Logo ROOT  
Reference Guide
triangle.c
Go to the documentation of this file.
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#include <stdint.h>
320#ifndef NO_TIMER
321#include <sys/time.h>
322#endif /* not NO_TIMER */
323#ifdef CPU86
324#include <float.h>
325#endif /* CPU86 */
326#ifdef LINUX
327#include <fenv.h>
328#endif /* LINUX */
329#ifdef TRILIBRARY
330#include "triangle.h"
331#endif /* TRILIBRARY */
332
333/* A few forward declarations. */
334
335#ifndef TRILIBRARY
336char *readline();
337char *findfield();
338#endif /* not TRILIBRARY */
339
340/* Labels that signify the result of point location. The result of a */
341/* search indicates that the point falls in the interior of a triangle, on */
342/* an edge, on a vertex, or outside the mesh. */
343
345
346/* Labels that signify the result of vertex insertion. The result indicates */
347/* that the vertex was inserted with complete success, was inserted but */
348/* encroaches upon a subsegment, was not inserted because it lies on a */
349/* segment, or was not inserted because another vertex occupies the same */
350/* location. */
351
354
355/* Labels that signify the result of direction finding. The result */
356/* indicates that a segment connecting the two query points falls within */
357/* the direction triangle, along the left edge of the direction triangle, */
358/* or along the right edge of the direction triangle. */
359
361
362/*****************************************************************************/
363/* */
364/* The basic mesh data structures */
365/* */
366/* There are three: vertices, triangles, and subsegments (abbreviated */
367/* `subseg'). These three data structures, linked by pointers, comprise */
368/* the mesh. A vertex simply represents a mesh vertex and its properties. */
369/* A triangle is a triangle. A subsegment is a special data structure used */
370/* to represent an impenetrable edge of the mesh (perhaps on the outer */
371/* boundary, on the boundary of a hole, or part of an internal boundary */
372/* separating two triangulated regions). Subsegments represent boundaries, */
373/* defined by the user, that triangles may not lie across. */
374/* */
375/* A triangle consists of a list of three vertices, a list of three */
376/* adjoining triangles, a list of three adjoining subsegments (when */
377/* segments exist), an arbitrary number of optional user-defined */
378/* floating-point attributes, and an optional area constraint. The latter */
379/* is an upper bound on the permissible area of each triangle in a region, */
380/* used for mesh refinement. */
381/* */
382/* For a triangle on a boundary of the mesh, some or all of the neighboring */
383/* triangles may not be present. For a triangle in the interior of the */
384/* mesh, often no neighboring subsegments are present. Such absent */
385/* triangles and subsegments are never represented by NULL pointers; they */
386/* are represented by two special records: `dummytri', the triangle that */
387/* fills "outer space", and `dummysub', the omnipresent subsegment. */
388/* `dummytri' and `dummysub' are used for several reasons; for instance, */
389/* they can be dereferenced and their contents examined without violating */
390/* protected memory. */
391/* */
392/* However, it is important to understand that a triangle includes other */
393/* information as well. The pointers to adjoining vertices, triangles, and */
394/* subsegments are ordered in a way that indicates their geometric relation */
395/* to each other. Furthermore, each of these pointers contains orientation */
396/* information. Each pointer to an adjoining triangle indicates which face */
397/* of that triangle is contacted. Similarly, each pointer to an adjoining */
398/* subsegment indicates which side of that subsegment is contacted, and how */
399/* the subsegment is oriented relative to the triangle. */
400/* */
401/* The data structure representing a subsegment may be thought to be */
402/* abutting the edge of one or two triangle data structures: either */
403/* sandwiched between two triangles, or resting against one triangle on an */
404/* exterior boundary or hole boundary. */
405/* */
406/* A subsegment consists of a list of four vertices--the vertices of the */
407/* subsegment, and the vertices of the segment it is a part of--a list of */
408/* two adjoining subsegments, and a list of two adjoining triangles. One */
409/* of the two adjoining triangles may not be present (though there should */
410/* always be one), and neighboring subsegments might not be present. */
411/* Subsegments also store a user-defined integer "boundary marker". */
412/* Typically, this integer is used to indicate what boundary conditions are */
413/* to be applied at that location in a finite element simulation. */
414/* */
415/* Like triangles, subsegments maintain information about the relative */
416/* orientation of neighboring objects. */
417/* */
418/* Vertices are relatively simple. A vertex is a list of floating-point */
419/* numbers, starting with the x, and y coordinates, followed by an */
420/* arbitrary number of optional user-defined floating-point attributes, */
421/* followed by an integer boundary marker. During the segment insertion */
422/* phase, there is also a pointer from each vertex to a triangle that may */
423/* contain it. Each pointer is not always correct, but when one is, it */
424/* speeds up segment insertion. These pointers are assigned values once */
425/* at the beginning of the segment insertion phase, and are not used or */
426/* updated except during this phase. Edge flipping during segment */
427/* insertion will render some of them incorrect. Hence, don't rely upon */
428/* them for anything. */
429/* */
430/* Other than the exception mentioned above, vertices have no information */
431/* about what triangles, subfacets, or subsegments they are linked to. */
432/* */
433/*****************************************************************************/
434
435/*****************************************************************************/
436/* */
437/* Handles */
438/* */
439/* The oriented triangle (`otri') and oriented subsegment (`osub') data */
440/* structures defined below do not themselves store any part of the mesh. */
441/* The mesh itself is made of `triangle's, `subseg's, and `vertex's. */
442/* */
443/* Oriented triangles and oriented subsegments will usually be referred to */
444/* as "handles." A handle is essentially a pointer into the mesh; it */
445/* allows you to "hold" one particular part of the mesh. Handles are used */
446/* to specify the regions in which one is traversing and modifying the mesh.*/
447/* A single `triangle' may be held by many handles, or none at all. (The */
448/* latter case is not a memory leak, because the triangle is still */
449/* connected to other triangles in the mesh.) */
450/* */
451/* An `otri' is a handle that holds a triangle. It holds a specific edge */
452/* of the triangle. An `osub' is a handle that holds a subsegment. It */
453/* holds either the left or right side of the subsegment. */
454/* */
455/* Navigation about the mesh is accomplished through a set of mesh */
456/* manipulation primitives, further below. Many of these primitives take */
457/* a handle and produce a new handle that holds the mesh near the first */
458/* handle. Other primitives take two handles and glue the corresponding */
459/* parts of the mesh together. The orientation of the handles is */
460/* important. For instance, when two triangles are glued together by the */
461/* bond() primitive, they are glued at the edges on which the handles lie. */
462/* */
463/* Because vertices have no information about which triangles they are */
464/* attached to, I commonly represent a vertex by use of a handle whose */
465/* origin is the vertex. A single handle can simultaneously represent a */
466/* triangle, an edge, and a vertex. */
467/* */
468/*****************************************************************************/
469
470/* The triangle data structure. Each triangle contains three pointers to */
471/* adjoining triangles, plus three pointers to vertices, plus three */
472/* pointers to subsegments (declared below; these pointers are usually */
473/* `dummysub'). It may or may not also contain user-defined attributes */
474/* and/or a floating-point "area constraint." It may also contain extra */
475/* pointers for nodes, when the user asks for high-order elements. */
476/* Because the size and structure of a `triangle' is not decided until */
477/* runtime, I haven't simply declared the type `triangle' as a struct. */
478
479typedef REAL **triangle; /* Really: typedef triangle *triangle */
480
481/* An oriented triangle: includes a pointer to a triangle and orientation. */
482/* The orientation denotes an edge of the triangle. Hence, there are */
483/* three possible orientations. By convention, each edge always points */
484/* counterclockwise about the corresponding triangle. */
485
486struct otri {
488 int orient; /* Ranges from 0 to 2. */
489};
490
491/* The subsegment data structure. Each subsegment contains two pointers to */
492/* adjoining subsegments, plus four pointers to vertices, plus two */
493/* pointers to adjoining triangles, plus one boundary marker, plus one */
494/* segment number. */
495
496typedef REAL **subseg; /* Really: typedef subseg *subseg */
497
498/* An oriented subsegment: includes a pointer to a subsegment and an */
499/* orientation. The orientation denotes a side of the edge. Hence, there */
500/* are two possible orientations. By convention, the edge is always */
501/* directed so that the "side" denoted is the right side of the edge. */
502
503struct osub {
505 int ssorient; /* Ranges from 0 to 1. */
506};
507
508/* The vertex data structure. Each vertex is actually an array of REALs. */
509/* The number of REALs is unknown until runtime. An integer boundary */
510/* marker, and sometimes a pointer to a triangle, is appended after the */
511/* REALs. */
512
513typedef REAL *vertex;
514
515/* A queue used to store encroached subsegments. Each subsegment's vertices */
516/* are stored so that we can check whether a subsegment is still the same. */
517
518struct badsubseg {
519 subseg encsubseg; /* An encroached subsegment. */
520 vertex subsegorg, subsegdest; /* Its two vertices. */
521};
522
523/* A queue used to store bad triangles. The key is the square of the cosine */
524/* of the smallest angle of the triangle. Each triangle's vertices are */
525/* stored so that one can check whether a triangle is still the same. */
526
527struct badtriang {
528 triangle poortri; /* A skinny or too-large triangle. */
529 REAL key; /* cos^2 of smallest (apical) angle. */
530 vertex triangorg, triangdest, triangapex; /* Its three vertices. */
531 struct badtriang *nexttriang; /* Pointer to next bad triangle. */
