt0, cdheight, u3, u[2], u[1], u[0]); u[3] = u3; finlength = fast_expansion_sum_zeroelim(finlength, finnow, 4, u, finother); finswap = finnow; finnow = finother; finother = finswap; if (cdheighttail != 0.0) { Two_One_Product(adxt_bdyt1, adxt_bdyt0, cdheighttail, u3, u[2], u[1], u[0]); u[3] = u3; finlength = fast_expansion_sum_zeroelim(finlength, finnow, 4, u, finother); finswap = finnow; finnow = finother; finother = finswap; } } if (cdytail != 0.0) { negate = -adxtail; Two_Product(negate, cdytail, adxt_cdyt1, adxt_cdyt0); Two_One_Product(adxt_cdyt1, adxt_cdyt0, bdheight, u3, u[2], u[1], u[0]); u[3] = u3; finlength = fast_expansion_sum_zeroelim(finlength, finnow, 4, u, finother); finswap = finnow; finnow = finother; finother = finswap; if (bdheighttail != 0.0) { Two_One_Product(adxt_cdyt1, adxt_cdyt0, bdheighttail, u3, u[2], u[1], u[0]); u[3] = u3; finlength = fast_expansion_sum_zeroelim(finlength, finnow, 4, u, finother); finswap = finnow; finnow = finother; finother = finswap; } } } if (bdxtail != 0.0) { if (cdytail != 0.0) { Two_Product(bdxtail, cdytail, bdxt_cdyt1, bdxt_cdyt0); Two_One_Product(bdxt_cdyt1, bdxt_cdyt0, adheight, u3, u[2], u[1], u[0]); u[3] = u3; finlength = fast_expansion_sum_zeroelim(finlength, finnow, 4, u, finother); finswap = finnow; finnow = finother; finother = finswap; if (adheighttail != 0.0) { Two_One_Product(bdxt_cdyt1, bdxt_cdyt0, adheighttail, u3, u[2], u[1], u[0]); u[3] = u3; finlength = fast_expansion_sum_zeroelim(finlength, finnow, 4, u, finother); finswap = finnow; finnow = finother; finother = finswap; } } if (adytail != 0.0) { negate = -bdxtail; Two_Product(negate, adytail, bdxt_adyt1, bdxt_adyt0); Two_One_Product(bdxt_adyt1, bdxt_adyt0, cdheight, u3, u[2], u[1], u[0]); u[3] = u3; finlength = fast_expansion_sum_zeroelim(finlength, finnow, 4, u, finother); finswap = finnow; finnow = finother; finother = finswap; if (cdheighttail != 0.0) { Two_One_Product(bdxt_adyt1, bdxt_adyt0, cdheighttail, u3, u[2], u[1], u[0]); u[3] = u3; finlength = fast_expansion_sum_zeroelim(finlength, finnow, 4, u, finother); finswap = finnow; finnow = finother; finother = finswap; } } } if (cdxtail != 0.0) { if (adytail != 0.0) { Two_Product(cdxtail, adytail, cdxt_adyt1, cdxt_adyt0); Two_One_Product(cdxt_adyt1, cdxt_adyt0, bdheight, u3, u[2], u[1], u[0]); u[3] = u3; finlength = fast_expansion_sum_zeroelim(finlength, finnow, 4, u, finother); finswap = finnow; finnow = finother; finother = finswap; if (bdheighttail != 0.0) { Two_One_Product(cdxt_adyt1, cdxt_adyt0, bdheighttail, u3, u[2], u[1], u[0]); u[3] = u3; finlength = fast_expansion_sum_zeroelim(finlength, finnow, 4, u, finother); finswap = finnow; finnow = finother; finother = finswap; } } if (bdytail != 0.0) { negate = -cdxtail; Two_Product(negate, bdytail, cdxt_bdyt1, cdxt_bdyt0); Two_One_Product(cdxt_bdyt1, cdxt_bdyt0, adheight, u3, u[2], u[1], u[0]); u[3] = u3; finlength = fast_expansion_sum_zeroelim(finlength, finnow, 4, u, finother); finswap = finnow; finnow = finother; finother = finswap; if (adheighttail != 0.0) { Two_One_Product(cdxt_bdyt1, cdxt_bdyt0, adheighttail, u3, u[2], u[1], u[0]); u[3] = u3; finlength = fast_expansion_sum_zeroelim(finlength, finnow, 4, u, finother); finswap = finnow; finnow = finother; finother = finswap; } } } if (adheighttail != 0.0) { wlength = scale_expansion_zeroelim(bctlen, bct, adheighttail, w); finlength = fast_expansion_sum_zeroelim(finlength, finnow, wlength, w, finother); finswap = finnow; finnow = finother; finother = finswap; } if (bdheighttail != 0.0) { wlength = scale_expansion_zeroelim(catlen, cat, bdheighttail, w); finlength = fast_expansion_sum_zeroelim(finlength, finnow, wlength, w, finother); finswap = finnow; finnow = finother; finother = finswap; } if (cdheighttail != 0.0) { wlength = scale_expansion_zeroelim(abtlen, abt, cdheighttail, w); finlength = fast_expansion_sum_zeroelim(finlength, finnow, wlength, w, finother); finswap = finnow; finnow = finother; finother = finswap; } return finnow[finlength - 1]; } REAL orient3d(struct mesh *m, struct behavior *b, vertex pa, vertex pb, vertex pc, vertex pd, REAL aheight, REAL bheight, REAL cheight, REAL dheight) { REAL adx, bdx, cdx, ady, bdy, cdy, adheight, bdheight, cdheight; REAL bdxcdy, cdxbdy, cdxady, adxcdy, adxbdy, bdxady; REAL det; REAL permanent, errbound; m->orient3dcount++; 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]; adheight = aheight - dheight; bdheight = bheight - dheight; cdheight = cheight - dheight; bdxcdy = bdx * cdy; cdxbdy = cdx * bdy; cdxady = cdx * ady; adxcdy = adx * cdy; adxbdy = adx * bdy; bdxady = bdx * ady; det = adheight * (bdxcdy - cdxbdy) + bdheight * (cdxady - adxcdy) + cdheight * (adxbdy - bdxady); if (b->noexact) { return det; } permanent = (Absolute(bdxcdy) + Absolute(cdxbdy)) * Absolute(adheight) + (Absolute(cdxady) + Absolute(adxcdy)) * Absolute(bdheight) + (Absolute(adxbdy) + Absolute(bdxady)) * Absolute(cdheight); errbound = o3derrboundA * permanent; if ((det > errbound) || (-det > errbound)) { return det; } return orient3dadapt(pa, pb, pc, pd, aheight, bheight, cheight, dheight, permanent); } /*****************************************************************************/ /* */ /* nonregular() Return a positive value if the point pd is incompatible */ /* with the circle or plane passing through pa, pb, and pc */ /* (meaning that pd is inside the circle or below the */ /* plane); a negative value if it is compatible; and zero if */ /* the four points are cocircular/coplanar. The points pa, */ /* pb, and pc must be in counterclockwise order, or the sign */ /* of the result will be reversed. */ /* */ /* If the -w switch is used, the points are lifted onto the parabolic */ /* lifting map, then they are dropped according to their weights, then the */ /* 3D orientation test is applied. If the -W switch is used, the points' */ /* heights are already provided, so the 3D orientation test is applied */ /* directly. If neither switch is used, the incircle test is applied. */ /* */ /*****************************************************************************/ REAL nonregular(struct mesh *m, struct behavior *b, vertex pa, vertex pb, vertex pc, vertex pd) { if (b->weighted == 0) { return incircle(m, b, pa, pb, pc, pd); } else if (b->weighted == 1) { return orient3d(m, b, pa, pb, pc, pd, pa[0] * pa[0] + pa[1] * pa[1] - pa[2], pb[0] * pb[0] + pb[1] * pb[1] - pb[2], pc[0] * pc[0] + pc[1] * pc[1] - pc[2], pd[0] * pd[0] + pd[1] * pd[1] - pd[2]); } else { return orient3d(m, b, pa, pb, pc, pd, pa[2], pb[2], pc[2], pd[2]); } } /*****************************************************************************/ /* */ /* findcircumcenter() Find the circumcenter of a triangle. */ /* */ /* The result is returned both in terms of x-y coordinates and xi-eta */ /* (barycentric) 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. */ /* This procedure also returns the square of the length of the triangle's */ /* shortest edge. */ /* */ /*****************************************************************************/ void findcircumcenter(struct mesh *m, struct behavior *b, vertex torg, vertex tdest, vertex tapex, vertex circumcenter, REAL *xi, REAL *eta, int offcenter) { REAL xdo, ydo, xao, yao; REAL dodist, aodist, dadist; REAL denominator; REAL dx, dy, dxoff, dyoff; m->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; dadist = (tdest[0] - tapex[0]) * (tdest[0] - tapex[0]) + (tdest[1] - tapex[1]) * (tdest[1] - tapex[1]); if (b->noexact) { denominator = 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 = 0.5 / counterclockwise(m, b, tdest, tapex, torg); /* Don't count the above as an orientation test. */ m->counterclockcount--; } dx = (yao * dodist - ydo * aodist) * denominator; dy = (xdo * aodist - xao * dodist) * denominator; /* Find the (squared) length of the triangle's shortest edge. This */ /* serves as a conservative estimate of the insertion radius of the */ /* circumcenter's parent. The estimate is used to ensure that */ /* the algorithm terminates even if very small angles appear in */ /* the input PSLG. */ if ((dodist < aodist) && (dodist < dadist)) { if (offcenter && (b->offconstant > 0.0)) { /* Find the position of the off-center, as described by Alper Ungor. */ dxoff = 0.5 * xdo - b->offconstant * ydo; dyoff = 0.5 * ydo + b->offconstant * xdo; /* If the off-center is closer to the origin than the */ /* circumcenter, use the off-center instead. */ if (dxoff * dxoff + dyoff * dyoff < dx * dx + dy * dy) { dx = dxoff; dy = dyoff; } } } else if (aodist < dadist) { if (offcenter && (b->offconstant > 0.0)) { dxoff = 0.5 * xao + b->offconstant * yao; dyoff = 0.5 * yao - b->offconstant * xao; /* If the off-center is closer to the origin than the */ /* circumcenter, use the off-center instead. */ if (dxoff * dxoff + dyoff * dyoff < dx * dx + dy * dy) { dx = dxoff; dy = dyoff; } } } else { if (offcenter && (b->offconstant > 0.0)) { dxoff = 0.5 * (tapex[0] - tdest[0]) - b->offconstant * (tapex[1] - tdest[1]); dyoff = 0.5 * (tapex[1] - tdest[1]) + b->offconstant * (tapex[0] - tdest[0]); /* If the off-center is closer to the destination than the */ /* circumcenter, use the off-center instead. */ if (dxoff * dxoff + dyoff * dyoff < (dx - xdo) * (dx - xdo) + (dy - ydo) * (dy - ydo)) { dx = xdo + dxoff; dy = ydo + dyoff; } } } circumcenter[0] = torg[0] + dx; circumcenter[1] = torg[1] + dy; /* To interpolate vertex attributes for the new vertex 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. */ *xi = (yao * dx - xao * dy) * (2.0 * denominator); *eta = (xdo * dy - ydo * dx) * (2.0 * denominator); } /** **/ /** **/ /********* Geometric primitives end here *********/ /*****************************************************************************/ /* */ /* triangleinit() Initialize some variables. */ /* */ /*****************************************************************************/ void triangleinit(struct mesh *m) { poolzero(&m->vertices); poolzero(&m->triangles); poolzero(&m->subsegs); poolzero(&m->viri); poolzero(&m->badsubsegs); poolzero(&m->badtriangles); poolzero(&m->flipstackers); poolzero(&m->splaynodes); m->recenttri.tri = (triangle *) NULL; /* No triangle has been visited yet. */ m->undeads = 0; /* No eliminated input vertices yet. */ m->samples = 1; /* Point location should take at least one sample. */ m->checksegments = 0; /* There are no segments in the triangulation yet. */ m->checkquality = 0; /* The quality triangulation stage has not begun. */ m->incirclecount = m->counterclockcount = m->orient3dcount = 0; m->hyperbolacount = m->circletopcount = m->circumcentercount = 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(unsigned int choices) { randomseed = (randomseed * 1366l + 150889l) % 714025l; return randomseed / (714025l / choices + 1); } /********* Point location routines begin here *********/ /** **/ /** **/ /*****************************************************************************/ /* */ /* makevertexmap() Construct a mapping from vertices 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 vertex is a corner */ /* of several triangles, but in the end every vertex will point to some */ /* triangle that contains it. */ /* */ /*****************************************************************************/ void makevertexmap(struct mesh *m, struct behavior *b) { struct otri triangleloop; vertex triorg; if (b->verbose) { printf(" Constructing mapping from vertices to triangles.\n"); } traversalinit(&m->triangles); triangleloop.tri = triangletraverse(m); while (triangleloop.tri != (triangle *) NULL) { /* Check all three vertices of the triangle. */ for (triangleloop.orient = 0; triangleloop.orient < 3; triangleloop.orient++) { org(triangleloop, triorg); setvertex2tri(triorg, encode(triangleloop)); } triangleloop.tri = triangletraverse(m); } } /*****************************************************************************/ /* */ /* 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. You must screen out the */ /* possibility that the vertex sought is an endpoint of the starting */ /* edge before you call preciselocate(). */ /* */ /* 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. */ /* */ /* If `stopatsubsegment' is nonzero, the search will stop if it tries to */ /* walk through a subsegment, and will return OUTSIDE. */ /* */ /* 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(struct mesh *m, struct behavior *b, vertex searchpoint, struct otri *searchtri, int stopatsubsegment) { struct otri backtracktri; struct osub checkedge; vertex forg, fdest, fapex; REAL orgorient, destorient; int moveleft; triangle ptr; /* Temporary variable used by sym(). */ subseg sptr; /* Temporary variable used by tspivot(). */ if (b->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 (b->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(m, b, 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(m, b, 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) and 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. */ if (moveleft) { lprev(*searchtri, backtracktri); fdest = fapex; } else { lnext(*searchtri, backtracktri); forg = fapex; } sym(backtracktri, *searchtri); if (m->checksegments && stopatsubsegment) { /* Check for walking through a subsegment. */ tspivot(backtracktri, checkedge); if (checkedge.ss != m->dummysub) { /* Go back to the last triangle. */ otricopy(backtracktri, *searchtri); return OUTSIDE; } } /* Check for walking right out of the triangulation. */ if (searchtri->tri == m->dummytri) { /* Go back to the last triangle. */ otricopy(backtracktri, *searchtri); return OUTSIDE; } 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 