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triangle.c
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1/*****************************************************************************/
2/* */
3/* 888888888 ,o, / 888 */
4/* 888 88o88o " o8888o 88o8888o o88888o 888 o88888o */
5/* 888 888 888 88b 888 888 888 888 888 d888 88b */
6/* 888 888 888 o88^o888 888 888 "88888" 888 8888oo888 */
7/* 888 888 888 C888 888 888 888 / 888 q888 */
8/* 888 888 888 "88o^888 888 888 Cb 888 "88oooo" */
9/* "8oo8D */
10/* */
11/* A Two-Dimensional Quality Mesh Generator and Delaunay Triangulator. */
12/* (triangle.c) */
13/* */
14/* Version 1.6 */
15/* July 28, 2005 */
16/* */
17/* Copyright 1993, 1995, 1997, 1998, 2002, 2005 */
18/* Jonathan Richard Shewchuk */
19/* 2360 Woolsey #H */
20/* Berkeley, California 94705-1927 */
21/* jrs@cs.berkeley.edu */
22/* */
23/* This program may be freely redistributed under the condition that the */
24/* copyright notices (including this entire header and the copyright */
25/* notice printed when the `-h' switch is selected) are not removed, and */
26/* no compensation is received. Private, research, and institutional */
27/* use is free. You may distribute modified versions of this code UNDER */
28/* THE CONDITION THAT THIS CODE AND ANY MODIFICATIONS MADE TO IT IN THE */
29/* SAME FILE REMAIN UNDER COPYRIGHT OF THE ORIGINAL AUTHOR, BOTH SOURCE */
30/* AND OBJECT CODE ARE MADE FREELY AVAILABLE WITHOUT CHARGE, AND CLEAR */
31/* NOTICE IS GIVEN OF THE MODIFICATIONS. Distribution of this code as */
32/* part of a commercial system is permissible ONLY BY DIRECT ARRANGEMENT */
33/* WITH THE AUTHOR. (If you are not directly supplying this code to a */
34/* customer, and you are instead telling them how they can obtain it for */
35/* free, then you are not required to make any arrangement with me.) */
36/* */
37/* Hypertext instructions for Triangle are available on the Web at */
38/* */
39/* http://www.cs.cmu.edu/~quake/triangle.html */
40/* */
41/* Disclaimer: Neither I nor Carnegie Mellon warrant this code in any way */
42/* whatsoever. This code is provided "as-is". Use at your own risk. */
43/* */
44/* Some of the references listed below are marked with an asterisk. [*] */
45/* These references are available for downloading from the Web page */
46/* */
47/* http://www.cs.cmu.edu/~quake/triangle.research.html */
48/* */
49/* Three papers discussing aspects of Triangle are available. A short */
50/* overview appears in "Triangle: Engineering a 2D Quality Mesh */
51/* Generator and Delaunay Triangulator," in Applied Computational */
52/* Geometry: Towards Geometric Engineering, Ming C. Lin and Dinesh */
53/* Manocha, editors, Lecture Notes in Computer Science volume 1148, */
54/* pages 203-222, Springer-Verlag, Berlin, May 1996 (from the First ACM */
55/* Workshop on Applied Computational Geometry). [*] */
56/* */
57/* The algorithms are discussed in the greatest detail in "Delaunay */
58/* Refinement Algorithms for Triangular Mesh Generation," Computational */
59/* Geometry: Theory and Applications 22(1-3):21-74, May 2002. [*] */
60/* */
61/* More detail about the data structures may be found in my dissertation: */
62/* "Delaunay Refinement Mesh Generation," Ph.D. thesis, Technical Report */
63/* CMU-CS-97-137, School of Computer Science, Carnegie Mellon University, */
64/* Pittsburgh, Pennsylvania, 18 May 1997. [*] */
65/* */
66/* Triangle was created as part of the Quake Project in the School of */
67/* Computer Science at Carnegie Mellon University. For further */
68/* information, see Hesheng Bao, Jacobo Bielak, Omar Ghattas, Loukas F. */
69/* Kallivokas, David R. O'Hallaron, Jonathan R. Shewchuk, and Jifeng Xu, */
70/* "Large-scale Simulation of Elastic Wave Propagation in Heterogeneous */
71/* Media on Parallel Computers," Computer Methods in Applied Mechanics */
72/* and Engineering 152(1-2):85-102, 22 January 1998. */
73/* */
74/* Triangle's Delaunay refinement algorithm for quality mesh generation is */
75/* a hybrid of one due to Jim Ruppert, "A Delaunay Refinement Algorithm */
76/* for Quality 2-Dimensional Mesh Generation," Journal of Algorithms */
77/* 18(3):548-585, May 1995 [*], and one due to L. Paul Chew, "Guaranteed- */
78/* Quality Mesh Generation for Curved Surfaces," Proceedings of the Ninth */
79/* Annual Symposium on Computational Geometry (San Diego, California), */
80/* pages 274-280, Association for Computing Machinery, May 1993, */
81/* http://portal.acm.org/citation.cfm?id=161150 . */
82/* */
83/* The Delaunay refinement algorithm has been modified so that it meshes */
84/* domains with small input angles well, as described in Gary L. Miller, */
85/* Steven E. Pav, and Noel J. Walkington, "When and Why Ruppert's */
86/* Algorithm Works," Twelfth International Meshing Roundtable, pages */
87/* 91-102, Sandia National Laboratories, September 2003. [*] */
88/* */
89/* My implementation of the divide-and-conquer and incremental Delaunay */
90/* triangulation algorithms follows closely the presentation of Guibas */
91/* and Stolfi, even though I use a triangle-based data structure instead */
92/* of their quad-edge data structure. (In fact, I originally implemented */
93/* Triangle using the quad-edge data structure, but the switch to a */
94/* triangle-based data structure sped Triangle by a factor of two.) The */
95/* mesh manipulation primitives and the two aforementioned Delaunay */
96/* triangulation algorithms are described by Leonidas J. Guibas and Jorge */
97/* Stolfi, "Primitives for the Manipulation of General Subdivisions and */
98/* the Computation of Voronoi Diagrams," ACM Transactions on Graphics */
99/* 4(2):74-123, April 1985, http://portal.acm.org/citation.cfm?id=282923 .*/
100/* */
101/* Their O(n log n) divide-and-conquer algorithm is adapted from Der-Tsai */
102/* Lee and Bruce J. Schachter, "Two Algorithms for Constructing the */
103/* Delaunay Triangulation," International Journal of Computer and */
104/* Information Science 9(3):219-242, 1980. Triangle's improvement of the */
105/* divide-and-conquer algorithm by alternating between vertical and */
106/* horizontal cuts was introduced by Rex A. Dwyer, "A Faster Divide-and- */
107/* Conquer Algorithm for Constructing Delaunay Triangulations," */
108/* Algorithmica 2(2):137-151, 1987. */
109/* */
110/* The incremental insertion algorithm was first proposed by C. L. Lawson, */
111/* "Software for C1 Surface Interpolation," in Mathematical Software III, */
112/* John R. Rice, editor, Academic Press, New York, pp. 161-194, 1977. */
113/* For point location, I use the algorithm of Ernst P. Mucke, Isaac */
114/* Saias, and Binhai Zhu, "Fast Randomized Point Location Without */
115/* Preprocessing in Two- and Three-Dimensional Delaunay Triangulations," */
116/* Proceedings of the Twelfth Annual Symposium on Computational Geometry, */
117/* ACM, May 1996. [*] If I were to randomize the order of vertex */
118/* insertion (I currently don't bother), their result combined with the */
119/* result of Kenneth L. Clarkson and Peter W. Shor, "Applications of */
120/* Random Sampling in Computational Geometry II," Discrete & */
121/* Computational Geometry 4(1):387-421, 1989, would yield an expected */
122/* O(n^{4/3}) bound on running time. */
123/* */
124/* The O(n log n) sweepline Delaunay triangulation algorithm is taken from */
125/* Steven Fortune, "A Sweepline Algorithm for Voronoi Diagrams", */
126/* Algorithmica 2(2):153-174, 1987. A random sample of edges on the */
127/* boundary of the triangulation are maintained in a splay tree for the */
128/* purpose of point location. Splay trees are described by Daniel */
129/* Dominic Sleator and Robert Endre Tarjan, "Self-Adjusting Binary Search */
130/* Trees," Journal of the ACM 32(3):652-686, July 1985, */
131/* http://portal.acm.org/citation.cfm?id=3835 . */
132/* */
133/* The algorithms for exact computation of the signs of determinants are */
134/* described in Jonathan Richard Shewchuk, "Adaptive Precision Floating- */
135/* Point Arithmetic and Fast Robust Geometric Predicates," Discrete & */
136/* Computational Geometry 18(3):305-363, October 1997. (Also available */
137/* as Technical Report CMU-CS-96-140, School of Computer Science, */
138/* Carnegie Mellon University, Pittsburgh, Pennsylvania, May 1996.) [*] */
139/* An abbreviated version appears as Jonathan Richard Shewchuk, "Robust */
140/* Adaptive Floating-Point Geometric Predicates," Proceedings of the */
141/* Twelfth Annual Symposium on Computational Geometry, ACM, May 1996. [*] */
142/* Many of the ideas for my exact arithmetic routines originate with */
143/* Douglas M. Priest, "Algorithms for Arbitrary Precision Floating Point */
144/* Arithmetic," Tenth Symposium on Computer Arithmetic, pp. 132-143, IEEE */
145/* Computer Society Press, 1991. [*] Many of the ideas for the correct */
146/* evaluation of the signs of determinants are taken from Steven Fortune */
147/* and Christopher J. Van Wyk, "Efficient Exact Arithmetic for Computa- */
148/* tional Geometry," Proceedings of the Ninth Annual Symposium on */
149/* Computational Geometry, ACM, pp. 163-172, May 1993, and from Steven */
150/* Fortune, "Numerical Stability of Algorithms for 2D Delaunay Triangu- */
151/* lations," International Journal of Computational Geometry & Applica- */
152/* tions 5(1-2):193-213, March-June 1995. */
153/* */
154/* The method of inserting new vertices off-center (not precisely at the */
155/* circumcenter of every poor-quality triangle) is from Alper Ungor, */
156/* "Off-centers: A New Type of Steiner Points for Computing Size-Optimal */
157/* Quality-Guaranteed Delaunay Triangulations," Proceedings of LATIN */
158/* 2004 (Buenos Aires, Argentina), April 2004. */
159/* */
160/* For definitions of and results involving Delaunay triangulations, */
161/* constrained and conforming versions thereof, and other aspects of */
162/* triangular mesh generation, see the excellent survey by Marshall Bern */
163/* and David Eppstein, "Mesh Generation and Optimal Triangulation," in */
164/* Computing and Euclidean Geometry, Ding-Zhu Du and Frank Hwang, */
165/* editors, World Scientific, Singapore, pp. 23-90, 1992. [*] */
166/* */
167/* The time for incrementally adding PSLG (planar straight line graph) */
168/* segments to create a constrained Delaunay triangulation is probably */
169/* O(t^2) per segment in the worst case and O(t) per segment in the */
170/* common case, where t is the number of triangles that intersect the */
171/* segment before it is inserted. This doesn't count point location, */
172/* which can be much more expensive. I could improve this to O(d log d) */
173/* time, but d is usually quite small, so it's not worth the bother. */
174/* (This note does not apply when the -s switch is used, invoking a */
175/* different method is used to insert segments.) */
176/* */
177/* The time for deleting a vertex from a Delaunay triangulation is O(d^2) */
178/* in the worst case and O(d) in the common case, where d is the degree */
179/* of the vertex being deleted. I could improve this to O(d log d) time, */
180/* but d is usually quite small, so it's not worth the bother. */
181/* */
182/* Ruppert's Delaunay refinement algorithm typically generates triangles */
183/* at a linear rate (constant time per triangle) after the initial */
184/* triangulation is formed. There may be pathological cases where */
185/* quadratic time is required, but these never arise in practice. */
186/* */
187/* The geometric predicates (circumcenter calculations, segment */
188/* intersection formulae, etc.) appear in my "Lecture Notes on Geometric */
189/* Robustness" at http://www.cs.berkeley.edu/~jrs/mesh . */
190/* */
191/* If you make any improvements to this code, please please please let me */
192/* know, so that I may obtain the improvements. Even if you don't change */
193/* the code, I'd still love to hear what it's being used for. */
194/* */
195/*****************************************************************************/
196
197/* If yours is not a Unix system, define the NO_TIMER compiler switch to */
198/* remove the Unix-specific timing code. */
199
200#define NO_TIMER
201
202/* To insert lots of self-checks for internal errors, define the SELF_CHECK */
203/* symbol. This will slow down the program significantly. It is best to */
204/* define the symbol using the -DSELF_CHECK compiler switch, but you could */
205/* write "#define SELF_CHECK" below. If you are modifying this code, I */
206/* recommend you turn self-checks on until your work is debugged. */
207
208/* #define SELF_CHECK */
209
210/* To compile Triangle as a callable object library (triangle.o), define the */
211/* TRILIBRARY symbol. Read the file triangle.h for details on how to call */
212/* the procedure triangulate() that results. */
213
214#define TRILIBRARY
215
216/* It is possible to generate a smaller version of Triangle using one or */
217/* both of the following symbols. Define the REDUCED symbol to eliminate */
218/* all features that are primarily of research interest; specifically, the */
219/* -i, -F, -s, and -C switches. Define the CDT_ONLY symbol to eliminate */
220/* all meshing algorithms above and beyond constrained Delaunay */
221/* triangulation; specifically, the -r, -q, -a, -u, -D, -S, and -s */
222/* switches. These reductions are most likely to be useful when */
223/* generating an object library (triangle.o) by defining the TRILIBRARY */
224/* symbol. */
225
226#define REDUCED
227#define CDT_ONLY
228
229/* On some machines, my exact arithmetic routines might be defeated by the */
230/* use of internal extended precision floating-point registers. The best */
231/* way to solve this problem is to set the floating-point registers to use */
232/* single or double precision internally. On 80x86 processors, this may */
233/* be accomplished by setting the CPU86 symbol for the Microsoft C */
234/* compiler, or the LINUX symbol for the gcc compiler running on Linux. */
235/* */
236/* An inferior solution is to declare certain values as `volatile', thus */
237/* forcing them to be stored to memory and rounded off. Unfortunately, */
238/* this solution might slow Triangle down quite a bit. To use volatile */
239/* values, write "#define INEXACT volatile" below. Normally, however, */
240/* INEXACT should be defined to be nothing. ("#define INEXACT".) */
241/* */
242/* For more discussion, see http://www.cs.cmu.edu/~quake/robust.pc.html . */
243/* For yet more discussion, see Section 5 of my paper, "Adaptive Precision */
244/* Floating-Point Arithmetic and Fast Robust Geometric Predicates" (also */
245/* available as Section 6.6 of my dissertation). */
246
247/* #define CPU86 */
248/* #define LINUX */
249
250#define INEXACT /* Nothing */
251/* #define INEXACT volatile */
252
253/* Maximum number of characters in a file name (including the null). */
254
255#define FILENAMESIZE 2048
256
257/* Maximum number of characters in a line read from a file (including the */
258/* null). */
259
260#define INPUTLINESIZE 1024
261
262/* For efficiency, a variety of data structures are allocated in bulk. The */
263/* following constants determine how many of each structure is allocated */
264/* at once. */
265
266#define TRIPERBLOCK 4092 /* Number of triangles allocated at once. */
267#define SUBSEGPERBLOCK 508 /* Number of subsegments allocated at once. */
268#define VERTEXPERBLOCK 4092 /* Number of vertices allocated at once. */
269#define VIRUSPERBLOCK 1020 /* Number of virus triangles allocated at once. */
270/* Number of encroached subsegments allocated at once. */
271#define BADSUBSEGPERBLOCK 252
272/* Number of skinny triangles allocated at once. */
273#define BADTRIPERBLOCK 4092
274/* Number of flipped triangles allocated at once. */
275#define FLIPSTACKERPERBLOCK 252
276/* Number of splay tree nodes allocated at once. */
277#define SPLAYNODEPERBLOCK 508
278
279/* The vertex types. A DEADVERTEX has been deleted entirely. An */
280/* UNDEADVERTEX is not part of the mesh, but is written to the output */
281/* .node file and affects the node indexing in the other output files. */
282
283#define INPUTVERTEX 0
284#define SEGMENTVERTEX 1
285#define FREEVERTEX 2
286#define DEADVERTEX -32768
287#define UNDEADVERTEX -32767
288
289/* Two constants for algorithms based on random sampling. Both constants */
290/* have been chosen empirically to optimize their respective algorithms. */
291
292/* Used for the point location scheme of Mucke, Saias, and Zhu, to decide */
293/* how large a random sample of triangles to inspect. */
294
295#define SAMPLEFACTOR 11
296
297/* Used in Fortune's sweepline Delaunay algorithm to determine what fraction */
298/* of boundary edges should be maintained in the splay tree for point */
299/* location on the front. */
300
301#define SAMPLERATE 10
302
303/* A number that speaks for itself, every kissable digit. */
304
305#define PI 3.141592653589793238462643383279502884197169399375105820974944592308
306
307/* Another fave. */
308
309#define SQUAREROOTTWO 1.4142135623730950488016887242096980785696718753769480732
310
311/* And here's one for those of you who are intimidated by math. */
312
313#define ONETHIRD 0.333333333333333333333333333333333333333333333333333333333333
314
315#include <stdio.h>
316#include <stdlib.h>
317#include <string.h>
318#include <math.h>
319#ifndef NO_TIMER
320#include <sys/time.h>
321#endif /* not NO_TIMER */
322#ifdef CPU86
323#include <float.h>
324#endif /* CPU86 */
325#ifdef LINUX
326#include <fpu_control.h>
327#endif /* LINUX */
328#ifdef TRILIBRARY
329#include "triangle.h"
330#endif /* TRILIBRARY */
331
332/* A few forward declarations. */
333
334#ifndef TRILIBRARY
335char *readline();
336char *findfield();
337#endif /* not TRILIBRARY */
338
339/* Labels that signify the result of point location. The result of a */
340/* search indicates that the point falls in the interior of a triangle, on */
341/* an edge, on a vertex, or outside the mesh. */
342
344
345/* Labels that signify the result of vertex insertion. The result indicates */
346/* that the vertex was inserted with complete success, was inserted but */
347/* encroaches upon a subsegment, was not inserted because it lies on a */
348/* segment, or was not inserted because another vertex occupies the same */
349/* location. */
350
353
354/* Labels that signify the result of direction finding. The result */
355/* indicates that a segment connecting the two query points falls within */
356/* the direction triangle, along the left edge of the direction triangle, */
357/* or along the right edge of the direction triangle. */
358
360
361/*****************************************************************************/
362/* */
363/* The basic mesh data structures */
364/* */
365/* There are three: vertices, triangles, and subsegments (abbreviated */
366/* `subseg'). These three data structures, linked by pointers, comprise */
367/* the mesh. A vertex simply represents a mesh vertex and its properties. */
368/* A triangle is a triangle. A subsegment is a special data structure used */
369/* to represent an impenetrable edge of the mesh (perhaps on the outer */
370/* boundary, on the boundary of a hole, or part of an internal boundary */
371/* separating two triangulated regions). Subsegments represent boundaries, */
372/* defined by the user, that triangles may not lie across. */
373/* */
374/* A triangle consists of a list of three vertices, a list of three */
375/* adjoining triangles, a list of three adjoining subsegments (when */
376/* segments exist), an arbitrary number of optional user-defined */
377/* floating-point attributes, and an optional area constraint. The latter */
378/* is an upper bound on the permissible area of each triangle in a region, */
379/* used for mesh refinement. */
380/* */
381/* For a triangle on a boundary of the mesh, some or all of the neighboring */
382/* triangles may not be present. For a triangle in the interior of the */
383/* mesh, often no neighboring subsegments are present. Such absent */
384/* triangles and subsegments are never represented by NULL pointers; they */
385/* are represented by two special records: `dummytri', the triangle that */
386/* fills "outer space", and `dummysub', the omnipresent subsegment. */
387/* `dummytri' and `dummysub' are used for several reasons; for instance, */
388/* they can be dereferenced and their contents examined without violating */
389/* protected memory. */
390/* */
391/* However, it is important to understand that a triangle includes other */
392/* information as well. The pointers to adjoining vertices, triangles, and */
393/* subsegments are ordered in a way that indicates their geometric relation */
394/* to each other. Furthermore, each of these pointers contains orientation */
395/* information. Each pointer to an adjoining triangle indicates which face */
396/* of that triangle is contacted. Similarly, each pointer to an adjoining */
397/* subsegment indicates which side of that subsegment is contacted, and how */
398/* the subsegment is oriented relative to the triangle. */
399/* */
400/* The data structure representing a subsegment may be thought to be */
401/* abutting the edge of one or two triangle data structures: either */
402/* sandwiched between two triangles, or resting against one triangle on an */
403/* exterior boundary or hole boundary. */
404/* */
405/* A subsegment consists of a list of four vertices--the vertices of the */
406/* subsegment, and the vertices of the segment it is a part of--a list of */
407/* two adjoining subsegments, and a list of two adjoining triangles. One */
408/* of the two adjoining triangles may not be present (though there should */
409/* always be one), and neighboring subsegments might not be present. */
410/* Subsegments also store a user-defined integer "boundary marker". */
411/* Typically, this integer is used to indicate what boundary conditions are */
412/* to be applied at that location in a finite element simulation. */
413/* */
414/* Like triangles, subsegments maintain information about the relative */
415/* orientation of neighboring objects. */
416/* */
417/* Vertices are relatively simple. A vertex is a list of floating-point */
418/* numbers, starting with the x, and y coordinates, followed by an */
419/* arbitrary number of optional user-defined floating-point attributes, */
420/* followed by an integer boundary marker. During the segment insertion */
421/* phase, there is also a pointer from each vertex to a triangle that may */
422/* contain it. Each pointer is not always correct, but when one is, it */
423/* speeds up segment insertion. These pointers are assigned values once */
424/* at the beginning of the segment insertion phase, and are not used or */
425/* updated except during this phase. Edge flipping during segment */
426/* insertion will render some of them incorrect. Hence, don't rely upon */
427/* them for anything. */
428/* */
429/* Other than the exception mentioned above, vertices have no information */
430/* about what triangles, subfacets, or subsegments they are linked to. */
431/* */
432/*****************************************************************************/
433
434/*****************************************************************************/
435/* */
436/* Handles */
437/* */
438/* The oriented triangle (`otri') and oriented subsegment (`osub') data */
439/* structures defined below do not themselves store any part of the mesh. */
440/* The mesh itself is made of `triangle's, `subseg's, and `vertex's. */
441/* */
442/* Oriented triangles and oriented subsegments will usually be referred to */
443/* as "handles." A handle is essentially a pointer into the mesh; it */
444/* allows you to "hold" one particular part of the mesh. Handles are used */
445/* to specify the regions in which one is traversing and modifying the mesh.*/
446/* A single `triangle' may be held by many handles, or none at all. (The */
447/* latter case is not a memory leak, because the triangle is still */
448/* connected to other triangles in the mesh.) */
449/* */
450/* An `otri' is a handle that holds a triangle. It holds a specific edge */
451/* of the triangle. An `osub' is a handle that holds a subsegment. It */
452/* holds either the left or right side of the subsegment. */
453/* */
454/* Navigation about the mesh is accomplished through a set of mesh */
455/* manipulation primitives, further below. Many of these primitives take */
456/* a handle and produce a new handle that holds the mesh near the first */
457/* handle. Other primitives take two handles and glue the corresponding */
458/* parts of the mesh together. The orientation of the handles is */
459/* important. For instance, when two triangles are glued together by the */
460/* bond() primitive, they are glued at the edges on which the handles lie. */
461/* */
462/* Because vertices have no information about which triangles they are */
463/* attached to, I commonly represent a vertex by use of a handle whose */
464/* origin is the vertex. A single handle can simultaneously represent a */
465/* triangle, an edge, and a vertex. */
466/* */
467/*****************************************************************************/
468
469/* The triangle data structure. Each triangle contains three pointers to */
470/* adjoining triangles, plus three pointers to vertices, plus three */
471/* pointers to subsegments (declared below; these pointers are usually */
472/* `dummysub'). It may or may not also contain user-defined attributes */
473/* and/or a floating-point "area constraint." It may also contain extra */
474/* pointers for nodes, when the user asks for high-order elements. */
475/* Because the size and structure of a `triangle' is not decided until */
476/* runtime, I haven't simply declared the type `triangle' as a struct. */
477
478typedef REAL **triangle; /* Really: typedef triangle *triangle */
479
480/* An oriented triangle: includes a pointer to a triangle and orientation. */
481/* The orientation denotes an edge of the triangle. Hence, there are */
482/* three possible orientations. By convention, each edge always points */
483/* counterclockwise about the corresponding triangle. */
484
485struct otri {
486 triangle *tri;
487 int orient; /* Ranges from 0 to 2. */
488};
489
490/* The subsegment data structure. Each subsegment contains two pointers to */
491/* adjoining subsegments, plus four pointers to vertices, plus two */
492/* pointers to adjoining triangles, plus one boundary marker, plus one */
493/* segment number. */
494
495typedef REAL **subseg; /* Really: typedef subseg *subseg */
496
497/* An oriented subsegment: includes a pointer to a subsegment and an */
498/* orientation. The orientation denotes a side of the edge. Hence, there */
499/* are two possible orientations. By convention, the edge is always */
500/* directed so that the "side" denoted is the right side of the edge. */
501
502struct osub {
503 subseg *ss;
504 int ssorient; /* Ranges from 0 to 1. */
505};
506
507/* The vertex data structure. Each vertex is actually an array of REALs. */
508/* The number of REALs is unknown until runtime. An integer boundary */
509/* marker, and sometimes a pointer to a triangle, is appended after the */
510/* REALs. */
511
512typedef REAL *vertex;
513
514/* A queue used to store encroached subsegments. Each subsegment's vertices */
515/* are stored so that we can check whether a subsegment is still the same. */
516
517struct badsubseg {
518 subseg encsubseg; /* An encroached subsegment. */
519 vertex subsegorg, subsegdest; /* Its two vertices. */
520};
521
522/* A queue used to store bad triangles. The key is the square of the cosine */
523/* of the smallest angle of the triangle. Each triangle's vertices are */