532};
533
534/* A stack of triangles flipped during the most recent vertex insertion. */
535/* The stack is used to undo the vertex insertion if the vertex encroaches */
536/* upon a subsegment. */
537
539 triangle flippedtri; /* A recently flipped triangle. */
540 struct flipstacker *prevflip; /* Previous flip in the stack. */
541};
542
543/* A node in a heap used to store events for the sweepline Delaunay */
544/* algorithm. Nodes do not point directly to their parents or children in */
545/* the heap. Instead, each node knows its position in the heap, and can */
546/* look up its parent and children in a separate array. The `eventptr' */
547/* points either to a `vertex' or to a triangle (in encoded format, so */
548/* that an orientation is included). In the latter case, the origin of */
549/* the oriented triangle is the apex of a "circle event" of the sweepline */
550/* algorithm. To distinguish site events from circle events, all circle */
551/* events are given an invalid (smaller than `xmin') x-coordinate `xkey'. */
552
553struct event {
554 REAL xkey, ykey; /* Coordinates of the event. */
555 VOID *eventptr; /* Can be a vertex or the location of a circle event. */
556 int heapposition; /* Marks this event's position in the heap. */
557};
558
559/* A node in the splay tree. Each node holds an oriented ghost triangle */
560/* that represents a boundary edge of the growing triangulation. When a */
561/* circle event covers two boundary edges with a triangle, so that they */
562/* are no longer boundary edges, those edges are not immediately deleted */
563/* from the tree; rather, they are lazily deleted when they are next */
564/* encountered. (Since only a random sample of boundary edges are kept */
565/* in the tree, lazy deletion is faster.) `keydest' is used to verify */
566/* that a triangle is still the same as when it entered the splay tree; if */
567/* it has been rotated (due to a circle event), it no longer represents a */
568/* boundary edge and should be deleted. */
569
570struct splaynode {
571 struct otri keyedge; /* Lprev of an edge on the front. */
572 vertex keydest; /* Used to verify that splay node is still live. */
573 struct splaynode *lchild, *rchild; /* Children in splay tree. */
574};
575
576/* A type used to allocate memory. firstblock is the first block of items. */
577/* nowblock is the block from which items are currently being allocated. */
578/* nextitem points to the next slab of free memory for an item. */
579/* deaditemstack is the head of a linked list (stack) of deallocated items */
580/* that can be recycled. unallocateditems is the number of items that */
581/* remain to be allocated from nowblock. */
582/* */
583/* Traversal is the process of walking through the entire list of items, and */
584/* is separate from allocation. Note that a traversal will visit items on */
585/* the "deaditemstack" stack as well as live items. pathblock points to */
586/* the block currently being traversed. pathitem points to the next item */
587/* to be traversed. pathitemsleft is the number of items that remain to */
588/* be traversed in pathblock. */
589/* */
590/* alignbytes determines how new records should be aligned in memory. */
591/* itembytes is the length of a record in bytes (after rounding up). */
592/* itemsperblock is the number of items allocated at once in a single */
593/* block. itemsfirstblock is the number of items in the first block, */
594/* which can vary from the others. items is the number of currently */
595/* allocated items. maxitems is the maximum number of items that have */
596/* been allocated at once; it is the current number of items plus the */
597/* number of records kept on deaditemstack. */
598
612};
613
614
615/* Global constants. */
616
617REAL splitter; /* Used to split REAL factors for exact multiplication. */
618REAL epsilon; /* Floating-point machine epsilon. */
623
624/* Random number seed is not constant, but I've made it global anyway. */
625
626uintptr_t randomseed; /* Current random number seed. */
627
628
629/* Mesh data structure. Triangle operates on only one mesh, but the mesh */
630/* structure is used (instead of global variables) to allow reentrancy. */
631
632struct mesh {
633
634/* Variables used to allocate memory for triangles, subsegments, vertices, */
635/* viri (triangles being eaten), encroached segments, bad (skinny or too */
636/* large) triangles, and splay tree nodes. */
637
646
647/* Variables that maintain the bad triangle queues. The queues are */
648/* ordered from 4095 (highest priority) to 0 (lowest priority). */
649
650 struct badtriang *queuefront[4096];
651 struct badtriang *queuetail[4096];
652 int nextnonemptyq[4096];
654
655/* Variable that maintains the stack of recently flipped triangles. */
656
658
659/* Other variables. */
660
661 REAL xmin, xmax, ymin, ymax; /* x and y bounds. */
662 REAL xminextreme; /* Nonexistent x value used as a flag in sweepline. */
663 int invertices; /* Number of input vertices. */
664 int inelements; /* Number of input triangles. */
665 int insegments; /* Number of input segments. */
666 int holes; /* Number of input holes. */
667 int regions; /* Number of input regions. */
668 int undeads; /* Number of input vertices that don't appear in the mesh. */
669 long edges; /* Number of output edges. */
670 int mesh_dim; /* Dimension (ought to be 2). */
671 int nextras; /* Number of attributes per vertex. */
672 int eextras; /* Number of attributes per triangle. */
673 long hullsize; /* Number of edges in convex hull. */
674 int steinerleft; /* Number of Steiner points not yet used. */
675 int vertexmarkindex; /* Index to find boundary marker of a vertex. */
676 int vertex2triindex; /* Index to find a triangle adjacent to a vertex. */
677 int highorderindex; /* Index to find extra nodes for high-order elements. */
678 int elemattribindex; /* Index to find attributes of a triangle. */
679 int areaboundindex; /* Index to find area bound of a triangle. */
680 int checksegments; /* Are there segments in the triangulation yet? */
681 int checkquality; /* Has quality triangulation begun yet? */
682 int readnodefile; /* Has a .node file been read? */
683 long samples; /* Number of random samples for point location. */
684
685 long incirclecount; /* Number of incircle tests performed. */
686 long counterclockcount; /* Number of counterclockwise tests performed. */
687 long orient3dcount; /* Number of 3D orientation tests performed. */
688 long hyperbolacount; /* Number of right-of-hyperbola tests performed. */
689 long circumcentercount; /* Number of circumcenter calculations performed. */
690 long circletopcount; /* Number of circle top calculations performed. */
691
692/* Triangular bounding box vertices. */
693
695
696/* Pointer to the `triangle' that occupies all of "outer space." */
697
699 triangle *dummytribase; /* Keep base address so we can free() it later. */
700
701/* Pointer to the omnipresent subsegment. Referenced by any triangle or */
702/* subsegment that isn't really connected to a subsegment at that */
703/* location. */
704
706 subseg *dummysubbase; /* Keep base address so we can free() it later. */
707
708/* Pointer to a recently visited triangle. Improves point location if */
709/* proximate vertices are inserted sequentially. */
710
712
713}; /* End of `struct mesh'. */
714
715
716/* Data structure for command line switches and file names. This structure */
717/* is used (instead of global variables) to allow reentrancy. */
718
719struct behavior {
720
721/* Switches for the triangulator. */
722/* poly: -p switch. refine: -r switch. */
723/* quality: -q switch. */
724/* minangle: minimum angle bound, specified after -q switch. */
725/* goodangle: cosine squared of minangle. */
726/* offconstant: constant used to place off-center Steiner points. */
727/* vararea: -a switch without number. */
728/* fixedarea: -a switch with number. */
729/* maxarea: maximum area bound, specified after -a switch. */
730/* usertest: -u switch. */
731/* regionattrib: -A switch. convex: -c switch. */
732/* weighted: 1 for -w switch, 2 for -W switch. jettison: -j switch */
733/* firstnumber: inverse of -z switch. All items are numbered starting */
734/* from `firstnumber'. */
735/* edgesout: -e switch. voronoi: -v switch. */
736/* neighbors: -n switch. geomview: -g switch. */
737/* nobound: -B switch. nopolywritten: -P switch. */
738/* nonodewritten: -N switch. noelewritten: -E switch. */
739/* noiterationnum: -I switch. noholes: -O switch. */
740/* noexact: -X switch. */
741/* order: element order, specified after -o switch. */
742/* nobisect: count of how often -Y switch is selected. */
743/* steiner: maximum number of Steiner points, specified after -S switch. */
744/* incremental: -i switch. sweepline: -F switch. */
745/* dwyer: inverse of -l switch. */
746/* splitseg: -s switch. */
747/* conformdel: -D switch. docheck: -C switch. */
748/* quiet: -Q switch. verbose: count of how often -V switch is selected. */
749/* usesegments: -p, -r, -q, or -c switch; determines whether segments are */
750/* used at all. */
751/* */
752/* Read the instructions to find out the meaning of these switches. */
753
765 int order;
770
771/* Variables for file names. */
772
773#ifndef TRILIBRARY
774 char innodefilename[FILENAMESIZE];
775 char inelefilename[FILENAMESIZE];
776 char inpolyfilename[FILENAMESIZE];
777 char areafilename[FILENAMESIZE];
778 char outnodefilename[FILENAMESIZE];
779 char outelefilename[FILENAMESIZE];
780 char outpolyfilename[FILENAMESIZE];
781 char edgefilename[FILENAMESIZE];
782 char vnodefilename[FILENAMESIZE];
783 char vedgefilename[FILENAMESIZE];
784 char neighborfilename[FILENAMESIZE];
785 char offfilename[FILENAMESIZE];
786#endif /* not TRILIBRARY */
787
788}; /* End of `struct behavior'. */
789
790
791/*****************************************************************************/
792/* */
793/* Mesh manipulation primitives. Each triangle contains three pointers to */
794/* other triangles, with orientations. Each pointer points not to the */
795/* first byte of a triangle, but to one of the first three bytes of a */
796/* triangle. It is necessary to extract both the triangle itself and the */
797/* orientation. To save memory, I keep both pieces of information in one */
798/* pointer. To make this possible, I assume that all triangles are aligned */
799/* to four-byte boundaries. The decode() routine below decodes a pointer, */
800/* extracting an orientation (in the range 0 to 2) and a pointer to the */
801/* beginning of a triangle. The encode() routine compresses a pointer to a */
802/* triangle and an orientation into a single pointer. My assumptions that */
803/* triangles are four-byte-aligned and that the `uintptr_t' type is */
804/* long enough to hold a pointer are two of the few kludges in this program.*/
805/* */