searching 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(struct mesh *m, struct behavior *b, vertex searchpoint, struct otri *searchtri) { VOID **sampleblock; char *firsttri; struct otri sampletri; vertex torg, tdest; unsigned long alignptr; REAL searchdist, dist; REAL ahead; long samplesperblock, totalsamplesleft, samplesleft; long population, totalpopulation; triangle ptr; /* Temporary variable used by sym(). */ if (b->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 (b->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 (m->recenttri.tri != (triangle *) NULL) { if (!deadtri(m->recenttri.tri)) { org(m->recenttri, torg); if ((torg[0] == searchpoint[0]) && (torg[1] == searchpoint[1])) { otricopy(m->recenttri, *searchtri); return ONVERTEX; } dist = (searchpoint[0] - torg[0]) * (searchpoint[0] - torg[0]) + (searchpoint[1] - torg[1]) * (searchpoint[1] - torg[1]); if (dist < searchdist) { otricopy(m->recenttri, *searchtri); searchdist = dist; if (b->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 (or at least */ /* doesn't decrease enough to matter). */ while (SAMPLEFACTOR * m->samples * m->samples * m->samples < m->triangles.items) { m->samples++; } /* We'll draw ceiling(samples * TRIPERBLOCK / maxitems) random samples */ /* from each block of triangles (except the first)--until we meet the */ /* sample quota. The ceiling means that blocks at the end might be */ /* neglected, but I don't care. */ samplesperblock = (m->samples * TRIPERBLOCK - 1) / m->triangles.maxitems + 1; /* We'll draw ceiling(samples * itemsfirstblock / maxitems) random samples */ /* from the first block of triangles. */ samplesleft = (m->samples * m->triangles.itemsfirstblock - 1) / m->triangles.maxitems + 1; totalsamplesleft = m->samples; population = m->triangles.itemsfirstblock; totalpopulation = m->triangles.maxitems; sampleblock = m->triangles.firstblock; sampletri.orient = 0; while (totalsamplesleft > 0) { /* If we're in the last block, `population' needs to be corrected. */ if (population > totalpopulation) { population = totalpopulation; } /* Find a pointer to the first triangle in the block. */ alignptr = (unsigned long) (sampleblock + 1); firsttri = (char *) (alignptr + (unsigned long) m->triangles.alignbytes - (alignptr % (unsigned long) m->triangles.alignbytes)); /* Choose `samplesleft' randomly sampled triangles in this block. */ do { sampletri.tri = (triangle *) (firsttri + (randomnation((unsigned int) population) * m->triangles.itembytes)); if (!deadtri(sampletri.tri)) { org(sampletri, torg); dist = (searchpoint[0] - torg[0]) * (searchpoint[0] - torg[0]) + (searchpoint[1] - torg[1]) * (searchpoint[1] - torg[1]); if (dist < searchdist) { otricopy(sampletri, *searchtri); searchdist = dist; if (b->verbose > 2) { printf(" Choosing triangle with origin (%.12g, %.12g).\n", torg[0], torg[1]); } } } samplesleft--; totalsamplesleft--; } while ((samplesleft > 0) && (totalsamplesleft > 0)); if (totalsamplesleft > 0) { sampleblock = (VOID **) *sampleblock; samplesleft = samplesperblock; totalpopulation -= population; population = TRIPERBLOCK; } } /* 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(m, b, 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(m, b, searchpoint, searchtri, 0); } /** **/ /** **/ /********* Point location routines end here *********/ /********* Mesh transformation routines begin here *********/ /** **/ /** **/ /*****************************************************************************/ /* */ /* insertsubseg() Create a new subsegment and insert it between two */ /* triangles. */ /* */ /* The new subsegment is inserted at the edge described by the handle */ /* `tri'. Its vertices are properly initialized. The marker `subsegmark' */ /* is applied to the subsegment and, if appropriate, its vertices. */ /* */ /*****************************************************************************/ void insertsubseg(struct mesh *m, struct behavior *b, struct otri *tri, int subsegmark) { struct otri oppotri; struct osub newsubseg; vertex triorg, tridest; triangle ptr; /* Temporary variable used by sym(). */ subseg sptr; /* Temporary variable used by tspivot(). */ org(*tri, triorg); dest(*tri, tridest); /* Mark vertices if possible. */ if (vertexmark(triorg) == 0) { setvertexmark(triorg, subsegmark); } if (vertexmark(tridest) == 0) { setvertexmark(tridest, subsegmark); } /* Check if there's already a subsegment here. */ tspivot(*tri, newsubseg); if (newsubseg.ss == m->dummysub) { /* Make new subsegment and initialize its vertices. */ makesubseg(m, &newsubseg); setsorg(newsubseg, tridest); setsdest(newsubseg, triorg); setsegorg(newsubseg, tridest); setsegdest(newsubseg, triorg); /* Bond new subsegment to the two triangles it is sandwiched between. */ /* Note that the facing triangle `oppotri' might be equal to */ /* `dummytri' (outer space), but the new subsegment is bonded to it */ /* all the same. */ tsbond(*tri, newsubseg); sym(*tri, oppotri); ssymself(newsubseg); tsbond(oppotri, newsubseg); setmark(newsubseg, subsegmark); if (b->verbose > 2) { printf(" Inserting new "); printsubseg(m, b, &newsubseg); } } else { if (mark(newsubseg) == 0) { setmark(newsubseg, subsegmark); } } } /*****************************************************************************/ /* */ /* 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 */ /* vertex. */ /* */ /* 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. */ /* */ /*****************************************************************************/ /*****************************************************************************/ /* */ /* flip() Transform two triangles to two different triangles by flipping */ /* an edge counterclockwise within a quadrilateral. */ /* */ /* Imagine the original triangles, abc and bad, oriented so that the */ /* shared edge ab lies in a horizontal plane, with the vertex b on the left */ /* and the vertex a on the right. The vertex c lies below the edge, and */ /* the vertex 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 subsegment 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(struct mesh *m, struct behavior *b, struct otri *flipedge) { struct otri botleft, botright; struct otri topleft, topright; struct otri top; struct otri botlcasing, botrcasing; struct otri toplcasing, toprcasing; struct osub botlsubseg, botrsubseg; struct osub toplsubseg, toprsubseg; vertex leftvertex, rightvertex, botvertex; vertex farvertex; triangle ptr; /* Temporary variable used by sym(). */ subseg sptr; /* Temporary variable used by tspivot(). */ /* Identify the vertices of the quadrilateral. */ org(*flipedge, rightvertex); dest(*flipedge, leftvertex); apex(*flipedge, botvertex); sym(*flipedge, top); #ifdef SELF_CHECK if (top.tri == m->dummytri) { printf("Internal error in flip(): Attempt to flip on boundary.\n"); lnextself(*flipedge); return; } if (m->checksegments) { tspivot(*flipedge, toplsubseg); if (toplsubseg.ss != m->dummysub) { printf("Internal error in flip(): Attempt to flip a segment.\n"); lnextself(*flipedge); return; } } #endif /* SELF_CHECK */ apex(top, farvertex); /* 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 (m->checksegments) { /* Check for subsegments and rebond them to the quadrilateral. */ tspivot(topleft, toplsubseg); tspivot(botleft, botlsubseg); tspivot(botright, botrsubseg); tspivot(topright, toprsubseg); if (toplsubseg.ss == m->dummysub) { tsdissolve(topright); } else { tsbond(topright, toplsubseg); } if (botlsubseg.ss == m->dummysub) { tsdissolve(topleft); } else { tsbond(topleft, botlsubseg); } if (botrsubseg.ss == m->dummysub) { tsdissolve(botleft); } else { tsbond(botleft, botrsubseg); } if (toprsubseg.ss == m->dummysub) { tsdissolve(botright); } else { tsbond(botright, toprsubseg); } } /* New vertex assignments for the rotated quadrilateral. */ setorg(*flipedge, farvertex); setdest(*flipedge, botvertex); setapex(*flipedge, rightvertex); setorg(top, botvertex); setdest(top, farvertex); setapex(top, leftvertex); if (b->verbose > 2) { printf(" Edge flip results in left "); printtriangle(m, b, &top); printf(" and right "); printtriangle(m, b, flipedge); } } /*****************************************************************************/ /* */ /* unflip() Transform two triangles to two different triangles by */ /* flipping an edge clockwise within a quadrilateral. Reverses */ /* the flip() operation so that the data structures representing */ /* the triangles are back where they were before the flip(). */ /* */ /* Imagine the original triangles, abc and bad, oriented so that the */ /* shared edge ab lies in a horizontal plane, with the vertex b on the left */ /* and the vertex a on the right. The vertex c lies below the edge, and */ /* the vertex 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 cdb and dca, 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 */ /* cd of triangle cdb, and is directed up, from vertex c to vertex d. */ /* (Hence, the two triangles have rotated clockwise.) */ /* */ /* WARNING: This transformation is geometrically valid only if the */ /* quadrilateral adbc is convex. Furthermore, this transformation is */ /* valid only if there is not a subsegment 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 unflip(struct mesh *m, struct behavior *b, struct otri *flipedge) { struct otri botleft, botright; struct otri topleft, topright; struct otri top; struct otri botlcasing, botrcasing; struct otri toplcasing, toprcasing; struct osub botlsubseg, botrsubseg; struct osub toplsubseg, toprsubseg; vertex leftvertex, rightvertex, botvertex; vertex farvertex; triangle ptr; /* Temporary variable used by sym(). */ subseg sptr; /* Temporary variable used by tspivot(). */ /* Identify the vertices of the quadrilateral. */ org(*flipedge, rightvertex); dest(*flipedge, leftvertex); apex(*flipedge, botvertex); sym(*flipedge, top); #ifdef SELF_CHECK if (top.tri == m->dummytri) { printf("Internal error in unflip(): Attempt to flip on boundary.\n"); lnextself(*flipedge); return; } if (m->checksegments) { tspivot(*flipedge, toplsubseg); if (toplsubseg.ss != m->dummysub) { printf("Internal error in unflip(): Attempt to flip a subsegment.\n"); lnextself(*flipedge); return; } } #endif /* SELF_CHECK */ apex(top, farvertex); /* 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 clockwise. */ bond(topleft, toprcasing); bond(botleft, toplcasing); bond(botright, botlcasing); bond(topright, botrcasing); if (m->checksegments) { /* Check for subsegments and rebond them to the quadrilateral. */ tspivot(topleft, toplsubseg); tspivot(botleft, botlsubseg); tspivot(botright, botrsubseg); tspivot(topright, toprsubseg); if (toplsubseg.ss == m->dummysub) { tsdissolve(botleft); } else { tsbond(botleft, toplsubseg); } if (botlsubseg.ss == m->dummysub) { tsdissolve(botright); } else { tsbond(botright, botlsubseg); } if (botrsubseg.ss == m->dummysub) { tsdissolve(topright); } else { tsbond(topright, botrsubseg); } if (toprsubseg.ss == m->dummysub) { tsdissolve(topleft); } else { tsbond(topleft, toprsubseg); } } /* New vertex assignments for the rotated quadrilateral. */ setorg(*flipedge, botvertex); setdest(*flipedge, farvertex); setapex(*flipedge, leftvertex); setorg(top, farvertex); setdest(top, botvertex); setapex(top, rightvertex); if (b->verbose > 2) { printf(" Edge unflip results in left "); printtriangle(m, b, flipedge); printf(" and right "); printtriangle(m, b, &top); } } /*****************************************************************************/ /* */ /* insertvertex() Insert a vertex into a Delaunay triangulation, */ /* performing flips as necessary to maintain the Delaunay */ /* property. */ /* */ /* The point `insertvertex' 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 `insertvertex' is found inside a triangle, the triangle is split into */ /* three; if `insertvertex' 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 `insertvertex' lies on an */ /* existing vertex, no action is taken, and the value DUPLICATEVERTEX is */ /* returned. On return, `searchtri' is set to a handle whose origin is the */ /* existing vertex. */ /* */ /* Normally, the parameter `splitseg' is set to NULL, implying that no */ /* subsegment should be split. In this case, if `insertvertex' is found to */ /* lie on a segment, no action is taken, and the value VIOLATINGVERTEX is */ /* returned. On return, `searchtri' is set to a handle whose primary edge */ /* is the violated subsegment. */ /* */ /* If the calling routine wishes to split a subsegment by inserting a */ /* vertex in it, the parameter `splitseg' should be that subsegment. In */ /* this case, `searchtri' MUST be the triangle handle reached by pivoting */ /* from that subsegment; no point location is done. */ /* */ /* `segmentflaws' and `triflaws' are flags that indicate whether or not */ /* there should be checks for the creation of encroached subsegments or bad */ /* quality triangles. If a newly inserted vertex encroaches upon */ /* subsegments, these subsegments are added to the list of subsegments 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 vertex or violated segment does not prevent the vertex */ /* from being inserted, the return value will be ENCROACHINGVERTEX if the */ /* vertex encroaches upon a subsegment (and checking is enabled), or */ /* SUCCESSFULVERTEX otherwise. In either case, `searchtri' is set to a */ /* handle whose origin is the newly inserted vertex. */ /* */ /* insertvertex() does not use flip() for reasons of speed; some */ /* information can be reused from edge flip to edge flip, like the */ /* locations of subsegments. */ /* */ /*****************************************************************************/ enum insertvertexresult insertvertex(struct mesh *m, struct behavior *b, vertex newvertex, struct otri *searchtri, struct osub *splitseg, int segmentflaws, int triflaws) { struct otri horiz; struct otri top; struct otri botleft, botright; struct otri topleft, topright; struct otri newbotleft, newbotright; struct otri newtopright; struct otri botlcasing, botrcasing; struct otri toplcasing, toprcasing; struct otri testtri; struct osub botlsubseg, botrsubseg; struct osub toplsubseg, toprsubseg; struct osub brokensubseg; struct osub checksubseg; struct osub rightsubseg; struct osub newsubseg; struct badsubseg *encroached; struct flipstacker *newflip; vertex first; vertex leftvertex, rightvertex, botvertex, topvertex, farvertex; vertex segmentorg, segmentdest; REAL attrib; REAL area; enum insertvertexresult success; enum locateresult intersect; int doflip; int mirrorflag; int enq; int i; triangle ptr; /* Temporary variable used by sym(). */ subseg sptr; /* Temporary variable used by spivot() and tspivot(). */ if (b->verbose > 1) { printf(" Inserting (%.12g, %.12g).