524/* stored so that one can check whether a triangle is still the same. */
525
526struct badtriang {
527 triangle poortri; /* A skinny or too-large triangle. */
528 REAL key; /* cos^2 of smallest (apical) angle. */
529 vertex triangorg, triangdest, triangapex; /* Its three vertices. */
530 struct badtriang *nexttriang; /* Pointer to next bad triangle. */
531};
532
533/* A stack of triangles flipped during the most recent vertex insertion. */
534/* The stack is used to undo the vertex insertion if the vertex encroaches */
535/* upon a subsegment. */
536
537struct flipstacker {
538 triangle flippedtri; /* A recently flipped triangle. */
539 struct flipstacker *prevflip; /* Previous flip in the stack. */
540};
541
542/* A node in a heap used to store events for the sweepline Delaunay */
543/* algorithm. Nodes do not point directly to their parents or children in */
544/* the heap. Instead, each node knows its position in the heap, and can */
545/* look up its parent and children in a separate array. The `eventptr' */
546/* points either to a `vertex' or to a triangle (in encoded format, so */
547/* that an orientation is included). In the latter case, the origin of */
548/* the oriented triangle is the apex of a "circle event" of the sweepline */
549/* algorithm. To distinguish site events from circle events, all circle */
550/* events are given an invalid (smaller than `xmin') x-coordinate `xkey'. */
551
552struct event {
553 REAL xkey, ykey; /* Coordinates of the event. */
554 VOID *eventptr; /* Can be a vertex or the location of a circle event. */
555 int heapposition; /* Marks this event's position in the heap. */
556};
557
558/* A node in the splay tree. Each node holds an oriented ghost triangle */
559/* that represents a boundary edge of the growing triangulation. When a */
560/* circle event covers two boundary edges with a triangle, so that they */
561/* are no longer boundary edges, those edges are not immediately deleted */
562/* from the tree; rather, they are lazily deleted when they are next */
563/* encountered. (Since only a random sample of boundary edges are kept */
564/* in the tree, lazy deletion is faster.) `keydest' is used to verify */
565/* that a triangle is still the same as when it entered the splay tree; if */
566/* it has been rotated (due to a circle event), it no longer represents a */
567/* boundary edge and should be deleted. */
568
569struct splaynode {
570 struct otri keyedge; /* Lprev of an edge on the front. */
571 vertex keydest; /* Used to verify that splay node is still live. */
572 struct splaynode *lchild, *rchild; /* Children in splay tree. */
573};
574
575/* A type used to allocate memory. firstblock is the first block of items. */
576/* nowblock is the block from which items are currently being allocated. */
577/* nextitem points to the next slab of free memory for an item. */
578/* deaditemstack is the head of a linked list (stack) of deallocated items */
579/* that can be recycled. unallocateditems is the number of items that */
580/* remain to be allocated from nowblock. */
581/* */
582/* Traversal is the process of walking through the entire list of items, and */
583/* is separate from allocation. Note that a traversal will visit items on */
584/* the "deaditemstack" stack as well as live items. pathblock points to */
585/* the block currently being traversed. pathitem points to the next item */
586/* to be traversed. pathitemsleft is the number of items that remain to */
587/* be traversed in pathblock. */
588/* */
589/* alignbytes determines how new records should be aligned in memory. */
590/* itembytes is the length of a record in bytes (after rounding up). */
591/* itemsperblock is the number of items allocated at once in a single */
592/* block. itemsfirstblock is the number of items in the first block, */
593/* which can vary from the others. items is the number of currently */
594/* allocated items. maxitems is the maximum number of items that have */
595/* been allocated at once; it is the current number of items plus the */
596/* number of records kept on deaditemstack. */
597
598struct memorypool {
599 VOID **firstblock, **nowblock;
600 VOID *nextitem;
601 VOID *deaditemstack;
602 VOID **pathblock;
603 VOID *pathitem;
604 int alignbytes;
605 int itembytes;
606 int itemsperblock;
607 int itemsfirstblock;
608 long items, maxitems;
609 int unallocateditems;
610 int pathitemsleft;
611};
612
613
614/* Global constants. */
615
616REAL splitter; /* Used to split REAL factors for exact multiplication. */
617REAL epsilon; /* Floating-point machine epsilon. */
622
623/* Random number seed is not constant, but I've made it global anyway. */
624
625unsigned long randomseed; /* Current random number seed. */
626
627
628/* Mesh data structure. Triangle operates on only one mesh, but the mesh */
629/* structure is used (instead of global variables) to allow reentrancy. */
630
631struct mesh {
632
633/* Variables used to allocate memory for triangles, subsegments, vertices, */
634/* viri (triangles being eaten), encroached segments, bad (skinny or too */
635/* large) triangles, and splay tree nodes. */
636
637 struct memorypool triangles;
638 struct memorypool subsegs;
639 struct memorypool vertices;
640 struct memorypool viri;
641 struct memorypool badsubsegs;
642 struct memorypool badtriangles;
643 struct memorypool flipstackers;
644 struct memorypool splaynodes;
645
646/* Variables that maintain the bad triangle queues. The queues are */
647/* ordered from 4095 (highest priority) to 0 (lowest priority). */
648
649 struct badtriang *queuefront[4096];
650 struct badtriang *queuetail[4096];
651 int nextnonemptyq[4096];
652 int firstnonemptyq;
653
654/* Variable that maintains the stack of recently flipped triangles. */
655
656 struct flipstacker *lastflip;
657
658/* Other variables. */
659
660 REAL xmin, xmax, ymin, ymax; /* x and y bounds. */
661 REAL xminextreme; /* Nonexistent x value used as a flag in sweepline. */
662 int invertices; /* Number of input vertices. */
663 int inelements; /* Number of input triangles. */
664 int insegments; /* Number of input segments. */
665 int holes; /* Number of input holes. */
666 int regions; /* Number of input regions. */
667 int undeads; /* Number of input vertices that don't appear in the mesh. */
668 long edges; /* Number of output edges. */
669 int mesh_dim; /* Dimension (ought to be 2). */
670 int nextras; /* Number of attributes per vertex. */
671 int eextras; /* Number of attributes per triangle. */
672 long hullsize; /* Number of edges in convex hull. */
673 int steinerleft; /* Number of Steiner points not yet used. */
674 int vertexmarkindex; /* Index to find boundary marker of a vertex. */
675 int vertex2triindex; /* Index to find a triangle adjacent to a vertex. */
676 int highorderindex; /* Index to find extra nodes for high-order elements. */
677 int elemattribindex; /* Index to find attributes of a triangle. */
678 int areaboundindex; /* Index to find area bound of a triangle. */
679 int checksegments; /* Are there segments in the triangulation yet? */
680 int checkquality; /* Has quality triangulation begun yet? */
681 int readnodefile; /* Has a .node file been read? */
682 long samples; /* Number of random samples for point location. */
683
684 long incirclecount; /* Number of incircle tests performed. */
685 long counterclockcount; /* Number of counterclockwise tests performed. */
686 long orient3dcount; /* Number of 3D orientation tests performed. */
687 long hyperbolacount; /* Number of right-of-hyperbola tests performed. */
688 long circumcentercount; /* Number of circumcenter calculations performed. */
689 long circletopcount; /* Number of circle top calculations performed. */
690
691/* Triangular bounding box vertices. */
692
693 vertex infvertex1, infvertex2, infvertex3;
694
695/* Pointer to the `triangle' that occupies all of "outer space." */
696
697 triangle *dummytri;
698 triangle *dummytribase; /* Keep base address so we can free() it later. */
699
700/* Pointer to the omnipresent subsegment. Referenced by any triangle or */
701/* subsegment that isn't really connected to a subsegment at that */
702/* location. */
703
704 subseg *dummysub;
705 subseg *dummysubbase; /* Keep base address so we can free() it later. */
706
707/* Pointer to a recently visited triangle. Improves point location if */
708/* proximate vertices are inserted sequentially. */
709
710 struct otri recenttri;
711
712}; /* End of `struct mesh'. */
713
714
715/* Data structure for command line switches and file names. This structure */
716/* is used (instead of global variables) to allow reentrancy. */
717
718struct behavior {
719
720/* Switches for the triangulator. */
721/* poly: -p switch. refine: -r switch. */
722/* quality: -q switch. */
723/* minangle: minimum angle bound, specified after -q switch. */
724/* goodangle: cosine squared of minangle. */
725/* offconstant: constant used to place off-center Steiner points. */
726/* vararea: -a switch without number. */
727/* fixedarea: -a switch with number. */
728/* maxarea: maximum area bound, specified after -a switch. */
729/* usertest: -u switch. */
730/* regionattrib: -A switch. convex: -c switch. */
731/* weighted: 1 for -w switch, 2 for -W switch. jettison: -j switch */
732/* firstnumber: inverse of -z switch. All items are numbered starting */
733/* from `firstnumber'. */
734/* edgesout: -e switch. voronoi: -v switch. */
735/* neighbors: -n switch. geomview: -g switch. */
736/* nobound: -B switch. nopolywritten: -P switch. */
737/* nonodewritten: -N switch. noelewritten: -E switch. */
738/* noiterationnum: -I switch. noholes: -O switch. */
739/* noexact: -X switch. */
740/* order: element order, specified after -o switch. */
741/* nobisect: count of how often -Y switch is selected. */
742/* steiner: maximum number of Steiner points, specified after -S switch. */
743/* incremental: -i switch. sweepline: -F switch. */
744/* dwyer: inverse of -l switch. */
745/* splitseg: -s switch. */
746/* conformdel: -D switch. docheck: -C switch. */
747/* quiet: -Q switch. verbose: count of how often -V switch is selected. */
748/* usesegments: -p, -r, -q, or -c switch; determines whether segments are */
749/* used at all. */
750/* */
751/* Read the instructions to find out the meaning of these switches. */
752
753 int poly, refine, quality, vararea, fixedarea, usertest;
754 int regionattrib, convex, weighted, jettison;
755 int firstnumber;
756 int edgesout, voronoi, neighbors, geomview;
757 int nobound, nopolywritten, nonodewritten, noelewritten, noiterationnum;
758 int noholes, noexact, conformdel;
759 int incremental, sweepline, dwyer;
760 int splitseg;
761 int docheck;
762 int quiet, verbose;
763 int usesegments;
764 int order;
765 int nobisect;
766 int steiner;
767 REAL minangle, goodangle, offconstant;
768 REAL maxarea;
769
770/* Variables for file names. */
771
772#ifndef TRILIBRARY
773 char innodefilename[FILENAMESIZE];
774 char inelefilename[FILENAMESIZE];
775 char inpolyfilename[FILENAMESIZE];
776 char areafilename[FILENAMESIZE];
777 char outnodefilename[FILENAMESIZE];
778 char outelefilename[FILENAMESIZE];
779 char outpolyfilename[FILENAMESIZE];
780 char edgefilename[FILENAMESIZE];
781 char vnodefilename[FILENAMESIZE];
782 char vedgefilename[FILENAMESIZE];
783 char neighborfilename[FILENAMESIZE];
784 char offfilename[FILENAMESIZE];
785#endif /* not TRILIBRARY */
786
787}; /* End of `struct behavior'. */
788
789
790/*****************************************************************************/
791/* */
792/* Mesh manipulation primitives. Each triangle contains three pointers to */
793/* other triangles, with orientations. Each pointer points not to the */
794/* first byte of a triangle, but to one of the first three bytes of a */
795/* triangle. It is necessary to extract both the triangle itself and the */
796/* orientation. To save memory, I keep both pieces of information in one */
797/* pointer. To make this possible, I assume that all triangles are aligned */
798/* to four-byte boundaries. The decode() routine below decodes a pointer, */
799/* extracting an orientation (in the range 0 to 2) and a pointer to the */
800/* beginning of a triangle. The encode() routine compresses a pointer to a */
801/* triangle and an orientation into a single pointer. My assumptions that */
802/* triangles are four-byte-aligned and that the `unsigned long' type is */
803/* long enough to hold a pointer are two of the few kludges in this program.*/
804/* */
805/* Subsegments are manipulated similarly. A pointer to a subsegment */
806/* carries both an address and an orientation in the range 0 to 1. */
807/* */
808/* The other primitives take an oriented triangle or oriented subsegment, */
809/* and return an oriented triangle or oriented subsegment or vertex; or */
810/* they change the connections in the data structure. */
811/* */
812/* Below, triangles and subsegments are denoted by their vertices. The */
813/* triangle abc has origin (org) a, destination (dest) b, and apex (apex) */
814/* c. These vertices occur in counterclockwise order about the triangle. */
815/* The handle abc may simultaneously denote vertex a, edge ab, and triangle */
816/* abc. */
817/* */
818/* Similarly, the subsegment ab has origin (sorg) a and destination (sdest) */
819/* b. If ab is thought to be directed upward (with b directly above a), */
820/* then the handle ab is thought to grasp the right side of ab, and may */
821/* simultaneously denote vertex a and edge ab. */
822/* */
823/* An asterisk (*) denotes a vertex whose identity is unknown. */
824/* */
825/* Given this notation, a partial list of mesh manipulation primitives */
826/* follows. */
827/* */
828/* */
829/* For triangles: */
830/* */
831/* sym: Find the abutting triangle; same edge. */
832/* sym(abc) -> ba* */
833/* */
834/* lnext: Find the next edge (counterclockwise) of a triangle. */
835/* lnext(abc) -> bca */
836/* */
837/* lprev: Find the previous edge (clockwise) of a triangle. */
838/* lprev(abc) -> cab */
839/* */
840/* onext: Find the next edge counterclockwise with the same origin. */
841/* onext(abc) -> ac* */
842/* */
843/* oprev: Find the next edge clockwise with the same origin. */
844/* oprev(abc) -> a*b */
845/* */
846/* dnext: Find the next edge counterclockwise with the same destination. */
847/* dnext(abc) -> *ba */
848/* */
849/* dprev: Find the next edge clockwise with the same destination. */
850/* dprev(abc) -> cb* */
851/* */
852/* rnext: Find the next edge (counterclockwise) of the adjacent triangle. */
853/* rnext(abc) -> *a* */
854/* */
855/* rprev: Find the previous edge (clockwise) of the adjacent triangle. */
856/* rprev(abc) -> b** */
857/* */
858/* org: Origin dest: Destination apex: Apex */
859/* org(abc) -> a dest(abc) -> b apex(abc) -> c */
860/* */
861/* bond: Bond two triangles together at the resepective handles. */
862/* bond(abc, bad) */
863/* */
864/* */
865/* For subsegments: */
866/* */
867/* ssym: Reverse the orientation of a subsegment. */
868/* ssym(ab) -> ba */
869/* */
870/* spivot: Find adjoining subsegment with the same origin. */
871/* spivot(ab) -> a* */
872/* */
873/* snext: Find next subsegment in sequence. */
874/* snext(ab) -> b* */
875/* */
876/* sorg: Origin sdest: Destination */
877/* sorg(ab) -> a sdest(ab) -> b */
878/* */
879/* sbond: Bond two subsegments together at the respective origins. */
880/* sbond(ab, ac) */
881/* */
882/* */
883/* For interacting tetrahedra and subfacets: */
884/* */
885/* tspivot: Find a subsegment abutting a triangle. */
886/* tspivot(abc) -> ba */
887/* */
888/* stpivot: Find a triangle abutting a subsegment. */
889/* stpivot(ab) -> ba* */
890/* */
891/* tsbond: Bond a triangle to a subsegment. */
892/* tsbond(abc, ba) */
893/* */
894/*****************************************************************************/
895
896/********* Mesh manipulation primitives begin here *********/
897/** **/
898/** **/
899
900/* Fast lookup arrays to speed some of the mesh manipulation primitives. */
901
902int plus1mod3[3] = {1, 2, 0};
903int minus1mod3[3] = {2, 0, 1};
904
905/********* Primitives for triangles *********/
906/* */
907/* */
908
909/* decode() converts a pointer to an oriented triangle. The orientation is */
910/* extracted from the two least significant bits of the pointer. */
911
912#define decode(ptr, otri) \
913 (otri).orient = (int) ((unsigned long) (ptr) & (unsigned long) 3l); \
914 (otri).tri = (triangle *) \
915 ((unsigned long) (ptr) ^ (unsigned long) (otri).orient)
916
917/* encode() compresses an oriented triangle into a single pointer. It */
918/* relies on the assumption that all triangles are aligned to four-byte */
919/* boundaries, so the two least significant bits of (otri).tri are zero. */
920
921#define encode(otri) \
922 (triangle) ((unsigned long) (otri).tri | (unsigned long) (otri).orient)
923
924/* The following handle manipulation primitives are all described by Guibas */
925/* and Stolfi. However, Guibas and Stolfi use an edge-based data */
926/* structure, whereas I use a triangle-based data structure. */
927
928/* sym() finds the abutting triangle, on the same edge. Note that the edge */
929/* direction is necessarily reversed, because the handle specified by an */
930/* oriented triangle is directed counterclockwise around the triangle. */
931
932#define sym(otri1, otri2) \
933 ptr = (otri1).tri[(otri1).orient]; \
934 decode(ptr, otri2);
935
936#define symself(otri) \
937 ptr = (otri).tri[(otri).orient]; \
938 decode(ptr, otri);
939
940/* lnext() finds the next edge (counterclockwise) of a triangle. */
941
942#define lnext(otri1, otri2) \
943 (otri2).tri = (otri1).tri; \
944 (otri2).orient = plus1mod3[(otri1).orient]
945
946#define lnextself(otri) \
947 (otri).orient = plus1mod3[(otri).orient]
948
949/* lprev() finds the previous edge (clockwise) of a triangle. */
950
951#define lprev(otri1, otri2) \
952 (otri2).tri = (otri1).tri; \
953 (otri2).orient = minus1mod3[(otri1).orient]
954
955#define lprevself(otri) \
956 (otri).orient = minus1mod3[(otri).orient]
957
958/* onext() spins counterclockwise around a vertex; that is, it finds the */
959/* next edge with the same origin in the counterclockwise direction. This */
960/* edge is part of a different triangle. */
961
962#define onext(otri1, otri2) \
963 lprev(otri1, otri2); \
964 symself(otri2);
965
966#define onextself(otri) \
967 lprevself(otri); \
968 symself(otri);
969
970/* oprev() spins clockwise around a vertex; that is, it finds the next edge */
971/* with the same origin in the clockwise direction. This edge is part of */
972/* a different triangle. */
973
974#define oprev(otri1, otri2) \
975 sym(otri1, otri2); \
976 lnextself(otri2);
977
978#define oprevself(otri) \
979 symself(otri); \
980 lnextself(otri);
981
982/* dnext() spins counterclockwise around a vertex; that is, it finds the */
983/* next edge with the same destination in the counterclockwise direction. */
984/* This edge is part of a different triangle. */
985
986#define dnext(otri1, otri2) \
987 sym(otri1, otri2); \
988 lprevself(otri2);
989
990#define dnextself(otri) \
991 symself(otri); \
992 lprevself(otri);
993
994/* dprev() spins clockwise around a vertex; that is, it finds the next edge */
995/* with the same destination in the clockwise direction. This edge is */
996/* part of a different triangle. */
997
998#define dprev(otri1, otri2) \
999 lnext(otri1, otri2); \
1000 symself(otri2);
1001
1002#define dprevself(otri) \
1003 lnextself(otri); \
1004 symself(otri);
1005
1006/* rnext() moves one edge counterclockwise about the adjacent triangle. */
1007/* (It's best understood by reading Guibas and Stolfi. It involves */
1008/* changing triangles twice.) */
1009
1010#define rnext(otri1, otri2) \
1011 sym(otri1, otri2); \
1012 lnextself(otri2); \
1013 symself(otri2);
1014
1015#define rnextself(otri) \
1016 symself(otri); \
1017 lnextself(otri); \
1018 symself(otri);
1019
1020/* rprev() moves one edge clockwise about the adjacent triangle. */
1021/* (It's best understood by reading Guibas and Stolfi. It involves */
1022/* changing triangles twice.) */
1023
1024#define rprev(otri1, otri2) \
1025 sym(otri1, otri2); \
1026 lprevself(otri2); \
1027 symself(otri2);
1028
1029#define rprevself(otri) \
1030 symself(otri); \
1031 lprevself(otri); \
1032 symself(otri);
1033
1034/* These primitives determine or set the origin, destination, or apex of a */
1035/* triangle. */
1036
1037#define org(otri, vertexptr) \
1038 vertexptr = (vertex) (otri).tri[plus1mod3[(otri).orient] + 3]
1039
1040#define dest(otri, vertexptr) \
1041 vertexptr = (vertex) (otri).tri[minus1mod3[(otri).orient] + 3]
1042
1043#define apex(otri, vertexptr) \
1044 vertexptr = (vertex) (otri).tri[(otri).orient + 3]
1045
1046#define setorg(otri, vertexptr) \
1047 (otri).tri[plus1mod3[(otri).orient] + 3] = (triangle) vertexptr
1048
1049#define setdest(otri, vertexptr) \
1050 (otri).tri[minus1mod3[(otri).orient] + 3] = (triangle) vertexptr
1051
1052#define setapex(otri, vertexptr) \
1053 (otri).tri[(otri).orient + 3] = (triangle) vertexptr
1054
1055/* Bond two triangles together. */
1056
1057#define bond(otri1, otri2) \
1058 (otri1).tri[(otri1).orient] = encode(otri2); \
1059 (otri2).tri[(otri2).orient] = encode(otri1)
1060
1061/* Dissolve a bond (from one side). Note that the other triangle will still */
1062/* think it's connected to this triangle. Usually, however, the other */
1063/* triangle is being deleted entirely, or bonded to another triangle, so */
1064/* it doesn't matter. */
1065
1066#define dissolve(otri) \
1067 (otri).tri[(otri).orient] = (triangle) m->dummytri
1068
1069/* Copy an oriented triangle. */
1070
1071#define otricopy(otri1, otri2) \
1072 (otri2).tri = (otri1).tri; \
1073 (otri2).orient = (otri1).orient
1074
1075/* Test for equality of oriented triangles. */
1076
1077#define otriequal(otri1, otri2) \
1078 (((otri1).tri == (otri2).tri) && \
1079 ((otri1).orient == (otri2).orient))
1080
1081/* Primitives to infect or cure a triangle with the virus. These rely on */
1082/* the assumption that all subsegments are aligned to four-byte boundaries.*/
1083
1084#define infect(otri) \
1085 (otri).tri[6] = (triangle) \
1086 ((unsigned long) (otri).tri[6] | (unsigned long) 2l)
1087
1088#define uninfect(otri) \
1089 (otri).tri[6] = (triangle) \
1090 ((unsigned long) (otri).tri[6] & ~ (unsigned long) 2l)
1091
1092/* Test a triangle for viral infection. */
1093
1094#define infected(otri) \
1095 (((unsigned long) (otri).tri[6] & (unsigned long) 2l) != 0l)
1096
1097/* Check or set a triangle's attributes. */
1098
1099#define elemattribute(otri, attnum) \
1100 ((REAL *) (otri).tri)[m->elemattribindex + (attnum)]
1101
1102#define setelemattribute(otri, attnum, value) \
1103 ((REAL *) (otri).tri)[m->elemattribindex + (attnum)] = value
1104
1105/* Check or set a triangle's maximum area bound. */
1106
1107#define areabound(otri) ((REAL *) (otri).tri)[m->areaboundindex]
1108
1109#define setareabound(otri, value) \
1110 ((REAL *) (otri).tri)[m->areaboundindex] = value
1111
1112/* Check or set a triangle's deallocation. Its second pointer is set to */
1113/* NULL to indicate that it is not allocated. (Its first pointer is used */
1114/* for the stack of dead items.) Its fourth pointer (its first vertex) */
1115/* is set to NULL in case a `badtriang' structure points to it. */
1116
1117#define deadtri(tria) ((tria)[1] == (triangle) NULL)
1118
1119#define killtri(tria) \
1120 (tria)[1] = (triangle) NULL; \
1121 (tria)[3] = (triangle) NULL
1122
1123/********* Primitives for subsegments *********/
1124/* */
1125/* */
1126
1127/* sdecode() converts a pointer to an oriented subsegment. The orientation */
1128/* is extracted from the least significant bit of the pointer. The two */
1129/* least significant bits (one for orientation, one for viral infection) */
1130/* are masked out to produce the real pointer. */
1131
1132#define sdecode(sptr, osub) \
1133 (osub).ssorient = (int) ((unsigned long) (sptr) & (unsigned long) 1l); \
1134 (osub).ss = (subseg *) \
1135 ((unsigned long) (sptr) & ~ (unsigned long) 3l)
1136
1137/* sencode() compresses an oriented subsegment into a single pointer. It */
1138/* relies on the assumption that all subsegments are aligned to two-byte */
1139/* boundaries, so the least significant bit of (osub).ss is zero. */
1140
1141#define sencode(osub) \
1142 (subseg) ((unsigned long) (osub).ss | (unsigned long) (osub).ssorient)
1143
1144/* ssym() toggles the orientation of a subsegment. */
1145
1146#define ssym(osub1, osub2) \
1147 (osub2).ss = (osub1).ss; \
1148 (osub2).ssorient = 1 - (osub1).ssorient
1149
1150#define ssymself(osub) \
1151 (osub).ssorient = 1 - (osub).ssorient
1152
1153/* spivot() finds the other subsegment (from the same segment) that shares */
1154/* the same origin. */
1155
1156#define spivot(osub1, osub2) \
1157 sptr = (osub1).ss[(osub1).ssorient]; \
1158 sdecode(sptr, osub2)
1159
1160#define spivotself(osub) \
1161 sptr = (osub).ss[(osub).ssorient]; \
1162 sdecode(sptr, osub)
1163
1164/* snext() finds the next subsegment (from the same segment) in sequence; */
1165/* one whose origin is the input subsegment's destination. */
1166
1167#define snext(osub1, osub2) \
1168 sptr = (osub1).ss[1 - (osub1).ssorient]; \
1169 sdecode(sptr, osub2)
1170
1171#define snextself(osub) \
1172 sptr = (osub).ss[1 - (osub).ssorient]; \
1173 sdecode(sptr, osub)
1174
1175/* These primitives determine or set the origin or destination of a */
1176/* subsegment or the segment that includes it. */
1177
1178#define sorg(osub, vertexptr) \
1179 vertexptr = (vertex) (osub).ss[2 + (osub).ssorient]
1180
1181#define sdest(osub, vertexptr) \
1182 vertexptr = (vertex) (osub).ss[3 - (osub).ssorient]
1183
1184#define setsorg(osub, vertexptr) \
1185 (osub).ss[2 + (osub).ssorient] = (subseg) vertexptr
1186
1187#define setsdest(osub, vertexptr) \
1188 (osub).ss[3 - (osub).ssorient] = (subseg) vertexptr
1189
1190#define segorg(osub, vertexptr) \
1191 vertexptr = (vertex) (osub).ss[4 + (osub).ssorient]
1192
1193#define segdest(osub, vertexptr) \
1194 vertexptr = (vertex) (osub).ss[5 - (osub).ssorient]
1195
1196#define setsegorg(osub, vertexptr) \
1197 (osub).ss[4 + (osub).ssorient] = (subseg) vertexptr
1198
1199#define setsegdest(osub, vertexptr) \
1200 (osub).ss[5 - (osub).ssorient] = (subseg) vertexptr
1201
1202/* These primitives read or set a boundary marker. Boundary markers are */
1203/* used to hold user-defined tags for setting boundary conditions in */
1204/* finite element solvers. */
1205
1206#define mark(osub) (* (int *) ((osub).ss + 8))
1207
1208#define setmark(osub, value) \
1209 * (int *) ((osub).ss + 8) = value
1210
1211/* Bond two subsegments together. */
1212
1213#define sbond(osub1, osub2) \
1214 (osub1).ss[(osub1).ssorient] = sencode(osub2); \
1215 (osub2).ss[(osub2).ssorient] = sencode(osub1)
1216
1217/* Dissolve a subsegment bond (from one side). Note that the other */
1218/* subsegment will still think it's connected to this subsegment. */
1219
1220#define sdissolve(osub) \
1221 (osub).ss[(osub).ssorient] = (subseg) m->dummysub
1222
1223/* Copy a subsegment. */
1224
1225#define subsegcopy(osub1, osub2) \
1226 (osub2).ss = (osub1).ss; \
1227 (osub2).ssorient = (osub1).ssorient
1228
1229/* Test for equality of subsegments. */
1230
1231#define subsegequal(osub1, osub2) \
1232 (((osub1).ss == (osub2).ss) && \
1233 ((osub1).ssorient == (osub2).ssorient))
1234
1235/* Check or set a subsegment's deallocation. Its second pointer is set to */
1236/* NULL to indicate that it is not allocated. (Its first pointer is used */
1237/* for the stack of dead items.) Its third pointer (its first vertex) */
1238/* is set to NULL in case a `badsubseg' structure points to it. */
1239
1240#define deadsubseg(sub) ((sub)[1] == (subseg) NULL)
1241
1242#define killsubseg(sub) \