806/* Subsegments are manipulated similarly. A pointer to a subsegment */
807/* carries both an address and an orientation in the range 0 to 1. */
808/* */
809/* The other primitives take an oriented triangle or oriented subsegment, */
810/* and return an oriented triangle or oriented subsegment or vertex; or */
811/* they change the connections in the data structure. */
812/* */
813/* Below, triangles and subsegments are denoted by their vertices. The */
814/* triangle abc has origin (org) a, destination (dest) b, and apex (apex) */
815/* c. These vertices occur in counterclockwise order about the triangle. */
816/* The handle abc may simultaneously denote vertex a, edge ab, and triangle */
817/* abc. */
818/* */
819/* Similarly, the subsegment ab has origin (sorg) a and destination (sdest) */
820/* b. If ab is thought to be directed upward (with b directly above a), */
821/* then the handle ab is thought to grasp the right side of ab, and may */
822/* simultaneously denote vertex a and edge ab. */
823/* */
824/* An asterisk (*) denotes a vertex whose identity is unknown. */
825/* */
826/* Given this notation, a partial list of mesh manipulation primitives */
827/* follows. */
828/* */
829/* */
830/* For triangles: */
831/* */
832/* sym: Find the abutting triangle; same edge. */
833/* sym(abc) -> ba* */
834/* */
835/* lnext: Find the next edge (counterclockwise) of a triangle. */
836/* lnext(abc) -> bca */
837/* */
838/* lprev: Find the previous edge (clockwise) of a triangle. */
839/* lprev(abc) -> cab */
840/* */
841/* onext: Find the next edge counterclockwise with the same origin. */
842/* onext(abc) -> ac* */
843/* */
844/* oprev: Find the next edge clockwise with the same origin. */
845/* oprev(abc) -> a*b */
846/* */
847/* dnext: Find the next edge counterclockwise with the same destination. */
848/* dnext(abc) -> *ba */
849/* */
850/* dprev: Find the next edge clockwise with the same destination. */
851/* dprev(abc) -> cb* */
852/* */
853/* rnext: Find the next edge (counterclockwise) of the adjacent triangle. */
854/* rnext(abc) -> *a* */
855/* */
856/* rprev: Find the previous edge (clockwise) of the adjacent triangle. */
857/* rprev(abc) -> b** */
858/* */
859/* org: Origin dest: Destination apex: Apex */
860/* org(abc) -> a dest(abc) -> b apex(abc) -> c */
861/* */
862/* bond: Bond two triangles together at the resepective handles. */
863/* bond(abc, bad) */
864/* */
865/* */
866/* For subsegments: */
867/* */
868/* ssym: Reverse the orientation of a subsegment. */
869/* ssym(ab) -> ba */
870/* */
871/* spivot: Find adjoining subsegment with the same origin. */
872/* spivot(ab) -> a* */
873/* */
874/* snext: Find next subsegment in sequence. */
875/* snext(ab) -> b* */
876/* */
877/* sorg: Origin sdest: Destination */
878/* sorg(ab) -> a sdest(ab) -> b */
879/* */
880/* sbond: Bond two subsegments together at the respective origins. */
881/* sbond(ab, ac) */
882/* */
883/* */
884/* For interacting tetrahedra and subfacets: */
885/* */
886/* tspivot: Find a subsegment abutting a triangle. */
887/* tspivot(abc) -> ba */
888/* */
889/* stpivot: Find a triangle abutting a subsegment. */
890/* stpivot(ab) -> ba* */
891/* */
892/* tsbond: Bond a triangle to a subsegment. */
893/* tsbond(abc, ba) */
894/* */
895/*****************************************************************************/
896
897/********* Mesh manipulation primitives begin here *********/
898/** **/
899/** **/
900
901/* Fast lookup arrays to speed some of the mesh manipulation primitives. */
902
903int plus1mod3[3] = {1, 2, 0};
904int minus1mod3[3] = {2, 0, 1};
905
906/********* Primitives for triangles *********/
907/* */
908/* */
909
910/* decode() converts a pointer to an oriented triangle. The orientation is */
911/* extracted from the two least significant bits of the pointer. */
912
913#define decode(ptr, otri) \
914 (otri).orient = (int) ((uintptr_t) (ptr) & (uintptr_t) 3l); \
915 (otri).tri = (triangle *) \
916 ((uintptr_t) (ptr) ^ (uintptr_t) (otri).orient)
917
918/* encode() compresses an oriented triangle into a single pointer. It */
919/* relies on the assumption that all triangles are aligned to four-byte */
920/* boundaries, so the two least significant bits of (otri).tri are zero. */
921
922#define encode(otri) \
923 (triangle) ((uintptr_t) (otri).tri | (uintptr_t) (otri).orient)
924
925/* The following handle manipulation primitives are all described by Guibas */
926/* and Stolfi. However, Guibas and Stolfi use an edge-based data */
927/* structure, whereas I use a triangle-based data structure. */
928
929/* sym() finds the abutting triangle, on the same edge. Note that the edge */
930/* direction is necessarily reversed, because the handle specified by an */
931/* oriented triangle is directed counterclockwise around the triangle. */
932
933#define sym(otri1, otri2) \
934 ptr = (otri1).tri[(otri1).orient]; \
935 decode(ptr, otri2);
936
937#define symself(otri) \
938 ptr = (otri).tri[(otri).orient]; \
939 decode(ptr, otri);
940
941/* lnext() finds the next edge (counterclockwise) of a triangle. */
942
943#define lnext(otri1, otri2) \
944 (otri2).tri = (otri1).tri; \
945 (otri2).orient = plus1mod3[(otri1).orient]
946
947#define lnextself(otri) \
948 (otri).orient = plus1mod3[(otri).orient]
949
950/* lprev() finds the previous edge (clockwise) of a triangle. */
951
952#define lprev(otri1, otri2) \
953 (otri2).tri = (otri1).tri; \
954 (otri2).orient = minus1mod3[(otri1).orient]
955
956#define lprevself(otri) \
957 (otri).orient = minus1mod3[(otri).orient]
958
959/* onext() spins counterclockwise around a vertex; that is, it finds the */
960/* next edge with the same origin in the counterclockwise direction. This */
961/* edge is part of a different triangle. */
962
963#define onext(otri1, otri2) \
964 lprev(otri1, otri2); \
965 symself(otri2);
966
967#define onextself(otri) \
968 lprevself(otri); \
969 symself(otri);
970
971/* oprev() spins clockwise around a vertex; that is, it finds the next edge */
972/* with the same origin in the clockwise direction. This edge is part of */
973/* a different triangle. */
974
975#define oprev(otri1, otri2) \
976 sym(otri1, otri2); \
977 lnextself(otri2);
978
979#define oprevself(otri) \
980 symself(otri); \
981 lnextself(otri);
982
983/* dnext() spins counterclockwise around a vertex; that is, it finds the */
984/* next edge with the same destination in the counterclockwise direction. */
985/* This edge is part of a different triangle. */
986
987#define dnext(otri1, otri2) \
988 sym(otri1, otri2); \
989 lprevself(otri2);
990
991#define dnextself(otri) \
992 symself(otri); \
993 lprevself(otri);
994
995/* dprev() spins clockwise around a vertex; that is, it finds the next edge */
996/* with the same destination in the clockwise direction. This edge is */
997/* part of a different triangle. */
998
999#define dprev(otri1, otri2) \
1000 lnext(otri1, otri2); \
1001 symself(otri2);
1002
1003#define dprevself(otri) \
1004 lnextself(otri); \
1005 symself(otri);
1006
1007/* rnext() moves one edge counterclockwise about the adjacent triangle. */
1008/* (It's best understood by reading Guibas and Stolfi. It involves */
1009/* changing triangles twice.) */
1010
1011#define rnext(otri1, otri2) \
1012 sym(otri1, otri2); \
1013 lnextself(otri2); \
1014 symself(otri2);
1015
1016#define rnextself(otri) \
1017 symself(otri); \
1018 lnextself(otri); \
1019 symself(otri);
1020
1021/* rprev() moves one edge clockwise about the adjacent triangle. */
1022/* (It's best understood by reading Guibas and Stolfi. It involves */
1023/* changing triangles twice.) */
1024
1025#define rprev(otri1, otri2) \
1026 sym(otri1, otri2); \
1027 lprevself(otri2); \
1028 symself(otri2);
1029
1030#define rprevself(otri) \
1031 symself(otri); \
1032 lprevself(otri); \
1033 symself(otri);
1034
1035/* These primitives determine or set the origin, destination, or apex of a */
1036/* triangle. */
1037
1038#define org(otri, vertexptr) \
1039 vertexptr = (vertex) (otri).tri[plus1mod3[(otri).orient] + 3]
1040
1041#define dest(otri, vertexptr) \
1042 vertexptr = (vertex) (otri).tri[minus1mod3[(otri).orient] + 3]
1043
1044#define apex(otri, vertexptr) \
1045 vertexptr = (vertex) (otri).tri[(otri).orient + 3]
1046
1047#define setorg(otri, vertexptr) \
1048 (otri).tri[plus1mod3[(otri).orient] + 3] = (triangle) vertexptr
1049
1050#define setdest(otri, vertexptr) \
1051 (otri).tri[minus1mod3[(otri).orient] + 3] = (triangle) vertexptr
1052
1053#define setapex(otri, vertexptr) \
1054 (otri).tri[(otri).orient + 3] = (triangle) vertexptr
1055
1056/* Bond two triangles together. */
1057
1058#define bond(otri1, otri2) \
1059 (otri1).tri[(otri1).orient] = encode(otri2); \
1060 (otri2).tri[(otri2).orient] = encode(otri1)
1061
1062/* Dissolve a bond (from one side). Note that the other triangle will still */
1063/* think it's connected to this triangle. Usually, however, the other */
1064/* triangle is being deleted entirely, or bonded to another triangle, so */
1065/* it doesn't matter. */
1066
1067#define dissolve(otri) \
1068 (otri).tri[(otri).orient] = (triangle) m->dummytri
1069
1070/* Copy an oriented triangle. */
1071
1072#define otricopy(otri1, otri2) \
1073 (otri2).tri = (otri1).tri; \
1074 (otri2).orient = (otri1).orient
1075
1076/* Test for equality of oriented triangles. */
1077
1078#define otriequal(otri1, otri2) \
1079 (((otri1).tri == (otri2).tri) && \
1080 ((otri1).orient == (otri2).orient))
1081
1082/* Primitives to infect or cure a triangle with the virus. These rely on */
1083/* the assumption that all subsegments are aligned to four-byte boundaries.*/
1084
1085#define infect(otri) \
1086 (otri).tri[6] = (triangle) \
1087 ((uintptr_t) (otri).tri[6] | (uintptr_t) 2l)
1088
1089#define uninfect(otri) \
1090 (otri).tri[6] = (triangle) \
1091 ((uintptr_t) (otri).tri[6] & ~ (uintptr_t) 2l)
1092
1093/* Test a triangle for viral infection. */
1094
1095#define infected(otri) \
1096 (((uintptr_t) (otri).tri[6] & (uintptr_t) 2l) != 0l)
1097
1098/* Check or set a triangle's attributes. */
1099
1100#define elemattribute(otri, attnum) \
1101 ((REAL *) (otri).tri)[m->elemattribindex + (attnum)]
1102
1103#define setelemattribute(otri, attnum, value) \
1104 ((REAL *) (otri).tri)[m->elemattribindex + (attnum)] = value
1105
1106/* Check or set a triangle's maximum area bound. */
1107
1108#define areabound(otri) ((REAL *) (otri).tri)[m->areaboundindex]
1109
1110#define setareabound(otri, value) \
1111 ((REAL *) (otri).tri)[m->areaboundindex] = value
1112
1113/* Check or set a triangle's deallocation. Its second pointer is set to */
1114/* NULL to indicate that it is not allocated. (Its first pointer is used */
1115/* for the stack of dead items.) Its fourth pointer (its first vertex) */
1116/* is set to NULL in case a `badtriang' structure points to it. */
1117
1118#define deadtri(tria) ((tria)[1] == (triangle) NULL)