\n", newvertex[0], newvertex[1]); } if (splitseg == (struct osub *) NULL) { /* Find the location of the vertex to be inserted. Check if a good */ /* starting triangle has already been provided by the caller. */ if (searchtri->tri == m->dummytri) { /* Find a boundary triangle. */ horiz.tri = m->dummytri; horiz.orient = 0; symself(horiz); /* Search for a triangle containing `newvertex'. */ intersect = locate(m, b, newvertex, &horiz); } else { /* Start searching from the triangle provided by the caller. */ otricopy(*searchtri, horiz); intersect = preciselocate(m, b, newvertex, &horiz, 1); } } else { /* The calling routine provides the subsegment in which */ /* the vertex is inserted. */ otricopy(*searchtri, horiz); intersect = ONEDGE; } if (intersect == ONVERTEX) { /* There's already a vertex there. Return in `searchtri' a triangle */ /* whose origin is the existing vertex. */ otricopy(horiz, *searchtri); otricopy(horiz, m->recenttri); return DUPLICATEVERTEX; } if ((intersect == ONEDGE) || (intersect == OUTSIDE)) { /* The vertex falls on an edge or boundary. */ if (m->checksegments && (splitseg == (struct osub *) NULL)) { /* Check whether the vertex falls on a subsegment. */ tspivot(horiz, brokensubseg); if (brokensubseg.ss != m->dummysub) { /* The vertex falls on a subsegment, and hence will not be inserted. */ if (segmentflaws) { enq = b->nobisect != 2; if (enq && (b->nobisect == 1)) { /* This subsegment may be split only if it is an */ /* internal boundary. */ sym(horiz, testtri); enq = testtri.tri != m->dummytri; } if (enq) { /* Add the subsegment to the list of encroached subsegments. */ encroached = (struct badsubseg *) poolalloc(&m->badsubsegs); encroached->encsubseg = sencode(brokensubseg); sorg(brokensubseg, encroached->subsegorg); sdest(brokensubseg, encroached->subsegdest); if (b->verbose > 2) { printf( " Queueing encroached subsegment (%.12g, %.12g) (%.12g, %.12g).\n", encroached->subsegorg[0], encroached->subsegorg[1], encroached->subsegdest[0], encroached->subsegdest[1]); } } } /* Return a handle whose primary edge contains the vertex, */ /* which has not been inserted. */ otricopy(horiz, *searchtri); otricopy(horiz, m->recenttri); return VIOLATINGVERTEX; } } /* Insert the vertex 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 != m->dummytri; if (mirrorflag) { lnextself(topright); sym(topright, toprcasing); maketriangle(m, b, &newtopright); } else { /* Splitting a boundary edge increases the number of boundary edges. */ m->hullsize++; } maketriangle(m, b, &newbotright); /* Set the vertices of changed and new triangles. */ org(horiz, rightvertex); dest(horiz, leftvertex); apex(horiz, botvertex); setorg(newbotright, botvertex); setdest(newbotright, rightvertex); setapex(newbotright, newvertex); setorg(horiz, newvertex); for (i = 0; i < m->eextras; i++) { /* Set the element attributes of a new triangle. */ setelemattribute(newbotright, i, elemattribute(botright, i)); } if (b->vararea) { /* Set the area constraint of a new triangle. */ setareabound(newbotright, areabound(botright)); } if (mirrorflag) { dest(topright, topvertex); setorg(newtopright, rightvertex); setdest(newtopright, topvertex); setapex(newtopright, newvertex); setorg(topright, newvertex); for (i = 0; i < m->eextras; i++) { /* Set the element attributes of another new triangle. */ setelemattribute(newtopright, i, elemattribute(topright, i)); } if (b->vararea) { /* Set the area constraint of another new triangle. */ setareabound(newtopright, areabound(topright)); } } /* There may be subsegments that need to be bonded */ /* to the new triangle(s). */ if (m->checksegments) { tspivot(botright, botrsubseg); if (botrsubseg.ss != m->dummysub) { tsdissolve(botright); tsbond(newbotright, botrsubseg); } if (mirrorflag) { tspivot(topright, toprsubseg); if (toprsubseg.ss != m->dummysub) { tsdissolve(topright); tsbond(newtopright, toprsubseg); } } } /* 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 (splitseg != (struct osub *) NULL) { /* Split the subsegment into two. */ setsdest(*splitseg, newvertex); segorg(*splitseg, segmentorg); segdest(*splitseg, segmentdest); ssymself(*splitseg); spivot(*splitseg, rightsubseg); insertsubseg(m, b, &newbotright, mark(*splitseg)); tspivot(newbotright, newsubseg); setsegorg(newsubseg, segmentorg); setsegdest(newsubseg, segmentdest); sbond(*splitseg, newsubseg); ssymself(newsubseg); sbond(newsubseg, rightsubseg); ssymself(*splitseg); /* Transfer the subsegment's boundary marker to the vertex */ /* if required. */ if (vertexmark(newvertex) == 0) { setvertexmark(newvertex, mark(*splitseg)); } } if (m->checkquality) { poolrestart(&m->flipstackers); m->lastflip = (struct flipstacker *) poolalloc(&m->flipstackers); m->lastflip->flippedtri = encode(horiz); m->lastflip->prevflip = (struct flipstacker *) &insertvertex; } #ifdef SELF_CHECK if (counterclockwise(m, b, rightvertex, leftvertex, botvertex) < 0.0) { printf("Internal error in insertvertex():\n"); printf( " Clockwise triangle prior to edge vertex insertion (bottom).\n"); } if (mirrorflag) { if (counterclockwise(m, b, leftvertex, rightvertex, topvertex) < 0.0) { printf("Internal error in insertvertex():\n"); printf(" Clockwise triangle prior to edge vertex insertion (top).\n"); } if (counterclockwise(m, b, rightvertex, topvertex, newvertex) < 0.0) { printf("Internal error in insertvertex():\n"); printf( " Clockwise triangle after edge vertex insertion (top right).\n"); } if (counterclockwise(m, b, topvertex, leftvertex, newvertex) < 0.0) { printf("Internal error in insertvertex():\n"); printf( " Clockwise triangle after edge vertex insertion (top left).\n"); } } if (counterclockwise(m, b, leftvertex, botvertex, newvertex) < 0.0) { printf("Internal error in insertvertex():\n"); printf( " Clockwise triangle after edge vertex insertion (bottom left).\n"); } if (counterclockwise(m, b, botvertex, rightvertex, newvertex) < 0.0) { printf("Internal error in insertvertex():\n"); printf( " Clockwise triangle after edge vertex insertion (bottom right).\n"); } #endif /* SELF_CHECK */ if (b->verbose > 2) { printf(" Updating bottom left "); printtriangle(m, b, &botright); if (mirrorflag) { printf(" Updating top left "); printtriangle(m, b, &topright); printf(" Creating top right "); printtriangle(m, b, &newtopright); } printf(" Creating bottom right "); printtriangle(m, b, &newbotright); } /* Position `horiz' on the first edge to check for */ /* the Delaunay property. */ lnextself(horiz); } else { /* Insert the vertex in a triangle, splitting it into three. */ lnext(horiz, botleft); lprev(horiz, botright); sym(botleft, botlcasing); sym(botright, botrcasing); maketriangle(m, b, &newbotleft); maketriangle(m, b, &newbotright); /* Set the vertices of changed and new triangles. */ org(horiz, rightvertex); dest(horiz, leftvertex); apex(horiz, botvertex); setorg(newbotleft, leftvertex); setdest(newbotleft, botvertex); setapex(newbotleft, newvertex); setorg(newbotright, botvertex); setdest(newbotright, rightvertex); setapex(newbotright, newvertex); setapex(horiz, newvertex); for (i = 0; i < m->eextras; i++) { /* Set the element attributes of the new triangles. */ attrib = elemattribute(horiz, i); setelemattribute(newbotleft, i, attrib); setelemattribute(newbotright, i, attrib); } if (b->vararea) { /* Set the area constraint of the new triangles. */ area = areabound(horiz); setareabound(newbotleft, area); setareabound(newbotright, area); } /* There may be subsegments that need to be bonded */ /* to the new triangles. */ if (m->checksegments) { tspivot(botleft, botlsubseg); if (botlsubseg.ss != m->dummysub) { tsdissolve(botleft); tsbond(newbotleft, botlsubseg); } tspivot(botright, botrsubseg); if (botrsubseg.ss != m->dummysub) { tsdissolve(botright); tsbond(newbotright, botrsubseg); } } /* 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); if (m->checkquality) { poolrestart(&m->flipstackers); m->lastflip = (struct flipstacker *) poolalloc(&m->flipstackers); m->lastflip->flippedtri = encode(horiz); m->lastflip->prevflip = (struct flipstacker *) NULL; } #ifdef SELF_CHECK if (counterclockwise(m, b, rightvertex, leftvertex, botvertex) < 0.0) { printf("Internal error in insertvertex():\n"); printf(" Clockwise triangle prior to vertex insertion.\n"); } if (counterclockwise(m, b, rightvertex, leftvertex, newvertex) < 0.0) { printf("Internal error in insertvertex():\n"); printf(" Clockwise triangle after vertex insertion (top).\n"); } if (counterclockwise(m, b, leftvertex, botvertex, newvertex) < 0.0) { printf("Internal error in insertvertex():\n"); printf(" Clockwise triangle after vertex insertion (left).\n"); } if (counterclockwise(m, b, botvertex, rightvertex, newvertex) < 0.0) { printf("Internal error in insertvertex():\n"); printf(" Clockwise triangle after vertex insertion (right).\n"); } #endif /* SELF_CHECK */ if (b->verbose > 2) { printf(" Updating top "); printtriangle(m, b, &horiz); printf(" Creating left "); printtriangle(m, b, &newbotleft); printf(" Creating right "); printtriangle(m, b, &newbotright); } } /* The insertion is successful by default, unless an encroached */ /* subsegment is found. */ success = SUCCESSFULVERTEX; /* 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); rightvertex = first; dest(horiz, leftvertex); /* Circle until finished. */ while (1) { /* By default, the edge will be flipped. */ doflip = 1; if (m->checksegments) { /* Check for a subsegment, which cannot be flipped. */ tspivot(horiz, checksubseg); if (checksubseg.ss != m->dummysub) { /* The edge is a subsegment and cannot be flipped. */ doflip = 0; #ifndef CDT_ONLY if (segmentflaws) { /* Does the new vertex encroach upon this subsegment? */ if (checkseg4encroach(m, b, &checksubseg)) { success = ENCROACHINGVERTEX; } } #endif /* not CDT_ONLY */ } } if (doflip) { /* Check if the edge is a boundary edge. */ sym(horiz, top); if (top.tri == m->dummytri) { /* The edge is a boundary edge and cannot be flipped. */ doflip = 0; } else { /* Find the vertex on the other side of the edge. */ apex(top, farvertex); /* In the incremental Delaunay triangulation algorithm, any of */ /* `leftvertex', `rightvertex', and `farvertex' 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 ((leftvertex == m->infvertex1) || (leftvertex == m->infvertex2) || (leftvertex == m->infvertex3)) { /* `leftvertex' is infinitely distant. Check the convexity of */ /* the boundary of the triangulation. 'farvertex' might be */ /* infinite as well, but trust me, this same condition should */ /* be applied. */ doflip = counterclockwise(m, b, newvertex, rightvertex, farvertex) > 0.0; } else if ((rightvertex == m->infvertex1) || (rightvertex == m->infvertex2) || (rightvertex == m->infvertex3)) { /* `rightvertex' is infinitely distant. Check the convexity of */ /* the boundary of the triangulation. 'farvertex' might be */ /* infinite as well, but trust me, this same condition should */ /* be applied. */ doflip = counterclockwise(m, b, farvertex, leftvertex, newvertex) > 0.0; } else if ((farvertex == m->infvertex1) || (farvertex == m->infvertex2) || (farvertex == m->infvertex3)) { /* `farvertex' 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(m, b, leftvertex, newvertex, rightvertex, farvertex) > 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 (m->checksegments) { /* Check for subsegments and rebond them to the quadrilateral. */ tspivot(topleft, toplsubseg); tspivot(botleft, botlsubseg); tspivot(botright, botrsubseg); tspivot(topright, toprsubseg); if (toplsubseg.ss == m->dummysub) { tsdissolve(topright); } else { tsbond(topright, toplsubseg); } if (botlsubseg.ss == m->dummysub) { tsdissolve(topleft); } else { tsbond(topleft, botlsubseg); } if (botrsubseg.ss == m->dummysub) { tsdissolve(botleft); } else { tsbond(botleft, botrsubseg); } if (toprsubseg.ss == m->dummysub) { tsdissolve(botright); } else { tsbond(botright, toprsubseg); } } /* New vertex assignments for the rotated quadrilateral. */ setorg(horiz, farvertex); setdest(horiz, newvertex); setapex(horiz, rightvertex); setorg(top, newvertex); setdest(top, farvertex); setapex(top, leftvertex); for (i = 0; i < m->eextras; i++) { /* Take the average of the two triangles' attributes. */ attrib = 0.5 * (elemattribute(top, i) + elemattribute(horiz, i)); setelemattribute(top, i, attrib); setelemattribute(horiz, i, attrib); } if (b->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 = 0.5 * (areabound(top) + areabound(horiz)); } setareabound(top, area); setareabound(horiz, area); } if (m->checkquality) { newflip = (struct flipstacker *) poolalloc(&m->flipstackers); newflip->flippedtri = encode(horiz); newflip->prevflip = m->lastflip; m->lastflip = newflip; } #ifdef SELF_CHECK if (newvertex != (vertex) NULL) { if (counterclockwise(m, b, leftvertex, newvertex, rightvertex) < 0.0) { printf("Internal error in insertvertex():\n"); printf(" Clockwise triangle prior to edge flip (bottom).