1243 (sub)[1] = (subseg) NULL; \
1244 (sub)[2] = (subseg) NULL
1245
1246/********* Primitives for interacting triangles and subsegments *********/
1247/* */
1248/* */
1249
1250/* tspivot() finds a subsegment abutting a triangle. */
1251
1252#define tspivot(otri, osub) \
1253 sptr = (subseg) (otri).tri[6 + (otri).orient]; \
1254 sdecode(sptr, osub)
1255
1256/* stpivot() finds a triangle abutting a subsegment. It requires that the */
1257/* variable `ptr' of type `triangle' be defined. */
1258
1259#define stpivot(osub, otri) \
1260 ptr = (triangle) (osub).ss[6 + (osub).ssorient]; \
1261 decode(ptr, otri)
1262
1263/* Bond a triangle to a subsegment. */
1264
1265#define tsbond(otri, osub) \
1266 (otri).tri[6 + (otri).orient] = (triangle) sencode(osub); \
1267 (osub).ss[6 + (osub).ssorient] = (subseg) encode(otri)
1268
1269/* Dissolve a bond (from the triangle side). */
1270
1271#define tsdissolve(otri) \
1272 (otri).tri[6 + (otri).orient] = (triangle) m->dummysub
1273
1274/* Dissolve a bond (from the subsegment side). */
1275
1276#define stdissolve(osub) \
1277 (osub).ss[6 + (osub).ssorient] = (subseg) m->dummytri
1278
1279/********* Primitives for vertices *********/
1280/* */
1281/* */
1282
1283#define vertexmark(vx) ((int *) (vx))[m->vertexmarkindex]
1284
1285#define setvertexmark(vx, value) \
1286 ((int *) (vx))[m->vertexmarkindex] = value
1287
1288#define vertextype(vx) ((int *) (vx))[m->vertexmarkindex + 1]
1289
1290#define setvertextype(vx, value) \
1291 ((int *) (vx))[m->vertexmarkindex + 1] = value
1292
1293#define vertex2tri(vx) ((triangle *) (vx))[m->vertex2triindex]
1294
1295#define setvertex2tri(vx, value) \
1296 ((triangle *) (vx))[m->vertex2triindex] = value
1297
1298/** **/
1299/** **/
1300/********* Mesh manipulation primitives end here *********/
1301
1302/********* User-defined triangle evaluation routine begins here *********/
1303/** **/
1304/** **/
1305
1306/*****************************************************************************/
1307/* */
1308/* triunsuitable() Determine if a triangle is unsuitable, and thus must */
1309/* be further refined. */
1310/* */
1311/* You may write your own procedure that decides whether or not a selected */
1312/* triangle is too big (and needs to be refined). There are two ways to do */
1313/* this. */
1314/* */
1315/* (1) Modify the procedure `triunsuitable' below, then recompile */
1316/* Triangle. */
1317/* */
1318/* (2) Define the symbol EXTERNAL_TEST (either by adding the definition */
1319/* to this file, or by using the appropriate compiler switch). This way, */
1320/* you can compile triangle.c separately from your test. Write your own */
1321/* `triunsuitable' procedure in a separate C file (using the same prototype */
1322/* as below). Compile it and link the object code with triangle.o. */
1323/* */
1324/* This procedure returns 1 if the triangle is too large and should be */
1325/* refined; 0 otherwise. */
1326/* */
1327/*****************************************************************************/
1328
1329#ifdef EXTERNAL_TEST
1330
1331int triunsuitable();
1332
1333#else /* not EXTERNAL_TEST */
1334
1335#ifdef ANSI_DECLARATORS
1336int triunsuitable(vertex triorg, vertex tridest, vertex triapex, REAL area )
1337#else /* not ANSI_DECLARATORS */
1338int triunsuitable(triorg, tridest, triapex, area)
1339vertex triorg; /* The triangle's origin vertex. */
1340vertex tridest; /* The triangle's destination vertex. */
1341vertex triapex; /* The triangle's apex vertex. */
1342REAL area; /* The area of the triangle. */
1343#endif /* not ANSI_DECLARATORS */
1344
1345{
1346 REAL dxoa, dxda, dxod;
1347 REAL dyoa, dyda, dyod;
1348 REAL oalen, dalen, odlen;
1349 REAL maxlen;
1350
1351 (void)area; /*LM: added to suppress warning */
1352
1353 dxoa = triorg[0] - triapex[0];
1354 dyoa = triorg[1] - triapex[1];
1355 dxda = tridest[0] - triapex[0];
1356 dyda = tridest[1] - triapex[1];
1357 dxod = triorg[0] - tridest[0];
1358 dyod = triorg[1] - tridest[1];
1359 /* Find the squares of the lengths of the triangle's three edges. */
1360 oalen = dxoa * dxoa + dyoa * dyoa;
1361 dalen = dxda * dxda + dyda * dyda;
1362 odlen = dxod * dxod + dyod * dyod;
1363 /* Find the square of the length of the longest edge. */
1364 maxlen = (dalen > oalen) ? dalen : oalen;
1365 maxlen = (odlen > maxlen) ? odlen : maxlen;
1366
1367 if (maxlen > 0.05 * (triorg[0] * triorg[0] + triorg[1] * triorg[1]) + 0.02) {
1368 return 1;
1369 } else {
1370 return 0;
1371 }
1372}
1373
1374#endif /* not EXTERNAL_TEST */
1375
1376/** **/
1377/** **/
1378/********* User-defined triangle evaluation routine ends here *********/
1379
1380/********* Memory allocation and program exit wrappers begin here *********/
1381/** **/
1382/** **/
1383
1384#ifdef ANSI_DECLARATORS
1385void triexit(int status)
1386#else /* not ANSI_DECLARATORS */
1387void triexit(status)
1388int status;
1389#endif /* not ANSI_DECLARATORS */
1390
1391{
1392 exit(status);
1393}
1394
1395#ifdef ANSI_DECLARATORS
1396VOID *trimalloc(int size)
1397#else /* not ANSI_DECLARATORS */
1398VOID *trimalloc(size)
1399int size;
1400#endif /* not ANSI_DECLARATORS */
1401
1402{
1403 VOID *memptr;
1404
1405 memptr = (VOID *) malloc((unsigned int) size);
1406 if (memptr == (VOID *) NULL) {
1407 printf("Error: Out of memory.\n");
1408 triexit(1);
1409 }
1410 return(memptr);
1411}
1412
1413#ifdef ANSI_DECLARATORS
1414void trifree(VOID *memptr)
1415#else /* not ANSI_DECLARATORS */
1416void trifree(memptr)
1417VOID *memptr;
1418#endif /* not ANSI_DECLARATORS */
1419
1420{
1421 free(memptr);
1422}
1423
1424/** **/
1425/** **/
1426/********* Memory allocation and program exit wrappers end here *********/
1427
1428/********* User interaction routines begin here *********/
1429/** **/
1430/** **/
1431
1432/*****************************************************************************/
1433/* */
1434/* syntax() Print list of command line switches. */
1435/* */
1436/*****************************************************************************/
1437
1438#ifndef TRILIBRARY
1439
1440void syntax()
1441{
1442#ifdef CDT_ONLY
1443#ifdef REDUCED
1444 printf("triangle [-pAcjevngBPNEIOXzo_lQVh] input_file\n");
1445#else /* not REDUCED */
1446 printf("triangle [-pAcjevngBPNEIOXzo_iFlCQVh] input_file\n");
1447#endif /* not REDUCED */
1448#else /* not CDT_ONLY */
1449#ifdef REDUCED
1450 printf("triangle [-prq__a__uAcDjevngBPNEIOXzo_YS__lQVh] input_file\n");
1451#else /* not REDUCED */
1452 printf("triangle [-prq__a__uAcDjevngBPNEIOXzo_YS__iFlsCQVh] input_file\n");
1453#endif /* not REDUCED */
1454#endif /* not CDT_ONLY */
1455
1456 printf(" -p Triangulates a Planar Straight Line Graph (.poly file).\n");
1457#ifndef CDT_ONLY
1458 printf(" -r Refines a previously generated mesh.\n");
1459 printf(
1460 " -q Quality mesh generation. A minimum angle may be specified.\n");
1461 printf(" -a Applies a maximum triangle area constraint.\n");
1462 printf(" -u Applies a user-defined triangle constraint.\n");
1463#endif /* not CDT_ONLY */
1464 printf(
1465 " -A Applies attributes to identify triangles in certain regions.\n");
1466 printf(" -c Encloses the convex hull with segments.\n");
1467#ifndef CDT_ONLY
1468 printf(" -D Conforming Delaunay: all triangles are truly Delaunay.\n");
1469#endif /* not CDT_ONLY */
1470/*
1471 printf(" -w Weighted Delaunay triangulation.\n");
1472 printf(" -W Regular triangulation (lower hull of a height field).\n");
1473*/
1474 printf(" -j Jettison unused vertices from output .node file.\n");
1475 printf(" -e Generates an edge list.\n");
1476 printf(" -v Generates a Voronoi diagram.\n");
1477 printf(" -n Generates a list of triangle neighbors.\n");
1478 printf(" -g Generates an .off file for Geomview.\n");
1479 printf(" -B Suppresses output of boundary information.\n");
1480 printf(" -P Suppresses output of .poly file.\n");
1481 printf(" -N Suppresses output of .node file.\n");
1482 printf(" -E Suppresses output of .ele file.\n");
1483 printf(" -I Suppresses mesh iteration numbers.\n");
1484 printf(" -O Ignores holes in .poly file.\n");
1485 printf(" -X Suppresses use of exact arithmetic.\n");
1486 printf(" -z Numbers all items starting from zero (rather than one).\n");
1487 printf(" -o2 Generates second-order subparametric elements.\n");
1488#ifndef CDT_ONLY
1489 printf(" -Y Suppresses boundary segment splitting.\n");
1490 printf(" -S Specifies maximum number of added Steiner points.\n");
1491#endif /* not CDT_ONLY */
1492#ifndef REDUCED
1493 printf(" -i Uses incremental method, rather than divide-and-conquer.\n");
1494 printf(" -F Uses Fortune's sweepline algorithm, rather than d-and-c.\n");
1495#endif /* not REDUCED */
1496 printf(" -l Uses vertical cuts only, rather than alternating cuts.\n");
1497#ifndef REDUCED
1498#ifndef CDT_ONLY
1499 printf(
1500 " -s Force segments into mesh by splitting (instead of using CDT).\n");
1501#endif /* not CDT_ONLY */
1502 printf(" -C Check consistency of final mesh.\n");
1503#endif /* not REDUCED */
1504 printf(" -Q Quiet: No terminal output except errors.\n");
1505 printf(" -V Verbose: Detailed information on what I'm doing.\n");
1506 printf(" -h Help: Detailed instructions for Triangle.\n");
1507 triexit(0);
1508}
1509
1510#endif /* not TRILIBRARY */
1511
1512/*****************************************************************************/
1513/* */
1514/* info() Print out complete instructions. */
1515/* */
1516/*****************************************************************************/
1517
1518#ifndef TRILIBRARY
1519
1520void info()
1521{
1522 printf("Triangle\n");
1523 printf(
1524"A Two-Dimensional Quality Mesh Generator and Delaunay Triangulator.\n");
1525 printf("Version 1.6\n\n");
1526 printf(
1527"Copyright 1993, 1995, 1997, 1998, 2002, 2005 Jonathan Richard Shewchuk\n");
1528 printf("2360 Woolsey #H / Berkeley, California 94705-1927\n");
1529 printf("Bugs/comments to jrs@cs.berkeley.edu\n");
1530 printf(
1531"Created as part of the Quake project (tools for earthquake simulation).\n");
1532 printf(
1533"Supported in part by NSF Grant CMS-9318163 and an NSERC 1967 Scholarship.\n");
1534 printf("There is no warranty whatsoever. Use at your own risk.\n");
1535#ifdef SINGLE
1536 printf("This executable is compiled for single precision arithmetic.\n\n\n");
1537#else /* not SINGLE */
1538 printf("This executable is compiled for double precision arithmetic.\n\n\n");
1539#endif /* not SINGLE */
1540 printf(
1541"Triangle generates exact Delaunay triangulations, constrained Delaunay\n");
1542 printf(
1543"triangulations, conforming Delaunay triangulations, Voronoi diagrams, and\n");
1544 printf(
1545"high-quality triangular meshes. The latter can be generated with no small\n"
1546);
1547 printf(
1548"or large angles, and are thus suitable for finite element analysis. If no\n"
1549);
1550 printf(
1551"command line switch is specified, your .node input file is read, and the\n");
1552 printf(
1553"Delaunay triangulation is returned in .node and .ele output files. The\n");
1554 printf("command syntax is:\n\n");
1555 printf("triangle [-prq__a__uAcDjevngBPNEIOXzo_YS__iFlsCQVh] input_file\n\n");
1556 printf(
1557"Underscores indicate that numbers may optionally follow certain switches.\n");
1558 printf(
1559"Do not leave any space between a switch and its numeric parameter.\n");
1560 printf(
1561"input_file must be a file with extension .node, or extension .poly if the\n");
1562 printf(
1563"-p switch is used. If -r is used, you must supply .node and .ele files,\n");
1564 printf(
1565"and possibly a .poly file and an .area file as well. The formats of these\n"
1566);
1567 printf("files are described below.\n\n");
1568 printf("Command Line Switches:\n\n");
1569 printf(
1570" -p Reads a Planar Straight Line Graph (.poly file), which can specify\n"
1571);
1572 printf(
1573" vertices, segments, holes, regional attributes, and regional area\n");
1574 printf(
1575" constraints. Generates a constrained Delaunay triangulation (CDT)\n"
1576);
1577 printf(
1578" fitting the input; or, if -s, -q, -a, or -u is used, a conforming\n");
1579 printf(
1580" constrained Delaunay triangulation (CCDT). If you want a truly\n");
1581 printf(
1582" Delaunay (not just constrained Delaunay) triangulation, use -D as\n");
1583 printf(
1584" well. When -p is not used, Triangle reads a .node file by default.\n"
1585);
1586 printf(
1587" -r Refines a previously generated mesh. The mesh is read from a .node\n"
1588);
1589 printf(
1590" file and an .ele file. If -p is also used, a .poly file is read\n");
1591 printf(
1592" and used to constrain segments in the mesh. If -a is also used\n");
1593 printf(
1594" (with no number following), an .area file is read and used to\n");
1595 printf(
1596" impose area constraints on the mesh. Further details on refinement\n"
1597);
1598 printf(" appear below.\n");
1599 printf(
1600" -q Quality mesh generation by Delaunay refinement (a hybrid of Paul\n");
1601 printf(
1602" Chew's and Jim Ruppert's algorithms). Adds vertices to the mesh to\n"
1603);
1604 printf(
1605" ensure that all angles are between 20 and 140 degrees. An\n");
1606 printf(
1607" alternative bound on the minimum angle, replacing 20 degrees, may\n");
1608 printf(
1609" be specified after the `q'. The specified angle may include a\n");
1610 printf(
1611" decimal point, but not exponential notation. Note that a bound of\n"
1612);
1613 printf(
1614" theta degrees on the smallest angle also implies a bound of\n");
1615 printf(
1616" (180 - 2 theta) on the largest angle. If the minimum angle is 28.6\n"
1617);
1618 printf(
1619" degrees or smaller, Triangle is mathematically guaranteed to\n");
1620 printf(
1621" terminate (assuming infinite precision arithmetic--Triangle may\n");
1622 printf(
1623" fail to terminate if you run out of precision). In practice,\n");
1624 printf(
1625" Triangle often succeeds for minimum angles up to 34 degrees. For\n");
1626 printf(
1627" some meshes, however, you might need to reduce the minimum angle to\n"
1628);
1629 printf(
1630" avoid problems associated with insufficient floating-point\n");
1631 printf(" precision.\n");
1632 printf(
1633" -a Imposes a maximum triangle area. If a number follows the `a', no\n");
1634 printf(
1635" triangle is generated whose area is larger than that number. If no\n"
1636);
1637 printf(
1638" number is specified, an .area file (if -r is used) or .poly file\n");
1639 printf(
1640" (if -r is not used) specifies a set of maximum area constraints.\n");
1641 printf(
1642" An .area file contains a separate area constraint for each\n");
1643 printf(
1644" triangle, and is useful for refining a finite element mesh based on\n"
1645);
1646 printf(
1647" a posteriori error estimates. A .poly file can optionally contain\n"
1648);
1649 printf(
1650" an area constraint for each segment-bounded region, thereby\n");
1651 printf(
1652" controlling triangle densities in a first triangulation of a PSLG.\n"
1653);
1654 printf(
1655" You can impose both a fixed area constraint and a varying area\n");
1656 printf(
1657" constraint by invoking the -a switch twice, once with and once\n");
1658 printf(
1659" without a number following. Each area specified may include a\n");
1660 printf(" decimal point.\n");
1661 printf(
1662" -u Imposes a user-defined constraint on triangle size. There are two\n"
1663);
1664 printf(
1665" ways to use this feature. One is to edit the triunsuitable()\n");
1666 printf(
1667" procedure in triangle.c to encode any constraint you like, then\n");
1668 printf(
1669" recompile Triangle. The other is to compile triangle.c with the\n");
1670 printf(
1671" EXTERNAL_TEST symbol set (compiler switch -DEXTERNAL_TEST), then\n");
1672 printf(
1673" link Triangle with a separate object file that implements\n");
1674 printf(
1675" triunsuitable(). In either case, the -u switch causes the user-\n");
1676 printf(" defined test to be applied to every triangle.\n");
1677 printf(
1678" -A Assigns an additional floating-point attribute to each triangle\n");
1679 printf(
1680" that identifies what segment-bounded region each triangle belongs\n");
1681 printf(
1682" to. Attributes are assigned to regions by the .poly file. If a\n");
1683 printf(
1684" region is not explicitly marked by the .poly file, triangles in\n");
1685 printf(
1686" that region are assigned an attribute of zero. The -A switch has\n");
1687 printf(
1688" an effect only when the -p switch is used and the -r switch is not.\n"
1689);
1690 printf(
1691" -c Creates segments on the convex hull of the triangulation. If you\n");
1692 printf(
1693" are triangulating a vertex set, this switch causes a .poly file to\n"
1694);
1695 printf(
1696" be written, containing all edges of the convex hull. If you are\n");
1697 printf(
1698" triangulating a PSLG, this switch specifies that the whole convex\n");
1699 printf(
1700" hull of the PSLG should be triangulated, regardless of what\n");
1701 printf(
1702" segments the PSLG has. If you do not use this switch when\n");
1703 printf(
1704" triangulating a PSLG, Triangle assumes that you have identified the\n"
1705);
1706 printf(
1707" region to be triangulated by surrounding it with segments of the\n");
1708 printf(
1709" input PSLG. Beware: if you are not careful, this switch can cause\n"
1710);
1711 printf(
1712" the introduction of an extremely thin angle between a PSLG segment\n"
1713);
1714 printf(
1715" and a convex hull segment, which can cause overrefinement (and\n");
1716 printf(
1717" possibly failure if Triangle runs out of precision). If you are\n");
1718 printf(
1719" refining a mesh, the -c switch works differently: it causes a\n");
1720 printf(
1721" .poly file to be written containing the boundary edges of the mesh\n"
1722);
1723 printf(" (useful if no .poly file was read).\n");
1724 printf(
1725" -D Conforming Delaunay triangulation: use this switch if you want to\n"
1726);
1727 printf(
1728" ensure that all the triangles in the mesh are Delaunay, and not\n");
1729 printf(
1730" merely constrained Delaunay; or if you want to ensure that all the\n"
1731);
1732 printf(
1733" Voronoi vertices lie within the triangulation. (Some finite volume\n"
1734);
1735 printf(
1736" methods have this requirement.) This switch invokes Ruppert's\n");
1737 printf(
1738" original algorithm, which splits every subsegment whose diametral\n");
1739 printf(
1740" circle is encroached. It usually increases the number of vertices\n"
1741);
1742 printf(" and triangles.\n");
1743 printf(
1744" -j Jettisons vertices that are not part of the final triangulation\n");
1745 printf(
1746" from the output .node file. By default, Triangle copies all\n");
1747 printf(
1748" vertices in the input .node file to the output .node file, in the\n");
1749 printf(
1750" same order, so their indices do not change. The -j switch prevents\n"
1751);
1752 printf(
1753" duplicated input vertices, or vertices `eaten' by holes, from\n");
1754 printf(
1755" appearing in the output .node file. Thus, if two input vertices\n");
1756 printf(
1757" have exactly the same coordinates, only the first appears in the\n");
1758 printf(
1759" output. If any vertices are jettisoned, the vertex numbering in\n");
1760 printf(
1761" the output .node file differs from that of the input .node file.\n");
1762 printf(
1763" -e Outputs (to an .edge file) a list of edges of the triangulation.\n");
1764 printf(
1765" -v Outputs the Voronoi diagram associated with the triangulation.\n");
1766 printf(
1767" Does not attempt to detect degeneracies, so some Voronoi vertices\n");
1768 printf(
1769" may be duplicated. See the discussion of Voronoi diagrams below.\n");
1770 printf(
1771" -n Outputs (to a .neigh file) a list of triangles neighboring each\n");
1772 printf(" triangle.\n");
1773 printf(
1774" -g Outputs the mesh to an Object File Format (.off) file, suitable for\n"
1775);
1776 printf(" viewing with the Geometry Center's Geomview package.\n");
1777 printf(
1778" -B No boundary markers in the output .node, .poly, and .edge output\n");
1779 printf(
1780" files. See the detailed discussion of boundary markers below.\n");
1781 printf(
1782" -P No output .poly file. Saves disk space, but you lose the ability\n");
1783 printf(
1784" to maintain constraining segments on later refinements of the mesh.\n"
1785);
1786 printf(" -N No output .node file.\n");
1787 printf(" -E No output .ele file.\n");
1788 printf(
1789" -I No iteration numbers. Suppresses the output of .node and .poly\n");
1790 printf(
1791" files, so your input files won't be overwritten. (If your input is\n"
1792);
1793 printf(
1794" a .poly file only, a .node file is written.) Cannot be used with\n");
1795 printf(
1796" the -r switch, because that would overwrite your input .ele file.\n");
1797 printf(
1798" Shouldn't be used with the -q, -a, -u, or -s switch if you are\n");
1799 printf(
1800" using a .node file for input, because no .node file is written, so\n"
1801);
1802 printf(" there is no record of any added Steiner points.\n");
1803 printf(" -O No holes. Ignores the holes in the .poly file.\n");
1804 printf(
1805" -X No exact arithmetic. Normally, Triangle uses exact floating-point\n"
1806);
1807 printf(
1808" arithmetic for certain tests if it thinks the inexact tests are not\n"
1809);
1810 printf(
1811" accurate enough. Exact arithmetic ensures the robustness of the\n");
1812 printf(
1813" triangulation algorithms, despite floating-point roundoff error.\n");
1814 printf(
1815" Disabling exact arithmetic with the -X switch causes a small\n");
1816 printf(
1817" improvement in speed and creates the possibility that Triangle will\n"
1818);
1819 printf(" fail to produce a valid mesh. Not recommended.\n");
1820 printf(
1821" -z Numbers all items starting from zero (rather than one). Note that\n"
1822);
1823 printf(
1824" this switch is normally overridden by the value used to number the\n"
1825);
1826 printf(
1827" first vertex of the input .node or .poly file. However, this\n");
1828 printf(
1829" switch is useful when calling Triangle from another program.\n");
1830 printf(
1831" -o2 Generates second-order subparametric elements with six nodes each.\n"
1832);
1833 printf(
1834" -Y No new vertices on the boundary. This switch is useful when the\n");
1835 printf(
1836" mesh boundary must be preserved so that it conforms to some\n");
1837 printf(
1838" adjacent mesh. Be forewarned that you will probably sacrifice much\n"
1839);
1840 printf(
1841" of the quality of the mesh; Triangle will try, but the resulting\n");
1842 printf(
1843" mesh may contain poorly shaped triangles. Works well if all the\n");
1844 printf(
1845" boundary vertices are closely spaced. Specify this switch twice\n");
1846 printf(
1847" (`-YY') to prevent all segment splitting, including internal\n");
1848 printf(" boundaries.\n");
1849 printf(
1850" -S Specifies the maximum number of Steiner points (vertices that are\n");
1851 printf(
1852" not in the input, but are added to meet the constraints on minimum\n"
1853);
1854 printf(
1855" angle and maximum area). The default is to allow an unlimited\n");
1856 printf(
1857" number. If you specify this switch with no number after it,\n");
1858 printf(
1859" the limit is set to zero. Triangle always adds vertices at segment\n"
1860);
1861 printf(
1862" intersections, even if it needs to use more vertices than the limit\n"
1863);
1864 printf(
1865" you set. When Triangle inserts segments by splitting (-s), it\n");
1866 printf(
1867" always adds enough vertices to ensure that all the segments of the\n"
1868);
1869 printf(" PLSG are recovered, ignoring the limit if necessary.\n");
1870 printf(
1871" -i Uses an incremental rather than a divide-and-conquer algorithm to\n");
1872 printf(
1873" construct a Delaunay triangulation. Try it if the divide-and-\n");
1874 printf(" conquer algorithm fails.\n");
1875 printf(
1876" -F Uses Steven Fortune's sweepline algorithm to construct a Delaunay\n");
1877 printf(
1878" triangulation. Warning: does not use exact arithmetic for all\n");
1879 printf(" calculations. An exact result is not guaranteed.\n");
1880 printf(
1881" -l Uses only vertical cuts in the divide-and-conquer algorithm. By\n");
1882 printf(
1883" default, Triangle alternates between vertical and horizontal cuts,\n"
1884);
1885 printf(
1886" which usually improve the speed except with vertex sets that are\n");
1887 printf(
1888" small or short and wide. This switch is primarily of theoretical\n");
1889 printf(" interest.\n");
1890 printf(
1891" -s Specifies that segments should be forced into the triangulation by\n"
1892);
1893 printf(
1894" recursively splitting them at their midpoints, rather than by\n");
1895 printf(
1896" generating a constrained Delaunay triangulation. Segment splitting\n"
1897);
1898 printf(
1899" is true to Ruppert's original algorithm, but can create needlessly\n"
1900);
1901 printf(
1902" small triangles. This switch is primarily of theoretical interest.\n"
1903);
1904 printf(
1905" -C Check the consistency of the final mesh. Uses exact arithmetic for\n"
1906);
1907 printf(
1908" checking, even if the -X switch is used. Useful if you suspect\n");
1909 printf(" Triangle is buggy.\n");
1910 printf(
1911" -Q Quiet: Suppresses all explanation of what Triangle is doing,\n");
1912 printf(" unless an error occurs.\n");
1913 printf(
1914" -V Verbose: Gives detailed information about what Triangle is doing.\n"
1915);
1916 printf(
1917" Add more `V's for increasing amount of detail. `-V' is most\n");
1918 printf(
1919" useful; itgives information on algorithmic progress and much more\n");
1920 printf(
1921" detailed statistics. `-VV' gives vertex-by-vertex details, and\n");
1922 printf(
1923" prints so much that Triangle runs much more slowly. `-VVVV' gives\n"
1924);
1925 printf(" information only a debugger could love.\n");
1926 printf(" -h Help: Displays these instructions.\n");
1927 printf("\n");
1928 printf("Definitions:\n");
1929 printf("\n");
1930 printf(
1931" A Delaunay triangulation of a vertex set is a triangulation whose\n");
1932 printf(
1933" vertices are the vertex set, that covers the convex hull of the vertex\n");
1934 printf(
1935" set. A Delaunay triangulation has the property that no vertex lies\n");
1936 printf(
1937" inside the circumscribing circle (circle that passes through all three\n");
1938 printf(" vertices) of any triangle in the triangulation.\n\n");
1939 printf(
1940" A Voronoi diagram of a vertex set is a subdivision of the plane into\n");
1941 printf(
1942" polygonal cells (some of which may be unbounded, meaning infinitely\n");
1943 printf(
1944" large), where each cell is the set of points in the plane that are closer\n"
1945);
1946 printf(
1947" to some input vertex than to any other input vertex. The Voronoi diagram\n"
1948);
1949 printf(" is a geometric dual of the Delaunay triangulation.\n\n");
1950 printf(
1951" A Planar Straight Line Graph (PSLG) is a set of vertices and segments.\n");
1952 printf(
1953" Segments are simply edges, whose endpoints are all vertices in the PSLG.\n"
1954);
1955 printf(
1956" Segments may intersect each other only at their endpoints. The file\n");
1957 printf(" format for PSLGs (.poly files) is described below.\n\n");
1958 printf(
1959" A constrained Delaunay triangulation (CDT) of a PSLG is similar to a\n");
1960 printf(
1961" Delaunay triangulation, but each PSLG segment is present as a single edge\n"
1962);
1963 printf(
1964" of the CDT. (A constrained Delaunay triangulation is not truly a\n");
1965 printf(
1966" Delaunay triangulation, because some of its triangles might not be\n");
1967 printf(
1968" Delaunay.) By definition, a CDT does not have any vertices other than\n");
1969 printf(
1970" those specified in the input PSLG. Depending on context, a CDT might\n");
1971 printf(
1972" cover the convex hull of the PSLG, or it might cover only a segment-\n");
1973 printf(" bounded region (e.g. a polygon).\n\n");