1119
1120#define killtri(tria) \
1121 (tria)[1] = (triangle) NULL; \
1122 (tria)[3] = (triangle) NULL
1123
1124/********* Primitives for subsegments *********/
1125/* */
1126/* */
1127
1128/* sdecode() converts a pointer to an oriented subsegment. The orientation */
1129/* is extracted from the least significant bit of the pointer. The two */
1130/* least significant bits (one for orientation, one for viral infection) */
1131/* are masked out to produce the real pointer. */
1132
1133#define sdecode(sptr, osub) \
1134 (osub).ssorient = (int) ((uintptr_t) (sptr) & (uintptr_t) 1l); \
1135 (osub).ss = (subseg *) \
1136 ((uintptr_t) (sptr) & ~ (uintptr_t) 3l)
1137
1138/* sencode() compresses an oriented subsegment into a single pointer. It */
1139/* relies on the assumption that all subsegments are aligned to two-byte */
1140/* boundaries, so the least significant bit of (osub).ss is zero. */
1141
1142#define sencode(osub) \
1143 (subseg) ((uintptr_t) (osub).ss | (uintptr_t) (osub).ssorient)
1144
1145/* ssym() toggles the orientation of a subsegment. */
1146
1147#define ssym(osub1, osub2) \
1148 (osub2).ss = (osub1).ss; \
1149 (osub2).ssorient = 1 - (osub1).ssorient
1150
1151#define ssymself(osub) \
1152 (osub).ssorient = 1 - (osub).ssorient
1153
1154/* spivot() finds the other subsegment (from the same segment) that shares */
1155/* the same origin. */
1156
1157#define spivot(osub1, osub2) \
1158 sptr = (osub1).ss[(osub1).ssorient]; \
1159 sdecode(sptr, osub2)
1160
1161#define spivotself(osub) \
1162 sptr = (osub).ss[(osub).ssorient]; \
1163 sdecode(sptr, osub)
1164
1165/* snext() finds the next subsegment (from the same segment) in sequence; */
1166/* one whose origin is the input subsegment's destination. */
1167
1168#define snext(osub1, osub2) \
1169 sptr = (osub1).ss[1 - (osub1).ssorient]; \
1170 sdecode(sptr, osub2)
1171
1172#define snextself(osub) \
1173 sptr = (osub).ss[1 - (osub).ssorient]; \
1174 sdecode(sptr, osub)
1175
1176/* These primitives determine or set the origin or destination of a */
1177/* subsegment or the segment that includes it. */
1178
1179#define sorg(osub, vertexptr) \
1180 vertexptr = (vertex) (osub).ss[2 + (osub).ssorient]
1181
1182#define sdest(osub, vertexptr) \
1183 vertexptr = (vertex) (osub).ss[3 - (osub).ssorient]
1184
1185#define setsorg(osub, vertexptr) \
1186 (osub).ss[2 + (osub).ssorient] = (subseg) vertexptr
1187
1188#define setsdest(osub, vertexptr) \
1189 (osub).ss[3 - (osub).ssorient] = (subseg) vertexptr
1190
1191#define segorg(osub, vertexptr) \
1192 vertexptr = (vertex) (osub).ss[4 + (osub).ssorient]
1193
1194#define segdest(osub, vertexptr) \
1195 vertexptr = (vertex) (osub).ss[5 - (osub).ssorient]
1196
1197#define setsegorg(osub, vertexptr) \
1198 (osub).ss[4 + (osub).ssorient] = (subseg) vertexptr
1199
1200#define setsegdest(osub, vertexptr) \
1201 (osub).ss[5 - (osub).ssorient] = (subseg) vertexptr
1202
1203/* These primitives read or set a boundary marker. Boundary markers are */
1204/* used to hold user-defined tags for setting boundary conditions in */
1205/* finite element solvers. */
1206
1207#define mark(osub) (* (int *) ((osub).ss + 8))
1208
1209#define setmark(osub, value) \
1210 * (int *) ((osub).ss + 8) = value
1211
1212/* Bond two subsegments together. */
1213
1214#define sbond(osub1, osub2) \
1215 (osub1).ss[(osub1).ssorient] = sencode(osub2); \
1216 (osub2).ss[(osub2).ssorient] = sencode(osub1)
1217
1218/* Dissolve a subsegment bond (from one side). Note that the other */
1219/* subsegment will still think it's connected to this subsegment. */
1220
1221#define sdissolve(osub) \
1222 (osub).ss[(osub).ssorient] = (subseg) m->dummysub
1223
1224/* Copy a subsegment. */
1225
1226#define subsegcopy(osub1, osub2) \
1227 (osub2).ss = (osub1).ss; \
1228 (osub2).ssorient = (osub1).ssorient
1229
1230/* Test for equality of subsegments. */
1231
1232#define subsegequal(osub1, osub2) \
1233 (((osub1).ss == (osub2).ss) && \
1234 ((osub1).ssorient == (osub2).ssorient))
1235
1236/* Check or set a subsegment's deallocation. Its second pointer is set to */
1237/* NULL to indicate that it is not allocated. (Its first pointer is used */
1238/* for the stack of dead items.) Its third pointer (its first vertex) */
1239/* is set to NULL in case a `badsubseg' structure points to it. */
1240
1241#define deadsubseg(sub) ((sub)[1] == (subseg) NULL)
1242
1243#define killsubseg(sub) \
1244 (sub)[1] = (subseg) NULL; \
1245 (sub)[2] = (subseg) NULL
1246
1247/********* Primitives for interacting triangles and subsegments *********/
1248/* */
1249/* */
1250
1251/* tspivot() finds a subsegment abutting a triangle. */
1252
1253#define tspivot(otri, osub) \
1254 sptr = (subseg) (otri).tri[6 + (otri).orient]; \
1255 sdecode(sptr, osub)
1256
1257/* stpivot() finds a triangle abutting a subsegment. It requires that the */
1258/* variable `ptr' of type `triangle' be defined. */
1259
1260#define stpivot(osub, otri) \
1261 ptr = (triangle) (osub).ss[6 + (osub).ssorient]; \
1262 decode(ptr, otri)
1263
1264/* Bond a triangle to a subsegment. */
1265
1266#define tsbond(otri, osub) \
1267 (otri).tri[6 + (otri).orient] = (triangle) sencode(osub); \
1268 (osub).ss[6 + (osub).ssorient] = (subseg) encode(otri)
1269
1270/* Dissolve a bond (from the triangle side). */
1271
1272#define tsdissolve(otri) \
1273 (otri).tri[6 + (otri).orient] = (triangle) m->dummysub
1274
1275/* Dissolve a bond (from the subsegment side). */
1276
1277#define stdissolve(osub) \
1278 (osub).ss[6 + (osub).ssorient] = (subseg) m->dummytri
1279
1280/********* Primitives for vertices *********/
1281/* */
1282/* */
1283
1284#define vertexmark(vx) ((int *) (vx))[m->vertexmarkindex]
1285
1286#define setvertexmark(vx, value) \
1287 ((int *) (vx))[m->vertexmarkindex] = value
1288
1289#define vertextype(vx) ((int *) (vx))[m->vertexmarkindex + 1]
1290
1291#define setvertextype(vx, value) \
1292 ((int *) (vx))[m->vertexmarkindex + 1] = value
1293
1294#define vertex2tri(vx) ((triangle *) (vx))[m->vertex2triindex]
1295
1296#define setvertex2tri(vx, value) \
1297 ((triangle *) (vx))[m->vertex2triindex] = value
1298
1299/** **/
1300/** **/
1301/********* Mesh manipulation primitives end here *********/
1302
1303/********* User-defined triangle evaluation routine begins here *********/
1304/** **/
1305/** **/
1306
1307/*****************************************************************************/
1308/* */
1309/* triunsuitable() Determine if a triangle is unsuitable, and thus must */
1310/* be further refined. */
1311/* */
1312/* You may write your own procedure that decides whether or not a selected */
1313/* triangle is too big (and needs to be refined). There are two ways to do */
1314/* this. */
1315/* */
1316/* (1) Modify the procedure `triunsuitable' below, then recompile */
1317/* Triangle. */
1318/* */
1319/* (2) Define the symbol EXTERNAL_TEST (either by adding the definition */
1320/* to this file, or by using the appropriate compiler switch). This way, */
1321/* you can compile triangle.c separately from your test. Write your own */
1322/* `triunsuitable' procedure in a separate C file (using the same prototype */
1323/* as below). Compile it and link the object code with triangle.o. */
1324/* */
1325/* This procedure returns 1 if the triangle is too large and should be */
1326/* refined; 0 otherwise. */
1327/* */
1328/*****************************************************************************/
1329
1330#ifdef EXTERNAL_TEST
1331
1332int triunsuitable();
1333
1334#else /* not EXTERNAL_TEST */
1335
1336#ifdef ANSI_DECLARATORS
1337int triunsuitable(vertex triorg, vertex tridest, vertex triapex, REAL area )
1338#else /* not ANSI_DECLARATORS */
1339int triunsuitable(triorg, tridest, triapex, area)
1340vertex triorg; /* The triangle's origin vertex. */
1341vertex tridest; /* The triangle's destination vertex. */
1342vertex triapex; /* The triangle's apex vertex. */
1343REAL area; /* The area of the triangle. */
1344#endif /* not ANSI_DECLARATORS */
1345
1346{
1347 REAL dxoa, dxda, dxod;
1348 REAL dyoa, dyda, dyod;
1349 REAL oalen, dalen, odlen;
1350 REAL maxlen;
1351
1352 (void)area; /*LM: added to suppress warning */
1353
1354 dxoa = triorg[0] - triapex[0];
1355 dyoa = triorg[1] - triapex[1];
1356 dxda = tridest[0] - triapex[0];
1357 dyda = tridest[1] - triapex[1];
1358 dxod = triorg[0] - tridest[0];
1359 dyod = triorg[1] - tridest[1];
1360 /* Find the squares of the lengths of the triangle's three edges. */
1361 oalen = dxoa * dxoa + dyoa * dyoa;
1362 dalen = dxda * dxda + dyda * dyda;
1363 odlen = dxod * dxod + dyod * dyod;
1364 /* Find the square of the length of the longest edge. */
1365 maxlen = (dalen > oalen) ? dalen : oalen;
1366 maxlen = (odlen > maxlen) ? odlen : maxlen;
1367
1368 if (maxlen > 0.05 * (triorg[0] * triorg[0] + triorg[1] * triorg[1]) + 0.02) {
1369 return 1;
1370 } else {
1371 return 0;
1372 }
1373}
1374
1375#endif /* not EXTERNAL_TEST */
1376
1377/** **/
1378/** **/
1379/********* User-defined triangle evaluation routine ends here *********/
1380
1381/********* Memory allocation and program exit wrappers begin here *********/
1382/** **/
1383/** **/
1384
1385#ifdef ANSI_DECLARATORS
1386void triexit(int status)
1387#else /* not ANSI_DECLARATORS */
1388void triexit(status)
1389int status;
1390#endif /* not ANSI_DECLARATORS */
1391
1392{
1393 exit(status);
1394}
1395
1396#ifdef ANSI_DECLARATORS
1398#else /* not ANSI_DECLARATORS */
1400int size;
1401#endif /* not ANSI_DECLARATORS */
1402
1403{
1404 VOID *memptr;
1405
1406 memptr = (VOID *) malloc((unsigned int) size);
1407 if (memptr == (VOID *) NULL) {
1408 printf("Error: Out of memory.\n");
1409 triexit(1);
1410 }
1411 return(memptr);
1412}
1413
1414#ifdef ANSI_DECLARATORS
1415void trifree(VOID *memptr)
1416#else /* not ANSI_DECLARATORS */
1417void trifree(memptr)
1418VOID *memptr;
1419#endif /* not ANSI_DECLARATORS */
1420
1421{
1422 free(memptr);
1423}
1424
1425/** **/
1426/** **/
1427/********* Memory allocation and program exit wrappers end here *********/
1428
1429/********* User interaction routines begin here *********/
1430/** **/
1431/** **/
1432
1433/*****************************************************************************/
1434/* */
1435/* syntax() Print list of command line switches. */
1436/* */
1437/*****************************************************************************/
1438
1439#ifndef TRILIBRARY
1440
1441void syntax()
1442{
1443#ifdef CDT_ONLY
1444#ifdef REDUCED
1445 printf("triangle [-pAcjevngBPNEIOXzo_lQVh] input_file\n");
1446#else /* not REDUCED */
1447 printf("triangle [-pAcjevngBPNEIOXzo_iFlCQVh] input_file\n");
1448#endif /* not REDUCED */
1449#else /* not CDT_ONLY */
1450#ifdef REDUCED
1451 printf("triangle [-prq__a__uAcDjevngBPNEIOXzo_YS__lQVh] input_file\n");
1452#else /* not REDUCED */
1453 printf("triangle [-prq__a__uAcDjevngBPNEIOXzo_YS__iFlsCQVh] input_file\n");
1454#endif /* not REDUCED */