\n"); } /* The following test has been removed because constrainededge() */ /* sometimes generates inverted triangles that insertvertex() */ /* removes. */ /* if (counterclockwise(m, b, rightvertex, farvertex, leftvertex) < 0.0) { printf("Internal error in insertvertex():\n"); printf(" Clockwise triangle prior to edge flip (top).\n"); } */ if (counterclockwise(m, b, farvertex, leftvertex, newvertex) < 0.0) { printf("Internal error in insertvertex():\n"); printf(" Clockwise triangle after edge flip (left).\n"); } if (counterclockwise(m, b, newvertex, rightvertex, farvertex) < 0.0) { printf("Internal error in insertvertex():\n"); printf(" Clockwise triangle after edge flip (right).\n"); } } #endif /* SELF_CHECK */ if (b->verbose > 2) { printf(" Edge flip results in left "); lnextself(topleft); printtriangle(m, b, &topleft); printf(" and right "); printtriangle(m, b, &horiz); } /* On the next iterations, consider the two edges that were */ /* exposed (this is, are now visible to the newly inserted */ /* vertex) by the edge flip. */ lprevself(horiz); leftvertex = farvertex; } } } if (!doflip) { /* The handle `horiz' is accepted as locally Delaunay. */ #ifndef CDT_ONLY if (triflaws) { /* Check the triangle `horiz' for quality. */ testtriangle(m, b, &horiz); } #endif /* not CDT_ONLY */ /* Look for the next edge around the newly inserted vertex. */ lnextself(horiz); sym(horiz, testtri); /* Check for finishing a complete revolution about the new vertex, or */ /* falling outside of the triangulation. The latter will happen */ /* when a vertex is inserted at a boundary. */ if ((leftvertex == first) || (testtri.tri == m->dummytri)) { /* We're done. Return a triangle whose origin is the new vertex. */ lnext(horiz, *searchtri); lnext(horiz, m->recenttri); return success; } /* Finish finding the next edge around the newly inserted vertex. */ lnext(testtri, horiz); rightvertex = leftvertex; dest(horiz, leftvertex); } } } /********* 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. */ /* */ /*****************************************************************************/ /*****************************************************************************/ /* */ /* vertexsort() Sort an array of vertices 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 vertexsort(vertex *sortarray, int arraysize) { int left, right; int pivot; REAL pivotx, pivoty; vertex 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((unsigned int) arraysize); pivotx = sortarray[pivot][0]; pivoty = sortarray[pivot][1]; /* Split the array. */ left = -1; right = arraysize; while (left < right) { /* Search for a vertex 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 vertex 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 vertices. */ temp = sortarray[left]; sortarray[left] = sortarray[right]; sortarray[right] = temp; } } if (left > 1) { /* Recursively sort the left subset. */ vertexsort(sortarray, left); } if (right < arraysize - 2) { /* Recursively sort the right subset. */ vertexsort(&sortarray[right + 1], arraysize - right - 1); } } /*****************************************************************************/ /* */ /* vertexmedian() An order statistic algorithm, almost. Shuffles an */ /* array of vertices so that the first `median' vertices */ /* occur lexicographically before the remaining vertices. */ /* */ /* Uses the x-coordinate as the primary key if axis == 0; the y-coordinate */ /* if axis == 1. Very similar to the vertexsort() procedure, but runs in */ /* randomized linear time. */ /* */ /*****************************************************************************/ void vertexmedian(vertex *sortarray, int arraysize, int median, int axis) { int left, right; int pivot; REAL pivot1, pivot2; vertex 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((unsigned int) arraysize); pivot1 = sortarray[pivot][axis]; pivot2 = sortarray[pivot][1 - axis]; /* Split the array. */ left = -1; right = arraysize; while (left < right) { /* Search for a vertex 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 vertex 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 vertices. */ temp = sortarray[left]; sortarray[left] = sortarray[right]; sortarray[right] = temp; } } /* Unlike in vertexsort(), at most one of the following */ /* conditionals is true. */ if (left > median) { /* Recursively shuffle the left subset. */ vertexmedian(sortarray, left, median, axis); } if (right < median - 1) { /* Recursively shuffle the right subset. */ vertexmedian(&sortarray[right + 1], arraysize - right - 1, median - right - 1, axis); } } /*****************************************************************************/ /* */ /* alternateaxes() Sorts the vertices 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 vertices are */ /* always sorted by x-coordinate. */ /* */ /*****************************************************************************/ void alternateaxes(vertex *sortarray, int arraysize, int axis) { int divider; divider = arraysize >> 1; if (arraysize <= 3) { /* Recursive base case: subsets of two or three vertices will be */ /* handled specially, and should always be sorted by x-coordinate. */ axis = 0; } /* Partition with a horizontal or vertical cut. */ vertexmedian(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 vertex. */ /* */ /* 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(struct mesh *m, struct behavior *b, struct otri *farleft, struct otri *innerleft, struct otri *innerright, struct otri *farright, int axis) { struct otri leftcand, rightcand; struct otri baseedge; struct otri nextedge; struct otri sidecasing, topcasing, outercasing; struct otri checkedge; vertex innerleftdest; vertex innerrightorg; vertex innerleftapex, innerrightapex; vertex farleftpt, farrightpt; vertex farleftapex, farrightapex; vertex lowerleft, lowerright; vertex upperleft, upperright; vertex nextapex; vertex 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 (b->dwyer && (axis == 1)) { org(*farleft, farleftpt); apex(*farleft, farleftapex); dest(*farright, farrightpt); apex(*farright, farrightapex); /* The pointers to the extremal vertices are shifted to point to the */ /* topmost and bottommost vertex of each hull, rather than the */ /* leftmost and rightmost vertices. */ 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" vertex of the left hull. */ if (counterclockwise(m, b, innerleftdest, innerleftapex, innerrightorg) > 0.0) { lprevself(*innerleft); symself(*innerleft); innerleftdest = innerleftapex; apex(*innerleft, innerleftapex); changemade = 1; } /* Make innerrightorg the "bottommost" vertex of the right hull. */ if (counterclockwise(m, b, 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(m, b, &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 (b->verbose > 2) { printf(" Creating base bounding "); printtriangle(m, b, &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 vertex of */ /* the left triangulation. And vice-versa.) */ leftfinished = counterclockwise(m, b, upperleft, lowerleft, lowerright) <= 0.0; rightfinished = counterclockwise(m, b, upperright, lowerleft, lowerright) <= 0.0; if (leftfinished && rightfinished) { /* Create the top new bounding triangle. */ maketriangle(m, b, &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 (b->verbose > 2) { printf(" Creating top bounding "); printtriangle(m, b, &nextedge); } /* Special treatment for horizontal cuts. */ if (b->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 vertices are restored to the */ /* leftmost and rightmost vertices (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 != (vertex) NULL) { /* Check whether the edge is Delaunay. */ badedge = incircle(m, b, 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? */ otricopy(sidecasing, nextedge); apex(nextedge, nextapex); if (nextapex != (vertex) NULL) { /* Check whether the edge is Delaunay. */ badedge = incircle(m, b, 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 != (vertex) NULL) { /* Check whether the edge is Delaunay. */ badedge = incircle(m, b, 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? */ otricopy(sidecasing, nextedge); apex(nextedge, nextapex); if (nextapex != (vertex) NULL) { /* Check whether the edge is Delaunay. */ badedge = incircle(m, b, lowerleft, lowerright, upperright, nextapex) > 0.0; } else { /* Avoid eating right through the triangulation. */ badedge = 0; } } } } if (leftfinished || (!rightfinished && (incircle(m, b, 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 (b->verbose > 2) { printf(" Connecting "); printtriangle(m, b, &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 vertices) 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(struct mesh *m, struct behavior *b, vertex *sortarray, int vertices, int axis, struct otri *farleft, struct otri *farright) { struct otri midtri, tri1, tri2, tri3; struct otri innerleft, innerright; REAL area; int divider; if (b->verbose > 2) { printf(" Triangulating %d vertices.\n", vertices); } if (vertices == 2) { /* The triangulation of two vertices is an edge. An edge is */ /* represented by two bounding triangles. */ maketriangle(m, b, farleft); setorg(*farleft, sortarray[0]); setdest(*farleft, sortarray[1]); /* The apex is intentionally left NULL. */ maketriangle(m, b, 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 (b->verbose > 2) { printf(" Creating "); printtriangle(m, b, farleft); printf(" Creating "); printtriangle(m, b, 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(m, b, &midtri); maketriangle(m, b, &tri1); maketriangle(m, b, &tri2); maketriangle(m, b, &tri3); area = counterclockwise(m, b, sortarray[0], sortarray[1], sortarray[2]); if (area == 0.0) { /* Three collinear vertices; 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]. */ otricopy(tri1, *farleft); /* Ensure that the destination of `farright' is sortarray[2]. */ otricopy(tri2, *farright); } else { /* The three vertices 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]. */ otricopy(tri1, *farleft); /* Ensure that the destination of `farright' is sortarray[2]. */ if (area > 0.0) { otricopy(tri2, *farright); } else { lnext(*farleft, *farright); } } if (b->verbose > 2) { printf(" Creating "); printtriangle(m, b, &midtri); printf(" Creating "); printtriangle(m, b, &tri1); printf(" Creating "); printtriangle(m, b, &tri2); printf(" Creating "); printtriangle(m, b, &tri3); } return; } else { /* Split the vertices in half. */ divider = vertices >> 1; /* Recursively triangulate each half. */ divconqrecurse(m, b, sortarray, divider, 1 - axis, farleft, &innerleft); divconqrecurse(m, b, &sortarray[divider], vertices - divider, 1 - axis, &innerright, farright); if (b->verbose > 1) { printf(" Joining triangulations with %d and %d vertices.\n", divider, vertices - divider); } /* Merge the two triangulations into one. */ mergehulls(m, b, farleft, &innerleft, &innerright, farright, axis); } } long removeghosts(struct mesh *m, struct behavior *b, struct otri *startghost) { struct otri searchedge; struct otri dissolveedge; struct otri deadtriangle; vertex markorg; long hullsize; triangle ptr; /* Temporary variable used by sym(). */ if (b->verbose) { printf(" Removing ghost triangles.\n"); } /* Find an edge on the convex hull to start point location from. */ lprev(*startghost, searchedge); symself(searchedge); m->dummytri[0] = encode(searchedge); /* Remove the bounding box and count the convex hull edges. */ otricopy(*startghost, dissolveedge); hullsize = 0; do { hullsize++; lnext(dissolveedge, deadtriangle); lprevself(dissolveedge); symself(dissolveedge); /* If no PSLG is involved, set the boundary markers of all the vertices */ /* on the convex hull. If a PSLG is used, this step is done later. */ if (!b->poly) { /* Watch out for the case where all the input vertices are collinear. */ if (dissolveedge.tri != m->dummytri) { org(dissolveedge, markorg); if (vertexmark(markorg) == 0) { setvertexmark(markorg, 1); } } } /* Remove a bounding triangle from a convex hull triangle. */ dissolve(dissolveedge); /* Find the next bounding triangle. */ sym(deadtriangle, dissolveedge); /* Delete the bounding triangle. */ triangledealloc(m, deadtriangle.tri); } while (!otriequal(dissolveedge, *startghost)); return hullsize; } /*****************************************************************************/ /* */ /* divconqdelaunay() Form a Delaunay triangulation by the divide-and- */ /* conquer method. */ /* */ /* Sorts the vertices, calls a recursive procedure to triangulate them, and */ /* removes the bounding box, setting boundary markers as appropriate. */ /* */ /*****************************************************************************/ long divconqdelaunay(struct mesh *m, struct behavior *b) { vertex *sortarray; struct otri hullleft, hullright; int divider; int i, j; if (b->verbose) { printf(" Sorting vertices.\n"); } /* Allocate an array of pointers to vertices for sorting. */ sortarray = (vertex *) trimalloc(m->invertices * (int) sizeof(vertex)); traversalinit(&m->vertices); for (i = 0; i < m->invertices; i++) { sortarray[i] = vertextraverse(m); } /* Sort the vertices. */ vertexsort(sortarray, m->invertices); /* Discard duplicate vertices, which can really mess up the algorithm. */ i = 0; for (j = 1; j < m->invertices; j++) { if ((sortarray[i][0] == sortarray[j][0]) && (sortarray[i][1] == sortarray[j][1])) { if (!b->quiet) { printf( "Warning: A duplicate vertex at (%.12g, %.12g) appeared and was ignored.