/*****************************************************************************/ /* */ /* 888888888 ,o, / 888 */ /* 888 88o88o " o8888o 88o8888o o88888o 888 o88888o */ /* 888 888 888 88b 888 888 888 888 888 d888 88b */ /* 888 888 888 o88^o888 888 888 "88888" 888 8888oo888 */ /* 888 888 888 C888 888 888 888 / 888 q888 */ /* 888 888 888 "88o^888 888 888 Cb 888 "88oooo" */ /* "8oo8D */ /* */ /* A Two-Dimensional Quality Mesh Generator and Delaunay Triangulator. */ /* (triangle.c) */ /* */ /* Version 1.3 */ /* July 19, 1996 */ /* */ /* Copyright 1996 */ /* Jonathan Richard Shewchuk */ /* School of Computer Science */ /* Carnegie Mellon University */ /* 5000 Forbes Avenue */ /* Pittsburgh, Pennsylvania 15213-3891 */ /* jrs@cs.cmu.edu */ /* Severely stripped by SG to reduce size ************************************/ /* */ /* This program may be freely redistributed under the condition that the */ /* copyright notices (including this entire header and the copyright */ /* notice printed when the `-h' switch is selected) are not removed, and */ /* no compensation is received. Private, research, and institutional */ /* use is free. You may distribute modified versions of this code UNDER */ /* THE CONDITION THAT THIS CODE AND ANY MODIFICATIONS MADE TO IT IN THE */ /* SAME FILE REMAIN UNDER COPYRIGHT OF THE ORIGINAL AUTHOR, BOTH SOURCE */ /* AND OBJECT CODE ARE MADE FREELY AVAILABLE WITHOUT CHARGE, AND CLEAR */ /* NOTICE IS GIVEN OF THE MODIFICATIONS. Distribution of this code as */ /* part of a commercial system is permissible ONLY BY DIRECT ARRANGEMENT */ /* WITH THE AUTHOR. (If you are not directly supplying this code to a */ /* customer, and you are instead telling them how they can obtain it for */ /* free, then you are not required to make any arrangement with me.) */ /* */ /* Hypertext instructions for Triangle are available on the Web at */ /* */ /* http://www.cs.cmu.edu/~quake/triangle.html */ /* */ /* Some of the references listed below are marked [*]. These are available */ /* for downloading from the Web page */ /* */ /* http://www.cs.cmu.edu/~quake/triangle.research.html */ /* */ /* A paper discussing some aspects of Triangle is available. See Jonathan */ /* Richard Shewchuk, "Triangle: Engineering a 2D Quality Mesh Generator */ /* and Delaunay Triangulator," First Workshop on Applied Computational */ /* Geometry, ACM, May 1996. [*] */ /* */ /* Triangle was created as part of the Archimedes project in the School of */ /* Computer Science at Carnegie Mellon University. Archimedes is a */ /* system for compiling parallel finite element solvers. For further */ /* information, see Anja Feldmann, Omar Ghattas, John R. Gilbert, Gary L. */ /* Miller, David R. O'Hallaron, Eric J. Schwabe, Jonathan R. Shewchuk, */ /* and Shang-Hua Teng, "Automated Parallel Solution of Unstructured PDE */ /* Problems." To appear in Communications of the ACM, we hope. */ /* */ /* The quality mesh generation algorithm is due to Jim Ruppert, "A */ /* Delaunay Refinement Algorithm for Quality 2-Dimensional Mesh */ /* Generation," Journal of Algorithms 18(3):548-585, May 1995. [*] */ /* */ /* My implementation of the divide-and-conquer and incremental Delaunay */ /* triangulation algorithms follows closely the presentation of Guibas */ /* and Stolfi, even though I use a triangle-based data structure instead */ /* of their quad-edge data structure. (In fact, I originally implemented */ /* Triangle using the quad-edge data structure, but switching to a */ /* triangle-based data structure sped Triangle by a factor of two.) The */ /* mesh manipulation primitives and the two aforementioned Delaunay */ /* triangulation algorithms are described by Leonidas J. Guibas and Jorge */ /* Stolfi, "Primitives for the Manipulation of General Subdivisions and */ /* the Computation of Voronoi Diagrams," ACM Transactions on Graphics */ /* 4(2):74-123, April 1985. */ /* */ /* Their O(n log n) divide-and-conquer algorithm is adapted from Der-Tsai */ /* Lee and Bruce J. Schachter, "Two Algorithms for Constructing the */ /* Delaunay Triangulation," International Journal of Computer and */ /* Information Science 9(3):219-242, 1980. The idea to improve the */ /* divide-and-conquer algorithm by alternating between vertical and */ /* horizontal cuts was introduced by Rex A. Dwyer, "A Faster Divide-and- */ /* Conquer Algorithm for Constructing Delaunay Triangulations," */ /* Algorithmica 2(2):137-151, 1987. */ /* */ /* The incremental insertion algorithm was first proposed by C. L. Lawson, */ /* "Software for C1 Surface Interpolation," in Mathematical Software III, */ /* John R. Rice, editor, Academic Press, New York, pp. 161-194, 1977. */ /* For point location, I use the algorithm of Ernst P. Mucke, Isaac */ /* Saias, and Binhai Zhu, "Fast Randomized Point Location Without */ /* Preprocessing in Two- and Three-dimensional Delaunay Triangulations," */ /* Proceedings of the Twelfth Annual Symposium on Computational Geometry, */ /* ACM, May 1996. [*] If I were to randomize the order of point */ /* insertion (I currently don't bother), their result combined with the */ /* result of Leonidas J. Guibas, Donald E. Knuth, and Micha Sharir, */ /* "Randomized Incremental Construction of Delaunay and Voronoi */ /* Diagrams," Algorithmica 7(4):381-413, 1992, would yield an expected */ /* O(n^{4/3}) bound on running time. */ /* */ /* The O(n log n) sweepline Delaunay triangulation algorithm is taken from */ /* Steven Fortune, "A Sweepline Algorithm for Voronoi Diagrams", */ /* Algorithmica 2(2):153-174, 1987. A random sample of edges on the */ /* boundary of the triangulation are maintained in a splay tree for the */ /* purpose of point location. Splay trees are described by Daniel */ /* Dominic Sleator and Robert Endre Tarjan, "Self-Adjusting Binary Search */ /* Trees," Journal of the ACM 32(3):652-686, July 1985. */ /* */ /* The algorithms for exact computation of the signs of determinants are */ /* described in Jonathan Richard Shewchuk, "Adaptive Precision Floating- */ /* Point Arithmetic and Fast Robust Geometric Predicates," Technical */ /* Report CMU-CS-96-140, School of Computer Science, Carnegie Mellon */ /* University, Pittsburgh, Pennsylvania, May 1996. [*] (Submitted to */ /* Discrete & Computational Geometry.) An abbreviated version appears as */ /* Jonathan Richard Shewchuk, "Robust Adaptive Floating-Point Geometric */ /* Predicates," Proceedings of the Twelfth Annual Symposium on Computa- */ /* tional Geometry, ACM, May 1996. [*] Many of the ideas for my exact */ /* arithmetic routines originate with Douglas M. Priest, "Algorithms for */ /* Arbitrary Precision Floating Point Arithmetic," Tenth Symposium on */ /* Computer Arithmetic, 132-143, IEEE Computer Society Press, 1991. [*] */ /* Many of the ideas for the correct evaluation of the signs of */ /* determinants are taken from Steven Fortune and Christopher J. Van Wyk, */ /* "Efficient Exact Arithmetic for Computational Geometry," Proceedings */ /* of the Ninth Annual Symposium on Computational Geometry, ACM, */ /* pp. 163-172, May 1993, and from Steven Fortune, "Numerical Stability */ /* of Algorithms for 2D Delaunay Triangulations," International Journal */ /* of Computational Geometry & Applications 5(1-2):193-213, March-June */ /* 1995. */ /* */ /* For definitions of and results involving Delaunay triangulations, */ /* constrained and conforming versions thereof, and other aspects of */ /* triangular mesh generation, see the excellent survey by Marshall Bern */ /* and David Eppstein, "Mesh Generation and Optimal Triangulation," in */ /* Computing and Euclidean Geometry, Ding-Zhu Du and Frank Hwang, */ /* editors, World Scientific, Singapore, pp. 23-90, 1992. */ /* */ /* The time for incrementally adding PSLG (planar straight line graph) */ /* segments to create a constrained Delaunay triangulation is probably */ /* O(n^2) per segment in the worst case and O(n) per edge in the common */ /* case, where n is the number of triangles that intersect the segment */ /* before it is inserted. This doesn't count point location, which can */ /* be much more expensive. (This note does not apply to conforming */ /* Delaunay triangulations, for which a different method is used to */ /* insert segments.) */ /* */ /* The time for adding segments to a conforming Delaunay triangulation is */ /* not clear, but does not depend upon n alone. In some cases, very */ /* small features (like a point lying next to a segment) can cause a */ /* single segment to be split an arbitrary number of times. Of course, */ /* floating-point precision is a practical barrier to how much this can */ /* happen. */ /* */ /* The time for deleting a point from a Delaunay triangulation is O(n^2) in */ /* the worst case and O(n) in the common case, where n is the degree of */ /* the point being deleted. I could improve this to expected O(n) time */ /* by "inserting" the neighboring vertices in random order, but n is */ /* usually quite small, so it's not worth the bother. (The O(n) time */ /* for random insertion follows from L. Paul Chew, "Building Voronoi */ /* Diagrams for Convex Polygons in Linear Expected Time," Technical */ /* Report PCS-TR90-147, Department of Mathematics and Computer Science, */ /* Dartmouth College, 1990. */ /* */ /* Ruppert's Delaunay refinement algorithm typically generates triangles */ /* at a linear rate (constant time per triangle) after the initial */ /* triangulation is formed. There may be pathological cases where more */ /* time is required, but these never arise in practice. */ /* */ /* The segment intersection formulae are straightforward. If you want to */ /* see them derived, see Franklin Antonio. "Faster Line Segment */ /* Intersection." In Graphics Gems III (David Kirk, editor), pp. 199- */ /* 202. Academic Press, Boston, 1992. */ /* */ /* If you make any improvements to this code, please please please let me */ /* know, so that I may obtain the improvements. Even if you don't change */ /* the code, I'd still love to hear what it's being used for. */ /* */ /* Disclaimer: Neither I nor Carnegie Mellon warrant this code in any way */ /* whatsoever. This code is provided "as-is". Use at your own risk. */ /* */ /*****************************************************************************/ /* On some machines, the exact arithmetic routines might be defeated by the */ /* use of internal extended precision floating-point registers. Sometimes */ /* this problem can be fixed by defining certain values to be volatile, */ /* thus forcing them to be stored to memory and rounded off. This isn't */ /* a great solution, though, as it slows Triangle down. */ /* */ /* To try this out, write "#define INEXACT volatile" below. Normally, */ /* however, INEXACT should be defined to be nothing. ("#define INEXACT".) */ #define INEXACT /* For efficiency, a variety of data structures are allocated in bulk. The */ /* following constants determine how many of each structure is allocated */ /* at once. */ #define TRIPERBLOCK 4092 /* Number of triangles allocated at once. */ #define SHELLEPERBLOCK 508 /* Number of shell edges allocated at once. */ #define POINTPERBLOCK 4092 /* Number of points allocated at once. */ #define VIRUSPERBLOCK 1020 /* Number of virus triangles allocated at once. */ /* The point marker DEADPOINT is an arbitrary number chosen large enough to */ /* (hopefully) not conflict with user boundary markers. Make sure that it */ /* is small enough to fit into your machine's integer size. */ #define DEADPOINT -1073741824 /* Used for the point location scheme of Mucke, Saias, and Zhu, to decide */ /* how large a random sample of triangles to inspect. */ #define SAMPLEFACTOR 11 /* A number that speaks for itself, every kissable digit. */ #define PI 3.141592653589793238462643383279502884197169399375105820974944592308 // Patch SG pour intégration MSVC6 / projet sKulpt #define STRICT extern void vTrace(char *Str, ...); #include // End patch #include #include #include #include "triangulator.h" /* A few forward declarations. */ void poolrestart(struct memorypool *pool); /* Labels that signify whether a record consists primarily of pointers or of */ /* floating-point words. Used to make decisions about data alignment. */ enum wordtype {POINTER, FLOATINGPOINT}; /* Labels that signify the result of point location. The result of a */ /* search indicates that the point falls in the interior of a triangle, on */ /* an edge, on a vertex, or outside the mesh. */ enum locateresult {INTRIANGLE, ONEDGE, ONVERTEX, OUTSIDE}; /* Labels that signify the result of site insertion. The result indicates */ /* that the point was inserted with complete success, was inserted but */ /* encroaches on a segment, was not inserted because it lies on a segment, */ /* or was not inserted because another point occupies the same location. */ enum insertsiteresult {SUCCESSFULPOINT, VIOLATINGPOINT, DUPLICATEPOINT}; /* Labels that signify the result of direction finding. The result */ /* indicates that a segment connecting the two query points falls within */ /* the direction triangle, along the left edge of the direction triangle, */ /* or along the right edge of the direction triangle. */ enum finddirectionresult {WITHIN, LEFTCOLLINEAR, RIGHTCOLLINEAR}; /*****************************************************************************/ /* */ /* The basic mesh data structures */ /* */ /* There are three: points, triangles, and shell edges (abbreviated */ /* `shelle'). These three data structures, linked by pointers, comprise */ /* the mesh. A point simply represents a point in space and its properties.*/ /* A triangle is a triangle. A shell edge is a special data structure used */ /* to represent impenetrable segments in the mesh (including the outer */ /* boundary, boundaries of holes, and internal boundaries separating two */ /* triangulated regions). Shell edges represent boundaries defined by the */ /* user that triangles may not lie across. */ /* */ /* A triangle consists of a list of three vertices, a list of three */ /* adjoining triangles, a list of three adjoining shell edges (when shell */ /* edges are used), an arbitrary number of optional user-defined floating- */ /* point attributes, and an optional area constraint. The latter is an */ /* upper bound on the permissible area of each triangle in a region, used */ /* for mesh refinement. */ /* */ /* For a triangle on a boundary of the mesh, some or all of the neighboring */ /* triangles may not be present. For a triangle in the interior of the */ /* mesh, often no neighboring shell edges are present. Such absent */ /* triangles and shell edges are never represented by NULL pointers; they */ /* are represented by two special records: `dummytri', the triangle that */ /* fills "outer space", and `dummysh', the omnipresent shell edge. */ /* `dummytri' and `dummysh' are used for several reasons; for instance, */ /* they can be dereferenced and their contents examined without causing the */ /* memory protection exception that would occur if NULL were dereferenced. */ /* */ /* However, it is important to understand that a triangle includes other */ /* information as well. The pointers to adjoining vertices, triangles, and */ /* shell edges are ordered in a way that indicates their geometric relation */ /* to each other. Furthermore, each of these pointers contains orientation */ /* information. Each pointer to an adjoining triangle indicates which face */ /* of that triangle is contacted. Similarly, each pointer to an adjoining */ /* shell edge indicates which side of that shell edge is contacted, and how */ /* the shell edge is oriented relative to the triangle. */ /* */ /* Shell edges are found abutting edges of triangles; either sandwiched */ /* between two triangles, or resting against one triangle on an exterior */ /* boundary or hole boundary. */ /* */ /* A shell edge consists of a list of two vertices, a list of two */ /* adjoining shell edges, and a list of two adjoining triangles. One of */ /* the two adjoining triangles may not be present (though there should */ /* always be one), and neighboring shell edges might not be present. */ /* Shell edges also store a user-defined integer "boundary marker". */ /* Typically, this integer is used to indicate what sort of boundary */ /* conditions are to be applied at that location in a finite element */ /* simulation. */ /* */ /* Like triangles, shell edges maintain information about the relative */ /* orientation of neighboring objects. */ /* */ /* Points are relatively simple. A point is a list of floating point */ /* numbers, starting with the x, and y coordinates, followed by an */ /* arbitrary number of optional user-defined floating-point attributes, */ /* followed by an integer boundary marker. During the segment insertion */ /* phase, there is also a pointer from each point to a triangle that may */ /* contain it. Each pointer is not always correct, but when one is, it */ /* speeds up segment insertion. These pointers are assigned values once */ /* at the beginning of the segment insertion phase, and are not used or */ /* updated at any other time. Edge swapping during segment insertion will */ /* render some of them incorrect. Hence, don't rely upon them for */ /* anything. For the most part, points do not have any information about */ /* what triangles or shell edges they are linked to. */ /* */ /*****************************************************************************/ /*****************************************************************************/ /* */ /* Handles */ /* */ /* The oriented triangle (`triedge') and oriented shell edge (`edge') data */ /* structures defined below do not themselves store any part of the mesh. */ /* The mesh itself is made of `triangle's, `shelle's, and `point's. */ /* */ /* Oriented triangles and oriented shell edges will usually be referred to */ /* as "handles". A handle is essentially a pointer into the mesh; it */ /* allows you to "hold" one particular part of the mesh. Handles are used */ /* to specify the regions in which one is traversing and modifying the mesh.*/ /* A single `triangle' may be held by many handles, or none at all. (The */ /* latter case is not a memory leak, because the triangle is still */ /* connected to other triangles in the mesh.) */ /* */ /* A `triedge' is a handle that holds a triangle. It holds a specific side */ /* of the triangle. An `edge' is a handle that holds a shell edge. It */ /* holds either the left or right side of the edge. */ /* */ /* Navigation about the mesh is accomplished through a set of mesh */ /* manipulation primitives, further below. Many of these primitives take */ /* a handle and produce a new handle that holds the mesh near the first */ /* handle. Other primitives take two handles and glue the corresponding */ /* parts of the mesh together. The exact position of the handles is */ /* important. For instance, when two triangles are glued together by the */ /* bond() primitive, they are glued by the sides on which the handles lie. */ /* */ /* Because points have no information about which triangles they are */ /* attached to, I commonly represent a point by use of a handle whose */ /* origin is the point. A single handle can simultaneously represent a */ /* triangle, an edge, and a point. */ /* */ /*****************************************************************************/ /* The triangle data structure. Each triangle contains three pointers to */ /* adjoining triangles, plus three pointers to vertex points, plus three */ /* pointers to shell edges (defined below; these pointers are usually */ /* `dummysh'). It may or may not also contain user-defined attributes */ /* and/or a floating-point "area constraint". It may also contain extra */ /* pointers for nodes, when the user asks for high-order elements. */ /* Because the size and structure of a `triangle' is not decided until */ /* runtime, I haven't simply defined the type `triangle' to be a struct. */ typedef double **triangle; /* Really: typedef triangle *triangle */ /* An oriented triangle: includes a pointer to a triangle and orientation. */ /* The orientation denotes an edge of the triangle. Hence, there are */ /* three possible orientations. By convention, each edge is always */ /* directed to point counterclockwise about the corresponding triangle. */ struct triedge { triangle *tri; int orient; /* Ranges from 0 to 2. */ }; /* The shell data structure. Each shell edge contains two pointers to */ /* adjoining shell edges, plus two pointers to vertex points, plus two */ /* pointers to adjoining triangles, plus one shell marker. */ typedef double **shelle; /* Really: typedef shelle *shelle */ /* An oriented shell edge: includes a pointer to a shell edge and an */ /* orientation. The orientation denotes a side of the edge. Hence, there */ /* are two possible orientations. By convention, the edge is always */ /* directed so that the "side" denoted is the right side of the edge. */ struct edge { shelle *sh; int shorient; /* Ranges from 0 to 1. */ }; /* The point data structure. Each point is actually an array of REALs. */ /* The number of REALs is unknown until runtime. An integer boundary */ /* marker, and sometimes a pointer to a triangle, is appended after the */ /* REALs. */ typedef double *point; /* A type used to allocate memory. firstblock is the first block of items. */ /* nowblock is the block from which items are currently being allocated. */ /* nextitem points to the next slab of free memory for an item. */ /* deaditemstack is the head of a linked list (stack) of deallocated items */ /* that can be recycled. unallocateditems is the number of items that */ /* remain to be allocated from nowblock. */ /* */ /* Traversal is the process of walking through the entire list of items, and */ /* is separate from allocation. Note that a traversal will visit items on */ /* the "deaditemstack" stack as well as live items. pathblock points to */ /* the block currently being traversed. pathitem points to the next item */ /* to be traversed. pathitemsleft is the number of items that remain to */ /* be traversed in pathblock. */ /* */ /* itemwordtype is set to POINTER or FLOATINGPOINT, and is used to suggest */ /* what sort of word the record is primarily made up of. alignbytes */ /* determines how new records should be aligned in memory. itembytes and */ /* itemwords are the length of a record in bytes (after rounding up) and */ /* words. itemsperblock is the