1455#endif /* not CDT_ONLY */
1456
1457 printf(" -p Triangulates a Planar Straight Line Graph (.poly file).\n");
1458#ifndef CDT_ONLY
1459 printf(" -r Refines a previously generated mesh.\n");
1460 printf(
1461 " -q Quality mesh generation. A minimum angle may be specified.\n");
1462 printf(" -a Applies a maximum triangle area constraint.\n");
1463 printf(" -u Applies a user-defined triangle constraint.\n");
1464#endif /* not CDT_ONLY */
1465 printf(
1466 " -A Applies attributes to identify triangles in certain regions.\n");
1467 printf(" -c Encloses the convex hull with segments.\n");
1468#ifndef CDT_ONLY
1469 printf(" -D Conforming Delaunay: all triangles are truly Delaunay.\n");
1470#endif /* not CDT_ONLY */
1471/*
1472 printf(" -w Weighted Delaunay triangulation.\n");
1473 printf(" -W Regular triangulation (lower hull of a height field).\n");
1474*/
1475 printf(" -j Jettison unused vertices from output .node file.\n");
1476 printf(" -e Generates an edge list.\n");
1477 printf(" -v Generates a Voronoi diagram.\n");
1478 printf(" -n Generates a list of triangle neighbors.\n");
1479 printf(" -g Generates an .off file for Geomview.\n");
1480 printf(" -B Suppresses output of boundary information.\n");
1481 printf(" -P Suppresses output of .poly file.\n");
1482 printf(" -N Suppresses output of .node file.\n");
1483 printf(" -E Suppresses output of .ele file.\n");
1484 printf(" -I Suppresses mesh iteration numbers.\n");
1485 printf(" -O Ignores holes in .poly file.\n");
1486 printf(" -X Suppresses use of exact arithmetic.\n");
1487 printf(" -z Numbers all items starting from zero (rather than one).\n");
1488 printf(" -o2 Generates second-order subparametric elements.\n");
1489#ifndef CDT_ONLY
1490 printf(" -Y Suppresses boundary segment splitting.\n");
1491 printf(" -S Specifies maximum number of added Steiner points.\n");
1492#endif /* not CDT_ONLY */
1493#ifndef REDUCED
1494 printf(" -i Uses incremental method, rather than divide-and-conquer.\n");
1495 printf(" -F Uses Fortune's sweepline algorithm, rather than d-and-c.\n");
1496#endif /* not REDUCED */
1497 printf(" -l Uses vertical cuts only, rather than alternating cuts.\n");
1498#ifndef REDUCED
1499#ifndef CDT_ONLY
1500 printf(
1501 " -s Force segments into mesh by splitting (instead of using CDT).\n");
1502#endif /* not CDT_ONLY */
1503 printf(" -C Check consistency of final mesh.\n");
1504#endif /* not REDUCED */
1505 printf(" -Q Quiet: No terminal output except errors.\n");
1506 printf(" -V Verbose: Detailed information on what I'm doing.\n");
1507 printf(" -h Help: Detailed instructions for Triangle.\n");
1508 triexit(0);
1509}
1510
1511#endif /* not TRILIBRARY */
1512
1513/*****************************************************************************/
1514/* */
1515/* info() Print out complete instructions. */
1516/* */
1517/*****************************************************************************/
1518
1519#ifndef TRILIBRARY
1520
1521void info()
1522{
1523 printf("Triangle\n");
1524 printf(
1525"A Two-Dimensional Quality Mesh Generator and Delaunay Triangulator.\n");
1526 printf("Version 1.6\n\n");
1527 printf(
1528"Copyright 1993, 1995, 1997, 1998, 2002, 2005 Jonathan Richard Shewchuk\n");
1529 printf("2360 Woolsey #H / Berkeley, California 94705-1927\n");
1530 printf("Bugs/comments to jrs@cs.berkeley.edu\n");
1531 printf(
1532"Created as part of the Quake project (tools for earthquake simulation).\n");
1533 printf(
1534"Supported in part by NSF Grant CMS-9318163 and an NSERC 1967 Scholarship.\n");
1535 printf("There is no warranty whatsoever. Use at your own risk.\n");
1536#ifdef SINGLE
1537 printf("This executable is compiled for single precision arithmetic.\n\n\n");
1538#else /* not SINGLE */
1539 printf("This executable is compiled for double precision arithmetic.\n\n\n");
1540#endif /* not SINGLE */
1541 printf(
1542"Triangle generates exact Delaunay triangulations, constrained Delaunay\n");
1543 printf(
1544"triangulations, conforming Delaunay triangulations, Voronoi diagrams, and\n");
1545 printf(
1546"high-quality triangular meshes. The latter can be generated with no small\n"
1547);
1548 printf(
1549"or large angles, and are thus suitable for finite element analysis. If no\n"
1550);
1551 printf(
1552"command line switch is specified, your .node input file is read, and the\n");
1553 printf(
1554"Delaunay triangulation is returned in .node and .ele output files. The\n");
1555 printf("command syntax is:\n\n");
1556 printf("triangle [-prq__a__uAcDjevngBPNEIOXzo_YS__iFlsCQVh] input_file\n\n");
1557 printf(
1558"Underscores indicate that numbers may optionally follow certain switches.\n");
1559 printf(
1560"Do not leave any space between a switch and its numeric parameter.\n");
1561 printf(
1562"input_file must be a file with extension .node, or extension .poly if the\n");
1563 printf(
1564"-p switch is used. If -r is used, you must supply .node and .ele files,\n");
1565 printf(
1566"and possibly a .poly file and an .area file as well. The formats of these\n"
1567);
1568 printf("files are described below.\n\n");
1569 printf("Command Line Switches:\n\n");
1570 printf(
1571" -p Reads a Planar Straight Line Graph (.poly file), which can specify\n"
1572);
1573 printf(
1574" vertices, segments, holes, regional attributes, and regional area\n");
1575 printf(
1576" constraints. Generates a constrained Delaunay triangulation (CDT)\n"
1577);
1578 printf(
1579" fitting the input; or, if -s, -q, -a, or -u is used, a conforming\n");
1580 printf(
1581" constrained Delaunay triangulation (CCDT). If you want a truly\n");
1582 printf(
1583" Delaunay (not just constrained Delaunay) triangulation, use -D as\n");
1584 printf(
1585" well. When -p is not used, Triangle reads a .node file by default.\n"
1586);
1587 printf(
1588" -r Refines a previously generated mesh. The mesh is read from a .node\n"
1589);
1590 printf(
1591" file and an .ele file. If -p is also used, a .poly file is read\n");
1592 printf(
1593" and used to constrain segments in the mesh. If -a is also used\n");
1594 printf(
1595" (with no number following), an .area file is read and used to\n");
1596 printf(
1597" impose area constraints on the mesh. Further details on refinement\n"
1598);
1599 printf(" appear below.\n");
1600 printf(
1601" -q Quality mesh generation by Delaunay refinement (a hybrid of Paul\n");
1602 printf(
1603" Chew's and Jim Ruppert's algorithms). Adds vertices to the mesh to\n"
1604);
1605 printf(
1606" ensure that all angles are between 20 and 140 degrees. An\n");
1607 printf(
1608" alternative bound on the minimum angle, replacing 20 degrees, may\n");
1609 printf(
1610" be specified after the `q'. The specified angle may include a\n");
1611 printf(
1612" decimal point, but not exponential notation. Note that a bound of\n"
1613);
1614 printf(
1615" theta degrees on the smallest angle also implies a bound of\n");
1616 printf(
1617" (180 - 2 theta) on the largest angle. If the minimum angle is 28.6\n"
1618);
1619 printf(
1620" degrees or smaller, Triangle is mathematically guaranteed to\n");
1621 printf(
1622" terminate (assuming infinite precision arithmetic--Triangle may\n");
1623 printf(
1624" fail to terminate if you run out of precision). In practice,\n");
1625 printf(
1626" Triangle often succeeds for minimum angles up to 34 degrees. For\n");
1627 printf(
1628" some meshes, however, you might need to reduce the minimum angle to\n"
1629);
1630 printf(
1631" avoid problems associated with insufficient floating-point\n");
1632 printf(" precision.\n");
1633 printf(
1634" -a Imposes a maximum triangle area. If a number follows the `a', no\n");
1635 printf(
1636" triangle is generated whose area is larger than that number. If no\n"
1637);
1638 printf(
1639" number is specified, an .area file (if -r is used) or .poly file\n");
1640 printf(
1641" (if -r is not used) specifies a set of maximum area constraints.\n");
1642 printf(
1643" An .area file contains a separate area constraint for each\n");
1644 printf(
1645" triangle, and is useful for refining a finite element mesh based on\n"
1646);
1647 printf(
1648" a posteriori error estimates. A .poly file can optionally contain\n"
1649);
1650 printf(
1651" an area constraint for each segment-bounded region, thereby\n");
1652 printf(
1653" controlling triangle densities in a first triangulation of a PSLG.\n"
1654);
1655 printf(
1656" You can impose both a fixed area constraint and a varying area\n");
1657 printf(
1658" constraint by invoking the -a switch twice, once with and once\n");
1659 printf(
1660" without a number following. Each area specified may include a\n");
1661 printf(" decimal point.\n");
1662 printf(
1663" -u Imposes a user-defined constraint on triangle size. There are two\n"
1664);
1665 printf(
1666" ways to use this feature. One is to edit the triunsuitable()\n");
1667 printf(
1668" procedure in triangle.c to encode any constraint you like, then\n");
1669 printf(
1670" recompile Triangle. The other is to compile triangle.c with the\n");
1671 printf(
1672" EXTERNAL_TEST symbol set (compiler switch -DEXTERNAL_TEST), then\n");
1673 printf(
1674" link Triangle with a separate object file that implements\n");
1675 printf(
1676" triunsuitable(). In either case, the -u switch causes the user-\n");
1677 printf(" defined test to be applied to every triangle.\n");
1678 printf(
1679" -A Assigns an additional floating-point attribute to each triangle\n");
1680 printf(
1681" that identifies what segment-bounded region each triangle belongs\n");
1682 printf(
1683" to. Attributes are assigned to regions by the .poly file. If a\n");
1684 printf(
1685" region is not explicitly marked by the .poly file, triangles in\n");
1686 printf(
1687" that region are assigned an attribute of zero. The -A switch has\n");
1688 printf(
1689" an effect only when the -p switch is used and the -r switch is not.\n"
1690);
1691 printf(
1692" -c Creates segments on the convex hull of the triangulation. If you\n");
1693 printf(
1694" are triangulating a vertex set, this switch causes a .poly file to\n"
1695);
1696 printf(
1697" be written, containing all edges of the convex hull. If you are\n");
1698 printf(
1699" triangulating a PSLG, this switch specifies that the whole convex\n");
1700 printf(
1701" hull of the PSLG should be triangulated, regardless of what\n");
1702 printf(
1703" segments the PSLG has. If you do not use this switch when\n");
1704 printf(
1705" triangulating a PSLG, Triangle assumes that you have identified the\n"
1706);
1707 printf(
1708" region to be triangulated by surrounding it with segments of the\n");
1709 printf(
1710" input PSLG. Beware: if you are not careful, this switch can cause\n"
1711);
1712 printf(
1713" the introduction of an extremely thin angle between a PSLG segment\n"
1714);
1715 printf(
1716" and a convex hull segment, which can cause overrefinement (and\n");
1717 printf(