1974 printf(
1975" A conforming Delaunay triangulation of a PSLG is a triangulation in which\n"
1976);
1977 printf(
1978" each triangle is truly Delaunay, and each PSLG segment is represented by\n"
1979);
1980 printf(
1981" a linear contiguous sequence of edges of the triangulation. New vertices\n"
1982);
1983 printf(
1984" (not part of the PSLG) may appear, and each input segment may have been\n");
1985 printf(
1986" subdivided into shorter edges (subsegments) by these additional vertices.\n"
1987);
1988 printf(
1989" The new vertices are frequently necessary to maintain the Delaunay\n");
1990 printf(" property while ensuring that every segment is represented.\n\n");
1991 printf(
1992" A conforming constrained Delaunay triangulation (CCDT) of a PSLG is a\n");
1993 printf(
1994" triangulation of a PSLG whose triangles are constrained Delaunay. New\n");
1995 printf(" vertices may appear, and input segments may be subdivided into\n");
1996 printf(
1997" subsegments, but not to guarantee that segments are respected; rather, to\n"
1998);
1999 printf(
2000" improve the quality of the triangles. The high-quality meshes produced\n");
2001 printf(
2002" by the -q switch are usually CCDTs, but can be made conforming Delaunay\n");
2003 printf(" with the -D switch.\n\n");
2004 printf("File Formats:\n\n");
2005 printf(
2006" All files may contain comments prefixed by the character '#'. Vertices,\n"
2007);
2008 printf(
2009" triangles, edges, holes, and maximum area constraints must be numbered\n");
2010 printf(
2011" consecutively, starting from either 1 or 0. Whichever you choose, all\n");
2012 printf(
2013" input files must be consistent; if the vertices are numbered from 1, so\n");
2014 printf(
2015" must be all other objects. Triangle automatically detects your choice\n");
2016 printf(
2017" while reading the .node (or .poly) file. (When calling Triangle from\n");
2018 printf(
2019" another program, use the -z switch if you wish to number objects from\n");
2020 printf(" zero.) Examples of these file formats are given below.\n\n");
2021 printf(" .node files:\n");
2022 printf(
2023" First line: <# of vertices> <dimension (must be 2)> <# of attributes>\n"
2024);
2025 printf(
2026" <# of boundary markers (0 or 1)>\n"
2027);
2028 printf(
2029" Remaining lines: <vertex #> <x> <y> [attributes] [boundary marker]\n");
2030 printf("\n");
2031 printf(
2032" The attributes, which are typically floating-point values of physical\n");
2033 printf(
2034" quantities (such as mass or conductivity) associated with the nodes of\n"
2035);
2036 printf(
2037" a finite element mesh, are copied unchanged to the output mesh. If -q,\n"
2038);
2039 printf(
2040" -a, -u, -D, or -s is selected, each new Steiner point added to the mesh\n"
2041);
2042 printf(" has attributes assigned to it by linear interpolation.\n\n");
2043 printf(
2044" If the fourth entry of the first line is `1', the last column of the\n");
2045 printf(
2046" remainder of the file is assumed to contain boundary markers. Boundary\n"
2047);
2048 printf(
2049" markers are used to identify boundary vertices and vertices resting on\n"
2050);
2051 printf(
2052" PSLG segments; a complete description appears in a section below. The\n"
2053);
2054 printf(
2055" .node file produced by Triangle contains boundary markers in the last\n");
2056 printf(" column unless they are suppressed by the -B switch.\n\n");
2057 printf(" .ele files:\n");
2058 printf(
2059" First line: <# of triangles> <nodes per triangle> <# of attributes>\n");
2060 printf(
2061" Remaining lines: <triangle #> <node> <node> <node> ... [attributes]\n");
2062 printf("\n");
2063 printf(
2064" Nodes are indices into the corresponding .node file. The first three\n");
2065 printf(
2066" nodes are the corner vertices, and are listed in counterclockwise order\n"
2067);
2068 printf(
2069" around each triangle. (The remaining nodes, if any, depend on the type\n"
2070);
2071 printf(" of finite element used.)\n\n");
2072 printf(
2073" The attributes are just like those of .node files. Because there is no\n"
2074);
2075 printf(
2076" simple mapping from input to output triangles, Triangle attempts to\n");
2077 printf(
2078" interpolate attributes, and may cause a lot of diffusion of attributes\n"
2079);
2080 printf(
2081" among nearby triangles as the triangulation is refined. Attributes do\n"
2082);
2083 printf(" not diffuse across segments, so attributes used to identify\n");
2084 printf(" segment-bounded regions remain intact.\n\n");
2085 printf(
2086" In .ele files produced by Triangle, each triangular element has three\n");
2087 printf(
2088" nodes (vertices) unless the -o2 switch is used, in which case\n");
2089 printf(
2090" subparametric quadratic elements with six nodes each are generated.\n");
2091 printf(
2092" The first three nodes are the corners in counterclockwise order, and\n");
2093 printf(
2094" the fourth, fifth, and sixth nodes lie on the midpoints of the edges\n");
2095 printf(
2096" opposite the first, second, and third vertices, respectively.\n");
2097 printf("\n");
2098 printf(" .poly files:\n");
2099 printf(
2100" First line: <# of vertices> <dimension (must be 2)> <# of attributes>\n"
2101);
2102 printf(
2103" <# of boundary markers (0 or 1)>\n"
2104);
2105 printf(
2106" Following lines: <vertex #> <x> <y> [attributes] [boundary marker]\n");
2107 printf(" One line: <# of segments> <# of boundary markers (0 or 1)>\n");
2108 printf(
2109" Following lines: <segment #> <endpoint> <endpoint> [boundary marker]\n");
2110 printf(" One line: <# of holes>\n");
2111 printf(" Following lines: <hole #> <x> <y>\n");
2112 printf(
2113" Optional line: <# of regional attributes and/or area constraints>\n");
2114 printf(
2115" Optional following lines: <region #> <x> <y> <attribute> <max area>\n");
2116 printf("\n");
2117 printf(
2118" A .poly file represents a PSLG, as well as some additional information.\n"
2119);
2120 printf(
2121" The first section lists all the vertices, and is identical to the\n");
2122 printf(
2123" format of .node files. <# of vertices> may be set to zero to indicate\n"
2124);
2125 printf(
2126" that the vertices are listed in a separate .node file; .poly files\n");
2127 printf(
2128" produced by Triangle always have this format. A vertex set represented\n"
2129);
2130 printf(
2131" this way has the advantage that it may easily be triangulated with or\n");
2132 printf(
2133" without segments (depending on whether the -p switch is invoked).\n");
2134 printf("\n");
2135 printf(
2136" The second section lists the segments. Segments are edges whose\n");
2137 printf(
2138" presence in the triangulation is enforced. (Depending on the choice of\n"
2139);
2140 printf(
2141" switches, segment might be subdivided into smaller edges). Each\n");
2142 printf(
2143" segment is specified by listing the indices of its two endpoints. This\n"
2144);
2145 printf(
2146" means that you must include its endpoints in the vertex list. Each\n");
2147 printf(" segment, like each point, may have a boundary marker.\n\n");
2148 printf(
2149" If -q, -a, -u, and -s are not selected, Triangle produces a constrained\n"
2150);
2151 printf(
2152" Delaunay triangulation (CDT), in which each segment appears as a single\n"
2153);
2154 printf(
2155" edge in the triangulation. If -q, -a, -u, or -s is selected, Triangle\n"
2156);
2157 printf(
2158" produces a conforming constrained Delaunay triangulation (CCDT), in\n");
2159 printf(
2160" which segments may be subdivided into smaller edges. If -D is\n");
2161 printf(
2162" selected, Triangle produces a conforming Delaunay triangulation, so\n");
2163 printf(
2164" that every triangle is Delaunay, and not just constrained Delaunay.\n");
2165 printf("\n");
2166 printf(
2167" The third section lists holes (and concavities, if -c is selected) in\n");
2168 printf(
2169" the triangulation. Holes are specified by identifying a point inside\n");
2170 printf(
2171" each hole. After the triangulation is formed, Triangle creates holes\n");
2172 printf(
2173" by eating triangles, spreading out from each hole point until its\n");
2174 printf(
2175" progress is blocked by segments in the PSLG. You must be careful to\n");
2176 printf(
2177" enclose each hole in segments, or your whole triangulation might be\n");
2178 printf(
2179" eaten away. If the two triangles abutting a segment are eaten, the\n");
2180 printf(
2181" segment itself is also eaten. Do not place a hole directly on a\n");
2182 printf(" segment; if you do, Triangle chooses one side of the segment\n");
2183 printf(" arbitrarily.\n\n");
2184 printf(
2185" The optional fourth section lists regional attributes (to be assigned\n");
2186 printf(
2187" to all triangles in a region) and regional constraints on the maximum\n");
2188 printf(
2189" triangle area. Triangle reads this section only if the -A switch is\n");
2190 printf(
2191" used or the -a switch is used without a number following it, and the -r\n"
2192);
2193 printf(
2194" switch is not used. Regional attributes and area constraints are\n");
2195 printf(
2196" propagated in the same manner as holes: you specify a point for each\n");
2197 printf(
2198" attribute and/or constraint, and the attribute and/or constraint\n");
2199 printf(
2200" affects the whole region (bounded by segments) containing the point.\n");
2201 printf(
2202" If two values are written on a line after the x and y coordinate, the\n");
2203 printf(
2204" first such value is assumed to be a regional attribute (but is only\n");
2205 printf(
2206" applied if the -A switch is selected), and the second value is assumed\n"
2207);
2208 printf(
2209" to be a regional area constraint (but is only applied if the -a switch\n"
2210);
2211 printf(
2212" is selected). You may specify just one value after the coordinates,\n");
2213 printf(
2214" which can serve as both an attribute and an area constraint, depending\n"
2215);
2216 printf(
2217" on the choice of switches. If you are using the -A and -a switches\n");
2218 printf(
2219" simultaneously and wish to assign an attribute to some region without\n");
2220 printf(" imposing an area constraint, use a negative maximum area.\n\n");
2221 printf(
2222" When a triangulation is created from a .poly file, you must either\n");
2223 printf(
2224" enclose the entire region to be triangulated in PSLG segments, or\n");
2225 printf(
2226" use the -c switch, which automatically creates extra segments that\n");
2227 printf(
2228" enclose the convex hull of the PSLG. If you do not use the -c switch,\n"
2229);
2230 printf(
2231" Triangle eats all triangles that are not enclosed by segments; if you\n");
2232 printf(
2233" are not careful, your whole triangulation may be eaten away. If you do\n"
2234);
2235 printf(
2236" use the -c switch, you can still produce concavities by the appropriate\n"
2237);
2238 printf(
2239" placement of holes just inside the boundary of the convex hull.\n");
2240 printf("\n");
2241 printf(
2242" An ideal PSLG has no intersecting segments, nor any vertices that lie\n");
2243 printf(
2244" upon segments (except, of course, the endpoints of each segment). You\n"
2245);
2246 printf(
2247" aren't required to make your .poly files ideal, but you should be aware\n"
2248);
2249 printf(
2250" of what can go wrong. Segment intersections are relatively safe--\n");
2251 printf(
2252" Triangle calculates the intersection points for you and adds them to\n");
2253 printf(
2254" the triangulation--as long as your machine's floating-point precision\n");
2255 printf(
2256" doesn't become a problem. You are tempting the fates if you have three\n"
2257);
2258 printf(
2259" segments that cross at the same location, and expect Triangle to figure\n"
2260);
2261 printf(
2262" out where the intersection point is. Thanks to floating-point roundoff\n"
2263);
2264 printf(
2265" error, Triangle will probably decide that the three segments intersect\n"
2266);
2267 printf(
2268" at three different points, and you will find a minuscule triangle in\n");
2269 printf(
2270" your output--unless Triangle tries to refine the tiny triangle, uses\n");
2271 printf(
2272" up the last bit of machine precision, and fails to terminate at all.\n");
2273 printf(
2274" You're better off putting the intersection point in the input files,\n");
2275 printf(
2276" and manually breaking up each segment into two. Similarly, if you\n");
2277 printf(
2278" place a vertex at the middle of a segment, and hope that Triangle will\n"
2279);
2280 printf(
2281" break up the segment at that vertex, you might get lucky. On the other\n"
2282);
2283 printf(
2284" hand, Triangle might decide that the vertex doesn't lie precisely on\n");
2285 printf(
2286" the segment, and you'll have a needle-sharp triangle in your output--or\n"
2287);
2288 printf(" a lot of tiny triangles if you're generating a quality mesh.\n");
2289 printf("\n");
2290 printf(
2291" When Triangle reads a .poly file, it also writes a .poly file, which\n");
2292 printf(
2293" includes all the subsegments--the edges that are parts of input\n");
2294 printf(
2295" segments. If the -c switch is used, the output .poly file also\n");
2296 printf(
2297" includes all of the edges on the convex hull. Hence, the output .poly\n"
2298);
2299 printf(
2300" file is useful for finding edges associated with input segments and for\n"
2301);
2302 printf(
2303" setting boundary conditions in finite element simulations. Moreover,\n");
2304 printf(
2305" you will need the output .poly file if you plan to refine the output\n");
2306 printf(
2307" mesh, and don't want segments to be missing in later triangulations.\n");
2308 printf("\n");
2309 printf(" .area files:\n");
2310 printf(" First line: <# of triangles>\n");
2311 printf(" Following lines: <triangle #> <maximum area>\n");
2312 printf("\n");
2313 printf(
2314" An .area file associates with each triangle a maximum area that is used\n"
2315);
2316 printf(
2317" for mesh refinement. As with other file formats, every triangle must\n");
2318 printf(
2319" be represented, and the triangles must be numbered consecutively. A\n");
2320 printf(
2321" triangle may be left unconstrained by assigning it a negative maximum\n");
2322 printf(" area.\n\n");
2323 printf(" .edge files:\n");
2324 printf(" First line: <# of edges> <# of boundary markers (0 or 1)>\n");
2325 printf(
2326" Following lines: <edge #> <endpoint> <endpoint> [boundary marker]\n");
2327 printf("\n");
2328 printf(
2329" Endpoints are indices into the corresponding .node file. Triangle can\n"
2330);
2331 printf(
2332" produce .edge files (use the -e switch), but cannot read them. The\n");
2333 printf(
2334" optional column of boundary markers is suppressed by the -B switch.\n");
2335 printf("\n");
2336 printf(
2337" In Voronoi diagrams, one also finds a special kind of edge that is an\n");
2338 printf(
2339" infinite ray with only one endpoint. For these edges, a different\n");
2340 printf(" format is used:\n\n");
2341 printf(" <edge #> <endpoint> -1 <direction x> <direction y>\n\n");
2342 printf(
2343" The `direction' is a floating-point vector that indicates the direction\n"
2344);
2345 printf(" of the infinite ray.\n\n");
2346 printf(" .neigh files:\n");
2347 printf(
2348" First line: <# of triangles> <# of neighbors per triangle (always 3)>\n"
2349);
2350 printf(
2351" Following lines: <triangle #> <neighbor> <neighbor> <neighbor>\n");
2352 printf("\n");
2353 printf(
2354" Neighbors are indices into the corresponding .ele file. An index of -1\n"
2355);
2356 printf(
2357" indicates no neighbor (because the triangle is on an exterior\n");
2358 printf(
2359" boundary). The first neighbor of triangle i is opposite the first\n");
2360 printf(" corner of triangle i, and so on.\n\n");
2361 printf(
2362" Triangle can produce .neigh files (use the -n switch), but cannot read\n"
2363);
2364 printf(" them.\n\n");
2365 printf("Boundary Markers:\n\n");
2366 printf(
2367" Boundary markers are tags used mainly to identify which output vertices\n");
2368 printf(
2369" and edges are associated with which PSLG segment, and to identify which\n");
2370 printf(
2371" vertices and edges occur on a boundary of the triangulation. A common\n");
2372 printf(
2373" use is to determine where boundary conditions should be applied to a\n");
2374 printf(
2375" finite element mesh. You can prevent boundary markers from being written\n"
2376);
2377 printf(" into files produced by Triangle by using the -B switch.\n\n");
2378 printf(
2379" The boundary marker associated with each segment in an output .poly file\n"
2380);
2381 printf(" and each edge in an output .edge file is chosen as follows:\n");
2382 printf(
2383" - If an output edge is part or all of a PSLG segment with a nonzero\n");
2384 printf(
2385" boundary marker, then the edge is assigned the same marker.\n");
2386 printf(
2387" - Otherwise, if the edge lies on a boundary of the triangulation\n");
2388 printf(
2389" (even the boundary of a hole), then the edge is assigned the marker\n");
2390 printf(" one (1).\n");
2391 printf(" - Otherwise, the edge is assigned the marker zero (0).\n");
2392 printf(
2393" The boundary marker associated with each vertex in an output .node file\n");
2394 printf(" is chosen as follows:\n");
2395 printf(
2396" - If a vertex is assigned a nonzero boundary marker in the input file,\n"
2397);
2398 printf(
2399" then it is assigned the same marker in the output .node file.\n");
2400 printf(
2401" - Otherwise, if the vertex lies on a PSLG segment (even if it is an\n");
2402 printf(
2403" endpoint of the segment) with a nonzero boundary marker, then the\n");
2404 printf(
2405" vertex is assigned the same marker. If the vertex lies on several\n");
2406 printf(" such segments, one of the markers is chosen arbitrarily.\n");
2407 printf(
2408" - Otherwise, if the vertex occurs on a boundary of the triangulation,\n");
2409 printf(" then the vertex is assigned the marker one (1).\n");
2410 printf(" - Otherwise, the vertex is assigned the marker zero (0).\n");
2411 printf("\n");
2412 printf(
2413" If you want Triangle to determine for you which vertices and edges are on\n"
2414);
2415 printf(
2416" the boundary, assign them the boundary marker zero (or use no markers at\n"
2417);
2418 printf(
2419" all) in your input files. In the output files, all boundary vertices,\n");
2420 printf(" edges, and segments will be assigned the value one.\n\n");
2421 printf("Triangulation Iteration Numbers:\n\n");
2422 printf(
2423" Because Triangle can read and refine its own triangulations, input\n");
2424 printf(
2425" and output files have iteration numbers. For instance, Triangle might\n");
2426 printf(
2427" read the files mesh.3.node, mesh.3.ele, and mesh.3.poly, refine the\n");
2428 printf(
2429" triangulation, and output the files mesh.4.node, mesh.4.ele, and\n");
2430 printf(" mesh.4.poly. Files with no iteration number are treated as if\n");
2431 printf(
2432" their iteration number is zero; hence, Triangle might read the file\n");
2433 printf(
2434" points.node, triangulate it, and produce the files points.1.node and\n");
2435 printf(" points.1.ele.\n\n");
2436 printf(
2437" Iteration numbers allow you to create a sequence of successively finer\n");
2438 printf(
2439" meshes suitable for multigrid methods. They also allow you to produce a\n"
2440);
2441 printf(
2442" sequence of meshes using error estimate-driven mesh refinement.\n");
2443 printf("\n");
2444 printf(
2445" If you're not using refinement or quality meshing, and you don't like\n");
2446 printf(
2447" iteration numbers, use the -I switch to disable them. This switch also\n");
2448 printf(
2449" disables output of .node and .poly files to prevent your input files from\n"
2450);
2451 printf(
2452" being overwritten. (If the input is a .poly file that contains its own\n");
2453 printf(
2454" points, a .node file is written. This can be quite convenient for\n");
2455 printf(" computing CDTs or quality meshes.)\n\n");
2456 printf("Examples of How to Use Triangle:\n\n");
2457 printf(
2458" `triangle dots' reads vertices from dots.node, and writes their Delaunay\n"
2459);
2460 printf(
2461" triangulation to dots.1.node and dots.1.ele. (dots.1.node is identical\n");
2462 printf(
2463" to dots.node.) `triangle -I dots' writes the triangulation to dots.ele\n");
2464 printf(
2465" instead. (No additional .node file is needed, so none is written.)\n");
2466 printf("\n");
2467 printf(
2468" `triangle -pe object.1' reads a PSLG from object.1.poly (and possibly\n");
2469 printf(
2470" object.1.node, if the vertices are omitted from object.1.poly) and writes\n"
2471);
2472 printf(
2473" its constrained Delaunay triangulation to object.2.node and object.2.ele.\n"
2474);
2475 printf(
2476" The segments are copied to object.2.poly, and all edges are written to\n");
2477 printf(" object.2.edge.\n\n");
2478 printf(
2479" `triangle -pq31.5a.1 object' reads a PSLG from object.poly (and possibly\n"
2480);
2481 printf(
2482" object.node), generates a mesh whose angles are all between 31.5 and 117\n"
2483);
2484 printf(
2485" degrees and whose triangles all have areas of 0.1 or less, and writes the\n"
2486);
2487 printf(
2488" mesh to object.1.node and object.1.ele. Each segment may be broken up\n");
2489 printf(" into multiple subsegments; these are written to object.1.poly.\n");
2490 printf("\n");
2491 printf(
2492" Here is a sample file `box.poly' describing a square with a square hole:\n"
2493);
2494 printf("\n");
2495 printf(
2496" # A box with eight vertices in 2D, no attributes, one boundary marker.\n"
2497);
2498 printf(" 8 2 0 1\n");
2499 printf(" # Outer box has these vertices:\n");
2500 printf(" 1 0 0 0\n");
2501 printf(" 2 0 3 0\n");
2502 printf(" 3 3 0 0\n");
2503 printf(" 4 3 3 33 # A special marker for this vertex.\n");
2504 printf(" # Inner square has these vertices:\n");
2505 printf(" 5 1 1 0\n");
2506 printf(" 6 1 2 0\n");
2507 printf(" 7 2 1 0\n");
2508 printf(" 8 2 2 0\n");
2509 printf(" # Five segments with boundary markers.\n");
2510 printf(" 5 1\n");
2511 printf(" 1 1 2 5 # Left side of outer box.\n");
2512 printf(" # Square hole has these segments:\n");
2513 printf(" 2 5 7 0\n");
2514 printf(" 3 7 8 0\n");
2515 printf(" 4 8 6 10\n");
2516 printf(" 5 6 5 0\n");
2517 printf(" # One hole in the middle of the inner square.\n");
2518 printf(" 1\n");
2519 printf(" 1 1.5 1.5\n");
2520 printf("\n");
2521 printf(
2522" Note that some segments are missing from the outer square, so you must\n");
2523 printf(
2524" use the `-c' switch. After `triangle -pqc box.poly', here is the output\n"
2525);
2526 printf(
2527" file `box.1.node', with twelve vertices. The last four vertices were\n");
2528 printf(
2529" added to meet the angle constraint. Vertices 1, 2, and 9 have markers\n");
2530 printf(
2531" from segment 1. Vertices 6 and 8 have markers from segment 4. All the\n");
2532 printf(
2533" other vertices but 4 have been marked to indicate that they lie on a\n");
2534 printf(" boundary.\n\n");
2535 printf(" 12 2 0 1\n");
2536 printf(" 1 0 0 5\n");
2537 printf(" 2 0 3 5\n");
2538 printf(" 3 3 0 1\n");
2539 printf(" 4 3 3 33\n");
2540 printf(" 5 1 1 1\n");
2541 printf(" 6 1 2 10\n");
2542 printf(" 7 2 1 1\n");
2543 printf(" 8 2 2 10\n");
2544 printf(" 9 0 1.5 5\n");
2545 printf(" 10 1.5 0 1\n");
2546 printf(" 11 3 1.5 1\n");
2547 printf(" 12 1.5 3 1\n");
2548 printf(" # Generated by triangle -pqc box.poly\n");
2549 printf("\n");
2550 printf(" Here is the output file `box.1.ele', with twelve triangles.\n");
2551 printf("\n");
2552 printf(" 12 3 0\n");
2553 printf(" 1 5 6 9\n");
2554 printf(" 2 10 3 7\n");
2555 printf(" 3 6 8 12\n");
2556 printf(" 4 9 1 5\n");
2557 printf(" 5 6 2 9\n");
2558 printf(" 6 7 3 11\n");
2559 printf(" 7 11 4 8\n");
2560 printf(" 8 7 5 10\n");
2561 printf(" 9 12 2 6\n");
2562 printf(" 10 8 7 11\n");
2563 printf(" 11 5 1 10\n");
2564 printf(" 12 8 4 12\n");
2565 printf(" # Generated by triangle -pqc box.poly\n\n");
2566 printf(
2567" Here is the output file `box.1.poly'. Note that segments have been added\n"
2568);
2569 printf(
2570" to represent the convex hull, and some segments have been subdivided by\n");
2571 printf(
2572" newly added vertices. Note also that <# of vertices> is set to zero to\n");
2573 printf(" indicate that the vertices should be read from the .node file.\n");
2574 printf("\n");
2575 printf(" 0 2 0 1\n");
2576 printf(" 12 1\n");
2577 printf(" 1 1 9 5\n");
2578 printf(" 2 5 7 1\n");
2579 printf(" 3 8 7 1\n");
2580 printf(" 4 6 8 10\n");
2581 printf(" 5 5 6 1\n");
2582 printf(" 6 3 10 1\n");
2583 printf(" 7 4 11 1\n");
2584 printf(" 8 2 12 1\n");
2585 printf(" 9 9 2 5\n");
2586 printf(" 10 10 1 1\n");
2587 printf(" 11 11 3 1\n");
2588 printf(" 12 12 4 1\n");
2589 printf(" 1\n");
2590 printf(" 1 1.5 1.5\n");
2591 printf(" # Generated by triangle -pqc box.poly\n");
2592 printf("\n");
2593 printf("Refinement and Area Constraints:\n");
2594 printf("\n");
2595 printf(
2596" The -r switch causes a mesh (.node and .ele files) to be read and\n");
2597 printf(
2598" refined. If the -p switch is also used, a .poly file is read and used to\n"
2599);
2600 printf(
2601" specify edges that are constrained and cannot be eliminated (although\n");
2602 printf(
2603" they can be subdivided into smaller edges) by the refinement process.\n");
2604 printf("\n");
2605 printf(
2606" When you refine a mesh, you generally want to impose tighter constraints.\n"
2607);
2608 printf(
2609" One way to accomplish this is to use -q with a larger angle, or -a\n");
2610 printf(
2611" followed by a smaller area than you used to generate the mesh you are\n");
2612 printf(
2613" refining. Another way to do this is to create an .area file, which\n");
2614 printf(
2615" specifies a maximum area for each triangle, and use the -a switch\n");
2616 printf(
2617" (without a number following). Each triangle's area constraint is applied\n"
2618);
2619 printf(
2620" to that triangle. Area constraints tend to diffuse as the mesh is\n");
2621 printf(
2622" refined, so if there are large variations in area constraint between\n");
2623 printf(
2624" adjacent triangles, you may not get the results you want. In that case,\n"
2625);
2626 printf(
2627" consider instead using the -u switch and writing a C procedure that\n");
2628 printf(" determines which triangles are too large.\n\n");
2629 printf(
2630" If you are refining a mesh composed of linear (three-node) elements, the\n"
2631);
2632 printf(
2633" output mesh contains all the nodes present in the input mesh, in the same\n"
2634);
2635 printf(
2636" order, with new nodes added at the end of the .node file. However, the\n");
2637 printf(
2638" refinement is not hierarchical: there is no guarantee that each output\n");
2639 printf(
2640" element is contained in a single input element. Often, an output element\n"
2641);
2642 printf(
2643" can overlap two or three input elements, and some input edges are not\n");
2644 printf(
2645" present in the output mesh. Hence, a sequence of refined meshes forms a\n"
2646);
2647 printf(
2648" hierarchy of nodes, but not a hierarchy of elements. If you refine a\n");
2649 printf(
2650" mesh of higher-order elements, the hierarchical property applies only to\n"
2651);
2652 printf(
2653" the nodes at the corners of an element; the midpoint nodes on each edge\n");
2654 printf(" are discarded before the mesh is refined.\n\n");
2655 printf(
2656" Maximum area constraints in .poly files operate differently from those in\n"
2657);
2658 printf(
2659" .area files. A maximum area in a .poly file applies to the whole\n");
2660 printf(
2661" (segment-bounded) region in which a point falls, whereas a maximum area\n");
2662 printf(
2663" in an .area file applies to only one triangle. Area constraints in .poly\n"
2664);
2665 printf(
2666" files are used only when a mesh is first generated, whereas area\n");