number of items allocated at once in a */ /* single block. items is the number of currently allocated items. */ /* maxitems is the maximum number of items that have been allocated at */ /* once; it is the current number of items plus the number of records kept */ /* on deaditemstack. */ struct memorypool { void **firstblock, **nowblock; void *nextitem; void *deaditemstack; void **pathblock; void *pathitem; enum wordtype itemwordtype; int alignbytes; int itembytes, itemwords; int itemsperblock; long items, maxitems; int unallocateditems; int pathitemsleft; }; /* Variables used to allocate memory for triangles, shell edges, points, */ /* viri (triangles being eaten), bad (encroached) segments, bad (skinny */ /* or too large) triangles, and splay tree nodes. */ struct memorypool triangles; struct memorypool shelles; struct memorypool points; struct memorypool viri; struct memorypool badsegments; struct memorypool badtriangles; struct memorypool splaynodes; /* Variables that maintain the bad triangle queues. The tails are pointers */ /* to the pointers that have to be filled in to enqueue an item. */ double xmin, xmax, ymin, ymax; /* x and y bounds. */ double xminextreme; /* Nonexistent x value used as a flag in sweepline. */ int inpoints; /* Number of input points. */ int insegments; /* Number of input segments. */ int holes; /* Number of input holes. */ int regions; /* Number of input regions. */ long edges; /* Number of output edges. */ int mesh_dim; /* Dimension (ought to be 2). */ int nextras; /* Number of attributes per point. */ int eextras; /* Number of attributes per triangle. */ long hullsize; /* Number of edges of convex hull. */ int triwords; /* Total words per triangle. */ int shwords; /* Total words per shell edge. */ int pointmarkindex; /* Index to find boundary marker of a point. */ int point2triindex; /* Index to find a triangle adjacent to a point. */ int highorderindex; /* Index to find extra nodes for high-order elements. */ int elemattribindex; /* Index to find attributes of a triangle. */ int areaboundindex; /* Index to find area bound of a triangle. */ int checksegments; /* Are there segments in the triangulation yet? */ long samples; /* Number of random samples for point location. */ unsigned long randomseed; /* Current random number seed. */ double splitter; /* Used to split double factors for exact multiplication. */ double epsilon; /* Floating-point machine epsilon. */ double resulterrbound; double ccwerrboundA, ccwerrboundB, ccwerrboundC; double iccerrboundA, iccerrboundB, iccerrboundC; long incirclecount; /* Number of incircle tests performed. */ long counterclockcount; /* Number of counterclockwise tests performed. */ long hyperbolacount; /* Number of right-of-hyperbola tests performed. */ long circumcentercount; /* Number of circumcenter calculations performed. */ long circletopcount; /* Number of circle top calculations performed. */ /* Switches for the triangulator. */ /* useshelles: -p, -r, -q, or -c switch; determines whether shell edges */ /* are used at all. */ int useshelles; int order; double minangle, goodangle; double maxarea; /* Triangular bounding box points. */ point infpoint1, infpoint2, infpoint3; /* Pointer to the `triangle' that occupies all of "outer space". */ triangle *dummytri; triangle *dummytribase; /* Keep base address so we can free() it later. */ /* Pointer to the omnipresent shell edge. Referenced by any triangle or */ /* shell edge that isn't really connected to a shell edge at that */ /* location. */ shelle *dummysh; shelle *dummyshbase; /* Keep base address so we can free() it later. */ /* Pointer to a recently visited triangle. Improves point location if */ /* proximate points are inserted sequentially. */ struct triedge recenttri; /*****************************************************************************/ /* */ /* Mesh manipulation primitives. Each triangle contains three pointers to */ /* other triangles, with orientations. Each pointer points not to the */ /* first byte of a triangle, but to one of the first three bytes of a */ /* triangle. It is necessary to extract both the triangle itself and the */ /* orientation. To save memory, I keep both pieces of information in one */ /* pointer. To make this possible, I assume that all triangles are aligned */ /* to four-byte boundaries. The `decode' routine below decodes a pointer, */ /* extracting an orientation (in the range 0 to 2) and a pointer to the */ /* beginning of a triangle. The `encode' routine compresses a pointer to a */ /* triangle and an orientation into a single pointer. My assumptions that */ /* triangles are four-byte-aligned and that the `unsigned long' type is */ /* long enough to hold a pointer are two of the few kludges in this program.*/ /* */ /* Shell edges are manipulated similarly. A pointer to a shell edge */ /* carries both an address and an orientation in the range 0 to 1. */ /* */ /* The other primitives take an oriented triangle or oriented shell edge, */ /* and return an oriented triangle or oriented shell edge or point; or they */ /* change the connections in the data structure. */ /* */ /*****************************************************************************/ /********* Mesh manipulation primitives begin here *********/ /** **/ /** **/ /* Fast lookup arrays to speed some of the mesh manipulation primitives. */ int plus1mod3[3] = {1, 2, 0}; int minus1mod3[3] = {2, 0, 1}; /********* Primitives for triangles *********/ /* */ /* */ /* decode() converts a pointer to an oriented triangle. The orientation is */ /* extracted from the two least significant bits of the pointer. */ #define decode(ptr, triedge) \ (triedge).orient = (int) ((unsigned long) (ptr) & (unsigned long) 3l); \ (triedge).tri = (triangle *) \ ((unsigned long) (ptr) ^ (unsigned long) (triedge).orient) /* encode() compresses an oriented triangle into a single pointer. It */ /* relies on the assumption that all triangles are aligned to four-byte */ /* boundaries, so the two least significant bits of (triedge).tri are zero.*/ #define encode(triedge) \ (triangle) ((unsigned long) (triedge).tri | (unsigned long) (triedge).orient) /* The following edge manipulation primitives are all described by Guibas */ /* and Stolfi. However, they use an edge-based data structure, whereas I */ /* am using a triangle-based data structure. */ /* sym() finds the abutting triangle, on the same edge. Note that the */ /* edge direction is necessarily reversed, because triangle/edge handles */ /* are always directed counterclockwise around the triangle. */ #define sym(triedge1, triedge2) \ ptr = (triedge1).tri[(triedge1).orient]; \ decode(ptr, triedge2); #define symself(triedge) \ ptr = (triedge).tri[(triedge).orient]; \ decode(ptr, triedge); /* lnext() finds the next edge (counterclockwise) of a triangle. */ #define lnext(triedge1, triedge2) \ (triedge2).tri = (triedge1).tri; \ (triedge2).orient = plus1mod3[(triedge1).orient] #define lnextself(triedge) \ (triedge).orient = plus1mod3[(triedge).orient] /* lprev() finds the previous edge (clockwise) of a triangle. */ #define lprev(triedge1, triedge2) \ (triedge2).tri = (triedge1).tri; \ (triedge2).orient = minus1mod3[(triedge1).orient] #define lprevself(triedge) \ (triedge).orient = minus1mod3[(triedge).orient] /* onext() spins counterclockwise around a point; that is, it finds the next */ /* edge with the same origin in the counterclockwise direction. This edge */ /* will be part of a different triangle. */ #define onext(triedge1, triedge2) \ lprev(triedge1, triedge2); \ symself(triedge2); #define onextself(triedge) \ lprevself(triedge); \ symself(triedge); /* oprev() spins clockwise around a point; that is, it finds the next edge */ /* with the same origin in the clockwise direction. This edge will be */ /* part of a different triangle. */ #define oprev(triedge1, triedge2) \ sym(triedge1, triedge2); \ lnextself(triedge2); #define oprevself(triedge) \ symself(triedge); \ lnextself(triedge); /* These primitives determine or set the origin, destination, or apex of a */ /* triangle. */ #define org(triedge, pointptr) \ pointptr = (point) (triedge).tri[plus1mod3[(triedge).orient] + 3] #define dest(triedge, pointptr) \ pointptr = (point) (triedge).tri[minus1mod3[(triedge).orient] + 3] #define apex(triedge, pointptr) \ pointptr = (point) (triedge).tri[(triedge).orient + 3] #define setorg(triedge, pointptr) \ (triedge).tri[plus1mod3[(triedge).orient] + 3] = (triangle) pointptr #define setdest(triedge, pointptr) \ (triedge).tri[minus1mod3[(triedge).orient] + 3] = (triangle) pointptr #define setapex(triedge, pointptr) \ (triedge).tri[(triedge).orient + 3] = (triangle) pointptr /* Bond two triangles together. */ #define bond(triedge1, triedge2) \ (triedge1).tri[(triedge1).orient] = encode(triedge2); \ (triedge2).tri[(triedge2).orient] = encode(triedge1) /* Dissolve a bond (from one side). Note that the other triangle will still */ /* think it's connected to this triangle. Usually, however, the other */ /* triangle is being deleted entirely, or bonded to another triangle, so */ /* it doesn't matter. */ #define dissolve(triedge) \ (triedge).tri[(triedge).orient] = (triangle) dummytri /* Copy a triangle/edge handle. */ #define triedgecopy(triedge1, triedge2) \ (triedge2).tri = (triedge1).tri; \ (triedge2).orient = (triedge1).orient /* Test for equality of triangle/edge handles. */ #define triedgeequal(triedge1, triedge2) \ (((triedge1).tri == (triedge2).tri) && \ ((triedge1).orient == (triedge2).orient)) /* Primitives to infect or cure a triangle with the virus. These rely on */ /* the assumption that all shell edges are aligned to four-byte boundaries.*/ #define infect(triedge) \ (triedge).tri[6] = (triangle) \ ((unsigned long) (triedge).tri[6] | (unsigned long) 2l) #define uninfect(triedge) \ (triedge).tri[6] = (triangle) \ ((unsigned long) (triedge).tri[6] & ~ (unsigned long) 2l) /* Test a triangle for viral infection. */ #define infected(triedge) \ (((unsigned long) (triedge).tri[6] & (unsigned long) 2l) != 0) /* Check or set a triangle's attributes. */ #define elemattribute(triedge, attnum) \ ((double *) (triedge).tri)[elemattribindex + (attnum)] #define setelemattribute(triedge, attnum, value) \ ((double *) (triedge).tri)[elemattribindex + (attnum)] = value /********* Primitives for shell edges *********/ /* */ /* */ /* sdecode() converts a pointer to an oriented shell edge. The orientation */ /* is extracted from the least significant bit of the pointer. The two */ /* least significant bits (one for orientation, one for viral infection) */ /* are masked out to produce the real pointer. */ #define sdecode(sptr, edge) \ (edge).shorient = (int) ((unsigned long) (sptr) & (unsigned long) 1l); \ (edge).sh = (shelle *) \ ((unsigned long) (sptr) & ~ (unsigned long) 3l) /* sencode() compresses an oriented shell edge into a single pointer. It */ /* relies on the assumption that all shell edges are aligned to two-byte */ /* boundaries, so the least significant bit of (edge).sh is zero. */ #define sencode(edge) \ (shelle) ((unsigned long) (edge).sh | (unsigned long) (edge).shorient) /* ssym() toggles the orientation of a shell edge. */ #define ssymself(edge) \ (edge).shorient = 1 - (edge).shorient /* spivot() finds the other shell edge (from the same segment) that shares */ /* the same origin. */ #define spivot(edge1, edge2) \ sptr = (edge1).sh[(edge1).shorient]; \ sdecode(sptr, edge2) /* These primitives determine or set the origin or destination of a shell */ /* edge. */ #define setsorg(edge, pointptr) \ (edge).sh[2 + (edge).shorient] = (shelle) pointptr #define setsdest(edge, pointptr) \ (edge).sh[3 - (edge).shorient] = (shelle) pointptr /* These primitives read or set a shell marker. Shell markers are used to */ /* hold user boundary information. */ #define mark(edge) (* (int *) ((edge).sh + 6)) #define setmark(edge, value) \ * (int *) ((edge).sh + 6) = value /* Bond two shell edges together. */ #define sbond(edge1, edge2) \ (edge1).sh[(edge1).shorient] = sencode(edge2); \ (edge2).sh[(edge2).shorient] = sencode(edge1) /* Copy a shell edge. */ #define shellecopy(edge1, edge2) \ (edge2).sh = (edge1).sh; \ (edge2).shorient = (edge1).shorient /********* Primitives for interacting triangles and shell edges *********/ /* */ /* */ /* tspivot() finds a shell edge abutting a triangle. */ #define tspivot(triedge, edge) \ sptr = (shelle) (triedge).tri[6 + (triedge).orient]; \ sdecode(sptr, edge) /* Bond a triangle to a shell edge. */ #define tsbond(triedge, edge) \ (triedge).tri[6 + (triedge).orient] = (triangle) sencode(edge); \ (edge).sh[4 + (edge).shorient] = (shelle) encode(triedge) /* Dissolve a bond (from the triangle side). */ #define tsdissolve(triedge) \ (triedge).tri[6 + (triedge).orient] = (triangle) dummysh /* Dissolve a bond (from the shell edge side). */ #define stdissolve(edge) \ (edge).sh[4 + (edge).shorient] = (shelle) dummytri /********* Primitives for points *********/ /* */ /* */ #define pointmark(pt) ((int *) (pt))[pointmarkindex] #define setpointmark(pt, value) \ ((int *) (pt))[pointmarkindex] = value #define point2tri(pt) ((triangle *) (pt))[point2triindex] #define setpoint2tri(pt, value) \ ((triangle *) (pt))[point2triindex] = value /** **/ /** **/ /********* Mesh manipulation primitives end here *********/ /********* User interaction routines begin here *********/ /** **/ /** **/ /*****************************************************************************/ /* */ /* internalerror() Ask the user to send me the defective product. Exit. */ /* */ /*****************************************************************************/ void internalerror(void) { vTrace("*** E0031 : Erreur interne. Communiquez le contexte à stephane.guillard@steria.fr"); exit(1); } /*****************************************************************************/ /* */ /* parsecommandline() Read the command line, identify switches, and set */ /* up options and file names. */ /* */ /* The effects of this routine are felt entirely through global variables. */ /* */ /*****************************************************************************/ void parsecommandline(int argc, char **argv) { order = 1; minangle = 0.0; maxarea = -1.0; useshelles = 1; goodangle = cos(minangle * PI / 180.0); goodangle *= goodangle; } /** **/ /** **/ /********* User interaction routines begin here *********/ /********* Memory management routines begin here *********/ /** **/ /** **/ /*****************************************************************************/ /* */ /* poolinit() Initialize a pool of memory for allocation of items. */ /* */ /* This routine initializes the machinery for allocating items. A `pool' */ /* is created whose records have size at least `bytecount'. Items will be */ /* allocated in `itemcount'-item blocks. Each item is assumed to be a */ /* collection of words, and either pointers or floating-point values are */ /* assumed to be the "primary" word type. (The "primary" word type is used */ /* to determine alignment of items.) If `alignment' isn't zero, all items */ /* will be `alignment'-byte aligned in memory. `alignment' must be either */ /* a multiple or a factor of the primary word size; powers of two are safe. */ /* `alignment' is normally used to create a few unused bits at the bottom */ /* of each item's pointer, in which information may be stored. */ /* */ /* Don't change this routine unless you understand it. */ /* */ /*****************************************************************************/ void poolinit(struct memorypool *pool,int bytecount, int itemcount, enum wordtype wtype, int alignment) { int wordsize; /* Initialize values in the pool. */ pool->itemwordtype = wtype; wordsize = (pool->itemwordtype == POINTER) ? sizeof(void *) : sizeof(double); /* Find the proper alignment, which must be at least as large as: */ /* - The parameter `alignment'. */ /* - The primary word type, to avoid unaligned accesses. */ /* - sizeof(void *), so the stack of dead items can be maintained */ /* without unaligned accesses. */ if (alignment > wordsize) { pool->alignbytes = alignment; } else { pool->alignbytes = wordsize; } if (sizeof(void *) > pool->alignbytes) { pool->alignbytes = sizeof(void *); } pool->itemwords = ((bytecount + pool->alignbytes - 1) / pool->alignbytes) * (pool->alignbytes / wordsize); pool->itembytes = pool->itemwords * wordsize; pool->itemsperblock = itemcount; /* Allocate a block of items. Space for `itemsperblock' items and one */ /* pointer (to point to the next block) are allocated, as well as space */ /* to ensure alignment of the items. */ pool->firstblock = (void **) malloc(pool->itemsperblock * pool->itembytes + sizeof(void *) + pool->alignbytes); if (pool->firstblock == (void **) NULL) { vTrace("*** E0051: Out of memory."); exit(1); } /* Set the next block pointer to NULL. */ *(pool->firstblock) = (void *) NULL; poolrestart(pool); } /*****************************************************************************/ /* */ /* poolrestart() Deallocate all items in a pool. */ /* */ /* The pool is returned to its starting state, except that no memory is */ /* freed to the operating system. Rather, the previously allocated blocks */ /* are ready to be reused. */ /* */ /*****************************************************************************/ void poolrestart(struct memorypool *pool) { unsigned long alignptr; pool->items = 0; pool->maxitems = 0; /* Set the currently active block. */ pool->nowblock = pool->firstblock; /* Find the first item in the pool. Increment by the size of (void *). */ alignptr = (unsigned long) (pool->nowblock + 1); /* Align the item on an `alignbytes'-byte boundary. */ pool->nextitem = (void *) (alignptr + (unsigned long) pool->alignbytes - (alignptr % (unsigned long) pool->alignbytes)); /* There are lots of unallocated items left in this block. */ pool->unallocateditems = pool->itemsperblock; /* The stack of deallocated items is empty. */ pool->deaditemstack = (void *) NULL; } /*****************************************************************************/ /* */ /* pooldeinit() Free to the operating system all memory taken by a pool. */ /* */ /*****************************************************************************/ void pooldeinit(struct memorypool *pool) { while (pool->firstblock != (void **) NULL) { pool->nowblock = (void **) *(pool->firstblock); free(pool->firstblock); pool->firstblock = pool->nowblock; } } /*****************************************************************************/ /* */ /* poolalloc() Allocate space for an item. */ /* */ /*****************************************************************************/ void *poolalloc(struct memorypool *pool) { void *newitem; void **newblock; unsigned long alignptr; /* First check the linked list of dead items. If the list is not */ /* empty, allocate an item from the list rather than a fresh one. */ if (pool->deaditemstack != (void *) NULL) { newitem = pool->deaditemstack; /* Take first item in list. */ pool->deaditemstack = * (void **) pool->deaditemstack; } else { /* Check if there are any free items left in the current block. */ if (pool->unallocateditems == 0) { /* Check if another block must be allocated. */ if (*(pool->nowblock) == (void *) NULL) { /* Allocate a new block of items, pointed to by the previous block. */ newblock = (void **) malloc(pool->itemsperblock * pool->itembytes + sizeof(void *) + pool->alignbytes); if (newblock == (void **) NULL) { vTrace("*** E0052 : Out of memory."); exit(1); } *(pool->nowblock) = (void *) newblock; /* The next block pointer is NULL. */ *newblock = (void *) NULL; } /* Move to the new block. */ pool->nowblock = (void **) *(pool->nowblock); /* Find the first item in the block. */ /* Increment by the size of (void *). */ alignptr = (unsigned long) (pool->nowblock + 1); /* Align the item on an `alignbytes'-byte boundary. */ pool->nextitem = (void *) (alignptr + (unsigned long) pool->alignbytes - (alignptr % (unsigned long) pool->alignbytes)); /* There are lots of unallocated items left in this block. */ pool->unallocateditems = pool->itemsperblock; } /* Allocate a new item. */ newitem = pool->nextitem; /* Advance `nextitem' pointer to next free item in block. */ if (pool->itemwordtype == POINTER) { pool->nextitem = (void *) ((void **) pool->nextitem + pool->itemwords); } else { pool->nextitem = (void *) ((double *) pool->nextitem + pool->itemwords); } pool->unallocateditems--; pool->maxitems++; } pool->items++; return newitem; } /*****************************************************************************/ /* */ /* pooldealloc() Deallocate space for an item. */ /* */ /* The deallocated space is stored in a queue for later reuse. */ /* */ /*****************************************************************************/ void pooldealloc(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->itemsperblock; } /*****************************************************************************/ /* */ /* traverse() Find the next item in the list. */ /* */ /* This routine is used in conjunction with traversalinit(). Be forewarned */ /* that this routine successively returns all items in the list, including */ /* deallocated ones on the deaditemqueue. It's up to you to figure out */ /* which ones are actually dead. Why? I don't want to allocate extra */ /* space just to demarcate dead items. It can usually be done more */ /* space-efficiently by a routine that knows something about the structure */ /* of the item. */ /* */ /*****************************************************************************/ void *traverse(struct memorypool *pool) { void *newitem; unsigned long alignptr; /* Stop upon exhausting the list of items. */ if (pool->pathitem == pool->nextitem) { return (void *) NULL; } /* Check whether any untraversed items remain in the current block. */ if (pool->pathitemsleft == 0) { /* Find the next block. */ pool->pathblock = (void **) *(pool->pathblock); /* Find the first item in the block. Increment by the size of (void *). */ alignptr = (unsigned long) (pool->pathblock + 1); /* Align with item on an `alignbytes'-byte boundary. */ pool->pathitem = (void *) (alignptr + (unsigned long) pool->alignbytes - (alignptr % (unsigned long) pool->alignbytes)); /* Set the number of items left in the current block. */ pool->pathitemsleft = pool->itemsperblock; } newitem = pool->pathitem; /* Find the next item in the block. */ if (pool->itemwordtype == POINTER) { pool->pathitem = (void *) ((void **) pool->pathitem + pool->itemwords); } else { pool->pathitem = (void *) ((double *) pool->pathitem + pool->itemwords); } pool->pathitemsleft--; return newitem; } /*****************************************************************************/ /* */ /* dummyinit() Initialize the triangle that fills "outer space" and the */ /* omnipresent shell edge. */ /* */ /* The triangle that fills "outer space", called `dummytri', is pointed to */ /* by every triangle and shell edge on a boundary (be it outer or inner) of */ /* the triangulation. Also, `dummytri' points to one of the triangles on */ /* the convex hull (until the holes and concavities are carved), making it */ /* possible to find a starting triangle for point location. */ /* */ /* The omnipresent shell edge, `dummysh', is pointed to by every triangle */ /* or shell edge that doesn't have a full complement of real shell edges */ /* to point to. */ /* */ /*****************************************************************************/ void dummyinit(int trianglewords, int shellewords) { unsigned long alignptr; /* `triwords' and `shwords' are used by the mesh manipulation primitives */ /* to extract orientations of triangles and shell edges from pointers. */ triwords = trianglewords; /* Initialize `triwords' once and for all. */ shwords = shellewords; /* Initialize `shwords' once and for all. */ /* Set up `dummytri', the `triangle' that occupies "outer space". */ dummytribase = (triangle *) malloc(triwords * sizeof(triangle) + triangles.alignbytes); if (dummytribase == (triangle *) NULL) { vTrace("*** E0053 : Out of memory."); exit(1); } /* Align `dummytri' on a `triangles.alignbytes'-byte boundary. */ alignptr = (unsigned long) dummytribase; dummytri = (triangle *) (alignptr + (unsigned long) triangles.alignbytes - (alignptr % (unsigned long) triangles.alignbytes)); /* Initialize the three adjoining triangles to be "outer space". These */ /* will eventually be changed by various bonding operations, but their */ /* values don't really matter, as long as they can legally be */ /* dereferenced. */ dummytri[0] = (triangle) dummytri; dummytri[1] = (triangle) dummytri; dummytri[2] = (triangle) dummytri; /* Three NULL vertex points. */ dummytri[3] = (triangle) NULL; dummytri[4] = (triangle) NULL; dummytri[5] = (triangle) NULL; if (useshelles) { /* Set up `dummysh', the omnipresent "shell edge" pointed to by any */ /* triangle side or shell edge end that isn't attached to a real shell */ /* edge. */ dummyshbase = (shelle *) malloc(shwords * sizeof(shelle) + shelles.alignbytes); if (dummyshbase == (shelle *) NULL) { vTrace("*** E0054 : Out of memory."); exit(1); } /* Align `dummysh' on a `shelles.alignbytes'-byte boundary. */ alignptr = (unsigned long) dummyshbase; dummysh = (shelle *) (alignptr + (unsigned long) shelles.alignbytes - (alignptr % (unsigned long) shelles.alignbytes)); /* Initialize the two adjoining shell edges to be the omnipresent shell */ /* edge. These will eventually be changed by various bonding */ /* operations, but their values don't really matter, as long as they */ /* can legally be dereferenced. */ dummysh[0] = (shelle) dummysh; dummysh[1] = (shelle) dummysh; /* Two NULL vertex points. */ dummysh[2] = (shelle) NULL; dummysh[3] = (shelle) NULL; /* Initialize the two adjoining triangles to be "outer space". */ dummysh[4] = (shelle) dummytri; dummysh[5] = (shelle) dummytri; /* Set the boundary marker to zero. */ * (int *) (dummysh + 6) = 0; /* Initialize the three adjoining shell edges of `dummytri' to be */ /* the omnipresent shell edge. */ dummytri[6] = (triangle) dummysh; dummytri[7] = (triangle) dummysh; dummytri[8] = (triangle) dummysh; } } /*****************************************************************************/ /* */ /* initializepointpool() Calculate the size of the point data structure */ /* and initialize its memory pool. */ /* */ /* This routine also computes the `pointmarkindex' and `point2triindex' */ /* indices used to find values within each point. */ /* */ /*****************************************************************************/ void initializepointpool(void) { int pointsize; /* The index within each point at which the boundary marker is found. */ /* Ensure the point marker is aligned to a sizeof(int)-byte address. */ pointmarkindex = ((mesh_dim + nextras) * sizeof(double) + sizeof(int) - 1) / sizeof(int); pointsize = (pointmarkindex + 1) * sizeof(int); /* The index within each point at which a triangle pointer is found. */ /* Ensure the pointer is aligned to a sizeof(triangle)-byte address. */ point2triindex = (pointsize + sizeof(triangle) - 1) / sizeof(triangle); pointsize = (point2triindex + 1) * sizeof(triangle); /* Initialize the pool of points. */ poolinit(&points, pointsize, POINTPERBLOCK, (sizeof(double) >= sizeof(triangle)) ? FLOATINGPOINT : POINTER, 0); } /*****************************************************************************/ /* */ /* initializetrisegpools() Calculate the sizes of the triangle and shell */ /* edge data structures and initialize their */ /* memory pools. */ /* */ /* This routine also computes the `highorderindex', `elemattribindex', and */ /* `areaboundindex' indices used to find values within each triangle. */ /* */ /*****************************************************************************/ void initializetrisegpools(void) { int trisize; /* The index within each triangle at which the extra nodes (above three) */ /* associated with high order elements are found. There are three */ /* pointers to other triangles, three pointers to corners, and possibly */ /* three pointers to shell edges before the extra nodes. */ highorderindex = 6 + (useshelles * 3); /* The number of bytes occupied by a triangle. */ trisize = ((order + 1) * (order + 2) / 2 + (highorderindex - 3)) * sizeof(triangle); /* The index within each triangle at which its attributes are found, */ /* where the index is measured in REALs. */ elemattribindex = (trisize + sizeof(double) - 1) / sizeof(double); /* The index within each triangle at which the maximum area constraint */ /* is found, where the index is measured in REALs.*/ areaboundindex = elemattribindex + eextras; /* If triangle attributes or an area bound are needed, increase the number */ /* of bytes occupied by a triangle. */ if (eextras > 0) { trisize = areaboundindex * sizeof(double); } /* Having determined the memory size of a triangle, initialize the pool. */ poolinit(&triangles, trisize, TRIPERBLOCK, POINTER, 4); if (useshelles) { /* Initialize the pool of shell edges. */ poolinit(&shelles, 6 * sizeof(triangle) + sizeof(int), SHELLEPERBLOCK, POINTER, 4); /* Initialize the "outer space" triangle and omnipresent shell edge. */ dummyinit(triangles.itemwords, shelles.itemwords); } else { /* Initialize the "outer space" triangle. */ dummyinit(triangles.itemwords, 0); } } /*****************************************************************************/ /* */ /* triangledealloc() Deallocate space for a triangle, marking it dead. */ /* */ /*****************************************************************************/ void triangledealloc(triangle *dyingtriangle) { /* Set triangle's vertices to NULL. This makes it possible to */ /* detect dead triangles when traversing the list of all triangles. */ dyingtriangle[3] = (triangle) NULL; dyingtriangle[4] = (triangle) NULL; dyingtriangle[5] = (triangle) NULL; pooldealloc(&triangles, (void *) dyingtriangle); } /*****************************************************************************/ /* */ /* triangletraverse() Traverse the triangles, skipping dead ones. */ /* */ /*****************************************************************************/ triangle *triangletraverse(void) { triangle *newtriangle; do { newtriangle = (triangle *) traverse(&triangles); if (newtriangle == (triangle *) NULL) { return (triangle *) NULL; } } while (newtriangle[3] == (triangle) NULL); /* Skip dead ones. */ return newtriangle; } /*****************************************************************************/ /* */ /* shelledealloc() Deallocate space for a shell edge, marking it dead. */ /* */ /*****************************************************************************/ void shelledealloc(shelle *dyingshelle) { /* Set shell edge's vertices to NULL. This makes it possible to */ /* detect dead shells when traversing the list of all shells. */ dyingshelle[2] = (shelle) NULL; dyingshelle[3] = (shelle) NULL; pooldealloc(&shelles, (void *) dyingshelle); } /*****************************************************************************/ /* */ /* pointdealloc() Deallocate space for a point, marking it dead. */ /* */ /*****************************************************************************/ void pointdealloc(point dyingpoint) { /* Mark the point as dead. This makes it possible to detect dead points */ /* when traversing the list of all points. */ setpointmark(dyingpoint, DEADPOINT); pooldealloc(&points, (void *) dyingpoint); } /*****************************************************************************/ /* */ /* pointtraverse() Traverse the points, skipping dead ones. */ /* */ /*****************************************************************************/ point pointtraverse(void) { point newpoint; do { newpoint = (point) traverse(&points); if (newpoint == (point) NULL) { return (point) NULL; } } while (pointmark(newpoint) == DEADPOINT); /* Skip dead ones. */ return newpoint; } /*****************************************************************************/ /* */ /* getpoint() Get a specific point, by number, from the list. */ /* */ /* The first point is number 0. */ /* */ /* Note that this takes O(n) time (with a small constant, if POINTPERBLOCK */ /* is large). I don't care to take the trouble to make it work in constant */ /* time. */ /* */ /*****************************************************************************/ point getpoint(int number) { void **getblock; point foundpoint; unsigned long alignptr; int current; getblock = points.firstblock; current = 0; /* Find the right block. */ while (current + points.itemsperblock <= number) { getblock = (void **) *getblock; current += points.itemsperblock; } /* Now find the right point. */ alignptr = (unsigned long) (getblock + 1); foundpoint = (point) (alignptr + (unsigned long) points.alignbytes - (alignptr % (unsigned long) points.alignbytes)); while (current < number) { foundpoint += points.itemwords; current++; } return foundpoint; } /*****************************************************************************/ /* */ /* triangledeinit() Free all remaining allocated memory. */ /* */ /*****************************************************************************/ void triangledeinit(void) { pooldeinit(&triangles); free(dummytribase); if (useshelles) { pooldeinit(&shelles); free(dummyshbase); } pooldeinit(&points); } /** **/ /** **/ /********* Memory management routines end here *********/ /********* Constructors begin here *********/ /** **/ /** **/ /*****************************************************************************/ /* */ /* maketriangle() Create a new triangle with orientation zero. */ /* */ /*****************************************************************************/ void maketriangle(struct triedge *newtriedge) { int i; newtriedge->tri = (triangle *) poolalloc(&triangles); /* Initialize the three adjoining triangles to be "outer space". */ newtriedge->tri[0] = (triangle) dummytri; newtriedge->tri[1] = (triangle) dummytri; newtriedge->tri[2] = (triangle) dummytri; /* Three NULL vertex points. */ newtriedge->tri[3] = (triangle) NULL; newtriedge->tri[4] = (triangle) NULL; newtriedge->tri[5] = (triangle) NULL; /* Initialize the three adjoining shell edges to be the omnipresent */ /* shell edge. */ if (useshelles) { newtriedge->tri[6] = (triangle) dummysh; newtriedge->tri[7] = (triangle) dummysh; newtriedge->tri[8] = (triangle) dummysh; } for (i = 0; i < eextras; i++) { setelemattribute(*newtriedge, i, 0.0); } newtriedge->orient = 0; } /*****************************************************************************/ /* */ /* makeshelle() Create a new shell edge with orientation zero. */ /* */ /*****************************************************************************/ void makeshelle(struct edge *newedge) { newedge->sh = (shelle *) poolalloc(&shelles); /* Initialize the two adjoining shell edges to be the omnipresent */ /* shell edge. */ newedge->sh[0] = (shelle) dummysh; newedge->sh[1] = (shelle) dummysh; /* Two NULL vertex points. */ newedge->sh[2] = (shelle) NULL; newedge->sh[3] = (shelle) NULL; /* Initialize the two adjoining triangles to be "outer space". */ newedge->sh[4] = (shelle) dummytri; newedge->sh[5] = (shelle) dummytri; /* Set the boundary marker to zero. */ setmark(*newedge, 0); newedge->shorient = 0; } /** **/ /** **/ /********* Constructors end here *********/ /********* Determinant evaluation routines begin here *********/ /** **/ /** **/ /* The adaptive exact arithmetic geometric predicates implemented herein are */ /* described in detail in my Technical Report CMU-CS-96-140. The complete */ /* reference is given in the header. */ /* Which of the following two methods of finding the absolute values is */ /* fastest is compiler-dependent. A few compilers can inline and optimize */ /* the fabs() call; but most will incur the overhead of a function call, */ /* which is disastrously slow. A faster way on IEEE machines might be to */ /* mask the appropriate bit, but that's difficult to do in C. */ // #define Absolute(a) ((a) >= 0.0 ? (a) : -(a)) #define Absolute(a) fabs(a) /* Many of the operations are broken up into two pieces, a main part that */ /* performs an approximate operation, and a "tail" that computes the */ /* roundoff error of that operation. */ /* */ /* The operations Fast_Two_Sum(), Fast_Two_Diff(), Two_Sum(), Two_Diff(), */ /* Split(), and Two_Product() are all implemented as described in the */ /* reference. Each of these macros requires certain variables to be */ /* defined in the calling routine. The variables `bvirt', `c', `abig', */ /* `_i', `_j', `_k', `_l', `_m', and `_n' are declared `INEXACT' because */ /* they store the result of an operation that may incur roundoff error. */ /* The input parameter `x' (or the highest numbered `x_' parameter) must */ /* also be declared `INEXACT'. */ #define Fast_Two_Sum_Tail(a, b, x, y) \ bvirt = x - a; \ y = b - bvirt #define Fast_Two_Sum(a, b, x, y) \ x = (double) (a + b); \ Fast_Two_Sum_Tail(a, b, x, y) #define Two_Sum_Tail(a, b, x, y) \ bvirt = (double) (x - a); \ avirt = x - bvirt; \ bround = b - bvirt; \ around = a - avirt; \ y = around + bround #define Two_Sum(a, b, x, y) \ x = (double) (a + b); \ Two_Sum_Tail(a, b, x, y) #define Two_Diff_Tail(a, b, x, y) \ bvirt = (double) (a - x); \ avirt = x + bvirt; \ bround = bvirt - b; \ around = a - avirt; \ y = around + bround #define Two_Diff(a, b, x, y) \ x = (double) (a - b); \ Two_Diff_Tail(a, b, x, y) #define Split(a, ahi, alo) \ c = (double) (splitter * a); \ abig = (double) (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 = (double) (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 = (double) (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 = (double) (a * a); \ Square_Tail(a, x, y) /* Macros for summing expansions of various fixed lengths. These are all */ /* unrolled versions of Expansion_Sum(). */ #define Two_One_Sum(a1, a0, b, x2, x1, x0) \ Two_Sum(a0, b , _i, x0); \ Two_Sum(a1, _i, x2, x1) #define Two_One_Diff(a1, a0, b, x2, x1, x0) \ Two_Diff(a0, b , _i, x0); \ Two_Sum( a1, _i, x2, x1) #define Two_Two_Sum(a1, a0, b1, b0, x3, x2, x1, x0) \ Two_One_Sum(a1, a0, b0, _j, _0, x0); \ Two_One_Sum(_j, _0, b1, x3, x2, x1) #define Two_Two_Diff(a1, a0, b1, b0, x3, x2, x1, x0) \ Two_One_Diff(a1, a0, b0, _j, _0, x0); \ Two_One_Diff(_j, _0, b1, x3, x2, x1) /*****************************************************************************/ /* */ /* exactinit() Initialize the variables used for exact arithmetic. */ /* */ /* `epsilon' is the largest power of two such that 1.0 + epsilon = 1.0 in */ /* floating-point arithmetic. `epsilon' bounds the relative roundoff */ /* error. It is used for floating-point error analysis. */ /* */ /* `splitter' is used to split floating-point numbers into two half- */ /* length significands for exact multiplication. */ /* */ /* I imagine that a highly optimizing compiler might be too smart for its */ /* own good, and somehow cause this routine to fail, if it pretends that */ /* floating-point arithmetic is too much like real arithmetic. */ /* */ /* Don't change this routine unless you fully understand it. */ /* */ /*****************************************************************************/ void exactinit(void) { double half; double check, lastcheck; int every_other; every_other = 1; half = 0.5; epsilon = 1.0; splitter = 1.0; check = 1.0; /* Repeatedly divide `epsilon' by two until it is too small to add to */ /* one without causing roundoff. (Also check if the sum is equal to */ /* the previous sum, for machines that round up instead of using exact */ /* rounding. Not that these routines will work on such machines anyway. */ do { lastcheck = check; epsilon *= half; if (every_other) { splitter *= 2.0; } every_other = !every_other; check = 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; } /*****************************************************************************/ /* */ /* 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,double *e,int flen,double *f,double *h) /* h cannot be e or f. */ { double Q; INEXACT double Qnew; INEXACT double hh; INEXACT double bvirt; double avirt, bround, around; int eindex, findex, hindex; double 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,double *e,double b,double *h) /* e and h cannot be the same. */ { INEXACT double Q, sum; double hh; INEXACT double product1; double product0; int eindex, hindex; double enow; INEXACT double bvirt; double avirt, bround, around; INEXACT double c; INEXACT double abig; double ahi, alo, bhi, blo; double 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. */ /* */ /*****************************************************************************/ double estimate(int elen, double *e) { double 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. */ /* */ /*****************************************************************************/ double counterclockwiseadapt( point pa, point pb, point pc, double detsum) { INEXACT double acx, acy, bcx, bcy; double acxtail, acytail, bcxtail, bcytail; INEXACT double detleft, detright; double detlefttail, detrighttail; double det, errbound; double B[4], C1[8], C2[12], D[16]; INEXACT double B3; int C1length, C2length, Dlength; double u[4]; INEXACT double u3; INEXACT double s1, t1; double s0, t0; INEXACT double bvirt; double avirt, bround, around; INEXACT double c; INEXACT double abig; double ahi, alo, bhi, blo; double err1, err2, err3; INEXACT double _i, _j; double _0; acx = (double) (pa[0] - pc[0]); bcx = (double) (pb[0] - pc[0]); acy = (double) (pa[1] - pc[1]); bcy = (double) (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]); } double counterclockwise( point pa, point pb, point pc) { double detleft, detright, det; double detsum, errbound; counterclockcount++; detleft = (pa[0] - pc[0]) * (pb[1] - pc[1]); detright = (pa[1] - pc[1]) * (pb[0] - pc[0]); det = detleft - detright; if (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. */ /* */ /*****************************************************************************/ double incircleadapt( point pa, point pb, point pc, point pd, double permanent) { INEXACT double adx, bdx, cdx, ady, bdy, cdy; double det, errbound; INEXACT double bdxcdy1, cdxbdy1, cdxady1, adxcdy1, adxbdy1, bdxady1; double bdxcdy0, cdxbdy0, cdxady0, adxcdy0, adxbdy0, bdxady0; double bc[4], ca[4], ab[4]; INEXACT double bc3, ca3, ab3; double axbc[8], axxbc[16], aybc[8], ayybc[16], adet[32]; int axbclen, axxbclen, aybclen, ayybclen, alen; double bxca[8], bxxca[16], byca[8], byyca[16], bdet[32]; int bxcalen, bxxcalen, bycalen, byycalen, blen; double cxab[8], cxxab[16], cyab[8], cyyab[16], cdet[32]; int cxablen, cxxablen, cyablen, cyyablen, clen; double abdet[64]; int ablen; double fin1[1152], fin2[1152]; double *finnow, *finother, *finswap; int finlength; double adxtail, bdxtail, cdxtail, adytail, bdytail, cdytail; INEXACT double adxadx1, adyady1, bdxbdx1, bdybdy1, cdxcdx1, cdycdy1; double adxadx0, adyady0, bdxbdx0, bdybdy0, cdxcdx0, cdycdy0; double aa[4], bb[4], cc[4]; INEXACT double aa3, bb3, cc3; INEXACT double ti1, tj1; double ti0, tj0; double u[4], v[4]; INEXACT double u3, v3; double temp8[8], temp16a[16], temp16b[16], temp16c[16]; double temp32a[32], temp32b[32], temp48[48], temp64[64]; int temp8len, temp16alen, temp16blen, temp16clen; int temp32alen, temp32blen, temp48len, temp64len; double axtbb[8], axtcc[8], aytbb[8], aytcc[8]; int axtbblen, axtcclen, aytbblen, aytcclen; double bxtaa[8], bxtcc[8], bytaa[8], bytcc[8]; int bxtaalen, bxtcclen, bytaalen, bytcclen; double cxtaa[8], cxtbb[8], cytaa[8], cytbb[8]; int cxtaalen, cxtbblen, cytaalen, cytbblen; double axtbc[8], aytbc[8], bxtca[8], bytca[8], cxtab[8], cytab[8]; int axtbclen, aytbclen, bxtcalen, bytcalen, cxtablen, cytablen; double axtbct[16], aytbct[16], bxtcat[16], bytcat[16], cxtabt[16], cytabt[16]; int axtbctlen, aytbctlen, bxtcatlen, bytcatlen, cxtabtlen, cytabtlen; double axtbctt[8], aytbctt[8], bxtcatt[8]; double bytcatt[8], cxtabtt[8], cytabtt[8]; int axtbcttlen, aytbcttlen, bxtcattlen, bytcattlen, cxtabttlen, cytabttlen; double abt[8], bct[8], cat[8]; int abtlen, bctlen, catlen; double abtt[4], bctt[4], catt[4]; int abttlen, bcttlen, cattlen; INEXACT double abtt3, bctt3, catt3; double negate; INEXACT double bvirt; double avirt, bround, around; INEXACT double c; INEXACT double abig; double ahi, alo, bhi, blo; double err1, err2, err3; INEXACT double _i, _j; double _0; adx = (double) (pa[0] - pd[0]); bdx = (double) (pb[0] - pd[0]); cdx = (double) (pc[0] - pd[0]); ady = (double) (pa[1] - pd[1]); bdy = (double) (pb[1] - pd[1]); cdy = (double) (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]; } double incircle( point pa, point pb, point pc, point pd) { double adx, bdx, cdx, ady, bdy, cdy; double bdxcdy, cdxbdy, cdxady, adxcdy, adxbdy, bdxady; double alift, blift, clift; double det; double permanent, errbound; incirclecount++; adx = pa[0] - pd[0]; bdx = pb[0] - pd[0]; cdx = pc[0] - pd[0]; ady = pa[1] - pd[1]; bdy = pb[1] - pd[1]; cdy = pc[1] - pd[1]; bdxcdy = bdx * cdy; cdxbdy = cdx * bdy; alift = adx * adx + ady * ady; cdxady = cdx * ady; adxcdy = adx * cdy; blift = bdx * bdx + bdy * bdy; adxbdy = adx * bdy; bdxady = bdx * ady; clift = cdx * cdx + cdy * cdy; det = alift * (bdxcdy - cdxbdy) + blift * (cdxady - adxcdy) + clift * (adxbdy - bdxady); permanent = (Absolute(bdxcdy) + Absolute(cdxbdy)) * alift + (Absolute(cdxady) + Absolute(adxcdy)) * blift + (Absolute(adxbdy) + Absolute(bdxady)) * clift; errbound = iccerrboundA * permanent; if ((det > errbound) || (-det > errbound)) { return det; } return incircleadapt(pa, pb, pc, pd, permanent); } /** **/ /** **/ /********* Determinant evaluation routines end here *********/ /*****************************************************************************/ /* */ /* triangleinit() Initialize some variables. */ /* */ /*****************************************************************************/ void triangleinit(void) { points.maxitems = triangles.maxitems = shelles.maxitems = viri.maxitems = badsegments.maxitems = badtriangles.maxitems = splaynodes.maxitems = 0l; points.itembytes = triangles.itembytes = shelles.itembytes = viri.itembytes = badsegments.itembytes = badtriangles.itembytes = splaynodes.itembytes = 0; recenttri.tri = (triangle *) NULL; /* No triangle has been visited yet. */ samples = 1; /* Point location should take at least one sample. */ checksegments = 0; /* There are no segments in the triangulation yet. */ incirclecount = counterclockcount = hyperbolacount = 0; circumcentercount = circletopcount = 0; randomseed = 1; exactinit(); /* Initialize exact arithmetic constants. */ } /*****************************************************************************/ /* */ /* randomnation() Generate a random number between 0 and `choices' - 1. */ /* */ /* This is a simple linear congruential random number generator. Hence, it */ /* is a bad random number generator, but good enough for most randomized */ /* geometric algorithms. */ /* */ /*****************************************************************************/ unsigned long randomnation(unsigned int choices) { randomseed = (randomseed * 1366l + 150889l) % 714025l; return randomseed / (714025l / choices + 1); } /********* Point location routines begin here *********/ /** **/ /** **/ /*****************************************************************************/ /* */ /* makepointmap() Construct a mapping from points to triangles to improve */ /* the speed of point location for segment insertion. */ /* */ /* Traverses all the triangles, and provides each corner of each triangle */ /* with a pointer to that triangle. Of course, pointers will be */ /* overwritten by other pointers because (almost) each point is a corner */ /* of several triangles, but in the end every point will point to some */ /* triangle that contains it. */ /* */ /*****************************************************************************/ void makepointmap(void) { struct triedge triangleloop; point triorg; traversalinit(&triangles); triangleloop.tri = triangletraverse(); while (triangleloop.tri != (triangle *) NULL) { /* Check all three points of the triangle. */ for (triangleloop.orient = 0; triangleloop.orient < 3; triangleloop.orient++) { org(triangleloop, triorg); setpoint2tri(triorg, encode(triangleloop)); } triangleloop.tri = triangletraverse(); } } /*****************************************************************************/ /* */ /* preciselocate() Find a triangle or edge containing a given point. */ /* */ /* Begins its search from `searchtri'. It is important that `searchtri' */ /* be a handle with the property that `searchpoint' is strictly to the left */ /* of the edge denoted by `searchtri', or is collinear with that edge and */ /* does not intersect that edge. (In particular, `searchpoint' should not */ /* be the origin or destination of that edge.) */ /* */ /* These conditions are imposed because preciselocate() is normally used in */ /* one of two situations: */ /* */ /* (1) To try to find the location to insert a new point. Normally, we */ /* know an edge that the point is strictly to the left of. In the */ /* incremental Delaunay algorithm, that edge is a bounding box edge. */ /* In Ruppert's Delaunay refinement algorithm for quality meshing, */ /* that edge is the shortest edge of the triangle whose circumcenter */ /* is being inserted. */ /* */ /* (2) To try to find an existing point. In this case, any edge on the */ /* convex hull is a good starting edge. The possibility that the */ /* vertex one seeks is an endpoint of the starting edge must be */ /* screened out before preciselocate() is called. */ /* */ /* On completion, `searchtri' is a triangle that contains `searchpoint'. */ /* */ /* This implementation differs from that given by Guibas and Stolfi. It */ /* walks from triangle to triangle, crossing an edge only if `searchpoint' */ /* is on the other side of the line containing that edge. After entering */ /* a triangle, there are two edges by which one can leave that triangle. */ /* If both edges are valid (`searchpoint' is on the other side of both */ /* edges), one of the two is chosen by drawing a line perpendicular to */ /* the entry edge (whose endpoints are `forg' and `fdest') passing through */ /* `fapex'. Depending on which side of this perpendicular `searchpoint' */ /* falls on, an exit edge is chosen. */ /* */ /* This implementation is empirically faster than the Guibas and Stolfi */ /* point location routine (which I originally used), which tends to spiral */ /* in toward its target. */ /* */ /* Returns ONVERTEX if the point lies on an existing vertex. `searchtri' */ /* is a handle whose origin is the existing vertex. */ /* */ /* Returns ONEDGE if the point lies on a mesh edge. `searchtri' is a */ /* handle whose primary edge is the edge on which the point lies. */ /* */ /* Returns INTRIANGLE if the point lies strictly within a triangle. */ /* `searchtri' is a handle on the triangle that contains the point. */ /* */ /* Returns OUTSIDE if the point lies outside the mesh. `searchtri' is a */ /* handle whose primary edge the point is to the right of. This might */ /* occur when the circumcenter of a triangle falls just slightly outside */ /* the mesh due to floating-point roundoff error. It also occurs when */ /* seeking a hole or region point that a foolish user has placed outside */ /* the mesh. */ /* */ /* WARNING: This routine is designed for convex triangulations, and will */ /* not generally work after the holes and concavities have been carved. */ /* However, it can still be used to find the circumcenter of a triangle, as */ /* long as the search is begun from the triangle in question. */ /* */ /*****************************************************************************/ enum locateresult preciselocate(point searchpoint, struct triedge *searchtri) { struct triedge backtracktri; point forg, fdest, fapex; point swappoint; double orgorient, destorient; int moveleft; triangle ptr; /* Temporary variable used by sym(). */ /* Where are we? */ org(*searchtri, forg); dest(*searchtri, fdest); apex(*searchtri, fapex); while (1) { /* Check whether the apex is the point we seek. */ if ((fapex[0] == searchpoint[0]) && (fapex[1] == searchpoint[1])) { lprevself(*searchtri); return ONVERTEX; } /* Does the point lie on the other side of the line defined by the */ /* triangle edge opposite the triangle's destination? */ destorient = counterclockwise(forg, fapex, searchpoint); /* Does the point lie on the other side of the line defined by the */ /* triangle edge opposite the triangle's origin? */ orgorient = counterclockwise(fapex, fdest, searchpoint); if (destorient > 0.0) { if (orgorient > 0.0) { /* Move left if the inner product of (fapex - searchpoint) and */ /* (fdest - forg) is positive. This is equivalent to drawing */ /* a line perpendicular to the line (forg, fdest) passing */ /* through `fapex', and determining which side of this line */ /* `searchpoint' falls on. */ moveleft = (fapex[0] - searchpoint[0]) * (fdest[0] - forg[0]) + (fapex[1] - searchpoint[1]) * (fdest[1] - forg[1]) > 0.0; } else { moveleft = 1; } } else { if (orgorient > 0.0) { moveleft = 0; } else { /* The point we seek must be on the boundary of or inside this */ /* triangle. */ if (destorient == 0.0) { lprevself(*searchtri); return ONEDGE; } if (orgorient == 0.0) { lnextself(*searchtri); return ONEDGE; } return INTRIANGLE; } } /* Move to another triangle. Leave a trace `backtracktri' in case */ /* floating-point roundoff or some such bogey causes us to walk */ /* off a boundary of the triangulation. We can just bounce off */ /* the boundary as if it were an elastic band. */ if (moveleft) { lprev(*searchtri, backtracktri); fdest = fapex; } else { lnext(*searchtri, backtracktri); forg = fapex; } sym(backtracktri, *searchtri); /* Check for walking off the edge. */ if (searchtri->tri == dummytri) { /* Turn around. */ triedgecopy(backtracktri, *searchtri); swappoint = forg; forg = fdest; fdest = swappoint; apex(*searchtri, fapex); /* Check if the point really is beyond the triangulation boundary. */ destorient = counterclockwise(forg, fapex, searchpoint); orgorient = counterclockwise(fapex, fdest, searchpoint); if ((orgorient < 0.0) && (destorient < 0.0)) { return OUTSIDE; } } else { apex(*searchtri, fapex); } } } /*****************************************************************************/ /* */ /* locate() Find a triangle or edge containing a given point. */ /* */ /* Searching begins from one of: the input `searchtri', a recently */ /* encountered triangle `recenttri', or from a triangle chosen from a */ /* random sample. The choice is made by determining which triangle's */ /* origin is closest to the point we are searcing for. Normally, */ /* `searchtri' should be a handle on the convex hull of the triangulation. */ /* */ /* Details on the random sampling method can be found in the Mucke, Saias, */ /* and Zhu paper cited in the header of this code. */ /* */ /* On completion, `searchtri' is a triangle that contains `searchpoint'. */ /* */ /* Returns ONVERTEX if the point lies on an existing vertex. `searchtri' */ /* is a handle whose origin is the existing vertex. */ /* */ /* Returns ONEDGE if the point lies on a mesh edge. `searchtri' is a */ /* handle whose primary edge is the edge on which the point lies. */ /* */ /* Returns INTRIANGLE if the point lies strictly within a triangle. */ /* `searchtri' is a handle on the triangle that contains the point. */ /* */ /* Returns OUTSIDE if the point lies outside the mesh. `searchtri' is a */ /* handle whose primary edge the point is to the right of. This might */ /* occur when the circumcenter of a triangle falls just slightly outside */ /* the mesh due to floating-point roundoff error. It also occurs when */ /* seeking a hole or region point that a foolish user has placed outside */ /* the mesh. */ /* */ /* WARNING: This routine is designed for convex triangulations, and will */ /* not generally work after the holes and concavities have been carved. */ /* */ /*****************************************************************************/ enum locateresult locate(point searchpoint, struct triedge *searchtri) { void **sampleblock; triangle *firsttri; struct triedge sampletri; point torg, tdest; unsigned long alignptr; double searchdist, dist; double ahead; long sampleblocks, samplesperblock, samplenum; long triblocks; long i, j; triangle ptr; /* Temporary variable used by sym(). */ /* 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 a recently encountered triangle has been recorded and has not been */ /* deallocated, test it as a good starting point. */ if (recenttri.tri != (triangle *) NULL) { if (recenttri.tri[3] != (triangle) NULL) { org(recenttri, torg); if ((torg[0] == searchpoint[0]) && (torg[1] == searchpoint[1])) { triedgecopy(recenttri, *searchtri); return ONVERTEX; } dist = (searchpoint[0] - torg[0]) * (searchpoint[0] - torg[0]) + (searchpoint[1] - torg[1]) * (searchpoint[1] - torg[1]); if (dist < searchdist) { triedgecopy(recenttri, *searchtri); searchdist = dist; } } } /* The number of random samples taken is proportional to the cube root of */ /* the number of triangles in the mesh. The next bit of code assumes */ /* that the number of triangles increases monotonically. */ while (SAMPLEFACTOR * samples * samples * samples < triangles.items) { samples++; } triblocks = (triangles.maxitems + TRIPERBLOCK - 1) / TRIPERBLOCK; samplesperblock = 1 + (samples / triblocks); sampleblocks = samples / samplesperblock; sampleblock = triangles.firstblock; sampletri.orient = 0; for (i = 0; i < sampleblocks; i++) { alignptr = (unsigned long) (sampleblock + 1); firsttri = (triangle *) (alignptr + (unsigned long) triangles.alignbytes - (alignptr % (unsigned long) triangles.alignbytes)); for (j = 0; j < samplesperblock; j++) { if (i == triblocks - 1) { samplenum = randomnation((int) (triangles.maxitems - (i * TRIPERBLOCK))); } else { samplenum = randomnation(TRIPERBLOCK); } sampletri.tri = (triangle *) (firsttri + (samplenum * triangles.itemwords)); if (sampletri.tri[3] != (triangle) NULL) { org(sampletri, torg); dist = (searchpoint[0] - torg[0]) * (searchpoint[0] - torg[0]) + (searchpoint[1] - torg[1]) * (searchpoint[1] - torg[1]); if (dist < searchdist) { triedgecopy(sampletri, *searchtri); searchdist = dist; } } } sampleblock = (void **) *sampleblock; } /* Where are we? */ org(*searchtri, torg); dest(*searchtri, tdest); /* Check the starting triangle's vertices. */ if ((torg[0] == searchpoint[0]) && (torg[1] == searchpoint[1])) { return ONVERTEX; } if ((tdest[0] == searchpoint[0]) && (tdest[1] == searchpoint[1])) { lnextself(*searchtri); return ONVERTEX; } /* Orient `searchtri' to fit the preconditions of calling preciselocate(). */ ahead = counterclockwise(torg, tdest, searchpoint); if (ahead < 0.0) { /* Turn around so that `searchpoint' is to the left of the */ /* edge specified by `searchtri'. */ symself(*searchtri); } else if (ahead == 0.0) { /* Check if `searchpoint' is between `torg' and `tdest'. */ if (((torg[0] < searchpoint[0]) == (searchpoint[0] < tdest[0])) && ((torg[1] < searchpoint[1]) == (searchpoint[1] < tdest[1]))) { return ONEDGE; } } return preciselocate(searchpoint, searchtri); } /** **/ /** **/ /********* Point location routines end here *********/ /********* Mesh transformation routines begin here *********/ /** **/ /** **/ /*****************************************************************************/ /* */ /* insertshelle() Create a new shell edge and insert it between two */ /* triangles. */ /* */ /* The new shell edge is inserted at the edge described by the handle */ /* `tri'. Its vertices are properly initialized. The marker `shellemark' */ /* is applied to the shell edge and, if appropriate, its vertices. */ /* */ /*****************************************************************************/ void insertshelle( struct triedge *tri, /* Edge at which to insert the new shell edge. */ int shellemark) /* Marker for the new shell edge. */ { struct triedge oppotri; struct edge newshelle; point triorg, tridest; triangle ptr; /* Temporary variable used by sym(). */ shelle sptr; /* Temporary variable used by tspivot(). */ /* Mark points if possible. */ org(*tri, triorg); dest(*tri, tridest); if (pointmark(triorg) == 0) { setpointmark(triorg, shellemark); } if (pointmark(tridest) == 0) { setpointmark(tridest, shellemark); } /* Check if there's already a shell edge here. */ tspivot(*tri, newshelle); if (newshelle.sh == dummysh) { /* Make new shell edge and initialize its vertices. */ makeshelle(&newshelle); setsorg(newshelle, tridest); setsdest(newshelle, triorg); /* Bond new shell edge to the two triangles it is sandwiched between. */ /* Note that the facing triangle `oppotri' might be equal to */ /* `dummytri' (outer space), but the new shell edge is bonded to it */ /* all the same. */ tsbond(*tri, newshelle); sym(*tri, oppotri); ssymself(newshelle); tsbond(oppotri, newshelle); setmark(newshelle, shellemark); } else { if (mark(newshelle) == 0) { setmark(newshelle, shellemark); } } } /*****************************************************************************/ /* */ /* Terminology */ /* */ /* A "local transformation" replaces a small set of triangles with another */ /* set of triangles. This may or may not involve inserting or deleting a */ /* point. */ /* */ /* The term "casing" is used to describe the set of triangles that are */ /* attached to the triangles being transformed, but are not transformed */ /* themselves. Think of the casing as a fixed hollow structure inside */ /* which all the action happens. A "casing" is only defined relative to */ /* a single transformation; each occurrence of a transformation will */ /* involve a different casing. */ /* */ /* A "shell" is similar to a "casing". The term "shell" describes the set */ /* of shell edges (if any) that are attached to the triangles being */ /* transformed. However, I sometimes use "shell" to refer to a single */ /* shell edge, so don't get confused. */ /* */ /*****************************************************************************/ /*****************************************************************************/ /* */ /* flip() Transform two triangles to two different triangles by flipping */ /* an edge within a quadrilateral. */ /* */ /* Imagine the original triangles, abc and bad, oriented so that the */ /* shared edge ab lies in a horizontal plane, with the point b on the left */ /* and the point a on the right. The point c lies below the edge, and the */ /* point d lies above the edge. The `flipedge' handle holds the edge ab */ /* of triangle abc, and is directed left, from vertex a to vertex b. */ /* */ /* The triangles abc and bad are deleted and replaced by the triangles cdb */ /* and dca. The triangles that represent abc and bad are NOT deallocated; */ /* they are reused for dca and cdb, respectively. Hence, any handles that */ /* may have held the original triangles are still valid, although not */ /* directed as they were before. */ /* */ /* Upon completion of this routine, the `flipedge' handle holds the edge */ /* dc of triangle dca, and is directed down, from vertex d to vertex c. */ /* (Hence, the two triangles have rotated counterclockwise.) */ /* */ /* WARNING: This transformation is geometrically valid only if the */ /* quadrilateral adbc is convex. Furthermore, this transformation is */ /* valid only if there is not a shell edge between the triangles abc and */ /* bad. This routine does not check either of these preconditions, and */ /* it is the responsibility of the calling routine to ensure that they are */ /* met. If they are not, the streets shall be filled with wailing and */ /* gnashing of teeth. */ /* */ /*****************************************************************************/ void flip( struct triedge *flipedge) /* Handle for the triangle abc. */ { struct triedge botleft, botright; struct triedge topleft, topright; struct triedge top; struct triedge botlcasing, botrcasing; struct triedge toplcasing, toprcasing; struct edge botlshelle, botrshelle; struct edge toplshelle, toprshelle; point leftpoint, rightpoint, botpoint; point farpoint; triangle ptr; /* Temporary variable used by sym(). */ shelle sptr; /* Temporary variable used by tspivot(). */ /* Identify the vertices of the quadrilateral. */ org(*flipedge, rightpoint); dest(*flipedge, leftpoint); apex(*flipedge, botpoint); sym(*flipedge, top); apex(top, farpoint); /* Identify the casing of the quadrilateral. */ lprev(top, topleft); sym(topleft, toplcasing); lnext(top, topright); sym(topright, toprcasing); lnext(*flipedge, botleft); sym(botleft, botlcasing); lprev(*flipedge, botright); sym(botright, botrcasing); /* Rotate the quadrilateral one-quarter turn counterclockwise. */ bond(topleft, botlcasing); bond(botleft, botrcasing); bond(botright, toprcasing); bond(topright, toplcasing); if (checksegments) { /* Check for shell edges and rebond them to the quadrilateral. */ tspivot(topleft, toplshelle); tspivot(botleft, botlshelle); tspivot(botright, botrshelle); tspivot(topright, toprshelle); if (toplshelle.sh == dummysh) { tsdissolve(topright); } else { tsbond(topright, toplshelle); } if (botlshelle.sh == dummysh) { tsdissolve(topleft); } else { tsbond(topleft, botlshelle); } if (botrshelle.sh == dummysh) { tsdissolve(botleft); } else { tsbond(botleft, botrshelle); } if (toprshelle.sh == dummysh) { tsdissolve(botright); } else { tsbond(botright, toprshelle); } } /* New point assignments for the rotated quadrilateral. */ setorg(*flipedge, farpoint); setdest(*flipedge, botpoint); setapex(*flipedge, rightpoint); setorg(top, botpoint); setdest(top, farpoint); setapex(top, leftpoint); } /*****************************************************************************/ /* */ /* insertsite() Insert a vertex into a Delaunay triangulation, */ /* performing flips as necessary to maintain the Delaunay */ /* property. */ /* */ /* The point `insertpoint' is located. If `searchtri.tri' is not NULL, */ /* the search for the containing triangle begins from `searchtri'. If */ /* `searchtri.tri' is NULL, a full point location procedure is called. */ /* If `insertpoint' is found inside a triangle, the triangle is split into */ /* three; if `insertpoint' lies on an edge, the edge is split in two, */ /* thereby splitting the two adjacent triangles into four. Edge flips are */ /* used to restore the Delaunay property. If `insertpoint' lies on an */ /* existing vertex, no action is taken, and the value DUPLICATEPOINT is */ /* returned. On return, `searchtri' is set to a handle whose origin is the */ /* existing vertex. */ /* */ /* Normally, the parameter `splitedge' is set to NULL, implying that no */ /* segment should be split. In this case, if `insertpoint' is found to */ /* lie on a segment, no action is taken, and the value VIOLATINGPOINT is */ /* returned. On return, `searchtri' is set to a handle whose primary edge */ /* is the violated segment. */ /* */ /* If the calling routine wishes to split a segment by inserting a point in */ /* it, the parameter `splitedge' should be that segment. In this case, */ /* `searchtri' MUST be the triangle handle reached by pivoting from that */ /* segment; no point location is done. */ /* */ /* `segmentflaws' and `triflaws' are flags that indicate whether or not */ /* there should be checks for the creation of encroached segments or bad */ /* quality faces. If a newly inserted point encroaches upon segments, */ /* these