1718" possibly failure if Triangle runs out of precision). If you are\n");
1719 printf(
1720" refining a mesh, the -c switch works differently: it causes a\n");
1721 printf(
1722" .poly file to be written containing the boundary edges of the mesh\n"
1723);
1724 printf(" (useful if no .poly file was read).\n");
1725 printf(
1726" -D Conforming Delaunay triangulation: use this switch if you want to\n"
1727);
1728 printf(
1729" ensure that all the triangles in the mesh are Delaunay, and not\n");
1730 printf(
1731" merely constrained Delaunay; or if you want to ensure that all the\n"
1732);
1733 printf(
1734" Voronoi vertices lie within the triangulation. (Some finite volume\n"
1735);
1736 printf(
1737" methods have this requirement.) This switch invokes Ruppert's\n");
1738 printf(
1739" original algorithm, which splits every subsegment whose diametral\n");
1740 printf(
1741" circle is encroached. It usually increases the number of vertices\n"
1742);
1743 printf(" and triangles.\n");
1744 printf(
1745" -j Jettisons vertices that are not part of the final triangulation\n");
1746 printf(
1747" from the output .node file. By default, Triangle copies all\n");
1748 printf(
1749" vertices in the input .node file to the output .node file, in the\n");
1750 printf(
1751" same order, so their indices do not change. The -j switch prevents\n"
1752);
1753 printf(
1754" duplicated input vertices, or vertices `eaten' by holes, from\n");
1755 printf(
1756" appearing in the output .node file. Thus, if two input vertices\n");
1757 printf(
1758" have exactly the same coordinates, only the first appears in the\n");
1759 printf(
1760" output. If any vertices are jettisoned, the vertex numbering in\n");
1761 printf(
1762" the output .node file differs from that of the input .node file.\n");
1763 printf(
1764" -e Outputs (to an .edge file) a list of edges of the triangulation.\n");
1765 printf(
1766" -v Outputs the Voronoi diagram associated with the triangulation.\n");
1767 printf(
1768" Does not attempt to detect degeneracies, so some Voronoi vertices\n");
1769 printf(
1770" may be duplicated. See the discussion of Voronoi diagrams below.\n");
1771 printf(
1772" -n Outputs (to a .neigh file) a list of triangles neighboring each\n");
1773 printf(" triangle.\n");
1774 printf(
1775" -g Outputs the mesh to an Object File Format (.off) file, suitable for\n"
1776);
1777 printf(" viewing with the Geometry Center's Geomview package.\n");
1778 printf(
1779" -B No boundary markers in the output .node, .poly, and .edge output\n");
1780 printf(
1781" files. See the detailed discussion of boundary markers below.\n");
1782 printf(
1783" -P No output .poly file. Saves disk space, but you lose the ability\n");
1784 printf(
1785" to maintain constraining segments on later refinements of the mesh.\n"
1786);
1787 printf(" -N No output .node file.\n");
1788 printf(" -E No output .ele file.\n");
1789 printf(
1790" -I No iteration numbers. Suppresses the output of .node and .poly\n");
1791 printf(
1792" files, so your input files won't be overwritten. (If your input is\n"
1793);
1794 printf(
1795" a .poly file only, a .node file is written.) Cannot be used with\n");
1796 printf(
1797" the -r switch, because that would overwrite your input .ele file.\n");
1798 printf(
1799" Shouldn't be used with the -q, -a, -u, or -s switch if you are\n");
1800 printf(
1801" using a .node file for input, because no .node file is written, so\n"
1802);
1803 printf(" there is no record of any added Steiner points.\n");
1804 printf(" -O No holes. Ignores the holes in the .poly file.\n");
1805 printf(
1806" -X No exact arithmetic. Normally, Triangle uses exact floating-point\n"
1807);
1808 printf(
1809" arithmetic for certain tests if it thinks the inexact tests are not\n"
1810);
1811 printf(
1812" accurate enough. Exact arithmetic ensures the robustness of the\n");
1813 printf(
1814" triangulation algorithms, despite floating-point roundoff error.\n");
1815 printf(
1816" Disabling exact arithmetic with the -X switch causes a small\n");
1817 printf(
1818" improvement in speed and creates the possibility that Triangle will\n"
1819);
1820 printf(" fail to produce a valid mesh. Not recommended.\n");
1821 printf(
1822" -z Numbers all items starting from zero (rather than one). Note that\n"
1823);
1824 printf(
1825" this switch is normally overridden by the value used to number the\n"
1826);
1827 printf(
1828" first vertex of the input .node or .poly file. However, this\n");
1829 printf(
1830" switch is useful when calling Triangle from another program.\n");
1831 printf(
1832" -o2 Generates second-order subparametric elements with six nodes each.\n"
1833);
1834 printf(
1835" -Y No new vertices on the boundary. This switch is useful when the\n");
1836 printf(
1837" mesh boundary must be preserved so that it conforms to some\n");
1838 printf(
1839" adjacent mesh. Be forewarned that you will probably sacrifice much\n"
1840);
1841 printf(
1842" of the quality of the mesh; Triangle will try, but the resulting\n");
1843 printf(
1844" mesh may contain poorly shaped triangles. Works well if all the\n");
1845 printf(
1846" boundary vertices are closely spaced. Specify this switch twice\n");
1847 printf(
1848" (`-YY') to prevent all segment splitting, including internal\n");
1849 printf(" boundaries.\n");
1850 printf(
1851" -S Specifies the maximum number of Steiner points (vertices that are\n");
1852 printf(
1853" not in the input, but are added to meet the constraints on minimum\n"
1854);
1855 printf(
1856" angle and maximum area). The default is to allow an unlimited\n");
1857 printf(
1858" number. If you specify this switch with no number after it,\n");
1859 printf(
1860" the limit is set to zero. Triangle always adds vertices at segment\n"
1861);
1862 printf(
1863" intersections, even if it needs to use more vertices than the limit\n"
1864);
1865 printf(
1866" you set. When Triangle inserts segments by splitting (-s), it\n");
1867 printf(
1868" always adds enough vertices to ensure that all the segments of the\n"
1869);
1870 printf(" PLSG are recovered, ignoring the limit if necessary.\n");
1871 printf(
1872" -i Uses an incremental rather than a divide-and-conquer algorithm to\n");
1873 printf(
1874" construct a Delaunay triangulation. Try it if the divide-and-\n");
1875 printf(" conquer algorithm fails.\n");
1876 printf(
1877" -F Uses Steven Fortune's sweepline algorithm to construct a Delaunay\n");
1878 printf(
1879" triangulation. Warning: does not use exact arithmetic for all\n");
1880 printf(" calculations. An exact result is not guaranteed.\n");
1881 printf(
1882" -l Uses only vertical cuts in the divide-and-conquer algorithm. By\n");
1883 printf(
1884" default, Triangle alternates between vertical and horizontal cuts,\n"
1885);
1886 printf(
1887" which usually improve the speed except with vertex sets that are\n");
1888 printf(
1889" small or short and wide. This switch is primarily of theoretical\n");
1890 printf(" interest.\n");
1891 printf(
1892" -s Specifies that segments should be forced into the triangulation by\n"
1893);
1894 printf(
1895" recursively splitting them at their midpoints, rather than by\n");
1896 printf(
1897" generating a constrained Delaunay triangulation. Segment splitting\n"
1898);
1899 printf(
1900" is true to Ruppert's original algorithm, but can create needlessly\n"
1901);
1902 printf(
1903" small triangles. This switch is primarily of theoretical interest.\n"
1904);
1905 printf(
1906" -C Check the consistency of the final mesh. Uses exact arithmetic for\n"
1907);
1908 printf(
1909" checking, even if the -X switch is used. Useful if you suspect\n");
1910 printf(" Triangle is buggy.\n");
1911 printf(
1912" -Q Quiet: Suppresses all explanation of what Triangle is doing,\n");
1913 printf(" unless an error occurs.\n");
1914 printf(
1915" -V Verbose: Gives detailed information about what Triangle is doing.\n"
1916);
1917 printf(
1918" Add more `V's for increasing amount of detail. `-V' is most\n");
1919 printf(
1920" useful; itgives information on algorithmic progress and much more\n");
1921 printf(
1922" detailed statistics. `-VV' gives vertex-by-vertex details, and\n");
1923 printf(
1924" prints so much that Triangle runs much more slowly. `-VVVV' gives\n"
1925);
1926 printf(" information only a debugger could love.\n");
1927 printf(" -h Help: Displays these instructions.\n");
1928 printf("\n");
1929 printf("Definitions:\n");
1930 printf("\n");
1931 printf(
1932" A Delaunay triangulation of a vertex set is a triangulation whose\n");
1933 printf(
1934" vertices are the vertex set, that covers the convex hull of the vertex\n");
1935 printf(
1936" set. A Delaunay triangulation has the property that no vertex lies\n");
1937 printf(
1938" inside the circumscribing circle (circle that passes through all three\n");
1939 printf(" vertices) of any triangle in the triangulation.\n\n");
1940 printf(
1941" A Voronoi diagram of a vertex set is a subdivision of the plane into\n");
1942 printf(
1943" polygonal cells (some of which may be unbounded, meaning infinitely\n");
1944 printf(
1945" large), where each cell is the set of points in the plane that are closer\n"
1946);
1947 printf(
1948" to some input vertex than to any other input vertex. The Voronoi diagram\n"
1949);
1950 printf(" is a geometric dual of the Delaunay triangulation.\n\n");
1951 printf(
1952" A Planar Straight Line Graph (PSLG) is a set of vertices and segments.\n");
1953 printf(
1954" Segments are simply edges, whose endpoints are all vertices in the PSLG.\n"
1955);
1956 printf(
1957" Segments may intersect each other only at their endpoints. The file\n");
1958 printf(" format for PSLGs (.poly files) is described below.\n\n");
1959 printf(
1960" A constrained Delaunay triangulation (CDT) of a PSLG is similar to a\n");
1961 printf(
1962" Delaunay triangulation, but each PSLG segment is present as a single edge\n"
1963);
1964 printf(
1965" of the CDT. (A constrained Delaunay triangulation is not truly a\n");
1966 printf(
1967" Delaunay triangulation, because some of its triangles might not be\n");
1968 printf(
1969" Delaunay.) By definition, a CDT does not have any vertices other than\n");
1970 printf(
1971" those specified in the input PSLG. Depending on context, a CDT might\n");
1972 printf(
1973" cover the convex hull of the PSLG, or it might cover only a segment-\n");
1974 printf(" bounded region (e.g. a polygon).\n\n");
1975 printf(
1976" A conforming Delaunay triangulation of a PSLG is a triangulation in which\n"
1977);
1978 printf(
1979" each triangle is truly Delaunay, and each PSLG segment is represented by\n"
1980);
1981 printf(
1982" a linear contiguous sequence of edges of the triangulation. New vertices\n"
1983);
1984 printf(
1985" (not part of the PSLG) may appear, and each input segment may have been\n");
1986 printf(
1987" subdivided into shorter edges (subsegments) by these additional vertices.\n"