2667 printf(
2668" constraints in .area files are used only to refine an existing mesh, and\n"
2669);
2670 printf(
2671" are typically based on a posteriori error estimates resulting from a\n");
2672 printf(" finite element simulation on that mesh.\n\n");
2673 printf(
2674" `triangle -rq25 object.1' reads object.1.node and object.1.ele, then\n");
2675 printf(
2676" refines the triangulation to enforce a 25 degree minimum angle, and then\n"
2677);
2678 printf(
2679" writes the refined triangulation to object.2.node and object.2.ele.\n");
2680 printf("\n");
2681 printf(
2682" `triangle -rpaa6.2 z.3' reads z.3.node, z.3.ele, z.3.poly, and z.3.area.\n"
2683);
2684 printf(
2685" After reconstructing the mesh and its subsegments, Triangle refines the\n");
2686 printf(
2687" mesh so that no triangle has area greater than 6.2, and furthermore the\n");
2688 printf(
2689" triangles satisfy the maximum area constraints in z.3.area. No angle\n");
2690 printf(
2691" bound is imposed at all. The output is written to z.4.node, z.4.ele, and\n"
2692);
2693 printf(" z.4.poly.\n\n");
2694 printf(
2695" The sequence `triangle -qa1 x', `triangle -rqa.3 x.1', `triangle -rqa.1\n");
2696 printf(
2697" x.2' creates a sequence of successively finer meshes x.1, x.2, and x.3,\n");
2698 printf(" suitable for multigrid.\n\n");
2699 printf("Convex Hulls and Mesh Boundaries:\n\n");
2700 printf(
2701" If the input is a vertex set (not a PSLG), Triangle produces its convex\n");
2702 printf(
2703" hull as a by-product in the output .poly file if you use the -c switch.\n");
2704 printf(
2705" There are faster algorithms for finding a two-dimensional convex hull\n");
2706 printf(" than triangulation, of course, but this one comes for free.\n\n");
2707 printf(
2708" If the input is an unconstrained mesh (you are using the -r switch but\n");
2709 printf(
2710" not the -p switch), Triangle produces a list of its boundary edges\n");
2711 printf(
2712" (including hole boundaries) as a by-product when you use the -c switch.\n");
2713 printf(
2714" If you also use the -p switch, the output .poly file contains all the\n");
2715 printf(" segments from the input .poly file as well.\n\n");
2716 printf("Voronoi Diagrams:\n\n");
2717 printf(
2718" The -v switch produces a Voronoi diagram, in files suffixed .v.node and\n");
2719 printf(
2720" .v.edge. For example, `triangle -v points' reads points.node, produces\n");
2721 printf(
2722" its Delaunay triangulation in points.1.node and points.1.ele, and\n");
2723 printf(
2724" produces its Voronoi diagram in points.1.v.node and points.1.v.edge. The\n"
2725);
2726 printf(
2727" .v.node file contains a list of all Voronoi vertices, and the .v.edge\n");
2728 printf(
2729" file contains a list of all Voronoi edges, some of which may be infinite\n"
2730);
2731 printf(
2732" rays. (The choice of filenames makes it easy to run the set of Voronoi\n");
2733 printf(" vertices through Triangle, if so desired.)\n\n");
2734 printf(
2735" This implementation does not use exact arithmetic to compute the Voronoi\n"
2736);
2737 printf(
2738" vertices, and does not check whether neighboring vertices are identical.\n"
2739);
2740 printf(
2741" Be forewarned that if the Delaunay triangulation is degenerate or\n");
2742 printf(
2743" near-degenerate, the Voronoi diagram may have duplicate vertices or\n");
2744 printf(" crossing edges.\n\n");
2745 printf(
2746" The result is a valid Voronoi diagram only if Triangle's output is a true\n"
2747);
2748 printf(
2749" Delaunay triangulation. The Voronoi output is usually meaningless (and\n");
2750 printf(
2751" may contain crossing edges and other pathology) if the output is a CDT or\n"
2752);
2753 printf(
2754" CCDT, or if it has holes or concavities. If the triangulated domain is\n");
2755 printf(
2756" convex and has no holes, you can use -D switch to force Triangle to\n");
2757 printf(
2758" construct a conforming Delaunay triangulation instead of a CCDT, so the\n");
2759 printf(" Voronoi diagram will be valid.\n\n");
2760 printf("Mesh Topology:\n\n");
2761 printf(
2762" You may wish to know which triangles are adjacent to a certain Delaunay\n");
2763 printf(
2764" edge in an .edge file, which Voronoi cells are adjacent to a certain\n");
2765 printf(
2766" Voronoi edge in a .v.edge file, or which Voronoi cells are adjacent to\n");
2767 printf(
2768" each other. All of this information can be found by cross-referencing\n");
2769 printf(
2770" output files with the recollection that the Delaunay triangulation and\n");
2771 printf(" the Voronoi diagram are planar duals.\n\n");
2772 printf(
2773" Specifically, edge i of an .edge file is the dual of Voronoi edge i of\n");
2774 printf(
2775" the corresponding .v.edge file, and is rotated 90 degrees counterclock-\n");
2776 printf(
2777" wise from the Voronoi edge. Triangle j of an .ele file is the dual of\n");
2778 printf(
2779" vertex j of the corresponding .v.node file. Voronoi cell k is the dual\n");
2780 printf(" of vertex k of the corresponding .node file.\n\n");
2781 printf(
2782" Hence, to find the triangles adjacent to a Delaunay edge, look at the\n");
2783 printf(
2784" vertices of the corresponding Voronoi edge. If the endpoints of a\n");
2785 printf(
2786" Voronoi edge are Voronoi vertices 2 and 6 respectively, then triangles 2\n"
2787);
2788 printf(
2789" and 6 adjoin the left and right sides of the corresponding Delaunay edge,\n"
2790);
2791 printf(
2792" respectively. To find the Voronoi cells adjacent to a Voronoi edge, look\n"
2793);
2794 printf(
2795" at the endpoints of the corresponding Delaunay edge. If the endpoints of\n"
2796);
2797 printf(
2798" a Delaunay edge are input vertices 7 and 12, then Voronoi cells 7 and 12\n"
2799);
2800 printf(
2801" adjoin the right and left sides of the corresponding Voronoi edge,\n");
2802 printf(
2803" respectively. To find which Voronoi cells are adjacent to each other,\n");
2804 printf(" just read the list of Delaunay edges.\n\n");
2805 printf(
2806" Triangle does not write a list of the edges adjoining each Voronoi cell,\n"
2807);
2808 printf(
2809" but you can reconstructed it straightforwardly. For instance, to find\n");
2810 printf(
2811" all the edges of Voronoi cell 1, search the output .edge file for every\n");
2812 printf(
2813" edge that has input vertex 1 as an endpoint. The corresponding dual\n");
2814 printf(
2815" edges in the output .v.edge file form the boundary of Voronoi cell 1.\n");
2816 printf("\n");
2817 printf(
2818" For each Voronoi vertex, the .neigh file gives a list of the three\n");
2819 printf(
2820" Voronoi vertices attached to it. You might find this more convenient\n");
2821 printf(" than the .v.edge file.\n\n");
2822 printf("Quadratic Elements:\n\n");
2823 printf(
2824" Triangle generates meshes with subparametric quadratic elements if the\n");
2825 printf(
2826" -o2 switch is specified. Quadratic elements have six nodes per element,\n"
2827);
2828 printf(
2829" rather than three. `Subparametric' means that the edges of the triangles\n"
2830);
2831 printf(
2832" are always straight, so that subparametric quadratic elements are\n");
2833 printf(
2834" geometrically identical to linear elements, even though they can be used\n"
2835);
2836 printf(
2837" with quadratic interpolating functions. The three extra nodes of an\n");
2838 printf(
2839" element fall at the midpoints of the three edges, with the fourth, fifth,\n"
2840);
2841 printf(
2842" and sixth nodes appearing opposite the first, second, and third corners\n");
2843 printf(" respectively.\n\n");
2844 printf("Domains with Small Angles:\n\n");
2845 printf(
2846" If two input segments adjoin each other at a small angle, clearly the -q\n"
2847);
2848 printf(
2849" switch cannot remove the small angle. Moreover, Triangle may have no\n");
2850 printf(
2851" choice but to generate additional triangles whose smallest angles are\n");
2852 printf(
2853" smaller than the specified bound. However, these triangles only appear\n");
2854 printf(
2855" between input segments separated by small angles. Moreover, if you\n");
2856 printf(
2857" request a minimum angle of theta degrees, Triangle will generally produce\n"
2858);
2859 printf(
2860" no angle larger than 180 - 2 theta, even if it is forced to compromise on\n"
2861);
2862 printf(" the minimum angle.\n\n");
2863 printf("Statistics:\n\n");
2864 printf(
2865" After generating a mesh, Triangle prints a count of entities in the\n");
2866 printf(
2867" output mesh, including the number of vertices, triangles, edges, exterior\n"
2868);
2869 printf(
2870" boundary edges (i.e. subsegments on the boundary of the triangulation,\n");
2871 printf(
2872" including hole boundaries), interior boundary edges (i.e. subsegments of\n"
2873);
2874 printf(
2875" input segments not on the boundary), and total subsegments. If you've\n");
2876 printf(
2877" forgotten the statistics for an existing mesh, run Triangle on that mesh\n"
2878);
2879 printf(
2880" with the -rNEP switches to read the mesh and print the statistics without\n"
2881);
2882 printf(
2883" writing any files. Use -rpNEP if you've got a .poly file for the mesh.\n");
2884 printf("\n");
2885 printf(
2886" The -V switch produces extended statistics, including a rough estimate\n");
2887 printf(
2888" of memory use, the number of calls to geometric predicates, and\n");
2889 printf(
2890" histograms of the angles and the aspect ratios of the triangles in the\n");
2891 printf(" mesh.\n\n");
2892 printf("Exact Arithmetic:\n\n");
2893 printf(
2894" Triangle uses adaptive exact arithmetic to perform what computational\n");
2895 printf(
2896" geometers call the `orientation' and `incircle' tests. If the floating-\n"
2897);
2898 printf(
2899" point arithmetic of your machine conforms to the IEEE 754 standard (as\n");
2900 printf(
2901" most workstations do), and does not use extended precision internal\n");
2902 printf(
2903" floating-point registers, then your output is guaranteed to be an\n");
2904 printf(
2905" absolutely true Delaunay or constrained Delaunay triangulation, roundoff\n"
2906);
2907 printf(
2908" error notwithstanding. The word `adaptive' implies that these arithmetic\n"
2909);
2910 printf(
2911" routines compute the result only to the precision necessary to guarantee\n"
2912);
2913 printf(
2914" correctness, so they are usually nearly as fast as their approximate\n");
2915 printf(" counterparts.\n\n");
2916 printf(
2917" May CPUs, including Intel x86 processors, have extended precision\n");
2918 printf(
2919" floating-point registers. These must be reconfigured so their precision\n"
2920);
2921 printf(
2922" is reduced to memory precision. Triangle does this if it is compiled\n");
2923 printf(" correctly. See the makefile for details.\n\n");
2924 printf(
2925" The exact tests can be disabled with the -X switch. On most inputs, this\n"
2926);
2927 printf(
2928" switch reduces the computation time by about eight percent--it's not\n");
2929 printf(
2930" worth the risk. There are rare difficult inputs (having many collinear\n");
2931 printf(
2932" and cocircular vertices), however, for which the difference in speed\n");
2933 printf(
2934" could be a factor of two. Be forewarned that these are precisely the\n");
2935 printf(
2936" inputs most likely to cause errors if you use the -X switch. Hence, the\n"
2937);
2938 printf(" -X switch is not recommended.\n\n");
2939 printf(
2940" Unfortunately, the exact tests don't solve every numerical problem.\n");
2941 printf(
2942" Exact arithmetic is not used to compute the positions of new vertices,\n");
2943 printf(
2944" because the bit complexity of vertex coordinates would grow without\n");
2945 printf(
2946" bound. Hence, segment intersections aren't computed exactly; in very\n");
2947 printf(
2948" unusual cases, roundoff error in computing an intersection point might\n");
2949 printf(
2950" actually lead to an inverted triangle and an invalid triangulation.\n");
2951 printf(
2952" (This is one reason to specify your own intersection points in your .poly\n"
2953);
2954 printf(
2955" files.) Similarly, exact arithmetic is not used to compute the vertices\n"
2956);
2957 printf(" of the Voronoi diagram.\n\n");
2958 printf(
2959" Another pair of problems not solved by the exact arithmetic routines is\n");
2960 printf(
2961" underflow and overflow. If Triangle is compiled for double precision\n");
2962 printf(
2963" arithmetic, I believe that Triangle's geometric predicates work correctly\n"
2964);
2965 printf(
2966" if the exponent of every input coordinate falls in the range [-148, 201].\n"
2967);
2968 printf(
2969" Underflow can silently prevent the orientation and incircle tests from\n");
2970 printf(
2971" being performed exactly, while overflow typically causes a floating\n");
2972 printf(" exception.\n\n");
2973 printf("Calling Triangle from Another Program:\n\n");
2974 printf(" Read the file triangle.h for details.\n\n");
2975 printf("Troubleshooting:\n\n");
2976 printf(" Please read this section before mailing me bugs.\n\n");
2977 printf(" `My output mesh has no triangles!'\n\n");
2978 printf(
2979" If you're using a PSLG, you've probably failed to specify a proper set\n"
2980);
2981 printf(
2982" of bounding segments, or forgotten to use the -c switch. Or you may\n");
2983 printf(
2984" have placed a hole badly, thereby eating all your triangles. To test\n");
2985 printf(" these possibilities, try again with the -c and -O switches.\n");
2986 printf(
2987" Alternatively, all your input vertices may be collinear, in which case\n"
2988);
2989 printf(" you can hardly expect to triangulate them.\n\n");
2990 printf(" `Triangle doesn't terminate, or just crashes.'\n\n");
2991 printf(
2992" Bad things can happen when triangles get so small that the distance\n");
2993 printf(
2994" between their vertices isn't much larger than the precision of your\n");
2995 printf(
2996" machine's arithmetic. If you've compiled Triangle for single-precision\n"
2997);
2998 printf(
2999" arithmetic, you might do better by recompiling it for double-precision.\n"
3000);
3001 printf(
3002" Then again, you might just have to settle for more lenient constraints\n"
3003);
3004 printf(
3005" on the minimum angle and the maximum area than you had planned.\n");
3006 printf("\n");
3007 printf(
3008" You can minimize precision problems by ensuring that the origin lies\n");
3009 printf(
3010" inside your vertex set, or even inside the densest part of your\n");
3011 printf(
3012" mesh. If you're triangulating an object whose x-coordinates all fall\n");
3013 printf(
3014" between 6247133 and 6247134, you're not leaving much floating-point\n");
3015 printf(" precision for Triangle to work with.\n\n");
3016 printf(
3017" Precision problems can occur covertly if the input PSLG contains two\n");
3018 printf(
3019" segments that meet (or intersect) at an extremely small angle, or if\n");
3020 printf(
3021" such an angle is introduced by the -c switch. If you don't realize\n");
3022 printf(
3023" that a tiny angle is being formed, you might never discover why\n");
3024 printf(
3025" Triangle is crashing. To check for this possibility, use the -S switch\n"
3026);
3027 printf(
3028" (with an appropriate limit on the number of Steiner points, found by\n");
3029 printf(
3030" trial-and-error) to stop Triangle early, and view the output .poly file\n"
3031);
3032 printf(
3033" with Show Me (described below). Look carefully for regions where dense\n"
3034);
3035 printf(
3036" clusters of vertices are forming and for small angles between segments.\n"
3037);
3038 printf(
3039" Zoom in closely, as such segments might look like a single segment from\n"
3040);
3041 printf(" a distance.\n\n");
3042 printf(
3043" If some of the input values are too large, Triangle may suffer a\n");
3044 printf(
3045" floating exception due to overflow when attempting to perform an\n");
3046 printf(
3047" orientation or incircle test. (Read the section on exact arithmetic\n");
3048 printf(
3049" above.) Again, I recommend compiling Triangle for double (rather\n");
3050 printf(" than single) precision arithmetic.\n\n");
3051 printf(
3052" Unexpected problems can arise if you use quality meshing (-q, -a, or\n");
3053 printf(
3054" -u) with an input that is not segment-bounded--that is, if your input\n");
3055 printf(
3056" is a vertex set, or you're using the -c switch. If the convex hull of\n"
3057);
3058 printf(
3059" your input vertices has collinear vertices on its boundary, an input\n");
3060 printf(
3061" vertex that you think lies on the convex hull might actually lie just\n");
3062 printf(
3063" inside the convex hull. If so, the vertex and the nearby convex hull\n");
3064 printf(
3065" edge form an extremely thin triangle. When Triangle tries to refine\n");
3066 printf(
3067" the mesh to enforce angle and area constraints, Triangle might generate\n"
3068);
3069 printf(
3070" extremely tiny triangles, or it might fail because of insufficient\n");
3071 printf(" floating-point precision.\n\n");
3072 printf(
3073" `The numbering of the output vertices doesn't match the input vertices.'\n"
3074);
3075 printf("\n");
3076 printf(
3077" You may have had duplicate input vertices, or you may have eaten some\n");
3078 printf(
3079" of your input vertices with a hole, or by placing them outside the area\n"
3080);
3081 printf(
3082" enclosed by segments. In any case, you can solve the problem by not\n");
3083 printf(" using the -j switch.\n\n");
3084 printf(
3085" `Triangle executes without incident, but when I look at the resulting\n");
3086 printf(
3087" mesh, it has overlapping triangles or other geometric inconsistencies.'\n");
3088 printf("\n");
3089 printf(
3090" If you select the -X switch, Triangle occasionally makes mistakes due\n");
3091 printf(
3092" to floating-point roundoff error. Although these errors are rare,\n");
3093 printf(
3094" don't use the -X switch. If you still have problems, please report the\n"
3095);
3096 printf(" bug.\n\n");
3097 printf(
3098" `Triangle executes without incident, but when I look at the resulting\n");
3099 printf(" Voronoi diagram, it has overlapping edges or other geometric\n");
3100 printf(" inconsistencies.'\n");
3101 printf("\n");
3102 printf(
3103" If your input is a PSLG (-p), you can only expect a meaningful Voronoi\n"
3104);
3105 printf(
3106" diagram if the domain you are triangulating is convex and free of\n");
3107 printf(
3108" holes, and you use the -D switch to construct a conforming Delaunay\n");
3109 printf(" triangulation (instead of a CDT or CCDT).\n\n");
3110 printf(
3111" Strange things can happen if you've taken liberties with your PSLG. Do\n");
3112 printf(
3113" you have a vertex lying in the middle of a segment? Triangle sometimes\n");
3114 printf(
3115" copes poorly with that sort of thing. Do you want to lay out a collinear\n"
3116);
3117 printf(
3118" row of evenly spaced, segment-connected vertices? Have you simply\n");
3119 printf(
3120" defined one long segment connecting the leftmost vertex to the rightmost\n"
3121);
3122 printf(
3123" vertex, and a bunch of vertices lying along it? This method occasionally\n"
3124);
3125 printf(
3126" works, especially with horizontal and vertical lines, but often it\n");
3127 printf(
3128" doesn't, and you'll have to connect each adjacent pair of vertices with a\n"
3129);
3130 printf(" separate segment. If you don't like it, tough.\n\n");
3131 printf(
3132" Furthermore, if you have segments that intersect other than at their\n");
3133 printf(
3134" endpoints, try not to let the intersections fall extremely close to PSLG\n"
3135);
3136 printf(" vertices or each other.\n\n");
3137 printf(
3138" If you have problems refining a triangulation not produced by Triangle:\n");
3139 printf(
3140" Are you sure the triangulation is geometrically valid? Is it formatted\n");
3141 printf(
3142" correctly for Triangle? Are the triangles all listed so the first three\n"
3143);
3144 printf(
3145" vertices are their corners in counterclockwise order? Are all of the\n");
3146 printf(
3147" triangles constrained Delaunay? Triangle's Delaunay refinement algorithm\n"
3148);
3149 printf(" assumes that it starts with a CDT.\n\n");
3150 printf("Show Me:\n\n");
3151 printf(
3152" Triangle comes with a separate program named `Show Me', whose primary\n");
3153 printf(
3154" purpose is to draw meshes on your screen or in PostScript. Its secondary\n"
3155);
3156 printf(
3157" purpose is to check the validity of your input files, and do so more\n");
3158 printf(
3159" thoroughly than Triangle does. Unlike Triangle, Show Me requires that\n");
3160 printf(
3161" you have the X Windows system. Sorry, Microsoft Windows users.\n");
3162 printf("\n");
3163 printf("Triangle on the Web:\n");
3164 printf("\n");
3165 printf(" To see an illustrated version of these instructions, check out\n");
3166 printf("\n");
3167 printf(" http://www.cs.cmu.edu/~quake/triangle.html\n");
3168 printf("\n");
3169 printf("A Brief Plea:\n");
3170 printf("\n");
3171 printf(
3172" If you use Triangle, and especially if you use it to accomplish real\n");
3173 printf(
3174" work, I would like very much to hear from you. A short letter or email\n");
3175 printf(
3176" (to jrs@cs.berkeley.edu) describing how you use Triangle will mean a lot\n"
3177);
3178 printf(
3179" to me. The more people I know are using this program, the more easily I\n"
3180);
3181 printf(
3182" can justify spending time on improvements, which in turn will benefit\n");
3183 printf(
3184" you. Also, I can put you on a list to receive email whenever a new\n");
3185 printf(" version of Triangle is available.\n\n");
3186 printf(
3187" If you use a mesh generated by Triangle in a publication, please include\n"
3188);
3189 printf(
3190" an acknowledgment as well. And please spell Triangle with a capital `T'!\n"
3191);
3192 printf(
3193" If you want to include a citation, use `Jonathan Richard Shewchuk,\n");
3194 printf(
3195" ``Triangle: Engineering a 2D Quality Mesh Generator and Delaunay\n");
3196 printf(
3197" Triangulator,'' in Applied Computational Geometry: Towards Geometric\n");
3198 printf(
3199" Engineering (Ming C. Lin and Dinesh Manocha, editors), volume 1148 of\n");
3200 printf(
3201" Lecture Notes in Computer Science, pages 203-222, Springer-Verlag,\n");
3202 printf(
3203" Berlin, May 1996. (From the First ACM Workshop on Applied Computational\n"
3204);
3205 printf(" Geometry.)'\n\n");
3206 printf("Research credit:\n\n");
3207 printf(
3208" Of course, I can take credit for only a fraction of the ideas that made\n");
3209 printf(
3210" this mesh generator possible. Triangle owes its existence to the efforts\n"
3211);
3212 printf(
3213" of many fine computational geometers and other researchers, including\n");
3214 printf(
3215" Marshall Bern, L. Paul Chew, Kenneth L. Clarkson, Boris Delaunay, Rex A.\n"
3216);
3217 printf(
3218" Dwyer, David Eppstein, Steven Fortune, Leonidas J. Guibas, Donald E.\n");
3219 printf(
3220" Knuth, Charles L. Lawson, Der-Tsai Lee, Gary L. Miller, Ernst P. Mucke,\n");
3221 printf(
3222" Steven E. Pav, Douglas M. Priest, Jim Ruppert, Isaac Saias, Bruce J.\n");
3223 printf(
3224" Schachter, Micha Sharir, Peter W. Shor, Daniel D. Sleator, Jorge Stolfi,\n"
3225);
3226 printf(" Robert E. Tarjan, Alper Ungor, Christopher J. Van Wyk, Noel J.\n");
3227 printf(
3228" Walkington, and Binhai Zhu. See the comments at the beginning of the\n");
3229 printf(" source code for references.\n\n");
3230 triexit(0);
3231}
3232
3233#endif /* not TRILIBRARY */
3234
3235/*****************************************************************************/
3236/* */
3237/* internalerror() Ask the user to send me the defective product. Exit. */
3238/* */
3239/*****************************************************************************/
3240
3242{
3243 printf(" Please report this bug to jrs@cs.berkeley.edu\n");
3244 printf(" Include the message above, your input data set, and the exact\n");
3245 printf(" command line you used to run Triangle.\n");
3246 triexit(1);
3247}
3248
3249/*****************************************************************************/
3250/* */
3251/* parsecommandline() Read the command line, identify switches, and set */
3252/* up options and file names. */
3253/* */
3254/*****************************************************************************/
3255
3256#ifdef ANSI_DECLARATORS
3257void parsecommandline(int argc, char **argv, struct behavior *b)
3258#else /* not ANSI_DECLARATORS */
3259void parsecommandline(argc, argv, b)
3260int argc;
3261char **argv;
3262struct behavior *b;
3263#endif /* not ANSI_DECLARATORS */
3264
3265{
3266#ifdef TRILIBRARY
3267#define STARTINDEX 0
3268#else /* not TRILIBRARY */
3269#define STARTINDEX 1
3270 int increment;
3271 int meshnumber;
3272#endif /* not TRILIBRARY */
3273 /* int i, j, k; */
3274 int i, j;
3275 /* char workstring[FILENAMESIZE]; */
3276
3277 b->poly = b->refine = b->quality = 0;
3278 b->vararea = b->fixedarea = b->usertest = 0;
3279 b->regionattrib = b->convex = b->weighted = b->jettison = 0;
3280 b->firstnumber = 1;
3281 b->edgesout = b->voronoi = b->neighbors = b->geomview = 0;
3282 b->nobound = b->nopolywritten = b->nonodewritten = b->noelewritten = 0;
3283 b->noiterationnum = 0;
3284 b->noholes = b->noexact = 0;
3285 b->incremental = b->sweepline = 0;
3286 b->dwyer = 1;
3287 b->splitseg = 0;
3288 b->docheck = 0;
3289 b->nobisect = 0;
3290 b->conformdel = 0;
3291 b->steiner = -1;
3292 b->order = 1;
3293 b->minangle = 0.0;
3294 b->maxarea = -1.0;
3295 b->quiet = b->verbose = 0;
3296#ifndef TRILIBRARY
3297 b->innodefilename[0] = '\0';
3298#endif /* not TRILIBRARY */
3299
3300 for (i = STARTINDEX; i < argc; i++) {
3301#ifndef TRILIBRARY
3302 if (argv[i][0] == '-') {
3303#endif /* not TRILIBRARY */
3304 for (j = STARTINDEX; argv[i][j] != '\0'; j++) {
3305 if (argv[i][j] == 'p') {
3306 b->poly = 1;
3307 }
3308#ifndef CDT_ONLY
3309 if (argv[i][j] == 'r') {
3310 b->refine = 1;
3311 }
3312 if (argv[i][j] == 'q') {
3313 b->quality = 1;
3314 if (((argv[i][j + 1] >= '0') && (argv[i][j + 1] <= '9')) ||
3315 (argv[i][j + 1] == '.')) {
3316 k = 0;
3317 while (((argv[i][j + 1] >= '0') && (argv[i][j + 1] <= '9')) ||
3318 (argv[i][j + 1] == '.')) {
3319 j++;
3320 workstring[k] = argv[i][j];
3321 k++;
3322 }
3323 workstring[k] = '\0';
3324 b->minangle = (REAL) strtod(workstring, (char **) NULL);
3325 } else {
3326 b->minangle = 20.0;
3327 }
3328 }
3329 if (argv[i][j] == 'a') {
3330 b->quality = 1;
3331 if (((argv[i][j + 1] >= '0') && (argv[i][j + 1] <= '9')) ||
3332 (argv[i][j + 1] == '.')) {
3333 b->fixedarea = 1;
3334 k = 0;
3335 while (((argv[i][j + 1] >= '0') && (argv[i][j + 1] <= '9')) ||
3336 (argv[i][j + 1] == '.')) {
3337 j++;
3338 workstring[k] = argv[i][j];
3339 k++;
3340 }
3341 workstring[k] = '\0';
3342 b->maxarea = (REAL) strtod(workstring, (char **) NULL);
3343 if (b->maxarea <= 0.0) {
3344 printf("Error: Maximum area must be greater than zero.\n");
3345 triexit(1);
3346 }
3347 } else {
3348 b->vararea = 1;
3349 }
3350 }
3351 if (argv[i][j] == 'u') {
3352 b->quality = 1;
3353 b->usertest = 1;
3354 }
3355#endif /* not CDT_ONLY */
3356 if (argv[i][j] == 'A') {
3357 b->regionattrib = 1;
3358 }
3359 if (argv[i][j] == 'c') {
3360 b->convex = 1;
3361 }
3362 if (argv[i][j] == 'w') {
3363 b->weighted = 1;
3364 }
3365 if (argv[i][j] == 'W') {
3366 b->weighted = 2;
3367 }
3368 if (argv[i][j] == 'j') {
3369 b->jettison = 1;
3370 }
3371 if (argv[i][j] == 'z') {
3372 b->firstnumber = 0;
3373 }
3374 if (argv[i][j] == 'e') {
3375 b->edgesout = 1;
3376 }
3377 if (argv[i][j] == 'v') {
3378 b->voronoi = 1;
3379 }
3380 if (argv[i][j] == 'n') {
3381 b->neighbors = 1;
3382 }
3383 if (argv[i][j] == 'g') {
3384 b->geomview = 1;
3385 }
3386 if (argv[i][j] == 'B') {
3387 b->nobound = 1;
3388 }
3389 if (argv[i][j] == 'P') {
3390 b->nopolywritten = 1;
3391 }