segments are added to the list of segments to be split if */ /* `segmentflaws' is set. If bad triangles are created, these are added */ /* to the queue if `triflaws' is set. */ /* */ /* If a duplicate point or violated segment does not prevent the point */ /* from being inserted, the return value will be ENCROACHINGPOINT if the */ /* point encroaches upon a segment (and checking is enabled), or */ /* SUCCESSFULPOINT otherwise. In either case, `searchtri' is set to a */ /* handle whose origin is the newly inserted vertex. */ /* */ /* insertsite() does not use flip() for reasons of speed; some */ /* information can be reused from edge flip to edge flip, like the */ /* locations of shell edges. */ /* */ /*****************************************************************************/ enum insertsiteresult insertsite( point insertpoint, struct triedge *searchtri, struct edge *splitedge, int segmentflaws, int triflaws) { struct triedge horiz; struct triedge top; struct triedge botleft, botright; struct triedge topleft, topright; struct triedge newbotleft, newbotright; struct triedge newtopright; struct triedge botlcasing, botrcasing; struct triedge toplcasing, toprcasing; struct triedge testtri; struct edge botlshelle, botrshelle; struct edge toplshelle, toprshelle; struct edge brokenshelle; struct edge checkshelle; struct edge rightedge; struct edge newedge; struct edge *encroached; point first; point leftpoint, rightpoint, botpoint, toppoint, farpoint; double attrib; enum insertsiteresult success; enum locateresult intersect; int doflip; int mirrorflag; int i; triangle ptr; /* Temporary variable used by sym(). */ shelle sptr; /* Temporary variable used by spivot() and tspivot(). */ if (splitedge == (struct edge *) NULL) { /* Find the location of the point to be inserted. Check if a good */ /* starting triangle has already been provided by the caller. */ if (searchtri->tri == (triangle *) NULL) { /* Find a boundary triangle. */ horiz.tri = dummytri; horiz.orient = 0; symself(horiz); /* Search for a triangle containing `insertpoint'. */ intersect = locate(insertpoint, &horiz); } else { /* Start searching from the triangle provided by the caller. */ triedgecopy(*searchtri, horiz); intersect = preciselocate(insertpoint, &horiz); } } else { /* The calling routine provides the edge in which the point is inserted. */ triedgecopy(*searchtri, horiz); intersect = ONEDGE; } if (intersect == ONVERTEX) { /* There's already a vertex there. Return in `searchtri' a triangle */ /* whose origin is the existing vertex. */ triedgecopy(horiz, *searchtri); triedgecopy(horiz, recenttri); return DUPLICATEPOINT; } if ((intersect == ONEDGE) || (intersect == OUTSIDE)) { /* The vertex falls on an edge or boundary. */ if (checksegments && (splitedge == (struct edge *) NULL)) { /* Check whether the vertex falls on a shell edge. */ tspivot(horiz, brokenshelle); if (brokenshelle.sh != dummysh) { /* The vertex falls on a shell edge. */ if (segmentflaws) { /* Add the shell edge to the list of encroached segments. */ encroached = (struct edge *) poolalloc(&badsegments); shellecopy(brokenshelle, *encroached); } /* Return a handle whose primary edge contains the point, */ /* which has not been inserted. */ triedgecopy(horiz, *searchtri); triedgecopy(horiz, recenttri); return VIOLATINGPOINT; } } /* Insert the point on an edge, dividing one triangle into two (if */ /* the edge lies on a boundary) or two triangles into four. */ lprev(horiz, botright); sym(botright, botrcasing); sym(horiz, topright); /* Is there a second triangle? (Or does this edge lie on a boundary?) */ mirrorflag = topright.tri != dummytri; if (mirrorflag) { lnextself(topright); sym(topright, toprcasing); maketriangle(&newtopright); } else { /* Splitting the boundary edge increases the number of boundary edges. */ hullsize++; } maketriangle(&newbotright); /* Set the vertices of changed and new triangles. */ org(horiz, rightpoint); dest(horiz, leftpoint); apex(horiz, botpoint); setorg(newbotright, botpoint); setdest(newbotright, rightpoint); setapex(newbotright, insertpoint); setorg(horiz, insertpoint); for (i = 0; i < eextras; i++) { /* Set the element attributes of a new triangle. */ setelemattribute(newbotright, i, elemattribute(botright, i)); } if (mirrorflag) { dest(topright, toppoint); setorg(newtopright, rightpoint); setdest(newtopright, toppoint); setapex(newtopright, insertpoint); setorg(topright, insertpoint); for (i = 0; i < eextras; i++) { /* Set the element attributes of another new triangle. */ setelemattribute(newtopright, i, elemattribute(topright, i)); } } /* There may be shell edges that need to be bonded */ /* to the new triangle(s). */ if (checksegments) { tspivot(botright, botrshelle); if (botrshelle.sh != dummysh) { tsdissolve(botright); tsbond(newbotright, botrshelle); } if (mirrorflag) { tspivot(topright, toprshelle); if (toprshelle.sh != dummysh) { tsdissolve(topright); tsbond(newtopright, toprshelle); } } } /* Bond the new triangle(s) to the surrounding triangles. */ bond(newbotright, botrcasing); lprevself(newbotright); bond(newbotright, botright); lprevself(newbotright); if (mirrorflag) { bond(newtopright, toprcasing); lnextself(newtopright); bond(newtopright, topright); lnextself(newtopright); bond(newtopright, newbotright); } if (splitedge != (struct edge *) NULL) { /* Split the shell edge into two. */ setsdest(*splitedge, insertpoint); ssymself(*splitedge); spivot(*splitedge, rightedge); insertshelle(&newbotright, mark(*splitedge)); tspivot(newbotright, newedge); sbond(*splitedge, newedge); ssymself(newedge); sbond(newedge, rightedge); ssymself(*splitedge); } /* Position `horiz' on the first edge to check for */ /* the Delaunay property. */ lnextself(horiz); } else { /* Insert the point in a triangle, splitting it into three. */ lnext(horiz, botleft); lprev(horiz, botright); sym(botleft, botlcasing); sym(botright, botrcasing); maketriangle(&newbotleft); maketriangle(&newbotright); /* Set the vertices of changed and new triangles. */ org(horiz, rightpoint); dest(horiz, leftpoint); apex(horiz, botpoint); setorg(newbotleft, leftpoint); setdest(newbotleft, botpoint); setapex(newbotleft, insertpoint); setorg(newbotright, botpoint); setdest(newbotright, rightpoint); setapex(newbotright, insertpoint); setapex(horiz, insertpoint); for (i = 0; i < eextras; i++) { /* Set the element attributes of the new triangles. */ attrib = elemattribute(horiz, i); setelemattribute(newbotleft, i, attrib); setelemattribute(newbotright, i, attrib); } /* There may be shell edges that need to be bonded */ /* to the new triangles. */ if (checksegments) { tspivot(botleft, botlshelle); if (botlshelle.sh != dummysh) { tsdissolve(botleft); tsbond(newbotleft, botlshelle); } tspivot(botright, botrshelle); if (botrshelle.sh != dummysh) { tsdissolve(botright); tsbond(newbotright, botrshelle); } } /* Bond the new triangles to the surrounding triangles. */ bond(newbotleft, botlcasing); bond(newbotright, botrcasing); lnextself(newbotleft); lprevself(newbotright); bond(newbotleft, newbotright); lnextself(newbotleft); bond(botleft, newbotleft); lprevself(newbotright); bond(botright, newbotright); } /* The insertion is successful by default, unless an encroached */ /* edge is found. */ success = SUCCESSFULPOINT; /* Circle around the newly inserted vertex, checking each edge opposite */ /* it for the Delaunay property. Non-Delaunay edges are flipped. */ /* `horiz' is always the edge being checked. `first' marks where to */ /* stop circling. */ org(horiz, first); rightpoint = first; dest(horiz, leftpoint); /* Circle until finished. */ while (1) { /* By default, the edge will be flipped. */ doflip = 1; if (checksegments) { /* Check for a segment, which cannot be flipped. */ tspivot(horiz, checkshelle); if (checkshelle.sh != dummysh) { /* The edge is a segment and cannot be flipped. */ doflip = 0; } } if (doflip) { /* Check if the edge is a boundary edge. */ sym(horiz, top); if (top.tri == dummytri) { /* The edge is a boundary edge and cannot be flipped. */ doflip = 0; } else { /* Find the point on the other side of the edge. */ apex(top, farpoint); /* In the incremental Delaunay triangulation algorithm, any of */ /* `leftpoint', `rightpoint', and `farpoint' could be vertices */ /* of the triangular bounding box. These vertices must be */ /* treated as if they are infinitely distant, even though their */ /* "coordinates" are not. */ if ((leftpoint == infpoint1) || (leftpoint == infpoint2) || (leftpoint == infpoint3)) { /* `leftpoint' is infinitely distant. Check the convexity of */ /* the boundary of the triangulation. 'farpoint' might be */ /* infinite as well, but trust me, this same condition */ /* should be applied. */ doflip = counterclockwise(insertpoint, rightpoint, farpoint) > 0.0; } else if ((rightpoint == infpoint1) || (rightpoint == infpoint2) || (rightpoint == infpoint3)) { /* `rightpoint' is infinitely distant. Check the convexity of */ /* the boundary of the triangulation. 'farpoint' might be */ /* infinite as well, but trust me, this same condition */ /* should be applied. */ doflip = counterclockwise(farpoint, leftpoint, insertpoint) > 0.0; } else if ((farpoint == infpoint1) || (farpoint == infpoint2) || (farpoint == infpoint3)) { /* `farpoint' is infinitely distant and cannot be inside */ /* the circumcircle of the triangle `horiz'. */ doflip = 0; } else { /* Test whether the edge is locally Delaunay. */ doflip = incircle(leftpoint, insertpoint, rightpoint, farpoint) > 0.0; } if (doflip) { /* We made it! Flip the edge `horiz' by rotating its containing */ /* quadrilateral (the two triangles adjacent to `horiz'). */ /* Identify the casing of the quadrilateral. */ lprev(top, topleft); sym(topleft, toplcasing); lnext(top, topright); sym(topright, toprcasing); lnext(horiz, botleft); sym(botleft, botlcasing); lprev(horiz, botright); sym(botright, botrcasing); /* Rotate the quadrilateral one-quarter turn counterclockwise. */ bond(topleft, botlcasing); bond(botleft, botrcasing); bond(botright, toprcasing); bond(topright, toplcasing); if (checksegments) { /* Check for shell edges and rebond them to the quadrilateral. */ tspivot(topleft, toplshelle); tspivot(botleft, botlshelle); tspivot(botright, botrshelle); tspivot(topright, toprshelle); if (toplshelle.sh == dummysh) { tsdissolve(topright); } else { tsbond(topright, toplshelle); } if (botlshelle.sh == dummysh) { tsdissolve(topleft); } else { tsbond(topleft, botlshelle); } if (botrshelle.sh == dummysh) { tsdissolve(botleft); } else { tsbond(botleft, botrshelle); } if (toprshelle.sh == dummysh) { tsdissolve(botright); } else { tsbond(botright, toprshelle); } } /* New point assignments for the rotated quadrilateral. */ setorg(horiz, farpoint); setdest(horiz, insertpoint); setapex(horiz, rightpoint); setorg(top, insertpoint); setdest(top, farpoint); setapex(top, leftpoint); for (i = 0; i < eextras; i++) { /* Take the average of the two triangles' attributes. */ attrib = 0.5 * (elemattribute(top, i) + elemattribute(horiz, i)); setelemattribute(top, i, attrib); setelemattribute(horiz, i, attrib); } /* On the next iterations, consider the two edges that were */ /* exposed (this is, are now visible to the newly inserted */ /* point) by the edge flip. */ lprevself(horiz); leftpoint = farpoint; } } } if (!doflip) { /* The handle `horiz' is accepted as locally Delaunay. */ /* Look for the next edge around the newly inserted point. */ lnextself(horiz); sym(horiz, testtri); /* Check for finishing a complete revolution about the new point, or */ /* falling off the edge of the triangulation. The latter will */ /* happen when a point is inserted at a boundary. */ if ((leftpoint == first) || (testtri.tri == dummytri)) { /* We're done. Return a triangle whose origin is the new point. */ lnext(horiz, *searchtri); lnext(horiz, recenttri); return success; } /* Finish finding the next edge around the newly inserted point. */ lnext(testtri, horiz); rightpoint = leftpoint; dest(horiz, leftpoint); } } } /*****************************************************************************/ /* */ /* triangulatepolygon() Find the Delaunay triangulation of a polygon that */ /* has a certain "nice" shape. This includes the */ /* polygons that result from deletion of a point or */ /* insertion of a segment. */ /* */ /* This is a conceptually difficult routine. The starting assumption is */ /* that we have a polygon with n sides. n - 1 of these sides are currently */ /* represented as edges in the mesh. One side, called the "base", need not */ /* be. */ /* */ /* Inside the polygon is a structure I call a "fan", consisting of n - 1 */ /* triangles that share a common origin. For each of these triangles, the */ /* edge opposite the origin is one of the sides of the polygon. The */ /* primary edge of each triangle is the edge directed from the origin to */ /* the destination; note that this is not the same edge that is a side of */ /* the polygon. `firstedge' is the primary edge of the first triangle. */ /* From there, the triangles follow in counterclockwise order about the */ /* polygon, until `lastedge', the primary edge of the last triangle. */ /* `firstedge' and `lastedge' are probably connected to other triangles */ /* beyond the extremes of the fan, but their identity is not important, as */ /* long as the fan remains connected to them. */ /* */ /* Imagine the polygon oriented so that its base is at the bottom. This */ /* puts `firstedge' on the far right, and `lastedge' on the far left. */ /* The right vertex of the base is the destination of `firstedge', and the */ /* left vertex of the base is the apex of `lastedge'. */ /* */ /* The challenge now is to find the right sequence of edge flips to */ /* transform the fan into a Delaunay triangulation of the polygon. Each */ /* edge flip effectively removes one triangle from the fan, committing it */ /* to the polygon. The resulting polygon has one fewer edge. If `doflip' */ /* is set, the final flip will be performed, resulting in a fan of one */ /* (useless?) triangle. If `doflip' is not set, the final flip is not */ /* performed, resulting in a fan of two triangles, and an unfinished */ /* triangular polygon that is not yet filled out with a single triangle. */ /* On completion of the routine, `lastedge' is the last remaining triangle, */ /* or the leftmost of the last two. */ /* */ /* Although the flips are performed in the order described above, the */ /* decisions about what flips to perform are made in precisely the reverse */ /* order. The recursive triangulatepolygon() procedure makes a decision, */ /* uses up to two recursive calls to triangulate the "subproblems" */ /* (polygons with fewer edges), and then performs an edge flip. */ /* */ /* The "decision" it makes is which vertex of the polygon should be */ /* connected to the base. This decision is made by testing every possible */ /* vertex. Once the best vertex is found, the two edges that connect this */ /* vertex to the base become the bases for two smaller polygons. These */ /* are triangulated recursively. Unfortunately, this approach can take */ /* O(n^2) time not only in the worst case, but in many common cases. It's */ /* rarely a big deal for point deletion, where n is rarely larger than ten, */ /* but it could be a big deal for segment insertion, especially if there's */ /* a lot of long segments that each cut many triangles. I ought to code */ /* a faster algorithm some time. */ /* */ /* The `edgecount' parameter is the number of sides of the polygon, */ /* including its base. `triflaws' is a flag that determines whether the */ /* new triangles should be tested for quality, and enqueued if they are */ /* bad. */ /* */ /*****************************************************************************/ void triangulatepolygon( struct triedge *firstedge, struct triedge *lastedge, int edgecount, int doflip, int triflaws) { struct triedge testtri; struct triedge besttri; struct triedge tempedge; point leftbasepoint, rightbasepoint; point testpoint; point bestpoint; int bestnumber; int i; triangle ptr; /* Temporary variable used by sym(), onext(), and oprev(). */ /* Identify the base vertices. */ apex(*lastedge, leftbasepoint); dest(*firstedge, rightbasepoint); /* Find the best vertex to connect the base to. */ onext(*firstedge, besttri); dest(besttri, bestpoint); triedgecopy(besttri, testtri); bestnumber = 1; for (i = 2; i <= edgecount - 2; i++) { onextself(testtri); dest(testtri, testpoint); /* Is this a better vertex? */ if (incircle(leftbasepoint, rightbasepoint, bestpoint, testpoint) > 0.0) { triedgecopy(testtri, besttri); bestpoint = testpoint; bestnumber = i; } } if (bestnumber > 1) { /* Recursively triangulate the smaller polygon on the right. */ oprev(besttri, tempedge); triangulatepolygon(firstedge, &tempedge, bestnumber + 1, 1, triflaws); } if (bestnumber < edgecount - 2) { /* Recursively triangulate the smaller polygon on the left. */ sym(besttri, tempedge); triangulatepolygon(&besttri, lastedge, edgecount - bestnumber, 1, triflaws); /* Find `besttri' again; it may have been lost to edge flips. */ sym(tempedge, besttri); } if (doflip) { /* Do one final edge flip. */ flip(&besttri); } /* Return the base triangle. */ triedgecopy(besttri, *lastedge); } /** **/ /** **/ /********* Mesh transformation routines end here *********/ /********* Divide-and-conquer Delaunay triangulation begins here *********/ /** **/ /** **/ /*****************************************************************************/ /* */ /* The divide-and-conquer bounding box */ /* */ /* I originally implemented the divide-and-conquer and incremental Delaunay */ /* triangulations using the edge-based data structure presented by Guibas */ /* and Stolfi. Switching to a triangle-based data structure doubled the */ /* speed. However, I had to think of a few extra tricks to maintain the */ /* elegance of the original algorithms. */ /* */ /* The "bounding box" used by my variant of the divide-and-conquer */ /* algorithm uses one triangle for each edge of the convex hull of the */ /* triangulation. These bounding triangles all share a common apical */ /* vertex, which is represented by NULL and which represents nothing. */ /* The bounding triangles are linked in a circular fan about this NULL */ /* vertex, and the edges on the convex hull of the triangulation appear */ /* opposite the NULL vertex. You might find it easiest to imagine that */ /* the NULL vertex is a point in 3D space behind the center of the */ /* triangulation, and that the bounding triangles form a sort of cone. */ /* */ /* This bounding box makes it easy to represent degenerate cases. For */ /* instance, the triangulation of two vertices is a single edge. This edge */ /* is represented by two bounding box triangles, one on each "side" of the */ /* edge. These triangles are also linked together in a fan about the NULL */ /* vertex. */ /* */ /* The bounding box also makes it easy to traverse the convex hull, as the */ /* divide-and-conquer algorithm needs to do. */ /* */ /*****************************************************************************/ /*****************************************************************************/ /* */ /* pointsort() Sort an array of points by x-coordinate, using the */ /* y-coordinate as a secondary key. */ /* */ /* Uses quicksort. Randomized O(n log n) time. No, I did not make any of */ /* the usual quicksort mistakes. */ /* */ /*****************************************************************************/ void pointsort( point *sortarray, int arraysize) { int left, right; int pivot; double pivotx, pivoty; point temp; if (arraysize == 2) { /* Recursive base case. */ if ((sortarray[0][0] > sortarray[1][0]) || ((sortarray[0][0] == sortarray[1][0]) && (sortarray[0][1] > sortarray[1][1]))) { temp = sortarray[1]; sortarray[1] = sortarray[0]; sortarray[0] = temp; } return; } /* Choose a random pivot to split the array. */ pivot = (int) randomnation(arraysize); pivotx = sortarray[pivot][0]; pivoty = sortarray[pivot][1]; /* Split the array. */ left = -1; right = arraysize; while (left < right) { /* Search for a point whose x-coordinate is too large for the left. */ do { left++; } while ((left <= right) && ((sortarray[left][0] < pivotx) || ((sortarray[left][0] == pivotx) && (sortarray[left][1] < pivoty)))); /* Search for a point whose x-coordinate is too small for the right. */ do { right--; } while ((left <= right) && ((sortarray[right][0] > pivotx) || ((sortarray[right][0] == pivotx) && (sortarray[right][1] > pivoty)))); if (left < right) { /* Swap the left and right points. */ temp = sortarray[left]; sortarray[left] = sortarray[right]; sortarray[right] = temp; } } if (left > 1) { /* Recursively sort the left subset. */ pointsort(sortarray, left); } if (right < arraysize - 2) { /* Recursively sort the right subset. */ pointsort(&sortarray[right + 1], arraysize - right - 1); } } /*****************************************************************************/ /* */ /* pointmedian() An order statistic algorithm, almost. Shuffles an array */ /* of points so that the first `median' points occur */ /* lexicographically before the remaining points. */ /* */ /* Uses the x-coordinate as the primary key if axis == 0; the y-coordinate */ /* if axis == 1. Very similar to the pointsort() procedure, but runs in */ /* randomized linear time. */ /* */ /*****************************************************************************/ void pointmedian( point *sortarray, int arraysize, int median, int axis) { int left, right; int pivot; double pivot1, pivot2; point temp; if (arraysize == 2) { /* Recursive base case. */ if ((sortarray[0][axis] > sortarray[1][axis]) || ((sortarray[0][axis] == sortarray[1][axis]) && (sortarray[0][1 - axis] > sortarray[1][1 - axis]))) { temp = sortarray[1]; sortarray[1] = sortarray[0]; sortarray[0] = temp; } return; } /* Choose a random pivot to split the array. */ pivot = (int) randomnation(arraysize); pivot1 = sortarray[pivot][axis]; pivot2 = sortarray[pivot][1 - axis]; /* Split the array. */ left = -1; right = arraysize; while (left < right) { /* Search for a point whose x-coordinate is too large for the left. */ do { left++; } while ((left <= right) && ((sortarray[left][axis] < pivot1) || ((sortarray[left][axis] == pivot1) && (sortarray[left][1 - axis] < pivot2)))); /* Search for a point whose x-coordinate is too small for the right. */ do { right--; } while ((left <= right) && ((sortarray[right][axis] > pivot1) || ((sortarray[right][axis] == pivot1) && (sortarray[right][1 - axis] > pivot2)))); if (left < right) { /* Swap the left and right points. */ temp = sortarray[left]; sortarray[left] = sortarray[right]; sortarray[right] = temp; } } /* Unlike in pointsort(), at most one of the following */ /* conditionals is true. */ if (left > median) { /* Recursively shuffle the left subset. */ pointmedian(sortarray, left, median, axis); } if (right < median - 1) { /* Recursively shuffle the right subset. */ pointmedian(&sortarray[right + 1], arraysize - right - 1, median - right - 1, axis); } } /*****************************************************************************/ /* */ /* alternateaxes() Sorts the points as appropriate for the divide-and- */ /* conquer algorithm with alternating cuts. */ /* */ /* Partitions by x-coordinate if axis == 0; by y-coordinate if axis == 1. */ /* For the base case, subsets containing only two or three points are */ /* always sorted by x-coordinate. */ /* */ /*****************************************************************************/ void alternateaxes(point *sortarray, int arraysize, int axis) { int divider; divider = arraysize >> 1; if (arraysize <= 3) { /* Recursive base case: subsets of two or three points will be */ /* handled specially, and should always be sorted by x-coordinate. */ axis = 0; } /* Partition with a horizontal or vertical cut. */ pointmedian(sortarray, arraysize, divider, axis); /* Recursively partition the subsets with a cross cut. */ if (arraysize - divider >= 2) { if (divider >= 2) { alternateaxes(sortarray, divider, 1 - axis); } alternateaxes(&sortarray[divider], arraysize - divider, 1 - axis); } } /*****************************************************************************/ /* */ /* mergehulls() Merge two adjacent Delaunay triangulations into a */ /* single Delaunay triangulation. */ /* */ /* This is similar to the algorithm given by Guibas and Stolfi, but uses */ /* a triangle-based, rather than edge-based, data structure. */ /* */ /* The algorithm walks up the gap between the two triangulations, knitting */ /* them together. As they are merged, some of their bounding triangles */ /* are converted into real triangles of the triangulation. The procedure */ /* pulls each hull's bounding triangles apart, then knits them together */ /* like the teeth of two gears. The Delaunay property determines, at each */ /* step, whether the next "tooth" is a bounding triangle of the left hull */ /* or the right. When a bounding triangle becomes real, its apex is */ /* changed from NULL to a real point. */ /* */ /* Only two new triangles need to be allocated. These become new bounding */ /* triangles at the top and bottom of the seam. They are used to connect */ /* the remaining bounding triangles (those that have not been converted */ /* into real triangles) into a single fan. */ /* */ /* On entry, `farleft' and `innerleft' are bounding triangles of the left */ /* triangulation. The origin of `farleft' is the leftmost vertex, and */ /* the destination of `innerleft' is the rightmost vertex of the */ /* triangulation. Similarly, `innerright' and `farright' are bounding */ /* triangles of the right triangulation. The origin of `innerright' and */ /* destination of `farright' are the leftmost and rightmost vertices. */ /* */ /* On completion, the origin of `farleft' is the leftmost vertex of the */ /* merged triangulation, and the destination of `farright' is the rightmost */ /* vertex. */ /* */ /*****************************************************************************/ void mergehulls( struct triedge *farleft, struct triedge *innerleft, struct triedge *innerright, struct triedge *farright, int axis) { struct triedge leftcand, rightcand; struct triedge baseedge; struct triedge nextedge; struct triedge sidecasing, topcasing, outercasing; struct triedge checkedge; point innerleftdest; point innerrightorg; point innerleftapex, innerrightapex; point farleftpt, farrightpt; point farleftapex, farrightapex; point lowerleft, lowerright; point upperleft, upperright; point nextapex; point checkvertex; int changemade; int badedge; int leftfinished, rightfinished; triangle ptr; /* Temporary variable used by sym(). */ dest(*innerleft, innerleftdest); apex(*innerleft, innerleftapex); org(*innerright, innerrightorg); apex(*innerright, innerrightapex); /* Special treatment for horizontal cuts. */ if (axis == 1) { org(*farleft, farleftpt); apex(*farleft, farleftapex); dest(*farright, farrightpt); apex(*farright, farrightapex); /* The pointers to the extremal points are shifted to point to the */ /* topmost and bottommost point of each hull, rather than the */ /* leftmost and rightmost points. */ while (farleftapex[1] < farleftpt[1]) { lnextself(*farleft); symself(*farleft); farleftpt = farleftapex; apex(*farleft, farleftapex); } sym(*innerleft, checkedge); apex(checkedge, checkvertex); while (checkvertex[1] > innerleftdest[1]) { lnext(checkedge, *innerleft); innerleftapex = innerleftdest; innerleftdest = checkvertex; sym(*innerleft, checkedge); apex(checkedge, checkvertex); } while (innerrightapex[1] < innerrightorg[1]) { lnextself(*innerright); symself(*innerright); innerrightorg = innerrightapex; apex(*innerright, innerrightapex); } sym(*farright, checkedge); apex(checkedge, checkvertex); while (checkvertex[1] > farrightpt[1]) { lnext(checkedge, *farright); farrightapex = farrightpt; farrightpt = checkvertex; sym(*farright, checkedge); apex(checkedge, checkvertex); } } /* Find a line tangent to and below both hulls. */ do { changemade = 0; /* Make innerleftdest the "bottommost" point of the left hull. */ if (counterclockwise(innerleftdest, innerleftapex, innerrightorg) > 0.0) { lprevself(*innerleft); symself(*innerleft); innerleftdest = innerleftapex; apex(*innerleft, innerleftapex); changemade = 1; } /* Make innerrightorg the "bottommost" point of the right hull. */ if (counterclockwise(innerrightapex, innerrightorg, innerleftdest) > 0.0) { lnextself(*innerright); symself(*innerright); innerrightorg = innerrightapex; apex(*innerright, innerrightapex); changemade = 1; } } while (changemade); /* Find the two candidates to be the next "gear tooth". */ sym(*innerleft, leftcand); sym(*innerright, rightcand); /* Create the bottom new bounding triangle. */ maketriangle(&baseedge); /* Connect it to the bounding boxes of the left and right triangulations. */ bond(baseedge, *innerleft); lnextself(baseedge); bond(baseedge, *innerright); lnextself(baseedge); setorg(baseedge, innerrightorg); setdest(baseedge, innerleftdest); /* Apex is intentionally left NULL. */ /* Fix the extreme triangles if necessary. */ org(*farleft, farleftpt); if (innerleftdest == farleftpt) { lnext(baseedge, *farleft); } dest(*farright, farrightpt); if (innerrightorg == farrightpt) { lprev(baseedge, *farright); } /* The vertices of the current knitting edge. */ lowerleft = innerleftdest; lowerright = innerrightorg; /* The candidate vertices for knitting. */ apex(leftcand, upperleft); apex(rightcand, upperright); /* Walk up the gap between the two triangulations, knitting them together. */ while (1) { /* Have we reached the top? (This isn't quite the right question, */ /* because even though the left triangulation might seem finished now, */ /* moving up on the right triangulation might reveal a new point of */ /* the left triangulation. And vice-versa.) */ leftfinished = counterclockwise(upperleft, lowerleft, lowerright) <= 0.0; rightfinished = counterclockwise(upperright, lowerleft, lowerright) <= 0.0; if (leftfinished && rightfinished) { /* Create the top new bounding triangle. */ maketriangle(&nextedge); setorg(nextedge, lowerleft); setdest(nextedge, lowerright); /* Apex is intentionally left NULL. */ /* Connect it to the bounding boxes of the two triangulations. */ bond(nextedge, baseedge); lnextself(nextedge); bond(nextedge, rightcand); lnextself(nextedge); bond(nextedge, leftcand); /* Special treatment for horizontal cuts. */ if (axis == 1) { org(*farleft, farleftpt); apex(*farleft, farleftapex); dest(*farright, farrightpt); apex(*farright, farrightapex); sym(*farleft, checkedge); apex(checkedge, checkvertex); /* The pointers to the extremal points are restored to the leftmost */ /* and rightmost points (rather than topmost and bottommost). */ while (checkvertex[0] < farleftpt[0]) { lprev(checkedge, *farleft); farleftapex = farleftpt; farleftpt = checkvertex; sym(*farleft, checkedge); apex(checkedge, checkvertex); } while (farrightapex[0] > farrightpt[0]) { lprevself(*farright); symself(*farright); farrightpt = farrightapex; apex(*farright, farrightapex); } } return; } /* Consider eliminating edges from the left triangulation. */ if (!leftfinished) { /* What vertex would be exposed if an edge were deleted? */ lprev(leftcand, nextedge); symself(nextedge); apex(nextedge, nextapex); /* If nextapex is NULL, then no vertex would be exposed; the */ /* triangulation would have been eaten right through. */ if (nextapex != (point) NULL) { /* Check whether the edge is Delaunay. */ badedge = incircle(lowerleft, lowerright, upperleft, nextapex) > 0.0; while (badedge) { /* Eliminate the edge with an edge flip. As a result, the */ /* left triangulation will have one more boundary triangle. */ lnextself(nextedge); sym(nextedge, topcasing); lnextself(nextedge); sym(nextedge, sidecasing); bond(nextedge, topcasing); bond(leftcand, sidecasing); lnextself(leftcand); sym(leftcand, outercasing); lprevself(nextedge); bond(nextedge, outercasing); /* Correct the vertices to reflect the edge flip. */ setorg(leftcand, lowerleft); setdest(leftcand, NULL); setapex(leftcand, nextapex); setorg(nextedge, NULL); setdest(nextedge, upperleft); setapex(nextedge, nextapex); /* Consider the newly exposed vertex. */ upperleft = nextapex; /* What vertex would be exposed if another edge were deleted? */ triedgecopy(sidecasing, nextedge); apex(nextedge, nextapex); if (nextapex != (point) NULL) { /* Check whether the edge is Delaunay. */ badedge = incircle(lowerleft, lowerright, upperleft, nextapex) > 0.0; } else { /* Avoid eating right through the triangulation. */ badedge = 0; } } } } /* Consider eliminating edges from the right triangulation. */ if (!rightfinished) { /* What vertex would be exposed if an edge were deleted? */ lnext(rightcand, nextedge); symself(nextedge); apex(nextedge, nextapex); /* If nextapex is NULL, then no vertex would be exposed; the */ /* triangulation would have been eaten right through. */ if (nextapex != (point) NULL) { /* Check whether the edge is Delaunay. */ badedge = incircle(lowerleft, lowerright, upperright, nextapex) > 0.0; while (badedge) { /* Eliminate the edge with an edge flip. As a result, the */ /* right triangulation will have one more boundary triangle. */ lprevself(nextedge); sym(nextedge, topcasing); lprevself(nextedge); sym(nextedge, sidecasing); bond(nextedge, topcasing); bond(rightcand, sidecasing); lprevself(rightcand); sym(rightcand, outercasing); lnextself(nextedge); bond(nextedge, outercasing); /* Correct the vertices to reflect the edge flip. */ setorg(rightcand, NULL); setdest(rightcand, lowerright); setapex(rightcand, nextapex); setorg(nextedge, upperright); setdest(nextedge, NULL); setapex(nextedge, nextapex); /* Consider the newly exposed vertex. */ upperright = nextapex; /* What vertex would be exposed if another edge were deleted? */ triedgecopy(sidecasing, nextedge); apex(nextedge, nextapex); if (nextapex != (point) NULL) { /* Check whether the edge is Delaunay. */ badedge = incircle(lowerleft, lowerright, upperright, nextapex) > 0.0; } else { /* Avoid eating right through the triangulation. */ badedge = 0; } } } } if (leftfinished || (!rightfinished && (incircle(upperleft, lowerleft, lowerright, upperright) > 0.0))) { /* Knit the triangulations, adding an edge from `lowerleft' */ /* to `upperright'. */ bond(baseedge, rightcand); lprev(rightcand, baseedge); setdest(baseedge, lowerleft); lowerright = upperright; sym(baseedge, rightcand); apex(rightcand, upperright); } else { /* Knit the triangulations, adding an edge from `upperleft' */ /* to `lowerright'. */ bond(baseedge, leftcand); lnext(leftcand, baseedge); setorg(baseedge, lowerright); lowerleft = upperleft; sym(baseedge, leftcand); apex(leftcand, upperleft); } } } /*****************************************************************************/ /* */ /* divconqrecurse() Recursively form a Delaunay triangulation by the */ /* divide-and-conquer method. */ /* */ /* Recursively breaks down the problem into smaller pieces, which are */ /* knitted together by mergehulls(). The base cases (problems of two or */ /* three points) are handled specially here. */ /* */ /* On completion, `farleft' and `farright' are bounding triangles such that */ /* the origin of `farleft' is the leftmost vertex (breaking ties by */ /* choosing the highest leftmost vertex), and the destination of */ /* `farright' is the rightmost vertex (breaking ties by choosing the */ /* lowest rightmost vertex). */ /* */ /*****************************************************************************/ void divconqrecurse( point *sortarray, int vertices, int axis, struct triedge *farleft, struct triedge *farright) { struct triedge midtri, tri1, tri2, tri3; struct triedge innerleft, innerright; double area; int divider; if (vertices == 2) { /* The triangulation of two vertices is an edge. An edge is */ /* represented by two bounding triangles. */ maketriangle(farleft); setorg(*farleft, sortarray[0]); setdest(*farleft, sortarray[1]); /* The apex is intentionally left NULL. */ maketriangle(farright); setorg(*farright, sortarray[1]); setdest(*farright, sortarray[0]); /* The apex is intentionally left NULL. */ bond(*farleft, *farright); lprevself(*farleft); lnextself(*farright); bond(*farleft, *farright); lprevself(*farleft); lnextself(*farright); bond(*farleft, *farright); /* Ensure that the origin of `farleft' is sortarray[0]. */ lprev(*farright, *farleft); return; } else if (vertices == 3) { /* The triangulation of three vertices is either a triangle (with */ /* three bounding triangles) or two edges (with four bounding */ /* triangles). In either case, four triangles are created. */ maketriangle(&midtri); maketriangle(&tri1); maketriangle(&tri2); maketriangle(&tri3); area = counterclockwise(sortarray[0], sortarray[1], sortarray[2]); if (area == 0.0) { /* Three collinear points; the triangulation is two edges. */ setorg(midtri, sortarray[0]); setdest(midtri, sortarray[1]); setorg(tri1, sortarray[1]); setdest(tri1, sortarray[0]); setorg(tri2, sortarray[2]); setdest(tri2, sortarray[1]); setorg(tri3, sortarray[1]); setdest(tri3, sortarray[2]); /* All apices are intentionally left NULL. */ bond(midtri, tri1); bond(tri2, tri3); lnextself(midtri); lprevself(tri1); lnextself(tri2); lprevself(tri3); bond(midtri, tri3); bond(tri1, tri2); lnextself(midtri); lprevself(tri1); lnextself(tri2); lprevself(tri3); bond(midtri, tri1); bond(tri2, tri3); /* Ensure that the origin of `farleft' is sortarray[0]. */ triedgecopy(tri1, *farleft); /* Ensure that the destination of `farright' is sortarray[2]. */ triedgecopy(tri2, *farright); } else { /* The three points are not collinear; the triangulation is one */ /* triangle, namely `midtri'. */ setorg(midtri, sortarray[0]); setdest(tri1, sortarray[0]); setorg(tri3, sortarray[0]); /* Apices of tri1, tri2, and tri3 are left NULL. */ if (area > 0.0) { /* The vertices are in counterclockwise order. */ setdest(midtri, sortarray[1]); setorg(tri1, sortarray[1]); setdest(tri2, sortarray[1]); setapex(midtri, sortarray[2]); setorg(tri2, sortarray[2]); setdest(tri3, sortarray[2]); } else { /* The vertices are in clockwise order. */ setdest(midtri, sortarray[2]); setorg(tri1, sortarray[2]); setdest(tri2, sortarray[2]); setapex(midtri, sortarray[1]); setorg(tri2, sortarray[1]); setdest(tri3, sortarray[1]); } /* The topology does not depend on how the vertices are ordered. */ bond(midtri, tri1); lnextself(midtri); bond(midtri, tri2); lnextself(midtri); bond(midtri, tri3); lprevself(tri1); lnextself(tri2); bond(tri1, tri2); lprevself(tri1); lprevself(tri3); bond(tri1, tri3); lnextself(tri2); lprevself(tri3); bond(tri2, tri3); /* Ensure that the origin of `farleft' is sortarray[0]. */ triedgecopy(tri1, *farleft); /* Ensure that the destination of `farright' is sortarray[2]. */ if (area > 0.0) { triedgecopy(tri2, *farright); } else { lnext(*farleft, *farright); } } return; } else { /* Split the vertices in half. */ divider = vertices >> 1; /* Recursively triangulate each half. */ divconqrecurse(sortarray, divider, 1 - axis, farleft, &innerleft); divconqrecurse(&sortarray[divider], vertices - divider, 1 - axis, &innerright, farright); /* Merge the two triangulations into one. */ mergehulls(farleft, &innerleft, &innerright, farright, axis); } } long removeghosts(struct triedge *startghost) { struct triedge searchedge; struct triedge dissolveedge; struct triedge deadtri; long hullsize; triangle ptr; /* Temporary variable used by sym(). */ /* Find an edge on the convex hull to start point location from. */ lprev(*startghost, searchedge); symself(searchedge); dummytri[0] = encode(searchedge); /* Remove the bounding box and count the convex hull edges. */ triedgecopy(*startghost, dissolveedge); hullsize = 0; do { hullsize++; lnext(dissolveedge, deadtri); lprevself(dissolveedge); symself(dissolveedge); /* Remove a bounding triangle from a convex hull triangle. */ dissolve(dissolveedge); /* Find the next bounding triangle. */ sym(deadtri, dissolveedge); /* Delete the bounding triangle. */ triangledealloc(deadtri.tri); } while (!triedgeequal(dissolveedge, *startghost)); return hullsize; } /*****************************************************************************/ /* */ /* divconqdelaunay() Form a Delaunay triangulation by the divide-and- */ /* conquer method. */ /* */ /* Sorts the points, calls a recursive procedure to triangulate them, and */ /* removes the bounding box, setting boundary markers as appropriate. */ /* */ /*****************************************************************************/ long divconqdelaunay(void) { point *sortarray; struct triedge hullleft, hullright; int divider; int i, j; /* Allocate an array of pointers to points for sorting. */ sortarray = (point *) malloc(inpoints * sizeof(point)); if (sortarray == (point *) NULL) { vTrace("*** E0055 : Out of memory."); exit(1); } traversalinit(&points); for (i = 0; i < inpoints; i++) { sortarray[i] = pointtraverse(); } /* Sort the points. */ pointsort(sortarray, inpoints); /* Discard duplicate points, which can really mess up the algorithm. */ i = 0; for (j = 1; j < inpoints; j++) { if ((sortarray[i][0] == sortarray[j][0]) && (sortarray[i][1] == sortarray[j][1])) { /* Commented out - would eliminate point from output .node file, but causes a failure if some segment has this point as an endpoint. setpointmark(sortarray[j], DEADPOINT); */ } else { i++; sortarray[i] = sortarray[j]; } } i++; /* Re-sort the array of points to accommodate alternating cuts. */ divider = i >> 1; if (i - divider >= 2) { if (divider >= 2) { alternateaxes(sortarray, divider, 1); } alternateaxes(&sortarray[divider], i - divider, 1); } /* Form the Delaunay triangulation. */ divconqrecurse(sortarray, i, 0, &hullleft, &hullright); free(sortarray); return removeghosts(&hullleft); } /** **/ /** **/ /********* Divide-and-conquer Delaunay triangulation ends here *********/ /********* General mesh construction routines begin here *********/ /** **/ /** **/ /*****************************************************************************/ /* */ /* delaunay() Form a Delaunay triangulation. */ /* */ /*****************************************************************************/ long delaunay(void) { eextras = 0; initializetrisegpools(); return divconqdelaunay(); } /** **/ /** **/ /********* General mesh construction routines end here *********/ /********* Segment (shell edge) insertion begins here *********/ /** **/ /** **/ /*****************************************************************************/ /* */ /* finddirection() Find the first triangle on the path from one point */ /* to another. */ /* */ /* Finds the triangle that intersects a line segment drawn from the */ /* origin of `searchtri' to the point `endpoint', and returns the result */ /* in `searchtri'. The origin of `searchtri' does not change, even though */ /* the triangle returned may differ from the one passed in. This routine */ /* is used to find the direction to move in to get from one point to */ /* another. */ /* */ /* The return value notes whether the destination or apex of the found */ /* triangle is collinear with the two points in question. */ /* */ /*****************************************************************************/ enum finddirectionresult finddirection( struct triedge *searchtri, point endpoint) { struct triedge checktri; point startpoint; point leftpoint, rightpoint; double leftccw, rightccw; int leftflag, rightflag; triangle ptr; /* Temporary variable used by onext() and oprev(). */ org(*searchtri, startpoint); dest(*searchtri, rightpoint); apex(*searchtri, leftpoint); /* Is `endpoint' to the left? */ leftccw = counterclockwise(endpoint, startpoint, leftpoint); leftflag = leftccw > 0.0; /* Is `endpoint' to the right? */ rightccw = counterclockwise(startpoint, endpoint, rightpoint); rightflag = rightccw > 0.0; if (leftflag && rightflag) { /* `searchtri' faces directly away from `endpoint'. We could go */ /* left or right. Ask whether it's a triangle or a boundary */ /* on the left. */ onext(*searchtri, checktri); if (checktri.tri == dummytri) { leftflag = 0; } else { rightflag = 0; } } while (leftflag) { /* Turn left until satisfied. */ onextself(*searchtri); if (searchtri->tri == dummytri) { vTrace("*** E0056 : Internal error in finddirection(): Unable to find a triangle leading from (%.12g, %.12g) to (%.12g, %.12g).", startpoint[0], startpoint[1], endpoint[0], endpoint[1]); internalerror(); } apex(*searchtri, leftpoint); rightccw = leftccw; leftccw = counterclockwise(endpoint, startpoint, leftpoint); leftflag = leftccw > 0.0; } while (rightflag) { /* Turn right until satisfied. */ oprevself(*searchtri); if (searchtri->tri == dummytri) { vTrace("*** E0057 : Internal error in finddirection(): Unable to find a triangle leading from (%.12g, %.12g) to (%.12g, %.12g).", startpoint[0], startpoint[1], endpoint[0], endpoint[1]); internalerror(); } dest(*searchtri, rightpoint); leftccw = rightccw; rightccw = counterclockwise(startpoint, endpoint, rightpoint); rightflag = rightccw > 0.0; } if (leftccw == 0.0) { return LEFTCOLLINEAR; } else if (rightccw == 0.0) { return RIGHTCOLLINEAR; } else { return WITHIN; } } /*****************************************************************************/ /* */ /* segmentintersection() Find the intersection of an existing segment */ /* and a segment that is being inserted. Insert */ /* a point at the intersection, splitting an */ /* existing shell edge. */ /* */ /* The segment being inserted connects the apex of splittri to endpoint2. */ /* splitshelle is the shell edge being split, and MUST be opposite */ /* splittri. Hence, the edge being split connects the origin and */ /* destination of splittri. */ /* */ /* On completion, splittri is a handle having the newly inserted */ /* intersection point as its origin, and endpoint1 as its destination. */ /* */ /*****************************************************************************/ void segmentintersection( struct triedge *splittri, struct edge *splitshelle, point endpoint2) { point endpoint1; point torg, tdest; point leftpoint, rightpoint; point newpoint; enum insertsiteresult success; enum finddirectionresult collinear; double ex, ey; double tx, ty; double etx, ety; double split, denom; int i; triangle ptr; /* Temporary variable used by onext(). */ /* Find the other three segment endpoints. */ apex(*splittri, endpoint1); org(*splittri, torg); dest(*splittri, tdest); /* Segment intersection formulae; see the Antonio reference. */ tx = tdest[0] - torg[0]; ty = tdest[1] - torg[1]; ex = endpoint2[0] - endpoint1[0]; ey = endpoint2[1] - endpoint1[1]; etx = torg[0] - endpoint2[0]; ety = torg[1] - endpoint2[1]; denom = ty * ex - tx * ey; if (denom == 0.0) { vTrace("*** E0058 : Internal error in segmentintersection(): Attempt to find intersection of parallel segments."); internalerror(); } split = (ey * etx - ex * ety) / denom; /* Create the new point. */ newpoint = (point) poolalloc(&points); /* Interpolate its coordinate and attributes. */ for (i = 0; i < 2 + nextras; i++) { newpoint[i] = torg[i] + split * (tdest[i] - torg[i]); } setpointmark(newpoint, mark(*splitshelle)); /* Insert the intersection point. This should always succeed. */ success = insertsite(newpoint, splittri, splitshelle, 0, 0); if (success != SUCCESSFULPOINT) { vTrace("*** E0059 : Internal error in segmentintersection(): Failure to split a segment."); internalerror(); } /* Inserting the point may have caused edge flips. We wish to rediscover */ /* the edge connecting endpoint1 to the new intersection point. */ collinear = finddirection(splittri, endpoint1); dest(*splittri, rightpoint); apex(*splittri, leftpoint); if ((leftpoint[0] == endpoint1[0]) && (leftpoint[1] == endpoint1[1])) { onextself(*splittri); } else if ((rightpoint[0] != endpoint1[0]) || (rightpoint[1] != endpoint1[1])) { vTrace("*** E0060 : Internal error in segmentintersection(): Topological inconsistency after splitting a segment."); internalerror(); } /* `splittri' should have destination endpoint1. */ } /*****************************************************************************/ /* */ /* scoutsegment() Scout the first triangle on the path from one endpoint */ /* to another, and check for completion (reaching the */ /* second endpoint), a collinear point, and the */ /* intersection of two segments. */ /* */ /* Returns one if the entire segment is successfully inserted, and zero if */ /* the job must be finished by conformingedge() or constrainededge(). */ /* */ /* If the first triangle on the path has the second endpoint as its */ /* destination or apex, a shell edge is inserted and the job is done. */ /* */ /* If the first triangle on the path has a destination or apex that lies on */ /* the segment, a shell edge is inserted connecting the first endpoint to */ /* the collinear point, and the search is continued from the collinear */ /* point. */ /* */ /* If the first triangle on the path has a shell edge opposite its origin, */ /* then there is a segment that intersects the segment being inserted. */ /* Their intersection point is inserted, splitting the shell edge. */ /* */ /* Otherwise, return zero. */ /* */ /*****************************************************************************/ int scoutsegment( struct triedge *searchtri, point endpoint2, int newmark) { struct triedge crosstri; struct edge crossedge; point leftpoint, rightpoint; point endpoint1; enum finddirectionresult collinear; shelle sptr; /* Temporary variable used by tspivot(). */ collinear = finddirection(searchtri, endpoint2); dest(*searchtri, rightpoint); apex(*searchtri, leftpoint); if (((leftpoint[0] == endpoint2[0]) && (leftpoint[1] == endpoint2[1])) || ((rightpoint[0] == endpoint2[0]) && (rightpoint[1] == endpoint2[1]))) { /* The segment is already an edge in the mesh. */ if ((leftpoint[0] == endpoint2[0]) && (leftpoint[1] == endpoint2[1])) { lprevself(*searchtri); } /* Insert a shell edge, if there isn't already one there. */ insertshelle(searchtri, newmark); return 1; } else if (collinear == LEFTCOLLINEAR) { /* We've collided with a point between the segment's endpoints. */ /* Make the collinear point be the triangle's origin. */ lprevself(*searchtri); insertshelle(searchtri, newmark); /* Insert the remainder of the segment. */ return scoutsegment(searchtri, endpoint2, newmark); } else if (collinear == RIGHTCOLLINEAR) { /* We've collided with a point between the segment's endpoints. */ insertshelle(searchtri, newmark); /* Make the collinear point be the triangle's origin. */ lnextself(*searchtri); /* Insert the remainder of the segment. */ return scoutsegment(searchtri, endpoint2, newmark); } else { lnext(*searchtri, crosstri); tspivot(crosstri, crossedge); /* Check for a crossing segment. */ if (crossedge.sh == dummysh) { return 0; } else { org(*searchtri, endpoint1); /* Insert a point at the intersection. */ segmentintersection(&crosstri, &crossedge, endpoint2); triedgecopy(crosstri, *searchtri); insertshelle(searchtri, newmark); /* Insert the remainder of the segment. */ return scoutsegment(searchtri, endpoint2, newmark); } } } /*****************************************************************************/ /* */ /* delaunayfixup() Enforce the Delaunay condition at an edge, fanning out */ /* recursively from an existing point. Pay special */ /* attention to stacking inverted triangles. */ /* */ /* This is a support routine for inserting segments into a constrained */ /* Delaunay triangulation. */ /* */ /* The origin of fixuptri is treated as if it has just been inserted, and */ /* the local Delaunay condition needs to be enforced. It is only enforced */ /* in one sector, however, that being the angular range defined by */ /* fixuptri. */ /* */ /* This routine also needs to make decisions regarding the "stacking" of */ /* triangles. (Read the description of constrainededge() below before */ /* reading on here, so you understand the algorithm.) If the position of */ /* the new point (the origin of fixuptri) indicates that the vertex before */ /* it on the polygon is a reflex vertex, then "stack" the triangle by */ /* doing nothing. (fixuptri is an inverted triangle, which is how stacked */ /* triangles are identified.) */ /* */ /* Otherwise, check whether the vertex before that was a reflex vertex. */ /* If so, perform an edge flip, thereby eliminating an inverted triangle */ /* (popping it off the stack). The edge flip may result in the creation */ /* of a new inverted triangle, depending on whether or not the new vertex */ /* is visible to the vertex three edges behind on the polygon. */ /* */ /* If neither of the two vertices behind the new vertex are reflex */ /* vertices, fixuptri and fartri, the triangle opposite it, are not */ /* inverted; hence, ensure that the edge between them is locally Delaunay. */ /* */ /* `leftside' indicates whether or not fixuptri is to the left of the */ /* segment being inserted. (Imagine that the segment is pointing up from */ /* endpoint1 to endpoint2.) */ /* */ /*****************************************************************************/ void delaunayfixup( struct triedge *fixuptri, int leftside) { struct triedge neartri; struct triedge fartri; struct edge faredge; point nearpoint, leftpoint, rightpoint, farpoint; triangle ptr; /* Temporary variable used by sym(). */ shelle sptr; /* Temporary variable used by tspivot(). */ lnext(*fixuptri, neartri); sym(neartri, fartri); /* Check if the edge opposite the origin of fixuptri can be flipped. */ if (fartri.tri == dummytri) { return; } tspivot(neartri, faredge); if (faredge.sh != dummysh) { return; } /* Find all the relevant vertices. */ apex(neartri, nearpoint); org(neartri, leftpoint); dest(neartri, rightpoint); apex(fartri, farpoint); /* Check whether the previous polygon vertex is a reflex vertex. */ if (leftside) { if (counterclockwise(nearpoint, leftpoint, farpoint) <= 0.0) { /* leftpoint is a reflex vertex too. Nothing can */ /* be done until a convex section is found. */ return; } } else { if (counterclockwise(farpoint, rightpoint, nearpoint) <= 0.0) { /* rightpoint is a reflex vertex too. Nothing can */ /* be done until a convex section is found. */ return; } } if (counterclockwise(rightpoint, leftpoint, farpoint) > 0.0) { /* fartri is not an inverted triangle, and farpoint is not a reflex */ /* vertex. As there are no reflex vertices, fixuptri isn't an */ /* inverted triangle, either. Hence, test the edge between the */ /* triangles to ensure it is locally Delaunay. */ if (incircle(leftpoint, farpoint, rightpoint, nearpoint) <= 0.0) { return; } /* Not locally Delaunay; go on to an edge flip. */ } /* else fartri is inverted; remove it from the stack by flipping. */ flip(&neartri); lprevself(*fixuptri); /* Restore the origin of fixuptri after the flip. */ /* Recursively process the two triangles that result from the flip. */ delaunayfixup(fixuptri, leftside); delaunayfixup(&fartri, leftside); } /*****************************************************************************/ /* */ /* constrainededge() Force a segment into a constrained Delaunay */ /* triangulation by deleting the triangles it */ /* intersects, and triangulating the polygons that */ /* form on each side of it. */ /* */ /* Generates a single edge connecting `endpoint1' to `endpoint2'. The */ /* triangle `starttri' has `endpoint1' as its origin. `newmark' is the */ /* boundary marker of the segment. */ /* */ /* To insert a segment, every triangle whose interior intersects the */ /* segment is deleted. The union of these deleted triangles is a polygon */ /* (which is not necessarily monotone, but is close enough), which is */ /* divided into two polygons by the new segment. This routine's task is */ /* to generate the Delaunay triangulation of these two polygons. */ /* */ /* You might think of this routine's behavior as a two-step process. The */ /* first step is to walk from endpoint1 to endpoint2, flipping each edge */ /* encountered. This step creates a fan of edges connected to endpoint1, */ /* including the desired edge to endpoint2. The second step enforces the */ /* Delaunay condition on each side of the segment in an incremental manner: */ /* proceeding along the polygon from endpoint1 to endpoint2 (this is done */ /* independently on each side of the segment), each vertex is "enforced" */ /* as if it had just been inserted, but affecting only the previous */ /* vertices. The result is the same as if the vertices had been inserted */ /* in the order they appear on the polygon, so the result is Delaunay. */ /* */ /* In truth, constrainededge() interleaves these two steps. The procedure */ /* walks from endpoint1 to endpoint2, and each time an edge is encountered */ /* and flipped, the newly exposed vertex (at the far end of the flipped */ /* edge) is "enforced" upon the previously flipped edges, usually affecting */ /* only one side of the polygon (depending upon which side of the segment */ /* the vertex falls on). */ /* */ /* The algorithm is complicated by the need to handle polygons that are not */ /* convex. Although the polygon is not necessarily monotone, it can be */ /* triangulated in a manner similar to the stack-based algorithms for */ /* monotone polygons. For each reflex vertex (local concavity) of the */ /* polygon, there will be an inverted triangle formed by one of the edge */ /* flips. (An inverted triangle is one with negative area - that is, its */ /* vertices are arranged in clockwise order - and is best thought of as a */ /* wrinkle in the fabric of the mesh.) Each inverted triangle can be */ /* thought of as a reflex vertex pushed on the stack, waiting to be fixed */ /* later. */ /* */ /* A reflex vertex is popped from the stack when a vertex is inserted that */ /* is visible to the reflex vertex. (However, if the vertex behind the */ /* reflex vertex is not visible to the reflex vertex, a new inverted */ /* triangle will take its place on the stack.) These details are handled */ /* by the delaunayfixup() routine above. */ /* */ /*****************************************************************************/ void constrainededge( struct triedge *starttri, point endpoint2, int newmark) { struct triedge fixuptri, fixuptri2; struct edge fixupedge; point endpoint1; point farpoint; double area; int collision; int done; triangle ptr; /* Temporary variable used by sym() and oprev(). */ shelle sptr; /* Temporary variable used by tspivot(). */ org(*starttri, endpoint1); lnext(*starttri, fixuptri); flip(&fixuptri); /* `collision' indicates whether we have found a point directly */ /* between endpoint1 and endpoint2. */ collision = 0; done = 0; do { org(fixuptri, farpoint); /* `farpoint' is the extreme point of the polygon we are "digging" */ /* to get from endpoint1 to endpoint2. */ if ((farpoint[0] == endpoint2[0]) && (farpoint[1] == endpoint2[1])) { oprev(fixuptri, fixuptri2); /* Enforce the Delaunay condition around endpoint2. */ delaunayfixup(&fixuptri, 0); delaunayfixup(&fixuptri2, 1); done = 1; } else { /* Check whether farpoint is to the left or right of the segment */ /* being inserted, to decide which edge of fixuptri to dig */ /* through next. */ area = counterclockwise(endpoint1, endpoint2, farpoint); if (area == 0.0) { /* We've collided with a point between endpoint1 and endpoint2. */ collision = 1; oprev(fixuptri, fixuptri2); /* Enforce the Delaunay condition around farpoint. */ delaunayfixup(&fixuptri, 0); delaunayfixup(&fixuptri2, 1); done = 1; } else { if (area > 0.0) { /* farpoint is to the left of the segment. */ oprev(fixuptri, fixuptri2); /* Enforce the Delaunay condition around farpoint, on the */ /* left side of the segment only. */ delaunayfixup(&fixuptri2, 1); /* Flip the edge that crosses the segment. After the edge is */ /* flipped, one of its endpoints is the fan vertex, and the */ /* destination of fixuptri is the fan vertex. */ lprevself(fixuptri); } else { /* farpoint is to the right of the segment. */ delaunayfixup(&fixuptri, 0); /* Flip the edge that crosses the segment. After the edge is */ /* flipped, one of its endpoints is the fan vertex, and the */ /* destination of fixuptri is the fan vertex. */ oprevself(fixuptri); } /* Check for two intersecting segments. */ tspivot(fixuptri, fixupedge); if (fixupedge.sh == dummysh) { flip(&fixuptri); /* May create an inverted triangle on the left. */ } else { /* We've collided with a segment between endpoint1 and endpoint2. */ collision = 1; /* Insert a point at the intersection. */ segmentintersection(&fixuptri, &fixupedge, endpoint2); done = 1; } } } } while (!done); /* Insert a shell edge to make the segment permanent. */ insertshelle(&fixuptri, newmark); /* If there was a collision with an interceding vertex, install another */ /* segment connecting that vertex with endpoint2. */ if (collision) { /* Insert the remainder of the segment. */ if (!scoutsegment(&fixuptri, endpoint2, newmark)) { constrainededge(&fixuptri, endpoint2, newmark); } } } /*****************************************************************************/ /* */ /* insertsegment() Insert a PSLG segment into a triangulation. */ /* */ /*****************************************************************************/ void insertsegment( point endpoint1, point endpoint2, int newmark) { struct triedge searchtri1, searchtri2; triangle encodedtri; point checkpoint; triangle ptr; /* Temporary variable used by sym(). */ /* Find a triangle whose origin is the segment's first endpoint. */ checkpoint = (point) NULL; encodedtri = point2tri(endpoint1); if (encodedtri != (triangle) NULL) { decode(encodedtri, searchtri1); org(searchtri1, checkpoint); } if (checkpoint != endpoint1) { /* Find a boundary triangle to search from. */ searchtri1.tri = dummytri; searchtri1.orient = 0; symself(searchtri1); /* Search for the segment's first endpoint by point location. */ if (locate(endpoint1, &searchtri1) != ONVERTEX) { vTrace("*** E0061 : Internal error in insertsegment(): Unable to locate PSLG point (%.12g, %.12g) in triangulation.", endpoint1[0], endpoint1[1]); internalerror(); } } /* Remember this triangle to improve subsequent point location. */ triedgecopy(searchtri1, recenttri); /* Scout the beginnings of a path from the first endpoint */ /* toward the second. */ if (scoutsegment(&searchtri1, endpoint2, newmark)) { /* The segment was easily inserted. */ return; } /* The first endpoint may have changed if a collision with an intervening */ /* vertex on the segment occurred. */ org(searchtri1, endpoint1); /* Find a triangle whose origin is the segment's second endpoint. */ checkpoint = (point) NULL; encodedtri = point2tri(endpoint2); if (encodedtri != (triangle) NULL) { decode(encodedtri, searchtri2); org(searchtri2, checkpoint); } if (checkpoint != endpoint2) { /* Find a boundary triangle to search from. */ searchtri2.tri = dummytri; searchtri2.orient = 0; symself(searchtri2); /* Search for the segment's second endpoint by point location. */ if (locate(endpoint2, &searchtri2) != ONVERTEX) { vTrace("*** E0062 : Internal error in insertsegment(): Unable to locate PSLG point (%.12g, %.12g) in triangulation.", endpoint2[0], endpoint2[1]); internalerror(); } } /* Remember this triangle to improve subsequent point location. */ triedgecopy(searchtri2, recenttri); /* Scout the beginnings of a path from the second endpoint */ /* toward the first. */ if (scoutsegment(&searchtri2, endpoint1, newmark)) { /* The segment was easily inserted. */ return; } /* The second endpoint may have changed if a collision with an intervening */ /* vertex on the segment occurred. */ org(searchtri2, endpoint2); /* Insert the segment directly into the triangulation. */ constrainededge(&searchtri1, endpoint2, newmark); } /*****************************************************************************/ /* */ /* formskeleton() Create the shell edges of a triangulation, including */ /* PSLG edges and edges on the convex hull. */ /* */ /* The PSLG edges are read from a .poly file. The return value is the */ /* number of segments in the file. */ /* */ /*****************************************************************************/ int formskeleton( int *segmentlist, int *segmentmarkerlist, int numberofsegments) { char polyfilename[6]; int index; point endpoint1, endpoint2; int segments; int segmentmarkers; int end1, end2; int boundmarker; int i; strcpy(polyfilename, "input"); segments = numberofsegments; segmentmarkers = segmentmarkerlist != (int *) NULL; index = 0; /* If segments are to be inserted, compute a mapping */ /* from points to triangles. */ if (segments > 0) { makepointmap(); boundmarker = 0; /* Read and insert the segments. */ for (i = 1; i <= segments; i++) { end1 = segmentlist[index++]; end2 = segmentlist[index++]; if (segmentmarkers) { boundmarker = segmentmarkerlist[i - 1]; } if ((end1 < 0) || (end1 >= inpoints)) { } else if ((end2 < 0) || (end2 >= inpoints)) { } else { endpoint1 = getpoint(end1); endpoint2 = getpoint(end2); if ((endpoint1[0] == endpoint2[0]) && (endpoint1[1] == endpoint2[1])) { } else { insertsegment(endpoint1, endpoint2, boundmarker); } } } } else { segments = 0; } return segments; } /** **/ /** **/ /********* Segment (shell edge) insertion ends here *********/ /********* Carving out holes and concavities begins here *********/ /** **/ /** **/ /*****************************************************************************/ /* */ /* infecthull() Virally infect all of the triangles of the convex hull */ /* that are not protected by shell edges. Where there are */ /* shell edges, set boundary markers as appropriate. */ /* */ /*****************************************************************************/ void infecthull(void) { struct triedge hulltri; struct triedge nexttri; struct triedge starttri; struct edge hulledge; triangle **deadtri; point horg, hdest; triangle ptr; /* Temporary variable used by sym(). */ shelle sptr; /* Temporary variable used by tspivot(). */ /* Find a triangle handle on the hull. */ hulltri.tri = dummytri; hulltri.orient = 0; symself(hulltri); /* Remember where we started so we know when to stop. */ triedgecopy(hulltri, starttri); /* Go once counterclockwise around the convex hull. */ do { /* Ignore triangles that are already infected. */ if (!infected(hulltri)) { /* Is the triangle protected by a shell edge? */ tspivot(hulltri, hulledge); if (hulledge.sh == dummysh) { /* The triangle is not protected; infect it. */ infect(hulltri); deadtri = (triangle **) poolalloc(&viri); *deadtri = hulltri.tri; } else { /* The triangle is protected; set boundary markers if appropriate. */ if (mark(hulledge) == 0) { setmark(hulledge, 1); org(hulltri, horg); dest(hulltri, hdest); if (pointmark(horg) == 0) { setpointmark(horg, 1); } if (pointmark(hdest) == 0) { setpointmark(hdest, 1); } } } } /* To find the next hull edge, go clockwise around the next vertex. */ lnextself(hulltri); oprev(hulltri, nexttri); while (nexttri.tri != dummytri) { triedgecopy(nexttri, hulltri); oprev(hulltri, nexttri); } } while (!triedgeequal(hulltri, starttri)); } /*****************************************************************************/ /* */ /* plague() Spread the virus from all infected triangles to any neighbors */ /* not protected by shell edges. Delete all infected triangles. */ /* */ /* This is the procedure that actually creates holes and concavities. */ /* */ /* This procedure operates in two phases. The first phase identifies all */ /* the triangles that will die, and marks them as infected. They are */ /* marked to ensure that each triangle is added to the virus pool only */ /* once, so the procedure will terminate. */ /* */ /* The second phase actually eliminates the infected triangles. It also */ /* eliminates orphaned points. */ /* */ /*****************************************************************************/ void plague(void) { struct triedge testtri; struct triedge neighbor; triangle **virusloop; triangle **deadtri; struct edge neighborshelle; point testpoint; point norg, ndest; int killorg; triangle ptr; /* Temporary variable used by sym() and onext(). */ shelle sptr; /* Temporary variable used by tspivot(). */ /* Loop through all the infected triangles, spreading the virus to */ /* their neighbors, then to their neighbors' neighbors. */ traversalinit(&viri); virusloop = (triangle **) traverse(&viri); while (virusloop != (triangle **) NULL) { testtri.tri = *virusloop; /* A triangle is marked as infected by messing with one of its shell */ /* edges, setting it to an illegal value. Hence, we have to */ /* temporarily uninfect this triangle so that we can examine its */ /* adjacent shell edges. */ uninfect(testtri); /* Check each of the triangle's three neighbors. */ for (testtri.orient = 0; testtri.orient < 3; testtri.orient++) { /* Find the neighbor. */ sym(testtri, neighbor); /* Check for a shell between the triangle and its neighbor. */ tspivot(testtri, neighborshelle); /* Check if the neighbor is nonexistent or already infected. */ if ((neighbor.tri == dummytri) || infected(neighbor)) { if (neighborshelle.sh != dummysh) { /* There is a shell edge separating the triangle from its */ /* neighbor, but both triangles are dying, so the shell */ /* edge dies too. */ shelledealloc(neighborshelle.sh); if (neighbor.tri != dummytri) { /* Make sure the shell edge doesn't get deallocated again */ /* later when the infected neighbor is visited. */ uninfect(neighbor); tsdissolve(neighbor); infect(neighbor); } } } else { /* The neighbor exists and is not infected. */ if (neighborshelle.sh == dummysh) { /* There is no shell edge protecting the neighbor, so */ /* the neighbor becomes infected. */ infect(neighbor); /* Ensure that the neighbor's neighbors will be infected. */ deadtri = (triangle **) poolalloc(&viri); *deadtri = neighbor.tri; } else { /* The neighbor is protected by a shell edge. */ /* Remove this triangle from the shell edge. */ stdissolve(neighborshelle); /* The shell edge becomes a boundary. Set markers accordingly. */ if (mark(neighborshelle) == 0) { setmark(neighborshelle, 1); } org(neighbor, norg); dest(neighbor, ndest); if (pointmark(norg) == 0) { setpointmark(norg, 1); } if (pointmark(ndest) == 0) { setpointmark(ndest, 1); } } } } /* Remark the triangle as infected, so it doesn't get added to the */ /* virus pool again. */ infect(testtri); virusloop = (triangle **) traverse(&viri); } traversalinit(&viri); virusloop = (triangle **) traverse(&viri); while (virusloop != (triangle **) NULL) { testtri.tri = *virusloop; /* Check each of the three corners of the triangle for elimination. */ /* This is done by walking around each point, checking if it is */ /* still connected to at least one live triangle. */ for (testtri.orient = 0; testtri.orient < 3; testtri.orient++) { org(testtri, testpoint); /* Check if the point has already been tested. */ if (testpoint != (point) NULL) { killorg = 1; /* Mark the corner of the triangle as having been tested. */ setorg(testtri, NULL); /* Walk counterclockwise about the point. */ onext(testtri, neighbor); /* Stop upon reaching a boundary or the starting triangle. */ while ((neighbor.tri != dummytri) && (!triedgeequal(neighbor, testtri))) { if (infected(neighbor)) { /* Mark the corner of this triangle as having been tested. */ setorg(neighbor, NULL); } else { /* A live triangle. The point survives. */ killorg = 0; } /* Walk counterclockwise about the point. */ onextself(neighbor); } /* If we reached a boundary, we must walk clockwise as well. */ if (neighbor.tri == dummytri) { /* Walk clockwise about the point. */ oprev(testtri, neighbor); /* Stop upon reaching a boundary. */ while (neighbor.tri != dummytri) { if (infected(neighbor)) { /* Mark the corner of this triangle as having been tested. */ setorg(neighbor, NULL); } else { /* A live triangle. The point survives. */ killorg = 0; } /* Walk clockwise about the point. */ oprevself(neighbor); } } if (killorg) { pointdealloc(testpoint); } } } /* Record changes in the number of boundary edges, and disconnect */ /* dead triangles from their neighbors. */ for (testtri.orient = 0; testtri.orient < 3; testtri.orient++) { sym(testtri, neighbor); if (neighbor.tri == dummytri) { /* There is no neighboring triangle on this edge, so this edge */ /* is a boundary edge. This triangle is being deleted, so this */ /* boundary edge is deleted. */ hullsize--; } else { /* Disconnect the triangle from its neighbor. */ dissolve(neighbor); /* There is a neighboring triangle on this edge, so this edge */ /* becomes a boundary edge when this triangle is deleted. */ hullsize++; } } /* Return the dead triangle to the pool of triangles. */ triangledealloc(testtri.tri); virusloop = (triangle **) traverse(&viri); } /* Empty the virus pool. */ poolrestart(&viri); } /*****************************************************************************/ /* */ /* regionplague() Spread regional attributes and/or area constraints */ /* (from a .poly file) throughout the mesh. */ /* */ /* This procedure operates in two phases. The first phase spreads an */ /* attribute and/or an area constraint through a (segment-bounded) region. */ /* The triangles are marked to ensure that each triangle is added to the */ /* virus pool only once, so the procedure will terminate. */ /* */ /* The second phase uninfects all infected triangles, returning them to */ /* normal. */ /* */ /*****************************************************************************/ void regionplague( double attribute, double area) { struct triedge testtri; struct triedge neighbor; triangle **virusloop; triangle **regiontri; struct edge neighborshelle; triangle ptr; /* Temporary variable used by sym() and onext(). */ shelle sptr; /* Temporary variable used by tspivot(). */ /* Loop through all the infected triangles, spreading the attribute */ /* and/or area constraint to their neighbors, then to their neighbors' */ /* neighbors. */ traversalinit(&viri); virusloop = (triangle **) traverse(&viri); while (virusloop != (triangle **) NULL) { testtri.tri = *virusloop; /* A triangle is marked as infected by messing with one of its shell */ /* edges, setting it to an illegal value. Hence, we have to */ /* temporarily uninfect this triangle so that we can examine its */ /* adjacent shell edges. */ uninfect(testtri); /* Check each of the triangle's three neighbors. */ for (testtri.orient = 0; testtri.orient < 3; testtri.orient++) { /* Find the neighbor. */ sym(testtri, neighbor); /* Check for a shell between the triangle and its neighbor. */ tspivot(testtri, neighborshelle); /* Make sure the neighbor exists, is not already infected, and */ /* isn't protected by a shell edge. */ if ((neighbor.tri != dummytri) && !infected(neighbor) && (neighborshelle.sh == dummysh)) { /* Infect the neighbor. */ infect(neighbor); /* Ensure that the neighbor's neighbors will be infected. */ regiontri = (triangle **) poolalloc(&viri); *regiontri = neighbor.tri; } } /* Remark the triangle as infected, so it doesn't get added to the */ /* virus pool again. */ infect(testtri); virusloop = (triangle **) traverse(&viri); } /* Uninfect all triangles. */ traversalinit(&viri); virusloop = (triangle **) traverse(&viri); while (virusloop != (triangle **) NULL) { testtri.tri = *virusloop; uninfect(testtri); virusloop = (triangle **) traverse(&viri); } /* Empty the virus pool. */ poolrestart(&viri); } /*****************************************************************************/ /* */ /* carveholes() Find the holes and infect them. Find the area */ /* constraints and infect them. Infect the convex hull. */ /* Spread the infection and kill triangles. Spread the */ /* area constraints. */ /* */ /* This routine mainly calls other routines to carry out all these */ /* functions. */ /* */ /*****************************************************************************/ void carveholes( double *holelist, int holes, double *regionlist, int regions) { struct triedge searchtri; struct triedge *regiontris; triangle **holetri; triangle **regiontri; point searchorg, searchdest; enum locateresult intersect; int i; triangle ptr; /* Temporary variable used by sym(). */ if (regions > 0) { /* Allocate storage for the triangles in which region points fall. */ regiontris = (struct triedge *) malloc(regions * sizeof(struct triedge)); if (regiontris == (struct triedge *) NULL) { vTrace("*** E0063 : Out of memory."); exit(1); } } /* Initialize a pool of viri to be used for holes, concavities, */ /* regional attributes, and/or regional area constraints. */ poolinit(&viri, sizeof(triangle *), VIRUSPERBLOCK, POINTER, 0); /* Mark as infected any unprotected triangles on the boundary. */ /* This is one way by which concavities are created. */ infecthull(); if (holes > 0) { /* Infect each triangle in which a hole lies. */ for (i = 0; i < 2 * holes; i += 2) { /* Ignore holes that aren't within the bounds of the mesh. */ if ((holelist[i] >= xmin) && (holelist[i] <= xmax) && (holelist[i + 1] >= ymin) && (holelist[i + 1] <= ymax)) { /* Start searching from some triangle on the outer boundary. */ searchtri.tri = dummytri; searchtri.orient = 0; symself(searchtri); /* Ensure that the hole is to the left of this boundary edge; */ /* otherwise, locate() will falsely report that the hole */ /* falls within the starting triangle. */ org(searchtri, searchorg); dest(searchtri, searchdest); if (counterclockwise(searchorg, searchdest, &holelist[i]) > 0.0) { /* Find a triangle that contains the hole. */ intersect = locate(&holelist[i], &searchtri); if ((intersect != OUTSIDE) && (!infected(searchtri))) { /* Infect the triangle. This is done by marking the triangle */ /* as infect and including the triangle in the virus pool. */ infect(searchtri); holetri = (triangle **) poolalloc(&viri); *holetri = searchtri.tri; } } } } } /* Now, we have to find all the regions BEFORE we carve the holes, because */ /* locate() won't work when the triangulation is no longer convex. */ /* (Incidentally, this is the reason why regional attributes and area */ /* constraints can't be used when refining a preexisting mesh, which */ /* might not be convex; they can only be used with a freshly */ /* triangulated PSLG.) */ if (regions > 0) { /* Find the starting triangle for each region. */ for (i = 0; i < regions; i++) { regiontris[i].tri = dummytri; /* Ignore region points that aren't within the bounds of the mesh. */ if ((regionlist[4 * i] >= xmin) && (regionlist[4 * i] <= xmax) && (regionlist[4 * i + 1] >= ymin) && (regionlist[4 * i + 1] <= ymax)) { /* Start searching from some triangle on the outer boundary. */ searchtri.tri = dummytri; searchtri.orient = 0; symself(searchtri); /* Ensure that the region point is to the left of this boundary */ /* edge; otherwise, locate() will falsely report that the */ /* region point falls within the starting triangle. */ org(searchtri, searchorg); dest(searchtri, searchdest); if (counterclockwise(searchorg, searchdest, ®ionlist[4 * i]) > 0.0) { /* Find a triangle that contains the region point. */ intersect = locate(®ionlist[4 * i], &searchtri); if ((intersect != OUTSIDE) && (!infected(searchtri))) { /* Record the triangle for processing after the */ /* holes have been carved. */ triedgecopy(searchtri, regiontris[i]); } } } } } if (viri.items > 0) { /* Carve the holes and concavities. */ plague(); } /* The virus pool should be empty now. */ for (i = 0; i < regions; i++) { if (regiontris[i].tri != dummytri) { /* Make sure the triangle under consideration still exists. */ /* It may have been eaten by the virus. */ if (regiontris[i].tri[3] != (triangle) NULL) { /* Put one triangle in the virus pool. */ infect(regiontris[i]); regiontri = (triangle **) poolalloc(&viri); *regiontri = regiontris[i].tri; /* Apply one region's attribute and/or area constraint. */ regionplague(regionlist[4 * i + 2], regionlist[4 * i + 3]); /* The virus pool should be empty now. */ } } } /* Free up memory. */ pooldeinit(&viri); if (regions > 0) { free(regiontris); } } /** **/ /** **/ /********* Carving out holes and concavities ends here *********/ /*****************************************************************************/ /* */ /* highorder() Create extra nodes for quadratic subparametric elements. */ /* */ /*****************************************************************************/ void highorder(void) { struct triedge triangleloop, trisym; struct edge checkmark; point newpoint; point torg, tdest; int i; triangle ptr; /* Temporary variable used by sym(). */ shelle sptr; /* Temporary variable used by tspivot(). */ /* The following line ensures that dead items in the pool of nodes */ /* cannot be allocated for the extra nodes associated with high */ /* order elements. This ensures that the primary nodes (at the */ /* corners of elements) will occur earlier in the output files, and */ /* have lower indices, than the extra nodes. */ points.deaditemstack = (void *) NULL; traversalinit(&triangles); triangleloop.tri = triangletraverse(); /* To loop over the set of edges, loop over all triangles, and look at */ /* the three edges of each triangle. If there isn't another triangle */ /* adjacent to the edge, operate on the edge. If there is another */ /* adjacent triangle, operate on the edge only if the current triangle */ /* has a smaller pointer than its neighbor. This way, each edge is */ /* considered only once. */ while (triangleloop.tri != (triangle *) NULL) { for (triangleloop.orient = 0; triangleloop.orient < 3; triangleloop.orient++) { sym(triangleloop, trisym); if ((triangleloop.tri < trisym.tri) || (trisym.tri == dummytri)) { org(triangleloop, torg); dest(triangleloop, tdest); /* Create a new node in the middle of the edge. Interpolate */ /* its attributes. */ newpoint = (point) poolalloc(&points); for (i = 0; i < 2 + nextras; i++) { newpoint[i] = 0.5 * (torg[i] + tdest[i]); } /* Set the new node's marker to zero or one, depending on */ /* whether it lies on a boundary. */ setpointmark(newpoint, trisym.tri == dummytri); if (useshelles) { tspivot(triangleloop, checkmark); /* If this edge is a segment, transfer the marker to the new node. */ if (checkmark.sh != dummysh) { setpointmark(newpoint, mark(checkmark)); } } /* Record the new node in the (one or two) adjacent elements. */ triangleloop.tri[highorderindex + triangleloop.orient] = (triangle) newpoint; if (trisym.tri != dummytri) { trisym.tri[highorderindex + trisym.orient] = (triangle) newpoint; } } } triangleloop.tri = triangletraverse(); } } /*****************************************************************************/ /* */ /* transfernodes() Read the points from memory. */ /* */ /*****************************************************************************/ void transfernodes( double *pointlist, double *pointattriblist, int *pointmarkerlist, int numberofpoints, int numberofpointattribs) { point pointloop; double x, y; int i, j; int coordindex; int attribindex; inpoints = numberofpoints; mesh_dim = 2; nextras = numberofpointattribs; if (inpoints < 3) { vTrace("*** E0064 : Input must have at least three input points."); exit(1); } initializepointpool(); /* Read the points. */ coordindex = 0; attribindex = 0; for (i = 0; i < inpoints; i++) { pointloop = (point) poolalloc(&points); /* Read the point coordinates. */ x = pointloop[0] = pointlist[coordindex++]; y = pointloop[1] = pointlist[coordindex++]; /* Read the point attributes. */ for (j = 0; j < numberofpointattribs; j++) { pointloop[2 + j] = pointattriblist[attribindex++]; } if (pointmarkerlist != (int *) NULL) { /* Read a point marker. */ setpointmark(pointloop, pointmarkerlist[i]); } else { /* If no markers are specified, they default to zero. */ setpointmark(pointloop, 0); } x = pointloop[0]; y = pointloop[1]; /* Determine the smallest and largest x and y coordinates. */ if (i == 0) { xmin = xmax = x; ymin = ymax = y; } else { xmin = (x < xmin) ? x : xmin; xmax = (x > xmax) ? x : xmax; ymin = (y < ymin) ? y : ymin; ymax = (y > ymax) ? y : ymax; } } /* Nonexistent x value used as a flag to mark circle events in sweepline */ /* Delaunay algorithm. */ xminextreme = 10 * xmin - 9 * xmax; } /*****************************************************************************/ /* */ /* numbernodes() Number the points. */ /* */ /* Each point is assigned a marker equal to its number. */ /* */ /* Used when writenodes() is not called because no .node file is written. */ /* */ /*****************************************************************************/ void numbernodes(void) { point pointloop; int pointnumber; traversalinit(&points); pointloop = pointtraverse(); pointnumber = 0; while (pointloop != (point) NULL) { setpointmark(pointloop, pointnumber); pointloop = pointtraverse(); pointnumber++; } } /*****************************************************************************/ /* */ /* writeelements() Write the triangles to an .ele file. */ /* */ /*****************************************************************************/ void writeelements( int **trianglelist, double **triangleattriblist) { int *tlist; double *talist; int pointindex; int attribindex; struct triedge triangleloop; point p1, p2, p3; point mid1, mid2, mid3; int elementnumber; int i; /* Allocate memory for output triangles if necessary. */ if (*trianglelist == (int *) NULL) { *trianglelist = (int *) malloc(triangles.items * ((order + 1) * (order + 2) / 2) * sizeof(int)); if (*trianglelist == (int *) NULL) { vTrace("*** E0065 : Out of memory."); exit(1); } } /* Allocate memory for output triangle attributes if necessary. */ if ((eextras > 0) && (*triangleattriblist == (double *) NULL)) { *triangleattriblist = (double *) malloc(triangles.items * eextras * sizeof(double)); if (*triangleattriblist == (double *) NULL) { vTrace("*** E0066 : Out of memory."); exit(1); } } tlist = *trianglelist; talist = *triangleattriblist; pointindex = 0; attribindex = 0; traversalinit(&triangles); triangleloop.tri = triangletraverse(); triangleloop.orient = 0; elementnumber = 0; while (triangleloop.tri != (triangle *) NULL) { org(triangleloop, p1); dest(triangleloop, p2); apex(triangleloop, p3); if (order == 1) { tlist[pointindex++] = pointmark(p1); tlist[pointindex++] = pointmark(p2); tlist[pointindex++] = pointmark(p3); } else { mid1 = (point) triangleloop.tri[highorderindex + 1]; mid2 = (point) triangleloop.tri[highorderindex + 2]; mid3 = (point) triangleloop.tri[highorderindex]; tlist[pointindex++] = pointmark(p1); tlist[pointindex++] = pointmark(p2); tlist[pointindex++] = pointmark(p3); tlist[pointindex++] = pointmark(mid1); tlist[pointindex++] = pointmark(mid2); tlist[pointindex++] = pointmark(mid3); } for (i = 0; i < eextras; i++) { talist[attribindex++] = elemattribute(triangleloop, i); } triangleloop.tri = triangletraverse(); elementnumber++; } } /*****************************************************************************/ /* */ /* main() or triangulate() Gosh, do everything. */ /* */ /* The sequence is roughly as follows. Many of these steps can be skipped, */ /* depending on the command line switches. */ /* */ /* - Initialize constants and parse the command line. */ /* - Read the points from a file and either */ /* - triangulate them (no -r), or */ /* - read an old mesh from files and reconstruct it (-r). */ /* - Insert the PSLG segments (-p), and possibly segments on the convex */ /* hull (-c). */ /* - Read the holes (-p), regional attributes (-pA), and regional area */ /* constraints (-pa). Carve the holes and concavities, and spread the */ /* regional attributes and area constraints. */ /* - Enforce the constraints on minimum angle (-q) and maximum area (-a). */ /* Also enforce the conforming Delaunay property (-q and -a). */ /* - Compute the number of edges in the resulting mesh. */ /* - Promote the mesh's linear triangles to higher order elements (-o). */ /* - Write the output files and print the statistics. */ /* - Check the consistency and Delaunay property of the mesh (-C). */ /* */ /*****************************************************************************/ void triangulate( char *triswitches, struct triangulateio *in, struct triangulateio *out, struct triangulateio *vorout) { double *holearray; /* Array of holes. */ double *regionarray; /* Array of regional attributes and area constraints. */ triangleinit(); parsecommandline(1, &triswitches); transfernodes(in->pointlist, in->pointattributelist, in->pointmarkerlist, in->numberofpoints, in->numberofpointattributes); hullsize = delaunay(); /* Triangulate the points. */ /* Ensure that no point can be mistaken for a triangular bounding */ /* box point in insertsite(). */ infpoint1 = (point) NULL; infpoint2 = (point) NULL; infpoint3 = (point) NULL; if (useshelles) { checksegments = 1; /* Segments will be introduced next. */ /* Insert PSLG segments and/or convex hull segments. */ insegments = formskeleton(in->segmentlist, in->segmentmarkerlist, in->numberofsegments); } holearray = in->holelist; holes = in->numberofholes; regionarray = in->regionlist; regions = in->numberofregions; /* Carve out holes and concavities. */ carveholes(holearray, holes, regionarray, regions); /* Compute the number of edges. */ edges = (3l * triangles.items + hullsize) / 2l; if (order > 1) { highorder(); /* Promote elements to higher polynomial order. */ } out->numberofpoints = points.items; out->numberofpointattributes = nextras; out->numberoftriangles = triangles.items; out->numberofcorners = (order + 1) * (order + 2) / 2; out->numberoftriangleattributes = eextras; out->numberofedges = edges; if (useshelles) { out->numberofsegments = shelles.items; } else { out->numberofsegments = hullsize; } if (vorout != (struct triangulateio *) NULL) { vorout->numberofpoints = triangles.items; vorout->numberofpointattributes = nextras; vorout->numberofedges = edges; } /* If not using iteration numbers, don't write a .node file if one was */ /* read, because the original one would be overwritten! */ numbernodes(); /* We must remember to number the points. */ writeelements(&out->trianglelist, &out->triangleattributelist); triangledeinit(); }