\n", sortarray[j][0], sortarray[j][1]); } setvertextype(sortarray[j], UNDEADVERTEX); m->undeads++; } else { i++; sortarray[i] = sortarray[j]; } } i++; if (b->dwyer) { /* Re-sort the array of vertices 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 (b->verbose) { printf(" Forming triangulation.\n"); } /* Form the Delaunay triangulation. */ divconqrecurse(m, b, sortarray, i, 0, &hullleft, &hullright); trifree((VOID *) sortarray); return removeghosts(m, b, &hullleft); } /** **/ /** **/ /********* Divide-and-conquer Delaunay triangulation ends here *********/ /********* General mesh construction routines begin here *********/ /** **/ /** **/ /*****************************************************************************/ /* */ /* delaunay() Form a Delaunay triangulation. */ /* */ /*****************************************************************************/ long delaunay(struct mesh *m, struct behavior *b) { long hulledges; m->eextras = 0; initializetrisubpools(m, b); #ifdef REDUCED if (!b->quiet) { printf( "Constructing Delaunay triangulation by divide-and-conquer method.\n"); } hulledges = divconqdelaunay(m, b); #else /* not REDUCED */ if (!b->quiet) { printf("Constructing Delaunay triangulation "); if (b->incremental) { printf("by incremental method.\n"); } else if (b->sweepline) { printf("by sweepline method.\n"); } else { printf("by divide-and-conquer method.\n"); } } if (b->incremental) { hulledges = incrementaldelaunay(m, b); } else if (b->sweepline) { hulledges = sweeplinedelaunay(m, b); } else { hulledges = divconqdelaunay(m, b); } #endif /* not REDUCED */ if (m->triangles.items == 0) { /* The input vertices were all collinear, so there are no triangles. */ return 0l; } else { return hulledges; } } /** **/ /** **/ /********* General mesh construction routines end here *********/ /********* Segment 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 `searchpoint', 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(struct mesh *m, struct behavior *b, struct otri *searchtri, vertex searchpoint) { struct otri checktri; vertex startvertex; vertex leftvertex, rightvertex; REAL leftccw, rightccw; int leftflag, rightflag; triangle ptr; /* Temporary variable used by onext() and oprev(). */ org(*searchtri, startvertex); dest(*searchtri, rightvertex); apex(*searchtri, leftvertex); /* Is `searchpoint' to the left? */ leftccw = counterclockwise(m, b, searchpoint, startvertex, leftvertex); leftflag = leftccw > 0.0; /* Is `searchpoint' to the right? */ rightccw = counterclockwise(m, b, startvertex, searchpoint, rightvertex); rightflag = rightccw > 0.0; if (leftflag && rightflag) { /* `searchtri' faces directly away from `searchpoint'. We could go left */ /* or right. Ask whether it's a triangle or a boundary on the left. */ onext(*searchtri, checktri); if (checktri.tri == m->dummytri) { leftflag = 0; } else { rightflag = 0; } } while (leftflag) { /* Turn left until satisfied. */ onextself(*searchtri); if (searchtri->tri == m->dummytri) { printf("Internal error in finddirection(): Unable to find a\n"); printf(" triangle leading from (%.12g, %.12g) to", startvertex[0], startvertex[1]); printf(" (%.12g, %.12g).\n", searchpoint[0], searchpoint[1]); internalerror(); } apex(*searchtri, leftvertex); rightccw = leftccw; leftccw = counterclockwise(m, b, searchpoint, startvertex, leftvertex); leftflag = leftccw > 0.0; } while (rightflag) { /* Turn right until satisfied. */ oprevself(*searchtri); if (searchtri->tri == m->dummytri) { printf("Internal error in finddirection(): Unable to find a\n"); printf(" triangle leading from (%.12g, %.12g) to", startvertex[0], startvertex[1]); printf(" (%.12g, %.12g).\n", searchpoint[0], searchpoint[1]); internalerror(); } dest(*searchtri, rightvertex); leftccw = rightccw; rightccw = counterclockwise(m, b, startvertex, searchpoint, rightvertex); 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 vertex at the intersection, splitting an */ /* existing subsegment. */ /* */ /* The segment being inserted connects the apex of splittri to endpoint2. */ /* splitsubseg is the subsegment being split, and MUST adjoin splittri. */ /* Hence, endpoints of the subsegment being split are 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(struct mesh *m, struct behavior *b, struct otri *splittri, struct osub *splitsubseg, vertex endpoint2) { struct osub opposubseg; vertex endpoint1; vertex torg, tdest; vertex leftvertex, rightvertex; vertex newvertex; enum insertvertexresult 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(). */ subseg sptr; /* Temporary variable used by snext(). */ /* 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(); return; } split = (ey * etx - ex * ety) / denom; /* Create the new vertex. */ newvertex = (vertex) poolalloc(&m->vertices); /* Interpolate its coordinate and attributes. */ for (i = 0; i < 2 + m->nextras; i++) { newvertex[i] = torg[i] + split * (tdest[i] - torg[i]); } setvertexmark(newvertex, mark(*splitsubseg)); setvertextype(newvertex, INPUTVERTEX); if (b->verbose > 1) { printf( " Splitting subsegment (%.12g, %.12g) (%.12g, %.12g) at (%.12g, %.12g).\n", torg[0], torg[1], tdest[0], tdest[1], newvertex[0], newvertex[1]); } /* Insert the intersection vertex. This should always succeed. */ success = insertvertex(m, b, newvertex, splittri, splitsubseg, 0, 0); if (success != SUCCESSFULVERTEX) { printf("Internal error in segmentintersection():\n"); printf(" Failure to split a segment.\n"); internalerror(); return; } /* Record a triangle whose origin is the new vertex. */ setvertex2tri(newvertex, encode(*splittri)); if (m->steinerleft > 0) { m->steinerleft--; } /* Divide the segment into two, and correct the segment endpoints. */ ssymself(*splitsubseg); spivot(*splitsubseg, opposubseg); sdissolve(*splitsubseg); sdissolve(opposubseg); do { setsegorg(*splitsubseg, newvertex); snextself(*splitsubseg); } while (splitsubseg->ss != m->dummysub); do { setsegorg(opposubseg, newvertex); snextself(opposubseg); } while (opposubseg.ss != m->dummysub); /* Inserting the vertex may have caused edge flips. We wish to rediscover */ /* the edge connecting endpoint1 to the new intersection vertex. */ collinear = finddirection(m, b, splittri, endpoint1); dest(*splittri, rightvertex); apex(*splittri, leftvertex); if ((leftvertex[0] == endpoint1[0]) && (leftvertex[1] == endpoint1[1])) { onextself(*splittri); } else if ((rightvertex[0] != endpoint1[0]) || (rightvertex[1] != endpoint1[1])) { printf("Internal error in segmentintersection():\n"); printf(" Topological inconsistency after splitting a segment.\n"); internalerror(); return; } /* `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 vertex, or 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 subsegment 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 subsegment is inserted connecting the first endpoint to */ /* the collinear vertex, and the search is continued from the collinear */ /* vertex. */ /* */ /* If the first triangle on the path has a subsegment opposite its origin, */ /* then there is a segment that intersects the segment being inserted. */ /* Their intersection vertex is inserted, splitting the subsegment. */ /* */ /*****************************************************************************/ int scoutsegment(struct mesh *m, struct behavior *b, struct otri *searchtri, vertex endpoint2, int newmark) { struct otri crosstri; struct osub crosssubseg; vertex leftvertex, rightvertex; enum finddirectionresult collinear; subseg sptr; /* Temporary variable used by tspivot(). */ collinear = finddirection(m, b, searchtri, endpoint2); dest(*searchtri, rightvertex); apex(*searchtri, leftvertex); if (((leftvertex[0] == endpoint2[0]) && (leftvertex[1] == endpoint2[1])) || ((rightvertex[0] == endpoint2[0]) && (rightvertex[1] == endpoint2[1]))) { /* The segment is already an edge in the mesh. */ if ((leftvertex[0] == endpoint2[0]) && (leftvertex[1] == endpoint2[1])) { lprevself(*searchtri); } /* Insert a subsegment, if there isn't already one there. */ insertsubseg(m, b, searchtri, newmark); return 1; } else if (collinear == LEFTCOLLINEAR) { /* We've collided with a vertex between the segment's endpoints. */ /* Make the collinear vertex be the triangle's origin. */ lprevself(*searchtri); insertsubseg(m, b, searchtri, newmark); /* Insert the remainder of the segment. */ return scoutsegment(m, b, searchtri, endpoint2, newmark); } else if (collinear == RIGHTCOLLINEAR) { /* We've collided with a vertex between the segment's endpoints. */ insertsubseg(m, b, searchtri, newmark); /* Make the collinear vertex be the triangle's origin. */ lnextself(*searchtri); /* Insert the remainder of the segment. */ return scoutsegment(m, b, searchtri, endpoint2, newmark); } else { lnext(*searchtri, crosstri); tspivot(crosstri, crosssubseg); /* Check for a crossing segment. */ if (crosssubseg.ss == m->dummysub) { return 0; } else { /* Insert a vertex at the intersection. */ segmentintersection(m, b, &crosstri, &crosssubseg, endpoint2); if (error_set) return -1; otricopy(crosstri, *searchtri); insertsubseg(m, b, searchtri, newmark); /* Insert the remainder of the segment. */ return scoutsegment(m, b, searchtri, endpoint2, newmark); } } } /*****************************************************************************/ /* */ /* delaunayfixup() Enforce the Delaunay condition at an edge, fanning out */ /* recursively from an existing vertex. 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 vertex (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(struct mesh *m, struct behavior *b, struct otri *fixuptri, int leftside) { struct otri neartri; struct otri fartri; struct osub faredge; vertex nearvertex, leftvertex, rightvertex, farvertex; triangle ptr; /* Temporary variable used by sym(). */ subseg 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 == m->dummytri) { return; } tspivot(neartri, faredge); if (faredge.ss != m->dummysub) { return; } /* Find all the relevant vertices. */ apex(neartri, nearvertex); org(neartri, leftvertex); dest(neartri, rightvertex); apex(fartri, farvertex); /* Check whether the previous polygon vertex is a reflex vertex. */ if (leftside) { if (counterclockwise(m, b, nearvertex, leftvertex, farvertex) <= 0.0) { /* leftvertex is a reflex vertex too. Nothing can */ /* be done until a convex section is found. */ return; } } else { if (counterclockwise(m, b, farvertex, rightvertex, nearvertex) <= 0.0) { /* rightvertex is a reflex vertex too. Nothing can */ /* be done until a convex section is found. */ return; } } if (counterclockwise(m, b, rightvertex, leftvertex, farvertex) > 0.0) { /* fartri is not an inverted triangle, and farvertex 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(m, b, leftvertex, farvertex, rightvertex, nearvertex) <= 0.0) { return; } /* Not locally Delaunay; go on to an edge flip. */ } /* else fartri is inverted; remove it from the stack by flipping. */ flip(m, b, &neartri); lprevself(*fixuptri); /* Restore the origin of fixuptri after the flip. */ /* Recursively process the two triangles that result from the flip. */ delaunayfixup(m, b, fixuptri, leftside); delaunayfixup(m, b, &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 subsegment 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(struct mesh *m, struct behavior *b, struct otri *starttri, vertex endpoint2, int newmark) { struct otri fixuptri, fixuptri2; struct osub crosssubseg; vertex endpoint1; vertex farvertex; REAL area; int collision; int done; triangle ptr; /* Temporary variable used by sym() and oprev(). */ subseg sptr; /* Temporary variable used by tspivot(). */ org(*starttri, endpoint1); lnext(*starttri, fixuptri); flip(m, b, &fixuptri); /* `collision' indicates whether we have found a vertex directly */ /* between endpoint1 and endpoint2. */ collision = 0; done = 0; do { org(fixuptri, farvertex); /* `farvertex' is the extreme point of the polygon we are "digging" */ /* to get from endpoint1 to endpoint2. */ if ((farvertex[0] == endpoint2[0]) && (farvertex[1] == endpoint2[1])) { oprev(fixuptri, fixuptri2); /* Enforce the Delaunay condition around endpoint2. */ delaunayfixup(m, b, &fixuptri, 0); delaunayfixup(m, b, &fixuptri2, 1); done = 1; } else { /* Check whether farvertex is to the left or right of the segment */ /* being inserted, to decide which edge of fixuptri to dig */ /* through next. */ area = counterclockwise(m, b, endpoint1, endpoint2, farvertex); if (area == 0.0) { /* We've collided with a vertex between endpoint1 and endpoint2. */ collision = 1; oprev(fixuptri, fixuptri2); /* Enforce the Delaunay condition around farvertex. */ delaunayfixup(m, b, &fixuptri, 0); delaunayfixup(m, b, &fixuptri2, 1); done = 1; } else { if (area > 0.0) { /* farvertex is to the left of the segment. */ oprev(fixuptri, fixuptri2); /* Enforce the Delaunay condition around farvertex, on the */ /* left side of the segment only. */ delaunayfixup(m, b, &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 { /* farvertex is to the right of the segment. */ delaunayfixup(m, b, &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, crosssubseg); if (crosssubseg.ss == m->dummysub) { flip(m, b, &fixuptri); /* May create inverted triangle at left. */ } else { /* We've collided with a segment between endpoint1 and endpoint2. */ collision = 1; /* Insert a vertex at the intersection. */ segmentintersection(m, b, &fixuptri, &crosssubseg, endpoint2); done = 1; } } } } while (!done); /* Insert a subsegment to make the segment permanent. */ insertsubseg(m, b, &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(m, b, &fixuptri, endpoint2, newmark)) { constrainededge(m, b, &fixuptri, endpoint2, newmark); } } } /*****************************************************************************/ /* */ /* insertsegment() Insert a PSLG segment into a triangulation. */ /* */ /*****************************************************************************/ void insertsegment(struct mesh *m, struct behavior *b, vertex endpoint1, vertex endpoint2, int newmark) { struct otri searchtri1, searchtri2; triangle encodedtri; vertex checkvertex; triangle ptr; /* Temporary variable used by sym(). */ if (b->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. */ checkvertex = (vertex) NULL; encodedtri = vertex2tri(endpoint1); if (encodedtri != (triangle) NULL) { decode(encodedtri, searchtri1); org(searchtri1, checkvertex); } if (checkvertex != endpoint1) { /* Find a boundary triangle to search from. */ searchtri1.tri = m->dummytri; searchtri1.orient = 0; symself(searchtri1); /* Search for the segment's first endpoint by point location. */ if (locate(m, b, endpoint1, &searchtri1) != ONVERTEX) { printf( "Internal error in insertsegment(): Unable to locate PSLG vertex\n"); printf(" (%.12g, %.12g) in triangulation.