3392 if (argv[i][j] == 'N') {
3393 b->nonodewritten = 1;
3394 }
3395 if (argv[i][j] == 'E') {
3396 b->noelewritten = 1;
3397 }
3398#ifndef TRILIBRARY
3399 if (argv[i][j] == 'I') {
3400 b->noiterationnum = 1;
3401 }
3402#endif /* not TRILIBRARY */
3403 if (argv[i][j] == 'O') {
3404 b->noholes = 1;
3405 }
3406 if (argv[i][j] == 'X') {
3407 b->noexact = 1;
3408 }
3409 if (argv[i][j] == 'o') {
3410 if (argv[i][j + 1] == '2') {
3411 j++;
3412 b->order = 2;
3413 }
3414 }
3415#ifndef CDT_ONLY
3416 if (argv[i][j] == 'Y') {
3417 b->nobisect++;
3418 }
3419 if (argv[i][j] == 'S') {
3420 b->steiner = 0;
3421 while ((argv[i][j + 1] >= '0') && (argv[i][j + 1] <= '9')) {
3422 j++;
3423 b->steiner = b->steiner * 10 + (int) (argv[i][j] - '0');
3424 }
3425 }
3426#endif /* not CDT_ONLY */
3427#ifndef REDUCED
3428 if (argv[i][j] == 'i') {
3429 b->incremental = 1;
3430 }
3431 if (argv[i][j] == 'F') {
3432 b->sweepline = 1;
3433 }
3434#endif /* not REDUCED */
3435 if (argv[i][j] == 'l') {
3436 b->dwyer = 0;
3437 }
3438#ifndef REDUCED
3439#ifndef CDT_ONLY
3440 if (argv[i][j] == 's') {
3441 b->splitseg = 1;
3442 }
3443 if ((argv[i][j] == 'D') || (argv[i][j] == 'L')) {
3444 b->quality = 1;
3445 b->conformdel = 1;
3446 }
3447#endif /* not CDT_ONLY */
3448 if (argv[i][j] == 'C') {
3449 b->docheck = 1;
3450 }
3451#endif /* not REDUCED */
3452 if (argv[i][j] == 'Q') {
3453 b->quiet = 1;
3454 }
3455 if (argv[i][j] == 'V') {
3456 b->verbose++;
3457 }
3458#ifndef TRILIBRARY
3459 if ((argv[i][j] == 'h') || (argv[i][j] == 'H') ||
3460 (argv[i][j] == '?')) {
3461 info();
3462 }
3463#endif /* not TRILIBRARY */
3464 }
3465#ifndef TRILIBRARY
3466 } else {
3467 strncpy(b->innodefilename, argv[i], FILENAMESIZE - 1);
3468 b->innodefilename[FILENAMESIZE - 1] = '\0';
3469 }
3470#endif /* not TRILIBRARY */
3471 }
3472#ifndef TRILIBRARY
3473 if (b->innodefilename[0] == '\0') {
3474 syntax();
3475 }
3476 if (!strcmp(&b->innodefilename[strlen(b->innodefilename) - 5], ".node")) {
3477 b->innodefilename[strlen(b->innodefilename) - 5] = '\0';
3478 }
3479 if (!strcmp(&b->innodefilename[strlen(b->innodefilename) - 5], ".poly")) {
3480 b->innodefilename[strlen(b->innodefilename) - 5] = '\0';
3481 b->poly = 1;
3482 }
3483#ifndef CDT_ONLY
3484 if (!strcmp(&b->innodefilename[strlen(b->innodefilename) - 4], ".ele")) {
3485 b->innodefilename[strlen(b->innodefilename) - 4] = '\0';
3486 b->refine = 1;
3487 }
3488 if (!strcmp(&b->innodefilename[strlen(b->innodefilename) - 5], ".area")) {
3489 b->innodefilename[strlen(b->innodefilename) - 5] = '\0';
3490 b->refine = 1;
3491 b->quality = 1;
3492 b->vararea = 1;
3493 }
3494#endif /* not CDT_ONLY */
3495#endif /* not TRILIBRARY */
3496 b->usesegments = b->poly || b->refine || b->quality || b->convex;
3497 b->goodangle = cos(b->minangle * PI / 180.0);
3498 if (b->goodangle == 1.0) {
3499 b->offconstant = 0.0;
3500 } else {
3501 b->offconstant = 0.475 * sqrt((1.0 + b->goodangle) / (1.0 - b->goodangle));
3502 }
3503 b->goodangle *= b->goodangle;
3504 if (b->refine && b->noiterationnum) {
3505 printf(
3506 "Error: You cannot use the -I switch when refining a triangulation.\n");
3507 triexit(1);
3508 }
3509 /* Be careful not to allocate space for element area constraints that */
3510 /* will never be assigned any value (other than the default -1.0). */
3511 if (!b->refine && !b->poly) {
3512 b->vararea = 0;
3513 }
3514 /* Be careful not to add an extra attribute to each element unless the */
3515 /* input supports it (PSLG in, but not refining a preexisting mesh). */
3516 if (b->refine || !b->poly) {
3517 b->regionattrib = 0;
3518 }
3519 /* Regular/weighted triangulations are incompatible with PSLGs */
3520 /* and meshing. */
3521 if (b->weighted && (b->poly || b->quality)) {
3522 b->weighted = 0;
3523 if (!b->quiet) {
3524 printf("Warning: weighted triangulations (-w, -W) are incompatible\n");
3525 printf(" with PSLGs (-p) and meshing (-q, -a, -u). Weights ignored.\n"
3526 );
3527 }
3528 }
3529 if (b->jettison && b->nonodewritten && !b->quiet) {
3530 printf("Warning: -j and -N switches are somewhat incompatible.\n");
3531 printf(" If any vertices are jettisoned, you will need the output\n");
3532 printf(" .node file to reconstruct the new node indices.");
3533 }
3534
3535#ifndef TRILIBRARY
3536 strcpy(b->inpolyfilename, b->innodefilename);
3537 strcpy(b->inelefilename, b->innodefilename);
3538 strcpy(b->areafilename, b->innodefilename);
3539 increment = 0;
3540 strcpy(workstring, b->innodefilename);
3541 j = 1;
3542 while (workstring[j] != '\0') {
3543 if ((workstring[j] == '.') && (workstring[j + 1] != '\0')) {
3544 increment = j + 1;
3545 }
3546 j++;
3547 }
3548 meshnumber = 0;
3549 if (increment > 0) {
3550 j = increment;
3551 do {
3552 if ((workstring[j] >= '0') && (workstring[j] <= '9')) {
3553 meshnumber = meshnumber * 10 + (int) (workstring[j] - '0');
3554 } else {
3555 increment = 0;
3556 }
3557 j++;
3558 } while (workstring[j] != '\0');
3559 }
3560 if (b->noiterationnum) {
3561 strcpy(b->outnodefilename, b->innodefilename);
3562 strcpy(b->outelefilename, b->innodefilename);
3563 strcpy(b->edgefilename, b->innodefilename);
3564 strcpy(b->vnodefilename, b->innodefilename);
3565 strcpy(b->vedgefilename, b->innodefilename);
3566 strcpy(b->neighborfilename, b->innodefilename);
3567 strcpy(b->offfilename, b->innodefilename);
3568 strcat(b->outnodefilename, ".node");
3569 strcat(b->outelefilename, ".ele");
3570 strcat(b->edgefilename, ".edge");
3571 strcat(b->vnodefilename, ".v.node");
3572 strcat(b->vedgefilename, ".v.edge");
3573 strcat(b->neighborfilename, ".neigh");
3574 strcat(b->offfilename, ".off");
3575 } else if (increment == 0) {
3576 strcpy(b->outnodefilename, b->innodefilename);
3577 strcpy(b->outpolyfilename, b->innodefilename);
3578 strcpy(b->outelefilename, b->innodefilename);
3579 strcpy(b->edgefilename, b->innodefilename);
3580 strcpy(b->vnodefilename, b->innodefilename);
3581 strcpy(b->vedgefilename, b->innodefilename);
3582 strcpy(b->neighborfilename, b->innodefilename);
3583 strcpy(b->offfilename, b->innodefilename);
3584 strcat(b->outnodefilename, ".1.node");
3585 strcat(b->outpolyfilename, ".1.poly");
3586 strcat(b->outelefilename, ".1.ele");
3587 strcat(b->edgefilename, ".1.edge");
3588 strcat(b->vnodefilename, ".1.v.node");
3589 strcat(b->vedgefilename, ".1.v.edge");
3590 strcat(b->neighborfilename, ".1.neigh");
3591 strcat(b->offfilename, ".1.off");
3592 } else {
3593 workstring[increment] = '%';
3594 workstring[increment + 1] = 'd';
3595 workstring[increment + 2] = '\0';
3596 sprintf(b->outnodefilename, workstring, meshnumber + 1);
3597 strcpy(b->outpolyfilename, b->outnodefilename);
3598 strcpy(b->outelefilename, b->outnodefilename);
3599 strcpy(b->edgefilename, b->outnodefilename);
3600 strcpy(b->vnodefilename, b->outnodefilename);
3601 strcpy(b->vedgefilename, b->outnodefilename);
3602 strcpy(b->neighborfilename, b->outnodefilename);
3603 strcpy(b->offfilename, b->outnodefilename);
3604 strcat(b->outnodefilename, ".node");
3605 strcat(b->outpolyfilename, ".poly");
3606 strcat(b->outelefilename, ".ele");
3607 strcat(b->edgefilename, ".edge");
3608 strcat(b->vnodefilename, ".v.node");
3609 strcat(b->vedgefilename, ".v.edge");
3610 strcat(b->neighborfilename, ".neigh");
3611 strcat(b->offfilename, ".off");
3612 }
3613 strcat(b->innodefilename, ".node");
3614 strcat(b->inpolyfilename, ".poly");
3615 strcat(b->inelefilename, ".ele");
3616 strcat(b->areafilename, ".area");
3617#endif /* not TRILIBRARY */
3618}
3619
3620/** **/
3621/** **/
3622/********* User interaction routines begin here *********/
3623
3624/********* Debugging routines begin here *********/
3625/** **/
3626/** **/
3627
3628/*****************************************************************************/
3629/* */
3630/* printtriangle() Print out the details of an oriented triangle. */
3631/* */
3632/* I originally wrote this procedure to simplify debugging; it can be */
3633/* called directly from the debugger, and presents information about an */
3634/* oriented triangle in digestible form. It's also used when the */
3635/* highest level of verbosity (`-VVV') is specified. */
3636/* */
3637/*****************************************************************************/
3638
3639#ifdef ANSI_DECLARATORS
3640void printtriangle(struct mesh *m, struct behavior *b, struct otri *t)
3641#else /* not ANSI_DECLARATORS */
3642void printtriangle(m, b, t)
3643struct mesh *m;
3644struct behavior *b;
3645struct otri *t;
3646#endif /* not ANSI_DECLARATORS */
3647
3648{
3649 struct otri printtri;
3650 struct osub printsh;
3651 vertex printvertex;
3652
3653 printf("triangle x%lx with orientation %d:\n", (unsigned long) t->tri,
3654 t->orient);
3655 decode(t->tri[0], printtri);
3656 if (printtri.tri == m->dummytri) {
3657 printf(" [0] = Outer space\n");
3658 } else {
3659 printf(" [0] = x%lx %d\n", (unsigned long) printtri.tri,
3660 printtri.orient);
3661 }
3662 decode(t->tri[1], printtri);
3663 if (printtri.tri == m->dummytri) {
3664 printf(" [1] = Outer space\n");
3665 } else {
3666 printf(" [1] = x%lx %d\n", (unsigned long) printtri.tri,
3667 printtri.orient);
3668 }
3669 decode(t->tri[2], printtri);
3670 if (printtri.tri == m->dummytri) {
3671 printf(" [2] = Outer space\n");
3672 } else {
3673 printf(" [2] = x%lx %d\n", (unsigned long) printtri.tri,
3674 printtri.orient);
3675 }
3676
3677 org(*t, printvertex);
3678 if (printvertex == (vertex) NULL)
3679 printf(" Origin[%d] = NULL\n", (t->orient + 1) % 3 + 3);
3680 else
3681 printf(" Origin[%d] = x%lx (%.12g, %.12g)\n",
3682 (t->orient + 1) % 3 + 3, (unsigned long) printvertex,
3683 printvertex[0], printvertex[1]);
3684 dest(*t, printvertex);
3685 if (printvertex == (vertex) NULL)
3686 printf(" Dest [%d] = NULL\n", (t->orient + 2) % 3 + 3);
3687 else
3688 printf(" Dest [%d] = x%lx (%.12g, %.12g)\n",
3689 (t->orient + 2) % 3 + 3, (unsigned long) printvertex,
3690 printvertex[0], printvertex[1]);
3691 apex(*t, printvertex);
3692 if (printvertex == (vertex) NULL)
3693 printf(" Apex [%d] = NULL\n", t->orient + 3);
3694 else
3695 printf(" Apex [%d] = x%lx (%.12g, %.12g)\n",
3696 t->orient + 3, (unsigned long) printvertex,
3697 printvertex[0], printvertex[1]);
3698
3699 if (b->usesegments) {
3700 sdecode(t->tri[6], printsh);
3701 if (printsh.ss != m->dummysub) {
3702 printf(" [6] = x%lx %d\n", (unsigned long) printsh.ss,
3703 printsh.ssorient);
3704 }
3705 sdecode(t->tri[7], printsh);
3706 if (printsh.ss != m->dummysub) {
3707 printf(" [7] = x%lx %d\n", (unsigned long) printsh.ss,
3708 printsh.ssorient);
3709 }
3710 sdecode(t->tri[8], printsh);
3711 if (printsh.ss != m->dummysub) {
3712 printf(" [8] = x%lx %d\n", (unsigned long) printsh.ss,
3713 printsh.ssorient);
3714 }
3715 }
3716
3717 if (b->vararea) {
3718 printf(" Area constraint: %.4g\n", areabound(*t));
3719 }
3720}
3721
3722/*****************************************************************************/
3723/* */
3724/* printsubseg() Print out the details of an oriented subsegment. */
3725/* */
3726/* I originally wrote this procedure to simplify debugging; it can be */
3727/* called directly from the debugger, and presents information about an */
3728/* oriented subsegment in digestible form. It's also used when the highest */
3729/* level of verbosity (`-VVV') is specified. */
3730/* */
3731/*****************************************************************************/
3732
3733#ifdef ANSI_DECLARATORS
3734void printsubseg(struct mesh *m, struct behavior *b, struct osub *s)
3735#else /* not ANSI_DECLARATORS */
3736void printsubseg(m, b, s)
3737struct mesh *m;
3738struct behavior *b;
3739struct osub *s;
3740#endif /* not ANSI_DECLARATORS */
3741
3742{
3743 struct osub printsh;
3744 struct otri printtri;
3745 vertex printvertex;
3746
3747 (void)b; /*LM: added to suppress warning */
3748
3749 printf("subsegment x%lx with orientation %d and mark %d:\n",
3750 (unsigned long) s->ss, s->ssorient, mark(*s));
3751 sdecode(s->ss[0], printsh);
3752 if (printsh.ss == m->dummysub) {
3753 printf(" [0] = No subsegment\n");
3754 } else {
3755 printf(" [0] = x%lx %d\n", (unsigned long) printsh.ss,
3756 printsh.ssorient);
3757 }
3758 sdecode(s->ss[1], printsh);
3759 if (printsh.ss == m->dummysub) {
3760 printf(" [1] = No subsegment\n");
3761 } else {
3762 printf(" [1] = x%lx %d\n", (unsigned long) printsh.ss,
3763 printsh.ssorient);
3764 }
3765
3766 sorg(*s, printvertex);
3767 if (printvertex == (vertex) NULL)
3768 printf(" Origin[%d] = NULL\n", 2 + s->ssorient);
3769 else
3770 printf(" Origin[%d] = x%lx (%.12g, %.12g)\n",
3771 2 + s->ssorient, (unsigned long) printvertex,
3772 printvertex[0], printvertex[1]);
3773 sdest(*s, printvertex);
3774 if (printvertex == (vertex) NULL)
3775 printf(" Dest [%d] = NULL\n", 3 - s->ssorient);
3776 else
3777 printf(" Dest [%d] = x%lx (%.12g, %.12g)\n",
3778 3 - s->ssorient, (unsigned long) printvertex,
3779 printvertex[0], printvertex[1]);
3780
3781 decode(s->ss[6], printtri);
3782 if (printtri.tri == m->dummytri) {
3783 printf(" [6] = Outer space\n");
3784 } else {
3785 printf(" [6] = x%lx %d\n", (unsigned long) printtri.tri,
3786 printtri.orient);
3787 }
3788 decode(s->ss[7], printtri);
3789 if (printtri.tri == m->dummytri) {
3790 printf(" [7] = Outer space\n");
3791 } else {
3792 printf(" [7] = x%lx %d\n", (unsigned long) printtri.tri,
3793 printtri.orient);
3794 }
3795
3796 segorg(*s, printvertex);
3797 if (printvertex == (vertex) NULL)
3798 printf(" Segment origin[%d] = NULL\n", 4 + s->ssorient);
3799 else
3800 printf(" Segment origin[%d] = x%lx (%.12g, %.12g)\n",
3801 4 + s->ssorient, (unsigned long) printvertex,
3802 printvertex[0], printvertex[1]);
3803 segdest(*s, printvertex);
3804 if (printvertex == (vertex) NULL)
3805 printf(" Segment dest [%d] = NULL\n", 5 - s->ssorient);
3806 else
3807 printf(" Segment dest [%d] = x%lx (%.12g, %.12g)\n",
3808 5 - s->ssorient, (unsigned long) printvertex,
3809 printvertex[0], printvertex[1]);
3810}
3811
3812/** **/
3813/** **/
3814/********* Debugging routines end here *********/
3815
3816/********* Memory management routines begin here *********/
3817/** **/
3818/** **/
3819
3820/*****************************************************************************/
3821/* */
3822/* poolzero() Set all of a pool's fields to zero. */
3823/* */
3824/* This procedure should never be called on a pool that has any memory */
3825/* allocated to it, as that memory would leak. */
3826/* */
3827/*****************************************************************************/
3828
3829#ifdef ANSI_DECLARATORS
3830void poolzero(struct memorypool *pool)
3831#else /* not ANSI_DECLARATORS */
3832void poolzero(pool)
3833struct memorypool *pool;
3834#endif /* not ANSI_DECLARATORS */
3835
3836{
3837 pool->firstblock = (VOID **) NULL;
3838 pool->nowblock = (VOID **) NULL;
3839 pool->nextitem = (VOID *) NULL;
3840 pool->deaditemstack = (VOID *) NULL;
3841 pool->pathblock = (VOID **) NULL;
3842 pool->pathitem = (VOID *) NULL;
3843 pool->alignbytes = 0;
3844 pool->itembytes = 0;
3845 pool->itemsperblock = 0;
3846 pool->itemsfirstblock = 0;
3847 pool->items = 0;
3848 pool->maxitems = 0;
3849 pool->unallocateditems = 0;
3850 pool->pathitemsleft = 0;
3851}
3852
3853/*****************************************************************************/
3854/* */
3855/* poolrestart() Deallocate all items in a pool. */
3856/* */
3857/* The pool is returned to its starting state, except that no memory is */
3858/* freed to the operating system. Rather, the previously allocated blocks */
3859/* are ready to be reused. */
3860/* */
3861/*****************************************************************************/
3862
3863#ifdef ANSI_DECLARATORS
3864void poolrestart(struct memorypool *pool)
3865#else /* not ANSI_DECLARATORS */
3866void poolrestart(pool)
3867struct memorypool *pool;
3868#endif /* not ANSI_DECLARATORS */
3869
3870{
3871 unsigned long alignptr;
3872
3873 pool->items = 0;
3874 pool->maxitems = 0;
3875
3876 /* Set the currently active block. */
3877 pool->nowblock = pool->firstblock;
3878 /* Find the first item in the pool. Increment by the size of (VOID *). */
3879 alignptr = (unsigned long) (pool->nowblock + 1);
3880 /* Align the item on an `alignbytes'-byte boundary. */
3881 pool->nextitem = (VOID *)
3882 (alignptr + (unsigned long) pool->alignbytes -
3883 (alignptr % (unsigned long) pool->alignbytes));
3884 /* There are lots of unallocated items left in this block. */
3885 pool->unallocateditems = pool->itemsfirstblock;
3886 /* The stack of deallocated items is empty. */
3887 pool->deaditemstack = (VOID *) NULL;
3888}
3889
3890/*****************************************************************************/
3891/* */
3892/* poolinit() Initialize a pool of memory for allocation of items. */
3893/* */
3894/* This routine initializes the machinery for allocating items. A `pool' */
3895/* is created whose records have size at least `bytecount'. Items will be */
3896/* allocated in `itemcount'-item blocks. Each item is assumed to be a */
3897/* collection of words, and either pointers or floating-point values are */
3898/* assumed to be the "primary" word type. (The "primary" word type is used */
3899/* to determine alignment of items.) If `alignment' isn't zero, all items */
3900/* will be `alignment'-byte aligned in memory. `alignment' must be either */
3901/* a multiple or a factor of the primary word size; powers of two are safe. */
3902/* `alignment' is normally used to create a few unused bits at the bottom */
3903/* of each item's pointer, in which information may be stored. */
3904/* */
3905/* Don't change this routine unless you understand it. */
3906/* */
3907/*****************************************************************************/
3908
3909#ifdef ANSI_DECLARATORS
3910void poolinit(struct memorypool *pool, int bytecount, int itemcount,
3911 int firstitemcount, int alignment)
3912#else /* not ANSI_DECLARATORS */
3913void poolinit(pool, bytecount, itemcount, firstitemcount, alignment)
3914struct memorypool *pool;
3915int bytecount;
3916int itemcount;
3917int firstitemcount;
3918int alignment;
3919#endif /* not ANSI_DECLARATORS */
3920
3921{
3922 /* Find the proper alignment, which must be at least as large as: */
3923 /* - The parameter `alignment'. */
3924 /* - sizeof(VOID *), so the stack of dead items can be maintained */
3925 /* without unaligned accesses. */
3926 if (alignment > (int) sizeof(VOID *)) {
3927 pool->alignbytes = alignment;
3928 } else {
3929 pool->alignbytes = sizeof(VOID *);
3930 }
3931 pool->itembytes = ((bytecount - 1) / pool->alignbytes + 1) *
3932 pool->alignbytes;
3933 pool->itemsperblock = itemcount;
3934 if (firstitemcount == 0) {
3935 pool->itemsfirstblock = itemcount;
3936 } else {
3937 pool->itemsfirstblock = firstitemcount;
3938 }
3939
3940 /* Allocate a block of items. Space for `itemsfirstblock' items and one */
3941 /* pointer (to point to the next block) are allocated, as well as space */
3942 /* to ensure alignment of the items. */
3943 pool->firstblock = (VOID **)
3944 trimalloc(pool->itemsfirstblock * pool->itembytes + (int) sizeof(VOID *) +
3945 pool->alignbytes);
3946 /* Set the next block pointer to NULL. */
3947 *(pool->firstblock) = (VOID *) NULL;
3948 poolrestart(pool);
3949}
3950
3951/*****************************************************************************/
3952/* */
3953/* pooldeinit() Free to the operating system all memory taken by a pool. */
3954/* */
3955/*****************************************************************************/
3956
3957#ifdef ANSI_DECLARATORS
3958void pooldeinit(struct memorypool *pool)
3959#else /* not ANSI_DECLARATORS */
3960void pooldeinit(pool)
3961struct memorypool *pool;
3962#endif /* not ANSI_DECLARATORS */
3963
3964{
3965 while (pool->firstblock != (VOID **) NULL) {
3966 pool->nowblock = (VOID **) *(pool->firstblock);
3967 trifree((VOID *) pool->firstblock);
3968 pool->firstblock = pool->nowblock;
3969 }
3970}
3971
3972/*****************************************************************************/
3973/* */
3974/* poolalloc() Allocate space for an item. */
3975/* */
3976/*****************************************************************************/
3977
3978#ifdef ANSI_DECLARATORS
3979VOID *poolalloc(struct memorypool *pool)
3980#else /* not ANSI_DECLARATORS */
3981VOID *poolalloc(pool)
3982struct memorypool *pool;
3983#endif /* not ANSI_DECLARATORS */
3984
3985{
3986 VOID *newitem;
3987 VOID **newblock;
3988 unsigned long alignptr;
3989
3990 /* First check the linked list of dead items. If the list is not */
3991 /* empty, allocate an item from the list rather than a fresh one. */
3992 if (pool->deaditemstack != (VOID *) NULL) {
3993 newitem = pool->deaditemstack; /* Take first item in list. */
3994 pool->deaditemstack = * (VOID **) pool->deaditemstack;
3995 } else {
3996 /* Check if there are any free items left in the current block. */
3997 if (pool->unallocateditems == 0) {
3998 /* Check if another block must be allocated. */
3999 if (*(pool->nowblock) == (VOID *) NULL) {
4000 /* Allocate a new block of items, pointed to by the previous block. */
4001 newblock = (VOID **) trimalloc(pool->itemsperblock * pool->itembytes +
4002 (int) sizeof(VOID *) +
4003 pool->alignbytes);
4004 *(pool->nowblock) = (VOID *) newblock;
4005 /* The next block pointer is NULL. */
4006 *newblock = (VOID *) NULL;
4007 }
4008
4009 /* Move to the new block. */
4010 pool->nowblock = (VOID **) *(pool->nowblock);
4011 /* Find the first item in the block. */
4012 /* Increment by the size of (VOID *). */
4013 alignptr = (unsigned long) (pool->nowblock + 1);
4014 /* Align the item on an `alignbytes'-byte boundary. */
4015 pool->nextitem = (VOID *)
4016 (alignptr + (unsigned long) pool->alignbytes -
4017 (alignptr % (unsigned long) pool->alignbytes));
4018 /* There are lots of unallocated items left in this block. */
4019 pool->unallocateditems = pool->itemsperblock;
4020 }
4021
4022 /* Allocate a new item. */
4023 newitem = pool->nextitem;
4024 /* Advance `nextitem' pointer to next free item in block. */
4025 pool->nextitem = (VOID *) ((char *) pool->nextitem + pool->itembytes);
4026 pool->unallocateditems--;
4027 pool->maxitems++;
4028 }
4029 pool->items++;
4030 return newitem;
4031}
4032
4033/*****************************************************************************/
4034/* */
4035/* pooldealloc() Deallocate space for an item. */
4036/* */
4037/* The deallocated space is stored in a queue for later reuse. */
4038/* */
4039/*****************************************************************************/
4040
4041#ifdef ANSI_DECLARATORS
4042void pooldealloc(struct memorypool *pool, VOID *dyingitem)
4043#else /* not ANSI_DECLARATORS */
4044void pooldealloc(pool, dyingitem)
4045struct memorypool *pool;
4046VOID *dyingitem;
4047#endif /* not ANSI_DECLARATORS */
4048
4049{
4050 /* Push freshly killed item onto stack. */
4051 *((VOID **) dyingitem) = pool->deaditemstack;
4052 pool->deaditemstack = dyingitem;
4053 pool->items--;
4054}
4055
4056/*****************************************************************************/
4057/* */
4058/* traversalinit() Prepare to traverse the entire list of items. */
4059/* */
4060/* This routine is used in conjunction with traverse(). */
4061/* */
4062/*****************************************************************************/
4063
4064#ifdef ANSI_DECLARATORS
4065void traversalinit(struct memorypool *pool)
4066#else /* not ANSI_DECLARATORS */
4067void traversalinit(pool)
4068struct memorypool *pool;
4069#endif /* not ANSI_DECLARATORS */
4070
4071{
4072 unsigned long alignptr;
4073
4074 /* Begin the traversal in the first block. */
4075 pool->pathblock = pool->firstblock;
4076 /* Find the first item in the block. Increment by the size of (VOID *). */
4077 alignptr = (unsigned long) (pool->pathblock + 1);
4078 /* Align with item on an `alignbytes'-byte boundary. */
4079 pool->pathitem = (VOID *)
4080 (alignptr + (unsigned long) pool->alignbytes -
4081 (alignptr % (unsigned long) pool->alignbytes));
4082 /* Set the number of items left in the current block. */
4083 pool->pathitemsleft = pool->itemsfirstblock;
4084}
4085
4086/*****************************************************************************/
4087/* */
4088/* traverse() Find the next item in the list. */
4089/* */
4090/* This routine is used in conjunction with traversalinit(). Be forewarned */
4091/* that this routine successively returns all items in the list, including */
4092/* deallocated ones on the deaditemqueue. It's up to you to figure out */
4093/* which ones are actually dead. Why? I don't want to allocate extra */
4094/* space just to demarcate dead items. It can usually be done more */
4095/* space-efficiently by a routine that knows something about the structure */
4096/* of the item. */
4097/* */
4098/*****************************************************************************/
4099
4100#ifdef ANSI_DECLARATORS
4101VOID *traverse(struct memorypool *pool)
4102#else /* not ANSI_DECLARATORS */
4103VOID *traverse(pool)
4104struct memorypool *pool;
4105#endif /* not ANSI_DECLARATORS */
4106
4107{
4108 VOID *newitem;
4109 unsigned long alignptr;
4110
4111 /* Stop upon exhausting the list of items. */
4112 if (pool->pathitem == pool->nextitem) {
4113 return (VOID *) NULL;
4114 }
4115
4116 /* Check whether any untraversed items remain in the current block. */
4117 if (pool->pathitemsleft == 0) {
4118 /* Find the next block. */
4119 pool->pathblock = (VOID **) *(pool->pathblock);
4120 /* Find the first item in the block. Increment by the size of (VOID *). */
4121 alignptr = (unsigned long) (pool->pathblock + 1);
4122 /* Align with item on an `alignbytes'-byte boundary. */
4123 pool->pathitem = (VOID *)
4124 (alignptr + (unsigned long) pool->alignbytes -
4125 (alignptr % (unsigned long) pool->alignbytes));
4126 /* Set the number of items left in the current block. */
4127 pool->pathitemsleft = pool->itemsperblock;
4128 }
4129
4130 newitem = pool->pathitem;
4131 /* Find the next item in the block. */
4132 pool->pathitem = (VOID *) ((char *) pool->pathitem + pool->itembytes);
4133 pool->pathitemsleft--;
4134 return newitem;
4135}
4136
4137/*****************************************************************************/
4138/* */
4139/* dummyinit() Initialize the triangle that fills "outer space" and the */
4140/* omnipresent subsegment. */
4141/* */
4142/* The triangle that fills "outer space," called `dummytri', is pointed to */
4143/* by every triangle and subsegment on a boundary (be it outer or inner) of */
4144/* the triangulation. Also, `dummytri' points to one of the triangles on */
4145/* the convex hull (until the holes and concavities are carved), making it */
4146/* possible to find a starting triangle for point location. */
4147/* */
4148/* The omnipresent subsegment, `dummysub', is pointed to by every triangle */
4149/* or subsegment that doesn't have a full complement of real subsegments */
4150/* to point to. */
4151/* */
4152/* `dummytri' and `dummysub' are generally required to fulfill only a few */
4153/* invariants: their vertices must remain NULL and `dummytri' must always */
4154/* be bonded (at offset zero) to some triangle on the convex hull of the */
4155/* mesh, via a boundary edge. Otherwise, the connections of `dummytri' and */
4156/* `dummysub' may change willy-nilly. This makes it possible to avoid */
4157/* writing a good deal of special-case code (in the edge flip, for example) */
4158/* for dealing with the boundary of the mesh, places where no subsegment is */
4159/* present, and so forth. Other entities are frequently bonded to */
4160/* `dummytri' and `dummysub' as if they were real mesh entities, with no */
4161/* harm done. */
4162/* */
4163/*****************************************************************************/
4164
4165#ifdef ANSI_DECLARATORS
4166void dummyinit(struct mesh *m, struct behavior *b, int trianglebytes,