1988);
1989 printf(
1990" The new vertices are frequently necessary to maintain the Delaunay\n");
1991 printf(" property while ensuring that every segment is represented.\n\n");
1992 printf(
1993" A conforming constrained Delaunay triangulation (CCDT) of a PSLG is a\n");
1994 printf(
1995" triangulation of a PSLG whose triangles are constrained Delaunay. New\n");
1996 printf(" vertices may appear, and input segments may be subdivided into\n");
1997 printf(
1998" subsegments, but not to guarantee that segments are respected; rather, to\n"
1999);
2000 printf(
2001" improve the quality of the triangles. The high-quality meshes produced\n");
2002 printf(
2003" by the -q switch are usually CCDTs, but can be made conforming Delaunay\n");
2004 printf(" with the -D switch.\n\n");
2005 printf("File Formats:\n\n");
2006 printf(
2007" All files may contain comments prefixed by the character '#'. Vertices,\n"
2008);
2009 printf(
2010" triangles, edges, holes, and maximum area constraints must be numbered\n");
2011 printf(
2012" consecutively, starting from either 1 or 0. Whichever you choose, all\n");
2013 printf(
2014" input files must be consistent; if the vertices are numbered from 1, so\n");
2015 printf(
2016" must be all other objects. Triangle automatically detects your choice\n");
2017 printf(
2018" while reading the .node (or .poly) file. (When calling Triangle from\n");
2019 printf(
2020" another program, use the -z switch if you wish to number objects from\n");
2021 printf(" zero.) Examples of these file formats are given below.\n\n");
2022 printf(" .node files:\n");
2023 printf(
2024" First line: <# of vertices> <dimension (must be 2)> <# of attributes>\n"
2025);
2026 printf(
2027" <# of boundary markers (0 or 1)>\n"
2028);
2029 printf(
2030" Remaining lines: <vertex #> <x> <y> [attributes] [boundary marker]\n");
2031 printf("\n");
2032 printf(
2033" The attributes, which are typically floating-point values of physical\n");
2034 printf(
2035" quantities (such as mass or conductivity) associated with the nodes of\n"
2036);
2037 printf(
2038" a finite element mesh, are copied unchanged to the output mesh. If -q,\n"
2039);
2040 printf(
2041" -a, -u, -D, or -s is selected, each new Steiner point added to the mesh\n"
2042);
2043 printf(" has attributes assigned to it by linear interpolation.\n\n");
2044 printf(
2045" If the fourth entry of the first line is `1', the last column of the\n");
2046 printf(
2047" remainder of the file is assumed to contain boundary markers. Boundary\n"
2048);
2049 printf(
2050" markers are used to identify boundary vertices and vertices resting on\n"
2051);
2052 printf(
2053" PSLG segments; a complete description appears in a section below. The\n"
2054);
2055 printf(
2056" .node file produced by Triangle contains boundary markers in the last\n");
2057 printf(" column unless they are suppressed by the -B switch.\n\n");
2058 printf(" .ele files:\n");
2059 printf(
2060" First line: <# of triangles> <nodes per triangle> <# of attributes>\n");
2061 printf(
2062" Remaining lines: <triangle #> <node> <node> <node> ... [attributes]\n");
2063 printf("\n");
2064 printf(
2065" Nodes are indices into the corresponding .node file. The first three\n");
2066 printf(
2067" nodes are the corner vertices, and are listed in counterclockwise order\n"
2068);
2069 printf(
2070" around each triangle. (The remaining nodes, if any, depend on the type\n"
2071);
2072 printf(" of finite element used.)\n\n");
2073 printf(
2074" The attributes are just like those of .node files. Because there is no\n"
2075);
2076 printf(
2077" simple mapping from input to output triangles, Triangle attempts to\n");
2078 printf(
2079" interpolate attributes, and may cause a lot of diffusion of attributes\n"
2080);
2081 printf(
2082" among nearby triangles as the triangulation is refined. Attributes do\n"
2083);
2084 printf(" not diffuse across segments, so attributes used to identify\n");
2085 printf(" segment-bounded regions remain intact.\n\n");
2086 printf(
2087" In .ele files produced by Triangle, each triangular element has three\n");
2088 printf(
2089" nodes (vertices) unless the -o2 switch is used, in which case\n");
2090 printf(
2091" subparametric quadratic elements with six nodes each are generated.\n");
2092 printf(
2093" The first three nodes are the corners in counterclockwise order, and\n");
2094 printf(
2095" the fourth, fifth, and sixth nodes lie on the midpoints of the edges\n");
2096 printf(
2097" opposite the first, second, and third vertices, respectively.\n");
2098 printf("\n");
2099 printf(" .poly files:\n");
2100 printf(
2101" First line: <# of vertices> <dimension (must be 2)> <# of attributes>\n"
2102);
2103 printf(
2104" <# of boundary markers (0 or 1)>\n"
2105);
2106 printf(
2107" Following lines: <vertex #> <x> <y> [attributes] [boundary marker]\n");
2108 printf(" One line: <# of segments> <# of boundary markers (0 or 1)>\n");
2109 printf(
2110" Following lines: <segment #> <endpoint> <endpoint> [boundary marker]\n");
2111 printf(" One line: <# of holes>\n");
2112 printf(" Following lines: <hole #> <x> <y>\n");
2113 printf(
2114" Optional line: <# of regional attributes and/or area constraints>\n");
2115 printf(
2116" Optional following lines: <region #> <x> <y> <attribute> <max area>\n");
2117 printf("\n");
2118 printf(
2119" A .poly file represents a PSLG, as well as some additional information.\n"
2120);
2121 printf(
2122" The first section lists all the vertices, and is identical to the\n");
2123 printf(
2124" format of .node files. <# of vertices> may be set to zero to indicate\n"
2125);
2126 printf(
2127" that the vertices are listed in a separate .node file; .poly files\n");
2128 printf(
2129" produced by Triangle always have this format. A vertex set represented\n"
2130);
2131 printf(
2132" this way has the advantage that it may easily be triangulated with or\n");
2133 printf(
2134" without segments (depending on whether the -p switch is invoked).\n");
2135 printf("\n");
2136 printf(
2137" The second section lists the segments. Segments are edges whose\n");
2138 printf(
2139" presence in the triangulation is enforced. (Depending on the choice of\n"
2140);
2141 printf(
2142" switches, segment might be subdivided into smaller edges). Each\n");
2143 printf(
2144" segment is specified by listing the indices of its two endpoints. This\n"
2145);
2146 printf(
2147" means that you must include its endpoints in the vertex list. Each\n");
2148 printf(" segment, like each point, may have a boundary marker.\n\n");
2149 printf(
2150" If -q, -a, -u, and -s are not selected, Triangle produces a constrained\n"
2151);
2152 printf(
2153" Delaunay triangulation (CDT), in which each segment appears as a single\n"
2154);
2155 printf(
2156" edge in the triangulation. If -q, -a, -u, or -s is selected, Triangle\n"
2157);
2158 printf(
2159" produces a conforming constrained Delaunay triangulation (CCDT), in\n");
2160 printf(
2161" which segments may be subdivided into smaller edges. If -D is\n");
2162 printf(
2163" selected, Triangle produces a conforming Delaunay triangulation, so\n");
2164 printf(
2165" that every triangle is Delaunay, and not just constrained Delaunay.\n");
2166 printf("\n");
2167 printf(
2168" The third section lists holes (and concavities, if -c is selected) in\n");
2169 printf(
2170" the triangulation. Holes are specified by identifying a point inside\n");
2171 printf(
2172" each hole. After the triangulation is formed, Triangle creates holes\n");
2173 printf(
2174" by eating triangles, spreading out from each hole point until its\n");
2175 printf(
2176" progress is blocked by segments in the PSLG. You must be careful to\n");
2177 printf(
2178" enclose each hole in segments, or your whole triangulation might be\n");
2179 printf(
2180" eaten away. If the two triangles abutting a segment are eaten, the\n");
2181 printf(
2182" segment itself is also eaten. Do not place a hole directly on a\n");
2183 printf(" segment; if you do, Triangle chooses one side of the segment\n");
2184 printf(" arbitrarily.\n\n");
2185 printf(
2186" The optional fourth section lists regional attributes (to be assigned\n");
2187 printf(
2188" to all triangles in a region) and regional constraints on the maximum\n");
2189 printf(
2190" triangle area. Triangle reads this section only if the -A switch is\n");
2191 printf(
2192" used or the -a switch is used without a number following it, and the -r\n"
2193);
2194 printf(
2195" switch is not used. Regional attributes and area constraints are\n");
2196 printf(
2197" propagated in the same manner as holes: you specify a point for each\n");
2198 printf(
2199" attribute and/or constraint, and the attribute and/or constraint\n");
2200 printf(
2201" affects the whole region (bounded by segments) containing the point.\n");
2202 printf(
2203" If two values are written on a line after the x and y coordinate, the\n");
2204 printf(
2205" first such value is assumed to be a regional attribute (but is only\n");
2206 printf(
2207" applied if the -A switch is selected), and the second value is assumed\n"
2208);
2209 printf(
2210" to be a regional area constraint (but is only applied if the -a switch\n"
2211);
2212 printf(
2213" is selected). You may specify just one value after the coordinates,\n");
2214 printf(
2215" which can serve as both an attribute and an area constraint, depending\n"
2216);
2217 printf(
2218" on the choice of switches. If you are using the -A and -a switches\n");
2219 printf(
2220" simultaneously and wish to assign an attribute to some region without\n");
2221 printf(" imposing an area constraint, use a negative maximum area.\n\n");
2222 printf(
2223" When a triangulation is created from a .poly file, you must either\n");
2224 printf(
2225" enclose the entire region to be triangulated in PSLG segments, or\n");
2226 printf(
2227" use the -c switch, which automatically creates extra segments that\n");
2228 printf(
2229" enclose the convex hull of the PSLG. If you do not use the -c switch,\n"
2230);
2231 printf(
2232" Triangle eats all triangles that are not enclosed by segments; if you\n");
2233 printf(
2234" are not careful, your whole triangulation may be eaten away. If you do\n"
2235);
2236 printf(
2237" use the -c switch, you can still produce concavities by the appropriate\n"
2238);
2239 printf(
2240" placement of holes just inside the boundary of the convex hull.\n");
2241 printf("\n");
2242 printf(
2243" An ideal PSLG has no intersecting segments, nor any vertices that lie\n");
2244 printf(
2245" upon segments (except, of course, the endpoints of each segment). You\n"
2246);
2247 printf(
2248" aren't required to make your .poly files ideal, but you should be aware\n"
2249);
2250 printf(
2251" of what can go wrong. Segment intersections are relatively safe--\n");
2252 printf(