\n", endpoint1[0], endpoint1[1]); internalerror(); } } /* Remember this triangle to improve subsequent point location. */ otricopy(searchtri1, m->recenttri); /* Scout the beginnings of a path from the first endpoint */ /* toward the second. */ if (scoutsegment(m, b, &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. */ checkvertex = (vertex) NULL; encodedtri = vertex2tri(endpoint2); if (encodedtri != (triangle) NULL) { decode(encodedtri, searchtri2); org(searchtri2, checkvertex); } if (checkvertex != endpoint2) { /* Find a boundary triangle to search from. */ searchtri2.tri = m->dummytri; searchtri2.orient = 0; symself(searchtri2); /* Search for the segment's second endpoint by point location. */ if (locate(m, b, endpoint2, &searchtri2) != ONVERTEX) { printf( "Internal error in insertsegment(): Unable to locate PSLG vertex\n"); printf(" (%.12g, %.12g) in triangulation.\n", endpoint2[0], endpoint2[1]); internalerror(); } } /* Remember this triangle to improve subsequent point location. */ otricopy(searchtri2, m->recenttri); /* Scout the beginnings of a path from the second endpoint */ /* toward the first. */ if (scoutsegment(m, b, &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 (b->splitseg) { /* Insert vertices to force the segment into the triangulation. */ conformingedge(m, b, endpoint1, endpoint2, newmark); } else { #endif /* not CDT_ONLY */ #endif /* not REDUCED */ /* Insert the segment directly into the triangulation. */ constrainededge(m, b, &searchtri1, endpoint2, newmark); #ifndef REDUCED #ifndef CDT_ONLY } #endif /* not CDT_ONLY */ #endif /* not REDUCED */ } /*****************************************************************************/ /* */ /* markhull() Cover the convex hull of a triangulation with subsegments. */ /* */ /*****************************************************************************/ void markhull(struct mesh *m, struct behavior *b) { struct otri hulltri; struct otri nexttri; struct otri starttri; triangle ptr; /* Temporary variable used by sym() and oprev(). */ /* Find a triangle handle on the hull. */ hulltri.tri = m->dummytri; hulltri.orient = 0; symself(hulltri); /* Remember where we started so we know when to stop. */ otricopy(hulltri, starttri); /* Go once counterclockwise around the convex hull. */ do { /* Create a subsegment if there isn't already one here. */ insertsubseg(m, b, &hulltri, 1); /* To find the next hull edge, go clockwise around the next vertex. */ lnextself(hulltri); oprev(hulltri, nexttri); while (nexttri.tri != m->dummytri) { otricopy(nexttri, hulltri); oprev(hulltri, nexttri); } } while (!otriequal(hulltri, starttri)); } /*****************************************************************************/ /* */ /* formskeleton() Create the segments of a triangulation, including PSLG */ /* segments and edges on the convex hull. */ /* */ /* The PSLG segments are read from a .poly file. The return value is the */ /* number of segments in the file. */ /* */ /*****************************************************************************/ void formskeleton(struct mesh *m, struct behavior *b, int *segmentlist, int *segmentmarkerlist, int numberofsegments) { char polyfilename[6]; int index; vertex endpoint1, endpoint2; int segmentmarkers; int end1, end2; int boundmarker; int i; if (b->poly) { if (!b->quiet) { printf("Recovering segments in Delaunay triangulation.\n"); } strcpy(polyfilename, "input"); m->insegments = numberofsegments; segmentmarkers = segmentmarkerlist != (int *) NULL; index = 0; /* If the input vertices are collinear, there is no triangulation, */ /* so don't try to insert segments. */ if (m->triangles.items == 0) { return; } /* If segments are to be inserted, compute a mapping */ /* from vertices to triangles. */ if (m->insegments > 0) { makevertexmap(m, b); if (b->verbose) { printf(" Recovering PSLG segments.\n"); } } boundmarker = 0; /* Read and insert the segments. */ for (i = 0; i < m->insegments; i++) { end1 = segmentlist[index++]; end2 = segmentlist[index++]; if (segmentmarkers) { boundmarker = segmentmarkerlist[i]; } if ((end1 < b->firstnumber) || (end1 >= b->firstnumber + m->invertices)) { if (!b->quiet) { printf( "Warning: Invalid first endpoint of segment %d in %s.\n", b->firstnumber + i, polyfilename); } } else if ((end2 < b->firstnumber) || (end2 >= b->firstnumber + m->invertices)) { if (!b->quiet) { printf( "Warning: Invalid second endpoint of segment %d in %s.\n", b->firstnumber + i, polyfilename); } } else { /* Find the vertices numbered `end1' and `end2'. */ endpoint1 = getvertex(m, b, end1); endpoint2 = getvertex(m, b, end2); if ((endpoint1[0] == endpoint2[0]) && (endpoint1[1] == endpoint2[1])) { if (!b->quiet) { printf( "Warning: Endpoints of segment %d are coincident in %s.\n", b->firstnumber + i, polyfilename); } } else { insertsegment(m, b, endpoint1, endpoint2, boundmarker); } } } } else { m->insegments = 0; } if (b->convex || !b->poly) { /* Enclose the convex hull with subsegments. */ if (b->verbose) { printf(" Enclosing convex hull with segments.\n"); } markhull(m, b); } } /** **/ /** **/ /********* Segment 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 subsegments. Where there are */ /* subsegments, set boundary markers as appropriate. */ /* */ /*****************************************************************************/ void infecthull(struct mesh *m, struct behavior *b) { struct otri hulltri; struct otri nexttri; struct otri starttri; struct osub hullsubseg; triangle **deadtriangle; vertex horg, hdest; triangle ptr; /* Temporary variable used by sym(). */ subseg sptr; /* Temporary variable used by tspivot(). */ if (b->verbose) { printf(" Marking concavities (external triangles) for elimination.\n"); } /* Find a triangle handle on the hull. */ hulltri.tri = m->dummytri; hulltri.orient = 0; symself(hulltri); /* Remember where we started so we know when to stop. */ otricopy(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 subsegment? */ tspivot(hulltri, hullsubseg); if (hullsubseg.ss == m->dummysub) { /* The triangle is not protected; infect it. */ if (!infected(hulltri)) { infect(hulltri); deadtriangle = (triangle **) poolalloc(&m->viri); *deadtriangle = hulltri.tri; } } else { /* The triangle is protected; set boundary markers if appropriate. */ if (mark(hullsubseg) == 0) { setmark(hullsubseg, 1); org(hulltri, horg); dest(hulltri, hdest); if (vertexmark(horg) == 0) { setvertexmark(horg, 1); } if (vertexmark(hdest) == 0) { setvertexmark(hdest, 1); } } } } /* To find the next hull edge, go clockwise around the next vertex. */ lnextself(hulltri); oprev(hulltri, nexttri); while (nexttri.tri != m->dummytri) { otricopy(nexttri, hulltri); oprev(hulltri, nexttri); } } while (!otriequal(hulltri, starttri)); } /*****************************************************************************/ /* */ /* plague() Spread the virus from all infected triangles to any neighbors */ /* not protected by subsegments. 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 vertices. */ /* */ /*****************************************************************************/ void plague(struct mesh *m, struct behavior *b) { struct otri testtri; struct otri neighbor; triangle **virusloop; triangle **deadtriangle; struct osub neighborsubseg; vertex testvertex; vertex norg, ndest; vertex deadorg, deaddest, deadapex; int killorg; triangle ptr; /* Temporary variable used by sym() and onext(). */ subseg sptr; /* Temporary variable used by tspivot(). */ if (b->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(&m->viri); virusloop = (triangle **) traverse(&m->viri); while (virusloop != (triangle **) NULL) { testtri.tri = *virusloop; /* A triangle is marked as infected by messing with one of its pointers */ /* to subsegments, setting it to an illegal value. Hence, we have to */ /* temporarily uninfect this triangle so that we can examine its */ /* adjacent subsegments. */ uninfect(testtri); if (b->verbose > 2) { /* Assign the triangle an orientation for convenience in */ /* checking its vertices. */ 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 subsegment between the triangle and its neighbor. */ tspivot(testtri, neighborsubseg); /* Check if the neighbor is nonexistent or already infected. */ if ((neighbor.tri == m->dummytri) || infected(neighbor)) { if (neighborsubseg.ss != m->dummysub) { /* There is a subsegment separating the triangle from its */ /* neighbor, but both triangles are dying, so the subsegment */ /* dies too. */ subsegdealloc(m, neighborsubseg.ss); if (neighbor.tri != m->dummytri) { /* Make sure the subsegment 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 (neighborsubseg.ss == m->dummysub) { /* There is no subsegment protecting the neighbor, so */ /* the neighbor becomes infected. */ if (b->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. */ deadtriangle = (triangle **) poolalloc(&m->viri); *deadtriangle = neighbor.tri; } else { /* The neighbor is protected by a subsegment. */ /* Remove this triangle from the subsegment. */ stdissolve(neighborsubseg); /* The subsegment becomes a boundary. Set markers accordingly. */ if (mark(neighborsubseg) == 0) { setmark(neighborsubseg, 1); } org(neighbor, norg); dest(neighbor, ndest); if (vertexmark(norg) == 0) { setvertexmark(norg, 1); } if (vertexmark(ndest) == 0) { setvertexmark(ndest, 1); } } } } /* Remark the triangle as infected, so it doesn't get added to the */ /* virus pool again. */ infect(testtri); virusloop = (triangle **) traverse(&m->viri); } if (b->verbose) { printf(" Deleting marked triangles.\n"); } traversalinit(&m->viri); virusloop = (triangle **) traverse(&m->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 vertex, checking if it is */ /* still connected to at least one live triangle. */ for (testtri.orient = 0; testtri.orient < 3; testtri.orient++) { org(testtri, testvertex); /* Check if the vertex has already been tested. */ if (testvertex != (vertex) NULL) { killorg = 1; /* Mark the corner of the triangle as having been tested. */ setorg(testtri, NULL); /* Walk counterclockwise about the vertex. */ onext(testtri, neighbor); /* Stop upon reaching a boundary or the starting triangle. */ while ((neighbor.tri != m->dummytri) && (!otriequal(neighbor, testtri))) { if (infected(neighbor)) { /* Mark the corner of this triangle as having been tested. */ setorg(neighbor, NULL); } else { /* A live triangle. The vertex survives. */ killorg = 0; } /* Walk counterclockwise about the vertex. */ onextself(neighbor); } /* If we reached a boundary, we must walk clockwise as well. */ if (neighbor.tri == m->dummytri) { /* Walk clockwise about the vertex. */ oprev(testtri, neighbor); /* Stop upon reaching a boundary. */ while (neighbor.tri != m->dummytri) { if (infected(neighbor)) { /* Mark the corner of this triangle as having been tested. */ setorg(neighbor, NULL); } else { /* A live triangle. The vertex survives. */ killorg = 0; } /* Walk clockwise about the vertex. */ oprevself(neighbor); } } if (killorg) { if (b->verbose > 1) { printf(" Deleting vertex (%.12g, %.12g)\n", testvertex[0], testvertex[1]); } setvertextype(testvertex, UNDEADVERTEX); m->undeads++; } } } /* 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 == m->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. */ m->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. */ m->hullsize++; } } /* Return the dead triangle to the pool of triangles. */ triangledealloc(m, testtri.tri); virusloop = (triangle **) traverse(&m->viri); } /* Empty the virus pool. */ poolrestart(&m->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(struct mesh *m, struct behavior *b, REAL attribute, REAL area) { struct otri testtri; struct otri neighbor; triangle **virusloop; triangle **regiontri; struct osub neighborsubseg; vertex regionorg, regiondest, regionapex; triangle ptr; /* Temporary variable used by sym() and onext(). */ subseg sptr; /* Temporary variable used by tspivot(). */ if (b->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(&m->viri); virusloop = (triangle **) traverse(&m->viri); while (virusloop != (triangle **) NULL) { testtri.tri = *virusloop; /* A triangle is marked as infected by messing with one of its pointers */ /* to subsegments, setting it to an illegal value. Hence, we have to */ /* temporarily uninfect this triangle so that we can examine its */ /* adjacent subsegments. */ uninfect(testtri); if (b->regionattrib) { /* Set an attribute. */ setelemattribute(testtri, m->eextras, attribute); } if (b->vararea) { /* Set an area constraint. */ setareabound(testtri, area); } if (b->verbose > 2) { /* Assign the triangle an orientation for convenience in */ /* checking its vertices. */ 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 subsegment between the triangle and its neighbor. */ tspivot(testtri, neighborsubseg); /* Make sure the neighbor exists, is not already infected, and */ /* isn't protected by a subsegment. */ if ((neighbor.tri != m->dummytri) && !infected(neighbor) && (neighborsubseg.ss == m->dummysub)) { if (b->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(&m->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(&m->viri); } /* Uninfect all triangles. */ if (b->verbose > 1) { printf(" Unmarking marked triangles.