4167 int subsegbytes)
4168#else /* not ANSI_DECLARATORS */
4169void dummyinit(m, b, trianglebytes, subsegbytes)
4170struct mesh *m;
4171struct behavior *b;
4172int trianglebytes;
4173int subsegbytes;
4174#endif /* not ANSI_DECLARATORS */
4175
4176{
4177 unsigned long alignptr;
4178
4179 /* Set up `dummytri', the `triangle' that occupies "outer space." */
4180 m->dummytribase = (triangle *) trimalloc(trianglebytes +
4181 m->triangles.alignbytes);
4182 /* Align `dummytri' on a `triangles.alignbytes'-byte boundary. */
4183 alignptr = (unsigned long) m->dummytribase;
4184 m->dummytri = (triangle *)
4185 (alignptr + (unsigned long) m->triangles.alignbytes -
4186 (alignptr % (unsigned long) m->triangles.alignbytes));
4187 /* Initialize the three adjoining triangles to be "outer space." These */
4188 /* will eventually be changed by various bonding operations, but their */
4189 /* values don't really matter, as long as they can legally be */
4190 /* dereferenced. */
4191 m->dummytri[0] = (triangle) m->dummytri;
4192 m->dummytri[1] = (triangle) m->dummytri;
4193 m->dummytri[2] = (triangle) m->dummytri;
4194 /* Three NULL vertices. */
4195 m->dummytri[3] = (triangle) NULL;
4196 m->dummytri[4] = (triangle) NULL;
4197 m->dummytri[5] = (triangle) NULL;
4198
4199 if (b->usesegments) {
4200 /* Set up `dummysub', the omnipresent subsegment pointed to by any */
4201 /* triangle side or subsegment end that isn't attached to a real */
4202 /* subsegment. */
4203 m->dummysubbase = (subseg *) trimalloc(subsegbytes +
4204 m->subsegs.alignbytes);
4205 /* Align `dummysub' on a `subsegs.alignbytes'-byte boundary. */
4206 alignptr = (unsigned long) m->dummysubbase;
4207 m->dummysub = (subseg *)
4208 (alignptr + (unsigned long) m->subsegs.alignbytes -
4209 (alignptr % (unsigned long) m->subsegs.alignbytes));
4210 /* Initialize the two adjoining subsegments to be the omnipresent */
4211 /* subsegment. These will eventually be changed by various bonding */
4212 /* operations, but their values don't really matter, as long as they */
4213 /* can legally be dereferenced. */
4214 m->dummysub[0] = (subseg) m->dummysub;
4215 m->dummysub[1] = (subseg) m->dummysub;
4216 /* Four NULL vertices. */
4217 m->dummysub[2] = (subseg) NULL;
4218 m->dummysub[3] = (subseg) NULL;
4219 m->dummysub[4] = (subseg) NULL;
4220 m->dummysub[5] = (subseg) NULL;
4221 /* Initialize the two adjoining triangles to be "outer space." */
4222 m->dummysub[6] = (subseg) m->dummytri;
4223 m->dummysub[7] = (subseg) m->dummytri;
4224 /* Set the boundary marker to zero. */
4225 * (int *) (m->dummysub + 8) = 0;
4226
4227 /* Initialize the three adjoining subsegments of `dummytri' to be */
4228 /* the omnipresent subsegment. */
4229 m->dummytri[6] = (triangle) m->dummysub;
4230 m->dummytri[7] = (triangle) m->dummysub;
4231 m->dummytri[8] = (triangle) m->dummysub;
4232 }
4233}
4234
4235/*****************************************************************************/
4236/* */
4237/* initializevertexpool() Calculate the size of the vertex data structure */
4238/* and initialize its memory pool. */
4239/* */
4240/* This routine also computes the `vertexmarkindex' and `vertex2triindex' */
4241/* indices used to find values within each vertex. */
4242/* */
4243/*****************************************************************************/
4244
4245#ifdef ANSI_DECLARATORS
4246void initializevertexpool(struct mesh *m, struct behavior *b)
4247#else /* not ANSI_DECLARATORS */
4249struct mesh *m;
4250struct behavior *b;
4251#endif /* not ANSI_DECLARATORS */
4252
4253{
4254 int vertexsize;
4255
4256 /* The index within each vertex at which the boundary marker is found, */
4257 /* followed by the vertex type. Ensure the vertex marker is aligned to */
4258 /* a sizeof(int)-byte address. */
4259 m->vertexmarkindex = ((m->mesh_dim + m->nextras) * sizeof(REAL) +
4260 sizeof(int) - 1) /
4261 sizeof(int);
4262 vertexsize = (m->vertexmarkindex + 2) * sizeof(int);
4263 if (b->poly) {
4264 /* The index within each vertex at which a triangle pointer is found. */
4265 /* Ensure the pointer is aligned to a sizeof(triangle)-byte address. */
4266 m->vertex2triindex = (vertexsize + sizeof(triangle) - 1) /
4267 sizeof(triangle);
4268 vertexsize = (m->vertex2triindex + 1) * sizeof(triangle);
4269 }
4270
4271 /* Initialize the pool of vertices. */
4272 poolinit(&m->vertices, vertexsize, VERTEXPERBLOCK,
4273 m->invertices > VERTEXPERBLOCK ? m->invertices : VERTEXPERBLOCK,
4274 sizeof(REAL));
4275}
4276
4277/*****************************************************************************/
4278/* */
4279/* initializetrisubpools() Calculate the sizes of the triangle and */
4280/* subsegment data structures and initialize */
4281/* their memory pools. */
4282/* */
4283/* This routine also computes the `highorderindex', `elemattribindex', and */
4284/* `areaboundindex' indices used to find values within each triangle. */
4285/* */
4286/*****************************************************************************/
4287
4288#ifdef ANSI_DECLARATORS
4289void initializetrisubpools(struct mesh *m, struct behavior *b)
4290#else /* not ANSI_DECLARATORS */
4292struct mesh *m;
4293struct behavior *b;
4294#endif /* not ANSI_DECLARATORS */
4295
4296{
4297 int trisize;
4298
4299 /* The index within each triangle at which the extra nodes (above three) */
4300 /* associated with high order elements are found. There are three */
4301 /* pointers to other triangles, three pointers to corners, and possibly */
4302 /* three pointers to subsegments before the extra nodes. */
4303 m->highorderindex = 6 + (b->usesegments * 3);
4304 /* The number of bytes occupied by a triangle. */
4305 trisize = ((b->order + 1) * (b->order + 2) / 2 + (m->highorderindex - 3)) *
4306 sizeof(triangle);
4307 /* The index within each triangle at which its attributes are found, */
4308 /* where the index is measured in REALs. */
4309 m->elemattribindex = (trisize + sizeof(REAL) - 1) / sizeof(REAL);
4310 /* The index within each triangle at which the maximum area constraint */
4311 /* is found, where the index is measured in REALs. Note that if the */
4312 /* `regionattrib' flag is set, an additional attribute will be added. */
4313 m->areaboundindex = m->elemattribindex + m->eextras + b->regionattrib;
4314 /* If triangle attributes or an area bound are needed, increase the number */
4315 /* of bytes occupied by a triangle. */
4316 if (b->vararea) {
4317 trisize = (m->areaboundindex + 1) * sizeof(REAL);
4318 } else if (m->eextras + b->regionattrib > 0) {
4319 trisize = m->areaboundindex * sizeof(REAL);
4320 }
4321 /* If a Voronoi diagram or triangle neighbor graph is requested, make */
4322 /* sure there's room to store an integer index in each triangle. This */
4323 /* integer index can occupy the same space as the subsegment pointers */
4324 /* or attributes or area constraint or extra nodes. */
4325 if ((b->voronoi || b->neighbors) &&
4326 (trisize < (int) ( 6 * sizeof(triangle) + sizeof(int)))) {
4327 trisize = 6 * sizeof(triangle) + sizeof(int);
4328 }
4329
4330 /* Having determined the memory size of a triangle, initialize the pool. */
4331 poolinit(&m->triangles, trisize, TRIPERBLOCK,
4332 (2 * m->invertices - 2) > TRIPERBLOCK ? (2 * m->invertices - 2) :
4333 TRIPERBLOCK, 4);
4334
4335 if (b->usesegments) {
4336 /* Initialize the pool of subsegments. Take into account all eight */
4337 /* pointers and one boundary marker. */
4338 poolinit(&m->subsegs, 8 * sizeof(triangle) + sizeof(int),
4340
4341 /* Initialize the "outer space" triangle and omnipresent subsegment. */
4342 dummyinit(m, b, m->triangles.itembytes, m->subsegs.itembytes);
4343 } else {
4344 /* Initialize the "outer space" triangle. */
4345 dummyinit(m, b, m->triangles.itembytes, 0);
4346 }
4347}
4348
4349/*****************************************************************************/
4350/* */
4351/* triangledealloc() Deallocate space for a triangle, marking it dead. */
4352/* */
4353/*****************************************************************************/
4354
4355#ifdef ANSI_DECLARATORS
4356void triangledealloc(struct mesh *m, triangle *dyingtriangle)
4357#else /* not ANSI_DECLARATORS */
4358void triangledealloc(m, dyingtriangle)
4359struct mesh *m;
4360triangle *dyingtriangle;
4361#endif /* not ANSI_DECLARATORS */
4362
4363{
4364 /* Mark the triangle as dead. This makes it possible to detect dead */
4365 /* triangles when traversing the list of all triangles. */
4366 killtri(dyingtriangle);
4367 pooldealloc(&m->triangles, (VOID *) dyingtriangle);
4368}
4369
4370/*****************************************************************************/
4371/* */
4372/* triangletraverse() Traverse the triangles, skipping dead ones. */
4373/* */
4374/*****************************************************************************/
4375
4376#ifdef ANSI_DECLARATORS
4378#else /* not ANSI_DECLARATORS */
4380struct mesh *m;
4381#endif /* not ANSI_DECLARATORS */
4382
4383{
4384 triangle *newtriangle;
4385
4386 do {
4387 newtriangle = (triangle *) traverse(&m->triangles);
4388 if (newtriangle == (triangle *) NULL) {
4389 return (triangle *) NULL;
4390 }
4391 } while (deadtri(newtriangle)); /* Skip dead ones. */
4392 return newtriangle;
4393}
4394
4395/*****************************************************************************/
4396/* */
4397/* subsegdealloc() Deallocate space for a subsegment, marking it dead. */
4398/* */
4399/*****************************************************************************/
4400
4401#ifdef ANSI_DECLARATORS
4402void subsegdealloc(struct mesh *m, subseg *dyingsubseg)
4403#else /* not ANSI_DECLARATORS */
4404void subsegdealloc(m, dyingsubseg)
4405struct mesh *m;
4406subseg *dyingsubseg;
4407#endif /* not ANSI_DECLARATORS */
4408
4409{
4410 /* Mark the subsegment as dead. This makes it possible to detect dead */
4411 /* subsegments when traversing the list of all subsegments. */
4412 killsubseg(dyingsubseg);
4413 pooldealloc(&m->subsegs, (VOID *) dyingsubseg);
4414}
4415
4416/*****************************************************************************/
4417/* */
4418/* subsegtraverse() Traverse the subsegments, skipping dead ones. */
4419/* */
4420/*****************************************************************************/
4421
4422#ifdef ANSI_DECLARATORS
4424#else /* not ANSI_DECLARATORS */
4426struct mesh *m;
4427#endif /* not ANSI_DECLARATORS */
4428
4429{
4430 subseg *newsubseg;
4431
4432 do {
4433 newsubseg = (subseg *) traverse(&m->subsegs);
4434 if (newsubseg == (subseg *) NULL) {
4435 return (subseg *) NULL;
4436 }
4437 } while (deadsubseg(newsubseg)); /* Skip dead ones. */
4438 return newsubseg;
4439}
4440
4441/*****************************************************************************/
4442/* */
4443/* vertexdealloc() Deallocate space for a vertex, marking it dead. */
4444/* */
4445/*****************************************************************************/
4446
4447#ifdef ANSI_DECLARATORS
4448void vertexdealloc(struct mesh *m, vertex dyingvertex)
4449#else /* not ANSI_DECLARATORS */
4450void vertexdealloc(m, dyingvertex)
4451struct mesh *m;
4452vertex dyingvertex;
4453#endif /* not ANSI_DECLARATORS */
4454
4455{
4456 /* Mark the vertex as dead. This makes it possible to detect dead */
4457 /* vertices when traversing the list of all vertices. */
4458 setvertextype(dyingvertex, DEADVERTEX);
4459 pooldealloc(&m->vertices, (VOID *) dyingvertex);
4460}
4461
4462/*****************************************************************************/
4463/* */
4464/* vertextraverse() Traverse the vertices, skipping dead ones. */
4465/* */
4466/*****************************************************************************/
4467
4468#ifdef ANSI_DECLARATORS
4470#else /* not ANSI_DECLARATORS */
4472struct mesh *m;
4473#endif /* not ANSI_DECLARATORS */
4474
4475{
4476 vertex newvertex;
4477
4478 do {
4479 newvertex = (vertex) traverse(&m->vertices);
4480 if (newvertex == (vertex) NULL) {
4481 return (vertex) NULL;
4482 }
4483 } while (vertextype(newvertex) == DEADVERTEX); /* Skip dead ones. */
4484 return newvertex;
4485}
4486
4487/*****************************************************************************/
4488/* */
4489/* badsubsegdealloc() Deallocate space for a bad subsegment, marking it */
4490/* dead. */
4491/* */
4492/*****************************************************************************/
4493
4494#ifndef CDT_ONLY
4495
4496#ifdef ANSI_DECLARATORS
4497void badsubsegdealloc(struct mesh *m, struct badsubseg *dyingseg)
4498#else /* not ANSI_DECLARATORS */
4499void badsubsegdealloc(m, dyingseg)
4500struct mesh *m;
4501struct badsubseg *dyingseg;
4502#endif /* not ANSI_DECLARATORS */
4503
4504{
4505 /* Set subsegment's origin to NULL. This makes it possible to detect dead */
4506 /* badsubsegs when traversing the list of all badsubsegs . */
4507 dyingseg->subsegorg = (vertex) NULL;
4508 pooldealloc(&m->badsubsegs, (VOID *) dyingseg);
4509}
4510
4511#endif /* not CDT_ONLY */
4512
4513/*****************************************************************************/
4514/* */
4515/* badsubsegtraverse() Traverse the bad subsegments, skipping dead ones. */
4516/* */
4517/*****************************************************************************/
4518
4519#ifndef CDT_ONLY
4520
4521#ifdef ANSI_DECLARATORS
4522struct badsubseg *badsubsegtraverse(struct mesh *m)
4523#else /* not ANSI_DECLARATORS */
4524struct badsubseg *badsubsegtraverse(m)
4525struct mesh *m;
4526#endif /* not ANSI_DECLARATORS */
4527
4528{
4529 struct badsubseg *newseg;
4530
4531 do {
4532 newseg = (struct badsubseg *) traverse(&m->badsubsegs);
4533 if (newseg == (struct badsubseg *) NULL) {
4534 return (struct badsubseg *) NULL;
4535 }
4536 } while (newseg->subsegorg == (vertex) NULL); /* Skip dead ones. */
4537 return newseg;
4538}
4539
4540#endif /* not CDT_ONLY */
4541
4542/*****************************************************************************/
4543/* */
4544/* getvertex() Get a specific vertex, by number, from the list. */
4545/* */
4546/* The first vertex is number 'firstnumber'. */
4547/* */
4548/* Note that this takes O(n) time (with a small constant, if VERTEXPERBLOCK */
4549/* is large). I don't care to take the trouble to make it work in constant */
4550/* time. */
4551/* */
4552/*****************************************************************************/
4553
4554#ifdef ANSI_DECLARATORS
4555vertex getvertex(struct mesh *m, struct behavior *b, int number)
4556#else /* not ANSI_DECLARATORS */
4557vertex getvertex(m, b, number)
4558struct mesh *m;
4559struct behavior *b;
4560int number;
4561#endif /* not ANSI_DECLARATORS */
4562
4563{
4564 VOID **getblock;
4565 char *foundvertex;
4566 unsigned long alignptr;
4567 int current;
4568
4569 getblock = m->vertices.firstblock;
4570 current = b->firstnumber;
4571
4572 /* Find the right block. */
4573 if (current + m->vertices.itemsfirstblock <= number) {
4574 getblock = (VOID **) *getblock;
4575 current += m->vertices.itemsfirstblock;
4576 while (current + m->vertices.itemsperblock <= number) {
4577 getblock = (VOID **) *getblock;
4578 current += m->vertices.itemsperblock;
4579 }
4580 }
4581
4582 /* Now find the right vertex. */
4583 alignptr = (unsigned long) (getblock + 1);
4584 foundvertex = (char *) (alignptr + (unsigned long) m->vertices.alignbytes -
4585 (alignptr % (unsigned long) m->vertices.alignbytes));
4586 return (vertex) (foundvertex + m->vertices.itembytes * (number - current));
4587}
4588
4589/*****************************************************************************/
4590/* */
4591/* triangledeinit() Free all remaining allocated memory. */
4592/* */
4593/*****************************************************************************/
4594
4595#ifdef ANSI_DECLARATORS
4596void triangledeinit(struct mesh *m, struct behavior *b)
4597#else /* not ANSI_DECLARATORS */
4598void triangledeinit(m, b)
4599struct mesh *m;
4600struct behavior *b;
4601#endif /* not ANSI_DECLARATORS */
4602
4603{
4604 pooldeinit(&m->triangles);
4605 trifree((VOID *) m->dummytribase);
4606 if (b->usesegments) {
4607 pooldeinit(&m->subsegs);
4608 trifree((VOID *) m->dummysubbase);
4609 }
4610 pooldeinit(&m->vertices);
4611#ifndef CDT_ONLY
4612 if (b->quality) {
4613 pooldeinit(&m->badsubsegs);
4614 if ((b->minangle > 0.0) || b->vararea || b->fixedarea || b->usertest) {
4615 pooldeinit(&m->badtriangles);
4616 pooldeinit(&m->flipstackers);
4617 }
4618 }
4619#endif /* not CDT_ONLY */
4620}
4621
4622/** **/
4623/** **/
4624/********* Memory management routines end here *********/
4625
4626/********* Constructors begin here *********/
4627/** **/
4628/** **/
4629
4630/*****************************************************************************/
4631/* */
4632/* maketriangle() Create a new triangle with orientation zero. */
4633/* */
4634/*****************************************************************************/
4635
4636#ifdef ANSI_DECLARATORS
4637void maketriangle(struct mesh *m, struct behavior *b, struct otri *newotri)
4638#else /* not ANSI_DECLARATORS */
4639void maketriangle(m, b, newotri)
4640struct mesh *m;
4641struct behavior *b;
4642struct otri *newotri;
4643#endif /* not ANSI_DECLARATORS */
4644
4645{
4646 int i;
4647
4648 newotri->tri = (triangle *) poolalloc(&m->triangles);
4649 /* Initialize the three adjoining triangles to be "outer space". */
4650 newotri->tri[0] = (triangle) m->dummytri;
4651 newotri->tri[1] = (triangle) m->dummytri;
4652 newotri->tri[2] = (triangle) m->dummytri;
4653 /* Three NULL vertices. */
4654 newotri->tri[3] = (triangle) NULL;
4655 newotri->tri[4] = (triangle) NULL;
4656 newotri->tri[5] = (triangle) NULL;
4657 if (b->usesegments) {
4658 /* Initialize the three adjoining subsegments to be the omnipresent */
4659 /* subsegment. */
4660 newotri->tri[6] = (triangle) m->dummysub;
4661 newotri->tri[7] = (triangle) m->dummysub;
4662 newotri->tri[8] = (triangle) m->dummysub;
4663 }
4664 for (i = 0; i < m->eextras; i++) {
4665 setelemattribute(*newotri, i, 0.0);
4666 }
4667 if (b->vararea) {
4668 setareabound(*newotri, -1.0);
4669 }
4670
4671 newotri->orient = 0;
4672}
4673
4674/*****************************************************************************/
4675/* */
4676/* makesubseg() Create a new subsegment with orientation zero. */
4677/* */
4678/*****************************************************************************/
4679
4680#ifdef ANSI_DECLARATORS
4681void makesubseg(struct mesh *m, struct osub *newsubseg)
4682#else /* not ANSI_DECLARATORS */
4683void makesubseg(m, newsubseg)
4684struct mesh *m;
4685struct osub *newsubseg;
4686#endif /* not ANSI_DECLARATORS */
4687
4688{
4689 newsubseg->ss = (subseg *) poolalloc(&m->subsegs);
4690 /* Initialize the two adjoining subsegments to be the omnipresent */
4691 /* subsegment. */
4692 newsubseg->ss[0] = (subseg) m->dummysub;
4693 newsubseg->ss[1] = (subseg) m->dummysub;
4694 /* Four NULL vertices. */
4695 newsubseg->ss[2] = (subseg) NULL;
4696 newsubseg->ss[3] = (subseg) NULL;
4697 newsubseg->ss[4] = (subseg) NULL;
4698 newsubseg->ss[5] = (subseg) NULL;
4699 /* Initialize the two adjoining triangles to be "outer space." */
4700 newsubseg->ss[6] = (subseg) m->dummytri;
4701 newsubseg->ss[7] = (subseg) m->dummytri;
4702 /* Set the boundary marker to zero. */
4703 setmark(*newsubseg, 0);
4704
4705 newsubseg->ssorient = 0;
4706}
4707
4708/** **/
4709/** **/
4710/********* Constructors end here *********/
4711
4712/********* Geometric primitives begin here *********/
4713/** **/
4714/** **/
4715
4716/* The adaptive exact arithmetic geometric predicates implemented herein are */
4717/* described in detail in my paper, "Adaptive Precision Floating-Point */
4718/* Arithmetic and Fast Robust Geometric Predicates." See the header for a */
4719/* full citation. */
4720
4721/* Which of the following two methods of finding the absolute values is */
4722/* fastest is compiler-dependent. A few compilers can inline and optimize */
4723/* the fabs() call; but most will incur the overhead of a function call, */
4724/* which is disastrously slow. A faster way on IEEE machines might be to */
4725/* mask the appropriate bit, but that's difficult to do in C without */
4726/* forcing the value to be stored to memory (rather than be kept in the */
4727/* register to which the optimizer assigned it). */
4728
4729#define Absolute(a) ((a) >= 0.0 ? (a) : -(a))
4730/* #define Absolute(a) fabs(a) */
4731
4732/* Many of the operations are broken up into two pieces, a main part that */
4733/* performs an approximate operation, and a "tail" that computes the */
4734/* roundoff error of that operation. */
4735/* */
4736/* The operations Fast_Two_Sum(), Fast_Two_Diff(), Two_Sum(), Two_Diff(), */
4737/* Split(), and Two_Product() are all implemented as described in the */
4738/* reference. Each of these macros requires certain variables to be */
4739/* defined in the calling routine. The variables `bvirt', `c', `abig', */
4740/* `_i', `_j', `_k', `_l', `_m', and `_n' are declared `INEXACT' because */
4741/* they store the result of an operation that may incur roundoff error. */
4742/* The input parameter `x' (or the highest numbered `x_' parameter) must */
4743/* also be declared `INEXACT'. */
4744
4745#define Fast_Two_Sum_Tail(a, b, x, y) \
4746 bvirt = x - a; \
4747 y = b - bvirt
4748
4749#define Fast_Two_Sum(a, b, x, y) \
4750 x = (REAL) (a + b); \
4751 Fast_Two_Sum_Tail(a, b, x, y)
4752
4753#define Two_Sum_Tail(a, b, x, y) \
4754 bvirt = (REAL) (x - a); \
4755 avirt = x - bvirt; \
4756 bround = b - bvirt; \
4757 around = a - avirt; \
4758 y = around + bround
4759
4760#define Two_Sum(a, b, x, y) \
4761 x = (REAL) (a + b); \
4762 Two_Sum_Tail(a, b, x, y)
4763
4764#define Two_Diff_Tail(a, b, x, y) \
4765 bvirt = (REAL) (a - x); \
4766 avirt = x + bvirt; \
4767 bround = bvirt - b; \
4768 around = a - avirt; \
4769 y = around + bround
4770
4771#define Two_Diff(a, b, x, y) \
4772 x = (REAL) (a - b); \
4773 Two_Diff_Tail(a, b, x, y)
4774
4775#define Split(a, ahi, alo) \
4776 c = (REAL) (splitter * a); \
4777 abig = (REAL) (c - a); \
4778 ahi = c - abig; \
4779 alo = a - ahi
4780
4781#define Two_Product_Tail(a, b, x, y) \
4782 Split(a, ahi, alo); \
4783 Split(b, bhi, blo); \
4784 err1 = x - (ahi * bhi); \
4785 err2 = err1 - (alo * bhi); \
4786 err3 = err2 - (ahi * blo); \
4787 y = (alo * blo) - err3
4788
4789#define Two_Product(a, b, x, y) \
4790 x = (REAL) (a * b); \
4791 Two_Product_Tail(a, b, x, y)
4792
4793/* Two_Product_Presplit() is Two_Product() where one of the inputs has */
4794/* already been split. Avoids redundant splitting. */
4795
4796#define Two_Product_Presplit(a, b, bhi, blo, x, y) \
4797 x = (REAL) (a * b); \
4798 Split(a, ahi, alo); \
4799 err1 = x - (ahi * bhi); \
4800 err2 = err1 - (alo * bhi); \
4801 err3 = err2 - (ahi * blo); \
4802 y = (alo * blo) - err3
4803
4804/* Square() can be done more quickly than Two_Product(). */
4805
4806#define Square_Tail(a, x, y) \
4807 Split(a, ahi, alo); \
4808 err1 = x - (ahi * ahi); \
4809 err3 = err1 - ((ahi + ahi) * alo); \
4810 y = (alo * alo) - err3
4811
4812#define Square(a, x, y) \
4813 x = (REAL) (a * a); \
4814 Square_Tail(a, x, y)
4815
4816/* Macros for summing expansions of various fixed lengths. These are all */
4817/* unrolled versions of Expansion_Sum(). */
4818
4819#define Two_One_Sum(a1, a0, b, x2, x1, x0) \
4820 Two_Sum(a0, b , _i, x0); \
4821 Two_Sum(a1, _i, x2, x1)
4822
4823#define Two_One_Diff(a1, a0, b, x2, x1, x0) \
4824 Two_Diff(a0, b , _i, x0); \
4825 Two_Sum( a1, _i, x2, x1)
4826
4827#define Two_Two_Sum(a1, a0, b1, b0, x3, x2, x1, x0) \
4828 Two_One_Sum(a1, a0, b0, _j, _0, x0); \
4829 Two_One_Sum(_j, _0, b1, x3, x2, x1)
4830
4831#define Two_Two_Diff(a1, a0, b1, b0, x3, x2, x1, x0) \
4832 Two_One_Diff(a1, a0, b0, _j, _0, x0); \
4833 Two_One_Diff(_j, _0, b1, x3, x2, x1)
4834
4835/* Macro for multiplying a two-component expansion by a single component. */
4836
4837#define Two_One_Product(a1, a0, b, x3, x2, x1, x0) \
4838 Split(b, bhi, blo); \
4839 Two_Product_Presplit(a0, b, bhi, blo, _i, x0); \
4840 Two_Product_Presplit(a1, b, bhi, blo, _j, _0); \
4841 Two_Sum(_i, _0, _k, x1); \
4842 Fast_Two_Sum(_j, _k, x3, x2)
4843
4844/*****************************************************************************/
4845/* */
4846/* exactinit() Initialize the variables used for exact arithmetic. */
4847/* */
4848/* `epsilon' is the largest power of two such that 1.0 + epsilon = 1.0 in */
4849/* floating-point arithmetic. `epsilon' bounds the relative roundoff */
4850/* error. It is used for floating-point error analysis. */
4851/* */
4852/* `splitter' is used to split floating-point numbers into two half- */
4853/* length significands for exact multiplication. */
4854/* */
4855/* I imagine that a highly optimizing compiler might be too smart for its */
4856/* own good, and somehow cause this routine to fail, if it pretends that */
4857/* floating-point arithmetic is too much like real arithmetic. */
4858/* */
4859/* Don't change this routine unless you fully understand it. */
4860/* */
4861/*****************************************************************************/
4862
4864{
4865 REAL half;
4866 REAL check, lastcheck;
4867 int every_other;
4868#ifdef LINUX
4869 int cword;
4870#endif /* LINUX */
4871
4872#ifdef CPU86
4873#ifdef SINGLE
4874 _control87(_PC_24, _MCW_PC); /* Set FPU control word for single precision. */
4875#else /* not SINGLE */
4876 _control87(_PC_53, _MCW_PC); /* Set FPU control word for double precision. */
4877#endif /* not SINGLE */
4878#endif /* CPU86 */
4879#ifdef LINUX
4880#ifdef SINGLE
4881 /* cword = 4223; */
4882 cword = 4210; /* set FPU control word for single precision */
4883#else /* not SINGLE */
4884 /* cword = 4735; */
4885 cword = 4722; /* set FPU control word for double precision */
4886#endif /* not SINGLE */
4887 _FPU_SETCW(cword);
4888#endif /* LINUX */
4889
4890 every_other = 1;
4891 half = 0.5;
4892 epsilon = 1.0;
4893 splitter = 1.0;
4894 check = 1.0;
4895 /* Repeatedly divide `epsilon' by two until it is too small to add to */
4896 /* one without causing roundoff. (Also check if the sum is equal to */
4897 /* the previous sum, for machines that round up instead of using exact */
4898 /* rounding. Not that these routines will work on such machines.) */
4899 do {
4900 lastcheck = check;
4901 epsilon *= half;
4902 if (every_other) {
4903 splitter *= 2.0;
4904 }
4905 every_other = !every_other;
4906 check = 1.0 + epsilon;
4907 } while ((check != 1.0) && (check != lastcheck));
4908 splitter += 1.0;
4909 /* Error bounds for orientation and incircle tests. */
4910 resulterrbound = (3.0 + 8.0 * epsilon) * epsilon;
4911 ccwerrboundA = (3.0 + 16.0 * epsilon) * epsilon;
4912 ccwerrboundB = (2.0 + 12.0 * epsilon) * epsilon;
4913 ccwerrboundC = (9.0 + 64.0 * epsilon) * epsilon * epsilon;
4914 iccerrboundA = (10.0 + 96.0 * epsilon) * epsilon;
4915 iccerrboundB = (4.0 + 48.0 * epsilon) * epsilon;
4916 iccerrboundC = (44.0 + 576.0 * epsilon) * epsilon * epsilon;
4917 o3derrboundA = (7.0 + 56.0 * epsilon) * epsilon;
4918 o3derrboundB = (3.0 + 28.0 * epsilon) * epsilon;
4919 o3derrboundC = (26.0 + 288.0 * epsilon) * epsilon * epsilon;
4920}
4921
4922/*****************************************************************************/
4923/* */
4924/* fast_expansion_sum_zeroelim() Sum two expansions, eliminating zero */
4925/* components from the output expansion. */
4926/* */
4927/* Sets h = e + f. See my Robust Predicates paper for details. */
4928/* */
4929/* If round-to-even is used (as with IEEE 754), maintains the strongly */
4930/* nonoverlapping property. (That is, if e is strongly nonoverlapping, h */
4931/* will be also.) Does NOT maintain the nonoverlapping or nonadjacent */
4932/* properties. */
4933/* */
4934/*****************************************************************************/
4935
4936#ifdef ANSI_DECLARATORS