\n"); } traversalinit(&m->viri); virusloop = (triangle **) traverse(&m->viri); while (virusloop != (triangle **) NULL) { testtri.tri = *virusloop; uninfect(testtri); virusloop = (triangle **) traverse(&m->viri); } /* Empty the virus pool. */ poolrestart(&m->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(struct mesh *m, struct behavior *b, REAL *holelist, int holes, REAL *regionlist, int regions) { struct otri searchtri; struct otri triangleloop; struct otri *regiontris; triangle **holetri; triangle **regiontri; vertex searchorg, searchdest; enum locateresult intersect; int i; triangle ptr; /* Temporary variable used by sym(). */ if (!(b->quiet || (b->noholes && b->convex))) { printf("Removing unwanted triangles.\n"); if (b->verbose && (holes > 0)) { printf(" Marking holes for elimination.\n"); } } if (regions > 0) { /* Allocate storage for the triangles in which region points fall. */ regiontris = (struct otri *) trimalloc(regions * (int) sizeof(struct otri)); } else { regiontris = (struct otri *) NULL; } if (((holes > 0) && !b->noholes) || !b->convex || (regions > 0)) { /* Initialize a pool of viri to be used for holes, concavities, */ /* regional attributes, and/or regional area constraints. */ poolinit(&m->viri, sizeof(triangle *), VIRUSPERBLOCK, VIRUSPERBLOCK, 0); } if (!b->convex) { /* Mark as infected any unprotected triangles on the boundary. */ /* This is one way by which concavities are created. */ infecthull(m, b); } if ((holes > 0) && !b->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] >= m->xmin) && (holelist[i] <= m->xmax) && (holelist[i + 1] >= m->ymin) && (holelist[i + 1] <= m->ymax)) { /* Start searching from some triangle on the outer boundary. */ searchtri.tri = m->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(m, b, searchorg, searchdest, &holelist[i]) > 0.0) { /* Find a triangle that contains the hole. */ intersect = locate(m, b, &holelist[i], &searchtri); if ((intersect != OUTSIDE) && (!infected(searchtri))) { /* Infect the triangle. This is done by marking the triangle */ /* as infected and including the triangle in the virus pool. */ infect(searchtri); holetri = (triangle **) poolalloc(&m->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 = m->dummytri; /* Ignore region points that aren't within the bounds of the mesh. */ if ((regionlist[4 * i] >= m->xmin) && (regionlist[4 * i] <= m->xmax) && (regionlist[4 * i + 1] >= m->ymin) && (regionlist[4 * i + 1] <= m->ymax)) { /* Start searching from some triangle on the outer boundary. */ searchtri.tri = m->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(m, b, searchorg, searchdest, ®ionlist[4 * i]) > 0.0) { /* Find a triangle that contains the region point. */ intersect = locate(m, b, ®ionlist[4 * i], &searchtri); if ((intersect != OUTSIDE) && (!infected(searchtri))) { /* Record the triangle for processing after the */ /* holes have been carved. */ otricopy(searchtri, regiontris[i]); } } } } } if (m->viri.items > 0) { /* Carve the holes and concavities. */ plague(m, b); } /* The virus pool should be empty now. */ if (regions > 0) { if (!b->quiet) { if (b->regionattrib) { if (b->vararea) { printf("Spreading regional attributes and area constraints.\n"); } else { printf("Spreading regional attributes.\n"); } } else { printf("Spreading regional area constraints.\n"); } } if (b->regionattrib && !b->refine) { /* Assign every triangle a regional attribute of zero. */ traversalinit(&m->triangles); triangleloop.orient = 0; triangleloop.tri = triangletraverse(m); while (triangleloop.tri != (triangle *) NULL) { setelemattribute(triangleloop, m->eextras, 0.0); triangleloop.tri = triangletraverse(m); } } for (i = 0; i < regions; i++) { if (regiontris[i].tri != m->dummytri) { /* Make sure the triangle under consideration still exists. */ /* It may have been eaten by the virus. */ if (!deadtri(regiontris[i].tri)) { /* Put one triangle in the virus pool. */ infect(regiontris[i]); regiontri = (triangle **) poolalloc(&m->viri); *regiontri = regiontris[i].tri; /* Apply one region's attribute and/or area constraint. */ regionplague(m, b, regionlist[4 * i + 2], regionlist[4 * i + 3]); /* The virus pool should be empty now. */ } } } if (b->regionattrib && !b->refine) { /* Note the fact that each triangle has an additional attribute. */ m->eextras++; } } /* Free up memory. */ if (((holes > 0) && !b->noholes) || !b->convex || (regions > 0)) { pooldeinit(&m->viri); } if (regions > 0) { trifree((VOID *) regiontris); } } /** **/ /** **/ /********* Carving out holes and concavities ends here *********/ /*****************************************************************************/ /* */ /* highorder() Create extra nodes for quadratic subparametric elements. */ /* */ /*****************************************************************************/ void highorder(struct mesh *m, struct behavior *b) { struct otri triangleloop, trisym; struct osub checkmark; vertex newvertex; vertex torg, tdest; int i; triangle ptr; /* Temporary variable used by sym(). */ subseg sptr; /* Temporary variable used by tspivot(). */ if (!b->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. */ m->vertices.deaditemstack = (VOID *) NULL; traversalinit(&m->triangles); triangleloop.tri = triangletraverse(m); /* 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 == m->dummytri)) { org(triangleloop, torg); dest(triangleloop, tdest); /* Create a new node in the middle of the edge. Interpolate */ /* its attributes. */ newvertex = (vertex) poolalloc(&m->vertices); for (i = 0; i < 2 + m->nextras; i++) { newvertex[i] = 0.5 * (torg[i] + tdest[i]); } /* Set the new node's marker to zero or one, depending on */ /* whether it lies on a boundary. */ setvertexmark(newvertex, trisym.tri == m->dummytri); setvertextype(newvertex, trisym.tri == m->dummytri ? FREEVERTEX : SEGMENTVERTEX); if (b->usesegments) { tspivot(triangleloop, checkmark); /* If this edge is a segment, transfer the marker to the new node. */ if (checkmark.ss != m->dummysub) { setvertexmark(newvertex, mark(checkmark)); setvertextype(newvertex, SEGMENTVERTEX); } } if (b->verbose > 1) { printf(" Creating (%.12g, %.12g).\n", newvertex[0], newvertex[1]); } /* Record the new node in the (one or two) adjacent elements. */ triangleloop.tri[m->highorderindex + triangleloop.orient] = (triangle) newvertex; if (trisym.tri != m->dummytri) { trisym.tri[m->highorderindex + trisym.orient] = (triangle) newvertex; } } } triangleloop.tri = triangletraverse(m); } } /*****************************************************************************/ /* */ /* transfernodes() Read the vertices from memory. */ /* */ /*****************************************************************************/ void transfernodes(struct mesh *m, struct behavior *b, REAL *pointlist, REAL *pointattriblist, int *pointmarkerlist, int numberofpoints, int numberofpointattribs) { vertex vertexloop; REAL x, y; int i, j; int coordindex; int attribindex; m->invertices = numberofpoints; m->mesh_dim = 2; m->nextras = numberofpointattribs; m->readnodefile = 0; if (m->invertices < 3) { printf("Error: Input must have at least three input vertices.\n"); triexit(1); } if (m->nextras == 0) { b->weighted = 0; } initializevertexpool(m, b); /* Read the vertices. */ coordindex = 0; attribindex = 0; for (i = 0; i < m->invertices; i++) { vertexloop = (vertex) poolalloc(&m->vertices); /* Read the vertex coordinates. */ x = vertexloop[0] = pointlist[coordindex++]; y = vertexloop[1] = pointlist[coordindex++]; /* Read the vertex attributes. */ for (j = 0; j < numberofpointattribs; j++) { vertexloop[2 + j] = pointattriblist[attribindex++]; } if (pointmarkerlist != (int *) NULL) { /* Read a vertex marker. */ setvertexmark(vertexloop, pointmarkerlist[i]); } else { /* If no markers are specified, they default to zero. */ setvertexmark(vertexloop, 0); } // ---------------------------------------------- for (j = (i - 1) * 2; j >= 0; j -= 2){ if (x == pointlist[j] && y == pointlist[j+1]){ printf("skip duplicate %d\n", j >> 1); setvertextype(vertexloop, UNDEADVERTEX); vertexloop[0] = 0xffffffff; vertexloop[1] = 0xffffffff; break; } } if (j >= 0) continue; // ---------------------------------------------- setvertextype(vertexloop, INPUTVERTEX); /* Determine the smallest and largest x and y coordinates. */ if (i == 0) { m->xmin = m->xmax = x; m->ymin = m->ymax = y; } else { m->xmin = (x < m->xmin) ? x : m->xmin; m->xmax = (x > m->xmax) ? x : m->xmax; m->ymin = (y < m->ymin) ? y : m->ymin; m->ymax = (y > m->ymax) ? y : m->ymax; } } /* Nonexistent x value used as a flag to mark circle events in sweepline */ /* Delaunay algorithm. */ m->xminextreme = 10 * m->xmin - 9 * m->xmax; } /*****************************************************************************/ /* */ /* writenodes() Number the vertices and write them to a .node file. */ /* */ /* To save memory, the vertex numbers are written over the boundary markers */ /* after the vertices are written to a file. */ /* */ /*****************************************************************************/ void writenodes(struct mesh *m, struct behavior *b, REAL **pointlist, REAL **pointattriblist, int **pointmarkerlist) { REAL *plist; REAL *palist; int *pmlist; int coordindex; int attribindex; vertex vertexloop; long outvertices; int vertexnumber; int i; if (b->jettison) { outvertices = m->vertices.items - m->undeads; } else { outvertices = m->vertices.items; } if (!b->quiet) { printf("Writing vertices.\n"); } /* Allocate memory for output vertices if necessary. */ if (*pointlist == (REAL *) NULL) { *pointlist = (REAL *) trimalloc((int) (outvertices * 2 * sizeof(REAL))); } /* Allocate memory for output vertex attributes if necessary. */ if ((m->nextras > 0) && (*pointattriblist == (REAL *) NULL)) { *pointattriblist = (REAL *) trimalloc((int) (outvertices * m->nextras * sizeof(REAL))); } /* Allocate memory for output vertex markers if necessary. */ if (!b->nobound && (*pointmarkerlist == (int *) NULL)) { *pointmarkerlist = (int *) trimalloc((int) (outvertices * sizeof(int))); } plist = *pointlist; palist = *pointattriblist; pmlist = *pointmarkerlist; coordindex = 0; attribindex = 0; traversalinit(&m->vertices); vertexnumber = b->firstnumber; vertexloop = vertextraverse(m); while (vertexloop != (vertex) NULL) { if (!b->jettison || (vertextype(vertexloop) != UNDEADVERTEX)) { /* X and y coordinates. */ plist[coordindex++] = vertexloop[0]; plist[coordindex++] = vertexloop[1]; /* Vertex attributes. */ for (i = 0; i < m->nextras; i++) { palist[attribindex++] = vertexloop[2 + i]; } if (!b->nobound) { /* Copy the boundary marker. */ pmlist[vertexnumber - b->firstnumber] = vertexmark(vertexloop); } setvertexmark(vertexloop, vertexnumber); vertexnumber++; } vertexloop = vertextraverse(m); } } /*****************************************************************************/ /* */ /* numbernodes() Number the vertices. */ /* */ /* Each vertex is assigned a marker equal to its number. */ /* */ /* Used when writenodes() is not called because no .node file is written. */ /* */ /*****************************************************************************/ void numbernodes(struct mesh *m, struct behavior *b) { vertex vertexloop; int vertexnumber; traversalinit(&m->vertices); vertexnumber = b->firstnumber; vertexloop = vertextraverse(m); while (vertexloop != (vertex) NULL) { setvertexmark(vertexloop, vertexnumber); if (!b->jettison || (vertextype(vertexloop) != UNDEADVERTEX)) { vertexnumber++; } vertexloop = vertextraverse(m); } } /*****************************************************************************/ /* */ /* writeelements() Write the triangles to an .ele file. */ /* */ /*****************************************************************************/ void writeelements(struct mesh *m, struct behavior *b, INDICE **trianglelist, REAL **triangleattriblist) { INDICE *tlist; REAL *talist; int vertexindex; int attribindex; struct otri triangleloop; vertex p1, p2, p3; vertex mid1, mid2, mid3; long elementnumber; int i; if (!b->quiet) { printf("Writing triangles.