4937int fast_expansion_sum_zeroelim(int elen, REAL *e, int flen, REAL *f, REAL *h)
4938#else /* not ANSI_DECLARATORS */
4939int fast_expansion_sum_zeroelim(elen, e, flen, f, h) /* h cannot be e or f. */
4940int elen;
4941REAL *e;
4942int flen;
4943REAL *f;
4944REAL *h;
4945#endif /* not ANSI_DECLARATORS */
4946
4947{
4948 REAL Q;
4949 INEXACT REAL Qnew;
4950 INEXACT REAL hh;
4951 INEXACT REAL bvirt;
4952 REAL avirt, bround, around;
4953 int eindex, findex, hindex;
4954 REAL enow, fnow;
4955
4956 enow = e[0];
4957 fnow = f[0];
4958 eindex = findex = 0;
4959 if ((fnow > enow) == (fnow > -enow)) {
4960 Q = enow;
4961 enow = e[++eindex];
4962 } else {
4963 Q = fnow;
4964 fnow = f[++findex];
4965 }
4966 hindex = 0;
4967 if ((eindex < elen) && (findex < flen)) {
4968 if ((fnow > enow) == (fnow > -enow)) {
4969 Fast_Two_Sum(enow, Q, Qnew, hh);
4970 enow = e[++eindex];
4971 } else {
4972 Fast_Two_Sum(fnow, Q, Qnew, hh);
4973 fnow = f[++findex];
4974 }
4975 Q = Qnew;
4976 if (hh != 0.0) {
4977 h[hindex++] = hh;
4978 }
4979 while ((eindex < elen) && (findex < flen)) {
4980 if ((fnow > enow) == (fnow > -enow)) {
4981 Two_Sum(Q, enow, Qnew, hh);
4982 enow = e[++eindex];
4983 } else {
4984 Two_Sum(Q, fnow, Qnew, hh);
4985 fnow = f[++findex];
4986 }
4987 Q = Qnew;
4988 if (hh != 0.0) {
4989 h[hindex++] = hh;
4990 }
4991 }
4992 }
4993 while (eindex < elen) {
4994 Two_Sum(Q, enow, Qnew, hh);
4995 enow = e[++eindex];
4996 Q = Qnew;
4997 if (hh != 0.0) {
4998 h[hindex++] = hh;
4999 }
5000 }
5001 while (findex < flen) {
5002 Two_Sum(Q, fnow, Qnew, hh);
5003 fnow = f[++findex];
5004 Q = Qnew;
5005 if (hh != 0.0) {
5006 h[hindex++] = hh;
5007 }
5008 }
5009 if ((Q != 0.0) || (hindex == 0)) {
5010 h[hindex++] = Q;
5011 }
5012 return hindex;
5013}
5014
5015/*****************************************************************************/
5016/* */
5017/* scale_expansion_zeroelim() Multiply an expansion by a scalar, */
5018/* eliminating zero components from the */
5019/* output expansion. */
5020/* */
5021/* Sets h = be. See my Robust Predicates paper for details. */
5022/* */
5023/* Maintains the nonoverlapping property. If round-to-even is used (as */
5024/* with IEEE 754), maintains the strongly nonoverlapping and nonadjacent */
5025/* properties as well. (That is, if e has one of these properties, so */
5026/* will h.) */
5027/* */
5028/*****************************************************************************/
5029
5030#ifdef ANSI_DECLARATORS
5032#else /* not ANSI_DECLARATORS */
5033int scale_expansion_zeroelim(elen, e, b, h) /* e and h cannot be the same. */
5034int elen;
5035REAL *e;
5036REAL b;
5037REAL *h;
5038#endif /* not ANSI_DECLARATORS */
5039
5040{
5041 INEXACT REAL Q, sum;
5042 REAL hh;
5043 INEXACT REAL product1;
5044 REAL product0;
5045 int eindex, hindex;
5046 REAL enow;
5047 INEXACT REAL bvirt;
5048 REAL avirt, bround, around;
5049 INEXACT REAL c;
5050 INEXACT REAL abig;
5051 REAL ahi, alo, bhi, blo;
5052 REAL err1, err2, err3;
5053
5054 Split(b, bhi, blo);
5055 Two_Product_Presplit(e[0], b, bhi, blo, Q, hh);
5056 hindex = 0;
5057 if (hh != 0) {
5058 h[hindex++] = hh;
5059 }
5060 for (eindex = 1; eindex < elen; eindex++) {
5061 enow = e[eindex];
5062 Two_Product_Presplit(enow, b, bhi, blo, product1, product0);
5063 Two_Sum(Q, product0, sum, hh);
5064 if (hh != 0) {
5065 h[hindex++] = hh;
5066 }
5067 Fast_Two_Sum(product1, sum, Q, hh);
5068 if (hh != 0) {
5069 h[hindex++] = hh;
5070 }
5071 }
5072 if ((Q != 0.0) || (hindex == 0)) {
5073 h[hindex++] = Q;
5074 }
5075 return hindex;
5076}
5077
5078/*****************************************************************************/
5079/* */
5080/* estimate() Produce a one-word estimate of an expansion's value. */
5081/* */
5082/* See my Robust Predicates paper for details. */
5083/* */
5084/*****************************************************************************/
5085
5086#ifdef ANSI_DECLARATORS
5087REAL estimate(int elen, REAL *e)
5088#else /* not ANSI_DECLARATORS */
5089REAL estimate(elen, e)
5090int elen;
5091REAL *e;
5092#endif /* not ANSI_DECLARATORS */
5093
5094{
5095 REAL Q;
5096 int eindex;
5097
5098 Q = e[0];
5099 for (eindex = 1; eindex < elen; eindex++) {
5100 Q += e[eindex];
5101 }
5102 return Q;
5103}
5104
5105/*****************************************************************************/
5106/* */
5107/* counterclockwise() Return a positive value if the points pa, pb, and */
5108/* pc occur in counterclockwise order; a negative */
5109/* value if they occur in clockwise order; and zero */
5110/* if they are collinear. The result is also a rough */
5111/* approximation of twice the signed area of the */
5112/* triangle defined by the three points. */
5113/* */
5114/* Uses exact arithmetic if necessary to ensure a correct answer. The */
5115/* result returned is the determinant of a matrix. This determinant is */
5116/* computed adaptively, in the sense that exact arithmetic is used only to */
5117/* the degree it is needed to ensure that the returned value has the */
5118/* correct sign. Hence, this function is usually quite fast, but will run */
5119/* more slowly when the input points are collinear or nearly so. */
5120/* */
5121/* See my Robust Predicates paper for details. */
5122/* */
5123/*****************************************************************************/
5124
5125#ifdef ANSI_DECLARATORS
5127#else /* not ANSI_DECLARATORS */
5128REAL counterclockwiseadapt(pa, pb, pc, detsum)
5129vertex pa;
5130vertex pb;
5131vertex pc;
5132REAL detsum;
5133#endif /* not ANSI_DECLARATORS */
5134
5135{
5136 INEXACT REAL acx, acy, bcx, bcy;
5137 REAL acxtail, acytail, bcxtail, bcytail;
5138 INEXACT REAL detleft, detright;
5139 REAL detlefttail, detrighttail;
5140 REAL det, errbound;
5141 REAL B[4], C1[8], C2[12], D[16];
5142 INEXACT REAL B3;
5143 int C1length, C2length, Dlength;
5144 REAL u[4];
5145 INEXACT REAL u3;
5146 INEXACT REAL s1, t1;
5147 REAL s0, t0;
5148
5149 INEXACT REAL bvirt;
5150 REAL avirt, bround, around;
5151 INEXACT REAL c;
5152 INEXACT REAL abig;
5153 REAL ahi, alo, bhi, blo;
5154 REAL err1, err2, err3;
5155 INEXACT REAL _i, _j;
5156 REAL _0;
5157
5158 acx = (REAL) (pa[0] - pc[0]);
5159 bcx = (REAL) (pb[0] - pc[0]);
5160 acy = (REAL) (pa[1] - pc[1]);
5161 bcy = (REAL) (pb[1] - pc[1]);
5162
5163 Two_Product(acx, bcy, detleft, detlefttail);
5164 Two_Product(acy, bcx, detright, detrighttail);
5165
5166 Two_Two_Diff(detleft, detlefttail, detright, detrighttail,
5167 B3, B[2], B[1], B[0]);
5168 B[3] = B3;
5169
5170 det = estimate(4, B);
5171 errbound = ccwerrboundB * detsum;
5172 if ((det >= errbound) || (-det >= errbound)) {
5173 return det;
5174 }
5175
5176 Two_Diff_Tail(pa[0], pc[0], acx, acxtail);
5177 Two_Diff_Tail(pb[0], pc[0], bcx, bcxtail);
5178 Two_Diff_Tail(pa[1], pc[1], acy, acytail);
5179 Two_Diff_Tail(pb[1], pc[1], bcy, bcytail);
5180
5181 if ((acxtail == 0.0) && (acytail == 0.0)
5182 && (bcxtail == 0.0) && (bcytail == 0.0)) {
5183 return det;
5184 }
5185
5186 errbound = ccwerrboundC * detsum + resulterrbound * Absolute(det);
5187 det += (acx * bcytail + bcy * acxtail)
5188 - (acy * bcxtail + bcx * acytail);
5189 if ((det >= errbound) || (-det >= errbound)) {
5190 return det;
5191 }
5192
5193 Two_Product(acxtail, bcy, s1, s0);
5194 Two_Product(acytail, bcx, t1, t0);
5195 Two_Two_Diff(s1, s0, t1, t0, u3, u[2], u[1], u[0]);
5196 u[3] = u3;
5197 C1length = fast_expansion_sum_zeroelim(4, B, 4, u, C1);
5198
5199 Two_Product(acx, bcytail, s1, s0);
5200 Two_Product(acy, bcxtail, t1, t0);
5201 Two_Two_Diff(s1, s0, t1, t0, u3, u[2], u[1], u[0]);
5202 u[3] = u3;
5203 C2length = fast_expansion_sum_zeroelim(C1length, C1, 4, u, C2);
5204
5205 Two_Product(acxtail, bcytail, s1, s0);
5206 Two_Product(acytail, bcxtail, t1, t0);
5207 Two_Two_Diff(s1, s0, t1, t0, u3, u[2], u[1], u[0]);
5208 u[3] = u3;
5209 Dlength = fast_expansion_sum_zeroelim(C2length, C2, 4, u, D);
5210
5211 return(D[Dlength - 1]);
5212}
5213
5214#ifdef ANSI_DECLARATORS
5215REAL counterclockwise(struct mesh *m, struct behavior *b,
5216 vertex pa, vertex pb, vertex pc)
5217#else /* not ANSI_DECLARATORS */
5218REAL counterclockwise(m, b, pa, pb, pc)
5219struct mesh *m;
5220struct behavior *b;
5221vertex pa;
5222vertex pb;
5223vertex pc;
5224#endif /* not ANSI_DECLARATORS */
5225
5226{
5227 REAL detleft, detright, det;
5228 REAL detsum, errbound;
5229
5230 m->counterclockcount++;
5231
5232 detleft = (pa[0] - pc[0]) * (pb[1] - pc[1]);
5233 detright = (pa[1] - pc[1]) * (pb[0] - pc[0]);
5234 det = detleft - detright;
5235
5236 if (b->noexact) {
5237 return det;
5238 }
5239
5240 if (detleft > 0.0) {
5241 if (detright <= 0.0) {
5242 return det;
5243 } else {
5244 detsum = detleft + detright;
5245 }
5246 } else if (detleft < 0.0) {
5247 if (detright >= 0.0) {
5248 return det;
5249 } else {
5250 detsum = -detleft - detright;
5251 }
5252 } else {
5253 return det;
5254 }
5255
5256 errbound = ccwerrboundA * detsum;
5257 if ((det >= errbound) || (-det >= errbound)) {
5258 return det;
5259 }
5260
5261 return counterclockwiseadapt(pa, pb, pc, detsum);
5262}
5263
5264/*****************************************************************************/
5265/* */
5266/* incircle() Return a positive value if the point pd lies inside the */
5267/* circle passing through pa, pb, and pc; a negative value if */
5268/* it lies outside; and zero if the four points are cocircular.*/
5269/* The points pa, pb, and pc must be in counterclockwise */
5270/* order, or the sign of the result will be reversed. */
5271/* */
5272/* Uses exact arithmetic if necessary to ensure a correct answer. The */
5273/* result returned is the determinant of a matrix. This determinant is */
5274/* computed adaptively, in the sense that exact arithmetic is used only to */
5275/* the degree it is needed to ensure that the returned value has the */
5276/* correct sign. Hence, this function is usually quite fast, but will run */
5277/* more slowly when the input points are cocircular or nearly so. */
5278/* */
5279/* See my Robust Predicates paper for details. */
5280/* */
5281/*****************************************************************************/
5282
5283#ifdef ANSI_DECLARATORS
5285#else /* not ANSI_DECLARATORS */
5286REAL incircleadapt(pa, pb, pc, pd, permanent)
5287vertex pa;
5288vertex pb;
5289vertex pc;
5290vertex pd;
5291REAL permanent;
5292#endif /* not ANSI_DECLARATORS */
5293
5294{
5295 INEXACT REAL adx, bdx, cdx, ady, bdy, cdy;
5296 REAL det, errbound;
5297
5298 INEXACT REAL bdxcdy1, cdxbdy1, cdxady1, adxcdy1, adxbdy1, bdxady1;
5299 REAL bdxcdy0, cdxbdy0, cdxady0, adxcdy0, adxbdy0, bdxady0;
5300 REAL bc[4], ca[4], ab[4];
5301 INEXACT REAL bc3, ca3, ab3;
5302 REAL axbc[8], axxbc[16], aybc[8], ayybc[16], adet[32];
5303 int axbclen, axxbclen, aybclen, ayybclen, alen;
5304 REAL bxca[8], bxxca[16], byca[8], byyca[16], bdet[32];
5305 int bxcalen, bxxcalen, bycalen, byycalen, blen;
5306 REAL cxab[8], cxxab[16], cyab[8], cyyab[16], cdet[32];
5307 int cxablen, cxxablen, cyablen, cyyablen, clen;
5308 REAL abdet[64];
5309 int ablen;
5310 REAL fin1[1152], fin2[1152];
5311 REAL *finnow, *finother, *finswap;
5312 int finlength;
5313
5314 REAL adxtail, bdxtail, cdxtail, adytail, bdytail, cdytail;
5315 INEXACT REAL adxadx1, adyady1, bdxbdx1, bdybdy1, cdxcdx1, cdycdy1;
5316 REAL adxadx0, adyady0, bdxbdx0, bdybdy0, cdxcdx0, cdycdy0;
5317 REAL aa[4], bb[4], cc[4];
5318 INEXACT REAL aa3, bb3, cc3;
5319 INEXACT REAL ti1, tj1;
5320 REAL ti0, tj0;
5321 REAL u[4], v[4];
5322 INEXACT REAL u3, v3;
5323 REAL temp8[8], temp16a[16], temp16b[16], temp16c[16];
5324 REAL temp32a[32], temp32b[32], temp48[48], temp64[64];
5325 int temp8len, temp16alen, temp16blen, temp16clen;
5326 int temp32alen, temp32blen, temp48len, temp64len;
5327 REAL axtbb[8], axtcc[8], aytbb[8], aytcc[8];
5328 int axtbblen, axtcclen, aytbblen, aytcclen;
5329 REAL bxtaa[8], bxtcc[8], bytaa[8], bytcc[8];
5330 int bxtaalen, bxtcclen, bytaalen, bytcclen;
5331 REAL cxtaa[8], cxtbb[8], cytaa[8], cytbb[8];
5332 int cxtaalen, cxtbblen, cytaalen, cytbblen;
5333 REAL axtbc[8], aytbc[8], bxtca[8], bytca[8], cxtab[8], cytab[8];
5334 int axtbclen, aytbclen, bxtcalen, bytcalen, cxtablen, cytablen;
5335 REAL axtbct[16], aytbct[16], bxtcat[16], bytcat[16], cxtabt[16], cytabt[16];
5336 int axtbctlen, aytbctlen, bxtcatlen, bytcatlen, cxtabtlen, cytabtlen;
5337 REAL axtbctt[8], aytbctt[8], bxtcatt[8];
5338 REAL bytcatt[8], cxtabtt[8], cytabtt[8];
5339 int axtbcttlen, aytbcttlen, bxtcattlen, bytcattlen, cxtabttlen, cytabttlen;
5340 REAL abt[8], bct[8], cat[8];
5341 int abtlen, bctlen, catlen;
5342 REAL abtt[4], bctt[4], catt[4];
5343 int abttlen, bcttlen, cattlen;
5344 INEXACT REAL abtt3, bctt3, catt3;
5345 REAL negate;
5346
5347 INEXACT REAL bvirt;
5348 REAL avirt, bround, around;
5349 INEXACT REAL c;
5350 INEXACT REAL abig;
5351 REAL ahi, alo, bhi, blo;
5352 REAL err1, err2, err3;
5353 INEXACT REAL _i, _j;
5354 REAL _0;
5355
5356 adx = (REAL) (pa[0] - pd[0]);
5357 bdx = (REAL) (pb[0] - pd[0]);
5358 cdx = (REAL) (pc[0] - pd[0]);
5359 ady = (REAL) (pa[1] - pd[1]);
5360 bdy = (REAL) (pb[1] - pd[1]);
5361 cdy = (REAL) (pc[1] - pd[1]);
5362
5363 Two_Product(bdx, cdy, bdxcdy1, bdxcdy0);
5364 Two_Product(cdx, bdy, cdxbdy1, cdxbdy0);
5365 Two_Two_Diff(bdxcdy1, bdxcdy0, cdxbdy1, cdxbdy0, bc3, bc[2], bc[1], bc[0]);
5366 bc[3] = bc3;
5367 axbclen = scale_expansion_zeroelim(4, bc, adx, axbc);
5368 axxbclen = scale_expansion_zeroelim(axbclen, axbc, adx, axxbc);
5369 aybclen = scale_expansion_zeroelim(4, bc, ady, aybc);
5370 ayybclen = scale_expansion_zeroelim(aybclen, aybc, ady, ayybc);
5371 alen = fast_expansion_sum_zeroelim(axxbclen, axxbc, ayybclen, ayybc, adet);
5372
5373 Two_Product(cdx, ady, cdxady1, cdxady0);
5374 Two_Product(adx, cdy, adxcdy1, adxcdy0);
5375 Two_Two_Diff(cdxady1, cdxady0, adxcdy1, adxcdy0, ca3, ca[2], ca[1], ca[0]);
5376 ca[3] = ca3;
5377 bxcalen = scale_expansion_zeroelim(4, ca, bdx, bxca);
5378 bxxcalen = scale_expansion_zeroelim(bxcalen, bxca, bdx, bxxca);
5379 bycalen = scale_expansion_zeroelim(4, ca, bdy, byca);
5380 byycalen = scale_expansion_zeroelim(bycalen, byca, bdy, byyca);
5381 blen = fast_expansion_sum_zeroelim(bxxcalen, bxxca, byycalen, byyca, bdet);
5382
5383 Two_Product(adx, bdy, adxbdy1, adxbdy0);
5384 Two_Product(bdx, ady, bdxady1, bdxady0);
5385 Two_Two_Diff(adxbdy1, adxbdy0, bdxady1, bdxady0, ab3, ab[2], ab[1], ab[0]);
5386 ab[3] = ab3;
5387 cxablen = scale_expansion_zeroelim(4, ab, cdx, cxab);
5388 cxxablen = scale_expansion_zeroelim(cxablen, cxab, cdx, cxxab);
5389 cyablen = scale_expansion_zeroelim(4, ab, cdy, cyab);
5390 cyyablen = scale_expansion_zeroelim(cyablen, cyab, cdy, cyyab);
5391 clen = fast_expansion_sum_zeroelim(cxxablen, cxxab, cyyablen, cyyab, cdet);
5392
5393 ablen = fast_expansion_sum_zeroelim(alen, adet, blen, bdet, abdet);
5394 finlength = fast_expansion_sum_zeroelim(ablen, abdet, clen, cdet, fin1);
5395
5396 det = estimate(finlength, fin1);
5397 errbound = iccerrboundB * permanent;
5398 if ((det >= errbound) || (-det >= errbound)) {
5399 return det;
5400 }
5401
5402 Two_Diff_Tail(pa[0], pd[0], adx, adxtail);
5403 Two_Diff_Tail(pa[1], pd[1], ady, adytail);
5404 Two_Diff_Tail(pb[0], pd[0], bdx, bdxtail);
5405 Two_Diff_Tail(pb[1], pd[1], bdy, bdytail);
5406 Two_Diff_Tail(pc[0], pd[0], cdx, cdxtail);
5407 Two_Diff_Tail(pc[1], pd[1], cdy, cdytail);
5408 if ((adxtail == 0.0) && (bdxtail == 0.0) && (cdxtail == 0.0)
5409 && (adytail == 0.0) && (bdytail == 0.0) && (cdytail == 0.0)) {
5410 return det;
5411 }
5412
5413 errbound = iccerrboundC * permanent + resulterrbound * Absolute(det);
5414 det += ((adx * adx + ady * ady) * ((bdx * cdytail + cdy * bdxtail)
5415 - (bdy * cdxtail + cdx * bdytail))
5416 + 2.0 * (adx * adxtail + ady * adytail) * (bdx * cdy - bdy * cdx))
5417 + ((bdx * bdx + bdy * bdy) * ((cdx * adytail + ady * cdxtail)
5418 - (cdy * adxtail + adx * cdytail))
5419 + 2.0 * (bdx * bdxtail + bdy * bdytail) * (cdx * ady - cdy * adx))
5420 + ((cdx * cdx + cdy * cdy) * ((adx * bdytail + bdy * adxtail)
5421 - (ady * bdxtail + bdx * adytail))
5422 + 2.0 * (cdx * cdxtail + cdy * cdytail) * (adx * bdy - ady * bdx));
5423 if ((det >= errbound) || (-det >= errbound)) {
5424 return det;
5425 }
5426
5427 finnow = fin1;
5428 finother = fin2;
5429
5430 if ((bdxtail != 0.0) || (bdytail != 0.0)
5431 || (cdxtail != 0.0) || (cdytail != 0.0)) {
5432 Square(adx, adxadx1, adxadx0);
5433 Square(ady, adyady1, adyady0);
5434 Two_Two_Sum(adxadx1, adxadx0, adyady1, adyady0, aa3, aa[2], aa[1], aa[0]);
5435 aa[3] = aa3;
5436 }
5437 if ((cdxtail != 0.0) || (cdytail != 0.0)
5438 || (adxtail != 0.0) || (adytail != 0.0)) {
5439 Square(bdx, bdxbdx1, bdxbdx0);
5440 Square(bdy, bdybdy1, bdybdy0);
5441 Two_Two_Sum(bdxbdx1, bdxbdx0, bdybdy1, bdybdy0, bb3, bb[2], bb[1], bb[0]);
5442 bb[3] = bb3;
5443 }
5444 if ((adxtail != 0.0) || (adytail != 0.0)
5445 || (bdxtail != 0.0) || (bdytail != 0.0)) {
5446 Square(cdx, cdxcdx1, cdxcdx0);
5447 Square(cdy, cdycdy1, cdycdy0);
5448 Two_Two_Sum(cdxcdx1, cdxcdx0, cdycdy1, cdycdy0, cc3, cc[2], cc[1], cc[0]);
5449 cc[3] = cc3;
5450 }
5451
5452 if (adxtail != 0.0) {
5453 axtbclen = scale_expansion_zeroelim(4, bc, adxtail, axtbc);
5454 temp16alen = scale_expansion_zeroelim(axtbclen, axtbc, 2.0 * adx,
5455 temp16a);
5456
5457 axtcclen = scale_expansion_zeroelim(4, cc, adxtail, axtcc);
5458 temp16blen = scale_expansion_zeroelim(axtcclen, axtcc, bdy, temp16b);
5459
5460 axtbblen = scale_expansion_zeroelim(4, bb, adxtail, axtbb);
5461 temp16clen = scale_expansion_zeroelim(axtbblen, axtbb, -cdy, temp16c);
5462
5463 temp32alen = fast_expansion_sum_zeroelim(temp16alen, temp16a,
5464 temp16blen, temp16b, temp32a);
5465 temp48len = fast_expansion_sum_zeroelim(temp16clen, temp16c,
5466 temp32alen, temp32a, temp48);
5467 finlength = fast_expansion_sum_zeroelim(finlength, finnow, temp48len,
5468 temp48, finother);
5469 finswap = finnow; finnow = finother; finother = finswap;
5470 }
5471 if (adytail != 0.0) {
5472 aytbclen = scale_expansion_zeroelim(4, bc, adytail, aytbc);
5473 temp16alen = scale_expansion_zeroelim(aytbclen, aytbc, 2.0 * ady,
5474 temp16a);
5475
5476 aytbblen = scale_expansion_zeroelim(4, bb, adytail, aytbb);
5477 temp16blen = scale_expansion_zeroelim(aytbblen, aytbb, cdx, temp16b);
5478
5479 aytcclen = scale_expansion_zeroelim(4, cc, adytail, aytcc);
5480 temp16clen = scale_expansion_zeroelim(aytcclen, aytcc, -bdx, temp16c);
5481
5482 temp32alen = fast_expansion_sum_zeroelim(temp16alen, temp16a,
5483 temp16blen, temp16b, temp32a);
5484 temp48len = fast_expansion_sum_zeroelim(temp16clen, temp16c,
5485 temp32alen, temp32a, temp48);
5486 finlength = fast_expansion_sum_zeroelim(finlength, finnow, temp48len,
5487 temp48, finother);
5488 finswap = finnow; finnow = finother; finother = finswap;
5489 }
5490 if (bdxtail != 0.0) {
5491 bxtcalen = scale_expansion_zeroelim(4, ca, bdxtail, bxtca);
5492 temp16alen = scale_expansion_zeroelim(bxtcalen, bxtca, 2.0 * bdx,
5493 temp16a);
5494
5495 bxtaalen = scale_expansion_zeroelim(4, aa, bdxtail, bxtaa);
5496 temp16blen = scale_expansion_zeroelim(bxtaalen, bxtaa, cdy, temp16b);
5497
5498 bxtcclen = scale_expansion_zeroelim(4, cc, bdxtail, bxtcc);
5499 temp16clen = scale_expansion_zeroelim(bxtcclen, bxtcc, -ady, temp16c);
5500
5501 temp32alen = fast_expansion_sum_zeroelim(temp16alen, temp16a,
5502 temp16blen, temp16b, temp32a);
5503 temp48len = fast_expansion_sum_zeroelim(temp16clen, temp16c,
5504 temp32alen, temp32a, temp48);
5505 finlength = fast_expansion_sum_zeroelim(finlength, finnow, temp48len,
5506 temp48, finother);
5507 finswap = finnow; finnow = finother; finother = finswap;
5508 }
5509 if (bdytail != 0.0) {
5510 bytcalen = scale_expansion_zeroelim(4, ca, bdytail, bytca);
5511 temp16alen = scale_expansion_zeroelim(bytcalen, bytca, 2.0 * bdy,
5512 temp16a);
5513
5514 bytcclen = scale_expansion_zeroelim(4, cc, bdytail, bytcc);
5515 temp16blen = scale_expansion_zeroelim(bytcclen, bytcc, adx, temp16b);
5516
5517 bytaalen = scale_expansion_zeroelim(4, aa, bdytail, bytaa);
5518 temp16clen = scale_expansion_zeroelim(bytaalen, bytaa, -cdx, temp16c);
5519
5520 temp32alen = fast_expansion_sum_zeroelim(temp16alen, temp16a,
5521 temp16blen, temp16b, temp32a);
5522 temp48len = fast_expansion_sum_zeroelim(temp16clen, temp16c,
5523 temp32alen, temp32a, temp48);
5524 finlength = fast_expansion_sum_zeroelim(finlength, finnow, temp48len,
5525 temp48, finother);
5526 finswap = finnow; finnow = finother; finother = finswap;
5527 }
5528 if (cdxtail != 0.0) {
5529 cxtablen = scale_expansion_zeroelim(4, ab, cdxtail, cxtab);
5530 temp16alen = scale_expansion_zeroelim(cxtablen, cxtab, 2.0 * cdx,
5531 temp16a);
5532
5533 cxtbblen = scale_expansion_zeroelim(4, bb, cdxtail, cxtbb);
5534 temp16blen = scale_expansion_zeroelim(cxtbblen, cxtbb, ady, temp16b);
5535
5536 cxtaalen = scale_expansion_zeroelim(4, aa, cdxtail, cxtaa);
5537 temp16clen = scale_expansion_zeroelim(cxtaalen, cxtaa, -bdy, temp16c);
5538
5539 temp32alen = fast_expansion_sum_zeroelim(temp16alen, temp16a,
5540 temp16blen, temp16b, temp32a);
5541 temp48len = fast_expansion_sum_zeroelim(temp16clen, temp16c,
5542 temp32alen, temp32a, temp48);
5543 finlength = fast_expansion_sum_zeroelim(finlength, finnow, temp48len,
5544 temp48, finother);
5545 finswap = finnow; finnow = finother; finother = finswap;
5546 }
5547 if (cdytail != 0.0) {
5548 cytablen = scale_expansion_zeroelim(4, ab, cdytail, cytab);
5549 temp16alen = scale_expansion_zeroelim(cytablen, cytab, 2.0 * cdy,
5550 temp16a);
5551
5552 cytaalen = scale_expansion_zeroelim(4, aa, cdytail, cytaa);
5553 temp16blen = scale_expansion_zeroelim(cytaalen, cytaa, bdx, temp16b);
5554
5555 cytbblen = scale_expansion_zeroelim(4, bb, cdytail, cytbb);
5556 temp16clen = scale_expansion_zeroelim(cytbblen, cytbb, -adx, temp16c);
5557
5558 temp32alen = fast_expansion_sum_zeroelim(temp16alen, temp16a,
5559 temp16blen, temp16b, temp32a);
5560 temp48len = fast_expansion_sum_zeroelim(temp16clen, temp16c,
5561 temp32alen, temp32a, temp48);
5562 finlength = fast_expansion_sum_zeroelim(finlength, finnow, temp48len,
5563 temp48, finother);
5564 finswap = finnow; finnow = finother; finother = finswap;
5565 }
5566
5567 if ((adxtail != 0.0) || (adytail != 0.0)) {
5568 if ((bdxtail != 0.0) || (bdytail != 0.0)
5569 || (cdxtail != 0.0) || (cdytail != 0.0)) {
5570 Two_Product(bdxtail, cdy, ti1, ti0);
5571 Two_Product(bdx, cdytail, tj1, tj0);
5572 Two_Two_Sum(ti1, ti0, tj1, tj0, u3, u[2], u[1], u[0]);
5573 u[3] = u3;
5574 negate = -bdy;
5575 Two_Product(cdxtail, negate, ti1, ti0);
5576 negate = -bdytail;
5577 Two_Product(cdx, negate, tj1, tj0);
5578 Two_Two_Sum(ti1, ti0, tj1, tj0, v3, v[2], v[1], v[0]);
5579 v[3] = v3;
5580 bctlen = fast_expansion_sum_zeroelim(4, u, 4, v, bct);
5581
5582 Two_Product(bdxtail, cdytail, ti1, ti0);
5583 Two_Product(cdxtail, bdytail, tj1, tj0);
5584 Two_Two_Diff(ti1, ti0, tj1, tj0, bctt3, bctt[2], bctt[1], bctt[0]);
5585 bctt[3] = bctt3;
5586 bcttlen = 4;
5587 } else {
5588 bct[0] = 0.0;
5589 bctlen = 1;
5590 bctt[0] = 0.0;
5591 bcttlen = 1;
5592 }
5593
5594 if (adxtail != 0.0) {
5595 temp16alen = scale_expansion_zeroelim(axtbclen, axtbc, adxtail, temp16a);
5596 axtbctlen = scale_expansion_zeroelim(bctlen, bct, adxtail, axtbct);
5597 temp32alen = scale_expansion_zeroelim(axtbctlen, axtbct, 2.0 * adx,
5598 temp32a);
5599 temp48len = fast_expansion_sum_zeroelim(temp16alen, temp16a,
5600 temp32alen, temp32a, temp48);
5601 finlength = fast_expansion_sum_zeroelim(finlength, finnow, temp48len,
5602 temp48, finother);
5603 finswap = finnow; finnow = finother; finother = finswap;
5604 if (bdytail != 0.0) {
5605 temp8len = scale_expansion_zeroelim(4, cc, adxtail, temp8);
5606 temp16alen = scale_expansion_zeroelim(temp8len, temp8, bdytail,
5607 temp16a);
5608 finlength = fast_expansion_sum_zeroelim(finlength, finnow, temp16alen,
5609 temp16a, finother);
5610 finswap = finnow; finnow = finother; finother = finswap;
5611 }
5612 if (cdytail != 0.0) {
5613 temp8len = scale_expansion_zeroelim(4, bb, -adxtail, temp8);
5614 temp16alen = scale_expansion_zeroelim(temp8len, temp8, cdytail,
5615 temp16a);
5616 finlength = fast_expansion_sum_zeroelim(finlength, finnow, temp16alen,
5617 temp16a, finother);
5618 finswap = finnow; finnow = finother; finother = finswap;
5619 }
5620
5621 temp32alen = scale_expansion_zeroelim(axtbctlen, axtbct, adxtail,
5622 temp32a);
5623 axtbcttlen = scale_expansion_zeroelim(bcttlen, bctt, adxtail, axtbctt);
5624 temp16alen = scale_expansion_zeroelim(axtbcttlen, axtbctt, 2.0 * adx,
5625 temp16a);
5626 temp16blen = scale_expansion_zeroelim(axtbcttlen, axtbctt, adxtail,
5627 temp16b);
5628 temp32blen = fast_expansion_sum_zeroelim(temp16alen, temp16a,
5629 temp16blen, temp16b, temp32b);
5630 temp64len = fast_expansion_sum_zeroelim(temp32alen, temp32a,
5631 temp32blen, temp32b, temp64);
5632 finlength = fast_expansion_sum_zeroelim(finlength, finnow, temp64len,
5633 temp64, finother);
5634 finswap = finnow; finnow = finother; finother = finswap;
5635 }
5636 if (adytail != 0.0) {
5637 temp16alen = scale_expansion_zeroelim(aytbclen, aytbc, adytail, temp16a);
5638 aytbctlen = scale_expansion_zeroelim(bctlen, bct, adytail, aytbct);
5639 temp32alen = scale_expansion_zeroelim(aytbctlen, aytbct, 2.0 * ady,
5640 temp32a);
5641 temp48len = fast_expansion_sum_zeroelim(temp16alen, temp16a,
5642 temp32alen, temp32a, temp48);
5643 finlength = fast_expansion_sum_zeroelim(finlength, finnow, temp48len,
5644 temp48, finother);
5645 finswap = finnow; finnow = finother; finother = finswap;
5646
5647
5648 temp32alen = scale_expansion_zeroelim(aytbctlen, aytbct, adytail,
5649 temp32a);
5650 aytbcttlen = scale_expansion_zeroelim(bcttlen, bctt, adytail, aytbctt);
5651 temp16alen = scale_expansion_zeroelim(aytbcttlen, aytbctt, 2.0 * ady,
5652 temp16a);
5653 temp16blen = scale_expansion_zeroelim(aytbcttlen, aytbctt, adytail,
5654 temp16b);
5655 temp32blen = fast_expansion_sum_zeroelim(temp16alen, temp16a,
5656 temp16blen, temp16b, temp32b);
5657 temp64len = fast_expansion_sum_zeroelim(temp32alen, temp32a,
5658 temp32blen, temp32b, temp64);
5659 finlength = fast_expansion_sum_zeroelim(finlength, finnow, temp64len,
5660 temp64, finother);
5661 finswap = finnow; finnow = finother; finother = finswap;
5662 }
5663 }
5664 if ((bdxtail != 0.0) || (bdytail != 0.0)) {
5665 if ((cdxtail != 0.0) || (cdytail != 0.0)
5666 || (adxtail != 0.0) || (adytail != 0.0)) {
5667 Two_Product(cdxtail, ady, ti1, ti0);
5668 Two_Product(cdx, adytail, tj1, tj0);
5669 Two_Two_Sum(ti1, ti0, tj1, tj0, u3, u[2], u[1], u[0]);
5670 u[3] = u3;
5671 negate = -cdy;
5672 Two_Product(adxtail, negate, ti1, ti0);
5673 negate = -cdytail;
5674 Two_Product(adx, negate, tj1, tj0);
5675 Two_Two_Sum(ti1, ti0, tj1, tj0, v3, v[2], v[1], v[0]);
5676 v[3] = v3;
5677 catlen = fast_expansion_sum_zeroelim(4, u, 4, v, cat);
5678
5679 Two_Product(cdxtail, adytail, ti1, ti0);
5680 Two_Product(adxtail, cdytail, tj1, tj0);
5681 Two_Two_Diff(ti1, ti0, tj1, tj0, catt3, catt[2], catt[1], catt[0]);
5682 catt[3] = catt3;
5683 cattlen = 4;
5684 } else {
5685 cat[0] = 0.0;
5686 catlen = 1;
5687 catt[0] = 0.0;
5688 cattlen = 1;
5689 }
5690
5691 if (bdxtail != 0.0) {
5692 temp16alen = scale_expansion_zeroelim(bxtcalen, bxtca, bdxtail, temp16a);
5693 bxtcatlen = scale_expansion_zeroelim(catlen, cat, bdxtail, bxtcat);
5694 temp32alen = scale_expansion_zeroelim(bxtcatlen, bxtcat, 2.0 * bdx,
5695 temp32a);
5696 temp48len = fast_expansion_sum_zeroelim(temp16alen, temp16a,
5697 temp32alen, temp32a, temp48);
5698 finlength = fast_expansion_sum_zeroelim(finlength, finnow, temp48len,
5699 temp48, finother);
5700 finswap = finnow; finnow = finother; finother = finswap;
5701 if (cdytail != 0.0) {
5702 temp8len = scale_expansion_zeroelim(4, aa, bdxtail, temp8);
5703 temp16alen = scale_expansion_zeroelim(temp8len, temp8, cdytail,
5704 temp16a);
5705 finlength = fast_expansion_sum_zeroelim(finlength, finnow, temp16alen,
5706 temp16a, finoth