\n"); } /* Allocate memory for output triangles if necessary. */ if (*trianglelist == (INDICE *) NULL) { *trianglelist = (INDICE *) trimalloc( (INDICE) (m->triangles.items * ((b->order + 1) * (b->order + 2) / 2) * sizeof(int))); } /* Allocate memory for output triangle attributes if necessary. */ if ((m->eextras > 0) && (*triangleattriblist == (REAL *) NULL)) { *triangleattriblist = (REAL *) trimalloc( (int) (m->triangles.items * m->eextras * sizeof(REAL))); } tlist = *trianglelist; talist = *triangleattriblist; vertexindex = 0; attribindex = 0; traversalinit(&m->triangles); triangleloop.tri = triangletraverse(m); triangleloop.orient = 0; elementnumber = b->firstnumber; while (triangleloop.tri != (triangle *) NULL) { org(triangleloop, p1); dest(triangleloop, p2); apex(triangleloop, p3); if (b->order == 1) { tlist[vertexindex++] = vertexmark(p1); tlist[vertexindex++] = vertexmark(p2); tlist[vertexindex++] = vertexmark(p3); } else { mid1 = (vertex) triangleloop.tri[m->highorderindex + 1]; mid2 = (vertex) triangleloop.tri[m->highorderindex + 2]; mid3 = (vertex) triangleloop.tri[m->highorderindex]; tlist[vertexindex++] = vertexmark(p1); tlist[vertexindex++] = vertexmark(p2); tlist[vertexindex++] = vertexmark(p3); tlist[vertexindex++] = vertexmark(mid1); tlist[vertexindex++] = vertexmark(mid2); tlist[vertexindex++] = vertexmark(mid3); } for (i = 0; i < m->eextras; i++) { talist[attribindex++] = elemattribute(triangleloop, i); } triangleloop.tri = triangletraverse(m); elementnumber++; } } /*****************************************************************************/ /* */ /* writepoly() Write the segments and holes to a .poly file. */ /* */ /*****************************************************************************/ void writepoly(struct mesh *m, struct behavior *b, int **segmentlist, int **segmentmarkerlist) { int *slist; int *smlist; int index; struct osub subsegloop; vertex endpoint1, endpoint2; long subsegnumber; if (!b->quiet) { printf("Writing segments.\n"); } /* Allocate memory for output segments if necessary. */ if (*segmentlist == (int *) NULL) { *segmentlist = (int *) trimalloc((int) (m->subsegs.items * 2 * sizeof(int))); } /* Allocate memory for output segment markers if necessary. */ if (!b->nobound && (*segmentmarkerlist == (int *) NULL)) { *segmentmarkerlist = (int *) trimalloc((int) (m->subsegs.items * sizeof(int))); } slist = *segmentlist; smlist = *segmentmarkerlist; index = 0; traversalinit(&m->subsegs); subsegloop.ss = subsegtraverse(m); subsegloop.ssorient = 0; subsegnumber = b->firstnumber; while (subsegloop.ss != (subseg *) NULL) { sorg(subsegloop, endpoint1); sdest(subsegloop, endpoint2); /* Copy indices of the segment's two endpoints. */ slist[index++] = vertexmark(endpoint1); slist[index++] = vertexmark(endpoint2); if (!b->nobound) { /* Copy the boundary marker. */ smlist[subsegnumber - b->firstnumber] = mark(subsegloop); } subsegloop.ss = subsegtraverse(m); subsegnumber++; } } /*****************************************************************************/ /* */ /* writeedges() Write the edges to an .edge file. */ /* */ /*****************************************************************************/ void writeedges(struct mesh *m, struct behavior *b, int **edgelist, int **edgemarkerlist) { int *elist; int *emlist; int index; struct otri triangleloop, trisym; struct osub checkmark; vertex p1, p2; long edgenumber; triangle ptr; /* Temporary variable used by sym(). */ subseg sptr; /* Temporary variable used by tspivot(). */ if (!b->quiet) { printf("Writing edges.\n"); } /* Allocate memory for edges if necessary. */ if (*edgelist == (int *) NULL) { *edgelist = (int *) trimalloc((int) (m->edges * 2 * sizeof(int))); } /* Allocate memory for edge markers if necessary. */ if (!b->nobound && (*edgemarkerlist == (int *) NULL)) { *edgemarkerlist = (int *) trimalloc((int) (m->edges * sizeof(int))); } elist = *edgelist; emlist = *edgemarkerlist; index = 0; traversalinit(&m->triangles); triangleloop.tri = triangletraverse(m); edgenumber = b->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 == m->dummytri)) { org(triangleloop, p1); dest(triangleloop, p2); elist[index++] = vertexmark(p1); elist[index++] = vertexmark(p2); if (b->nobound) { } else { /* Edge number, indices of two endpoints, and a boundary marker. */ /* If there's no subsegment, the boundary marker is zero. */ if (b->usesegments) { tspivot(triangleloop, checkmark); if (checkmark.ss == m->dummysub) { emlist[edgenumber - b->firstnumber] = 0; } else { emlist[edgenumber - b->firstnumber] = mark(checkmark); } } else { emlist[edgenumber - b->firstnumber] = trisym.tri == m->dummytri; } } edgenumber++; } } triangleloop.tri = triangletraverse(m); } } /*****************************************************************************/ /* */ /* 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 subsegments or the extra nodes of every */ /* element. Hence, you should call this procedure last. */ /* */ /*****************************************************************************/ void writevoronoi(struct mesh *m, struct behavior *b, REAL **vpointlist, REAL **vpointattriblist, int **vpointmarkerlist, int **vedgelist, int **vedgemarkerlist, REAL **vnormlist) { REAL *plist; REAL *palist; int *elist; REAL *normlist; int coordindex; int attribindex; struct otri triangleloop, trisym; vertex torg, tdest, tapex; REAL circumcenter[2]; REAL xi, eta; long vnodenumber, vedgenumber; int p1, p2; int i; triangle ptr; /* Temporary variable used by sym(). */ if (!b->quiet) { printf("Writing Voronoi vertices.\n"); } /* Allocate memory for Voronoi vertices if necessary. */ if (*vpointlist == (REAL *) NULL) { *vpointlist = (REAL *) trimalloc((int) (m->triangles.items * 2 * sizeof(REAL))); } /* Allocate memory for Voronoi vertex attributes if necessary. */ if (*vpointattriblist == (REAL *) NULL) { *vpointattriblist = (REAL *) trimalloc( (int) (m->triangles.items * m->nextras * sizeof(REAL))); } *vpointmarkerlist = (int *) NULL; plist = *vpointlist; palist = *vpointattriblist; coordindex = 0; attribindex = 0; traversalinit(&m->triangles); triangleloop.tri = triangletraverse(m); triangleloop.orient = 0; vnodenumber = b->firstnumber; while (triangleloop.tri != (triangle *) NULL) { org(triangleloop, torg); dest(triangleloop, tdest); apex(triangleloop, tapex); findcircumcenter(m, b, torg, tdest, tapex, circumcenter, &xi, &eta, 0); /* X and y coordinates. */ plist[coordindex++] = circumcenter[0]; plist[coordindex++] = circumcenter[1]; for (i = 2; i < 2 + m->nextras; i++) { /* Interpolate the vertex attributes at the circumcenter. */ palist[attribindex++] = torg[i] + xi * (tdest[i] - torg[i]) + eta * (tapex[i] - torg[i]); } *(int *) (triangleloop.tri + 6) = (int) vnodenumber; triangleloop.tri = triangletraverse(m); vnodenumber++; } if (!b->quiet) { printf("Writing Voronoi edges.\n"); } /* Allocate memory for output Voronoi edges if necessary. */ if (*vedgelist == (int *) NULL) { *vedgelist = (int *) trimalloc((int) (m->edges * 2 * sizeof(int))); } *vedgemarkerlist = (int *) NULL; /* Allocate memory for output Voronoi norms if necessary. */ if (*vnormlist == (REAL *) NULL) { *vnormlist = (REAL *) trimalloc((int) (m->edges * 2 * sizeof(REAL))); } elist = *vedgelist; normlist = *vnormlist; coordindex = 0; traversalinit(&m->triangles); triangleloop.tri = triangletraverse(m); vedgenumber = b->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 == m->dummytri)) { /* Find the number of this triangle (and Voronoi vertex). */ p1 = *(int *) (triangleloop.tri + 6); if (trisym.tri == m->dummytri) { org(triangleloop, torg); dest(triangleloop, tdest); /* 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 { /* Find the number of the adjacent triangle (and Voronoi vertex). */ p2 = *(int *) (trisym.tri + 6); /* Finite edge. Write indices of two endpoints. */ elist[coordindex] = p1; normlist[coordindex++] = 0.0; elist[coordindex] = p2; normlist[coordindex++] = 0.0; } vedgenumber++; } } triangleloop.tri = triangletraverse(m); } } void writeneighbors(struct mesh *m, struct behavior *b, int **neighborlist) { int *nlist; int index; struct otri triangleloop, trisym; long elementnumber; int neighbor1, neighbor2, neighbor3; triangle ptr; /* Temporary variable used by sym(). */ if (!b->quiet) { printf("Writing neighbors.\n"); } /* Allocate memory for neighbors if necessary. */ if (*neighborlist == (int *) NULL) { *neighborlist = (int *) trimalloc((int) (m->triangles.items * 3 * sizeof(int))); } nlist = *neighborlist; index = 0; traversalinit(&m->triangles); triangleloop.tri = triangletraverse(m); triangleloop.orient = 0; elementnumber = b->firstnumber; while (triangleloop.tri != (triangle *) NULL) { *(int *) (triangleloop.tri + 6) = (int) elementnumber; triangleloop.tri = triangletraverse(m); elementnumber++; } *(int *) (m->dummytri + 6) = -1; traversalinit(&m->triangles); triangleloop.tri = triangletraverse(m); elementnumber = b->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); nlist[index++] = neighbor1; nlist[index++] = neighbor2; nlist[index++] = neighbor3; triangleloop.tri = triangletraverse(m); elementnumber++; } } /** **/ /** **/ /********* File I/O routines end here *********/ /*****************************************************************************/ /* */ /* 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 vertices 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). */ /* */ /*****************************************************************************/ void triangulate(struct behavior *command, struct triangulateio *in, struct triangulateio *out, struct triangulateio *vorout) { struct mesh m; struct behavior *b = command; REAL *holearray; /* Array of holes. */ REAL *regionarray; /* Array of regional attributes and area constraints. */ triangleinit(&m); //parsecommandline(1, &triswitches, &b); m.steinerleft = b->steiner; transfernodes(&m, b, in->pointlist, in->pointattributelist, in->pointmarkerlist, in->numberofpoints, in->numberofpointattributes); #ifdef CDT_ONLY m.hullsize = delaunay(&m, b); /* Triangulate the vertices. */ #else /* not CDT_ONLY */ if (b->refine) { /* Read and reconstruct a mesh. */ m.hullsize = reconstruct(&m, b, in->trianglelist, in->triangleattributelist, in->trianglearealist, in->numberoftriangles, in->numberofcorners, in->numberoftriangleattributes, in->segmentlist, in->segmentmarkerlist, in->numberofsegments); } else { m.hullsize = delaunay(&m, b); /* Triangulate the vertices. */ } #endif /* not CDT_ONLY */ /* Ensure that no vertex can be mistaken for a triangular bounding */ /* box vertex in insertvertex(). */ m.infvertex1 = (vertex) NULL; m.infvertex2 = (vertex) NULL; m.infvertex3 = (vertex) NULL; if (b->usesegments) { m.checksegments = 1; /* Segments will be introduced next. */ if (!b->refine) { /* Insert PSLG segments and/or convex hull segments. */ formskeleton(&m, b, in->segmentlist, in->segmentmarkerlist, in->numberofsegments); } } if (b->poly && (m.triangles.items > 0)) { holearray = in->holelist; m.holes = in->numberofholes; regionarray = in->regionlist; m.regions = in->numberofregions; if (!b->refine) { /* Carve out holes and concavities. */ carveholes(&m, b, holearray, m.holes, regionarray, m.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. */ m.holes = 0; m.regions = 0; } #ifndef CDT_ONLY if (b->quality && (m.triangles.items > 0)) { enforcequality(&m, b); /* Enforce angle and area constraints. */ } #endif /* not CDT_ONLY */ #ifndef CDT_ONLY if (b->quality) { printf("Quality milliseconds: %ld\n", 1000l * (tv5.tv_sec - tv4.tv_sec) + (tv5.tv_usec - tv4.tv_usec) / 1000l); } #endif /* not CDT_ONLY */ /* Calculate the number of edges. */ m.edges = (3l * m.triangles.items + m.hullsize) / 2l; if (b->order > 1) { highorder(&m, b); /* Promote elements to higher polynomial order. */ } if (!b->quiet) { printf("\n"); } if (b->jettison) { out->numberofpoints = m.vertices.items - m.undeads; } else { out->numberofpoints = m.vertices.items; } out->numberofpointattributes = m.nextras; out->numberoftriangles = m.triangles.items; out->numberofcorners = (b->order + 1) * (b->order + 2) / 2; out->numberoftriangleattributes = m.eextras; out->numberofedges = m.edges; if (b->usesegments) { out->numberofsegments = m.subsegs.items; } else { out->numberofsegments = m.hullsize; } if (vorout != (struct triangulateio *) NULL) { vorout->numberofpoints = m.triangles.items; vorout->numberofpointattributes = m.nextras; vorout->numberofedges = m.edges; } /* If not using iteration numbers, don't write a .node file if one was */ /* read, because the original one would be overwritten! */ if (b->nonodewritten || (b->noiterationnum && m.readnodefile)) { if (!b->quiet) { printf("NOT writing vertices.\n"); } numbernodes(&m, b); /* We must remember to number the vertices. */ } else { /* writenodes() numbers the vertices too. */ writenodes(&m, b, &out->pointlist, &out->pointattributelist, &out->pointmarkerlist); } if (b->noelewritten) { if (!b->quiet) { printf("NOT writing triangles.\n"); } } else { writeelements(&m, b, &out->trianglelist, &out->triangleattributelist); } /* The -c switch (convex switch) causes a PSLG to be written */ /* even if none was read. */ if (b->poly || b->convex) { /* If not using iteration numbers, don't overwrite the .poly file. */ if (b->nopolywritten || b->noiterationnum) { if (!b->quiet) { printf("NOT writing segments.\n"); } } else { writepoly(&m, b, &out->segmentlist, &out->segmentmarkerlist); out->numberofholes = m.holes; out->numberofregions = m.regions; if (b->poly) { out->holelist = in->holelist; out->regionlist = in->regionlist; } else { out->holelist = (REAL *) NULL; out->regionlist = (REAL *) NULL; } } } if (b->edgesout) { writeedges(&m, b, &out->edgelist, &out->edgemarkerlist); } if (b->voronoi) { writevoronoi(&m, b, &vorout->pointlist, &vorout->pointattributelist, &vorout->pointmarkerlist, &vorout->edgelist, &vorout->edgemarkerlist, &vorout->normlist); } if (b->neighbors) { writeneighbors(&m, b, &out->neighborlist); } if (!b->quiet) { statistics(&m, b); } #ifndef REDUCED if (b->docheck) { checkmesh(&m, b); checkdelaunay(&m, b); } #endif /* not REDUCED */ triangledeinit(&m, b); }