2253" Triangle calculates the intersection points for you and adds them to\n");
2254 printf(
2255" the triangulation--as long as your machine's floating-point precision\n");
2256 printf(
2257" doesn't become a problem. You are tempting the fates if you have three\n"
2258);
2259 printf(
2260" segments that cross at the same location, and expect Triangle to figure\n"
2261);
2262 printf(
2263" out where the intersection point is. Thanks to floating-point roundoff\n"
2264);
2265 printf(
2266" error, Triangle will probably decide that the three segments intersect\n"
2267);
2268 printf(
2269" at three different points, and you will find a minuscule triangle in\n");
2270 printf(
2271" your output--unless Triangle tries to refine the tiny triangle, uses\n");
2272 printf(
2273" up the last bit of machine precision, and fails to terminate at all.\n");
2274 printf(
2275" You're better off putting the intersection point in the input files,\n");
2276 printf(
2277" and manually breaking up each segment into two. Similarly, if you\n");
2278 printf(
2279" place a vertex at the middle of a segment, and hope that Triangle will\n"
2280);
2281 printf(
2282" break up the segment at that vertex, you might get lucky. On the other\n"
2283);
2284 printf(
2285" hand, Triangle might decide that the vertex doesn't lie precisely on\n");
2286 printf(
2287" the segment, and you'll have a needle-sharp triangle in your output--or\n"
2288);
2289 printf(" a lot of tiny triangles if you're generating a quality mesh.\n");
2290 printf("\n");
2291 printf(
2292" When Triangle reads a .poly file, it also writes a .poly file, which\n");
2293 printf(
2294" includes all the subsegments--the edges that are parts of input\n");
2295 printf(
2296" segments. If the -c switch is used, the output .poly file also\n");
2297 printf(
2298" includes all of the edges on the convex hull. Hence, the output .poly\n"
2299);
2300 printf(
2301" file is useful for finding edges associated with input segments and for\n"
2302);
2303 printf(
2304" setting boundary conditions in finite element simulations. Moreover,\n");
2305 printf(
2306" you will need the output .poly file if you plan to refine the output\n");
2307 printf(
2308" mesh, and don't want segments to be missing in later triangulations.\n");
2309 printf("\n");
2310 printf(" .area files:\n");
2311 printf(" First line: <# of triangles>\n");
2312 printf(" Following lines: <triangle #> <maximum area>\n");
2313 printf("\n");
2314 printf(
2315" An .area file associates with each triangle a maximum area that is used\n"
2316);
2317 printf(
2318" for mesh refinement. As with other file formats, every triangle must\n");
2319 printf(
2320" be represented, and the triangles must be numbered consecutively. A\n");
2321 printf(
2322" triangle may be left unconstrained by assigning it a negative maximum\n");
2323 printf(" area.\n\n");
2324 printf(" .edge files:\n");
2325 printf(" First line: <# of edges> <# of boundary markers (0 or 1)>\n");
2326 printf(
2327" Following lines: <edge #> <endpoint> <endpoint> [boundary marker]\n");
2328 printf("\n");
2329 printf(
2330" Endpoints are indices into the corresponding .node file. Triangle can\n"
2331);
2332 printf(
2333" produce .edge files (use the -e switch), but cannot read them. The\n");
2334 printf(
2335" optional column of boundary markers is suppressed by the -B switch.\n");
2336 printf("\n");
2337 printf(
2338" In Voronoi diagrams, one also finds a special kind of edge that is an\n");
2339 printf(
2340" infinite ray with only one endpoint. For these edges, a different\n");
2341 printf(" format is used:\n\n");
2342 printf(" <edge #> <endpoint> -1 <direction x> <direction y>\n\n");
2343 printf(
2344" The `direction' is a floating-point vector that indicates the direction\n"
2345);
2346 printf(" of the infinite ray.\n\n");
2347 printf(" .neigh files:\n");
2348 printf(
2349" First line: <# of triangles> <# of neighbors per triangle (always 3)>\n"
2350);
2351 printf(
2352" Following lines: <triangle #> <neighbor> <neighbor> <neighbor>\n");
2353 printf("\n");
2354 printf(
2355" Neighbors are indices into the corresponding .ele file. An index of -1\n"
2356);
2357 printf(
2358" indicates no neighbor (because the triangle is on an exterior\n");
2359 printf(
2360" boundary). The first neighbor of triangle i is opposite the first\n");
2361 printf(" corner of triangle i, and so on.\n\n");
2362 printf(
2363" Triangle can produce .neigh files (use the -n switch), but cannot read\n"
2364);
2365 printf(" them.\n\n");
2366 printf("Boundary Markers:\n\n");
2367 printf(
2368" Boundary markers are tags used mainly to identify which output vertices\n");
2369 printf(
2370" and edges are associated with which PSLG segment, and to identify which\n");
2371 printf(
2372" vertices and edges occur on a boundary of the triangulation. A common\n");
2373 printf(
2374" use is to determine where boundary conditions should be applied to a\n");
2375 printf(
2376" finite element mesh. You can prevent boundary markers from being written\n"
2377);
2378 printf(" into files produced by Triangle by using the -B switch.\n\n");
2379 printf(
2380" The boundary marker associated with each segment in an output .poly file\n"
2381);
2382 printf(" and each edge in an output .edge file is chosen as follows:\n");
2383 printf(
2384" - If an output edge is part or all of a PSLG segment with a nonzero\n");
2385 printf(
2386" boundary marker, then the edge is assigned the same marker.\n");
2387 printf(
2388" - Otherwise, if the edge lies on a boundary of the triangulation\n");
2389 printf(
2390" (even the boundary of a hole), then the edge is assigned the marker\n");
2391 printf(" one (1).\n");
2392 printf(" - Otherwise, the edge is assigned the marker zero (0).\n");
2393 printf(
2394" The boundary marker associated with each vertex in an output .node file\n");
2395 printf(" is chosen as follows:\n");
2396 printf(
2397" - If a vertex is assigned a nonzero boundary marker in the input file,\n"
2398);
2399 printf(
2400" then it is assigned the same marker in the output .node file.\n");
2401 printf(
2402" - Otherwise, if the vertex lies on a PSLG segment (even if it is an\n");
2403 printf(
2404" endpoint of the segment) with a nonzero boundary marker, then the\n");
2405 printf(
2406" vertex is assigned the same marker. If the vertex lies on several\n");
2407 printf(" such segments, one of the markers is chosen arbitrarily.\n");
2408 printf(
2409" - Otherwise, if the vertex occurs on a boundary of the triangulation,\n");
2410 printf(" then the vertex is assigned the marker one (1).\n");
2411 printf(" - Otherwise, the vertex is assigned the marker zero (0).\n");
2412 printf("\n");
2413 printf(
2414" If you want Triangle to determine for you which vertices and edges are on\n"
2415);
2416 printf(
2417" the boundary, assign them the boundary marker zero (or use no markers at\n"
2418);
2419 printf(
2420" all) in your input files. In the output files, all boundary vertices,\n");
2421 printf(" edges, and segments will be assigned the value one.\n\n");
2422 printf("Triangulation Iteration Numbers:\n\n");
2423 printf(
2424" Because Triangle can read and refine its own triangulations, input\n");
2425 printf(
2426" and output files have iteration numbers. For instance, Triangle might\n");
2427 printf(
2428" read the files mesh.3.node, mesh.3.ele, and mesh.3.poly, refine the\n");
2429 printf(
2430" triangulation, and output the files mesh.4.node, mesh.4.ele, and\n");
2431 printf(" mesh.4.poly. Files with no iteration number are treated as if\n");
2432 printf(
2433" their iteration number is zero; hence, Triangle might read the file\n");
2434 printf(
2435" points.node, triangulate it, and produce the files points.1.node and\n");
2436 printf(" points.1.ele.\n\n");
2437 printf(
2438" Iteration numbers allow you to create a sequence of successively finer\n");
2439 printf(
2440" meshes suitable for multigrid methods. They also allow you to produce a\n"
2441);
2442 printf(
2443" sequence of meshes using error estimate-driven mesh refinement.\n");
2444 printf("\n");
2445 printf(
2446" If you're not using refinement or quality meshing, and you don't like\n");
2447 printf(
2448" iteration numbers, use the -I switch to disable them. This switch also\n");
2449 printf(
2450" disables output of .node and .poly files to prevent your input files from\n"
2451);
2452 printf(
2453" being overwritten. (If the input is a .poly file that contains its own\n");
2454 printf(
2455" points, a .node file is written. This can be quite convenient for\n");
2456 printf(" computing CDTs or quality meshes.)\n\n");
2457 printf("Examples of How to Use Triangle:\n\n");
2458 printf(
2459" `triangle dots' reads vertices from dots.node, and writes their Delaunay\n"
2460);
2461 printf(
2462" triangulation to dots.1.node and dots.1.ele. (dots.1.node is identical\n");
2463 printf(
2464" to dots.node.) `triangle -I dots' writes the triangulation to dots.ele\n");
2465 printf(
2466" instead. (No additional .node file is needed, so none is written.)\n");
2467 printf("\n");
2468 printf(
2469" `triangle -pe object.1' reads a PSLG from object.1.poly (and possibly\n");
2470 printf(
2471" object.1.node, if the vertices are omitted from object.1.poly) and writes\n"
2472);
2473 printf(
2474" its constrained Delaunay triangulation to object.2.node and object.2.ele.\n"
2475);
2476 printf(
2477" The segments are copied to object.2.poly, and all edges are written to\n");
2478 printf(" object.2.edge.\n\n");
2479 printf(
2480" `triangle -pq31.5a.1 object' reads a PSLG from object.poly (and possibly\n"
2481);
2482 printf(
2483" object.node), generates a mesh whose angles are all between 31.5 and 117\n"
2484);
2485 printf(
2486" degrees and whose triangles all have areas of 0.1 or less, and writes the\n"
2487);
2488 printf(
2489" mesh to object.1.node and object.1.ele. Each segment may be broken up\n");
2490 printf(" into multiple subsegments; these are written to object.1.poly.\n");
2491 printf("\n");
2492 printf(
2493" Here is a sample file `box.poly' describing a square with a square hole:\n"
2494);
2495 printf("\n");
2496 printf(
2497" # A box with eight vertices in 2D, no attributes, one boundary marker.\n"
2498);
2499 printf(" 8 2 0 1\n");
2500 printf(" # Outer box has these vertices:\n");
2501 printf(" 1 0 0 0\n");
2502 printf(" 2 0 3 0\n");
2503 printf(" 3 3 0 0\n");
2504 printf(" 4 3 3 33 # A special marker for this vertex.\n");
2505 printf(" # Inner square has these vertices:\n");
2506 printf(" 5 1 1 0\n");
2507 printf(" 6 1 2 0\n");
2508 printf(" 7 2 1 0\n");
2509 printf(" 8 2 2 0\n");
2510 printf(" # Five segments with boundary markers.\n");
2511 printf(" 5 1\n");
2512 printf(" 1 1 2 5 # Left side of outer box.\n");
2513 printf(" # Square hole has these segments:\n");
2514 printf(" 2 5 7 0\n");
2515 printf(" 3 7 8 0\n");
2516 printf(" 4 8 6 10\n");
2517 printf(" 5 6 5 0\n");
2518 printf(" # One hole in the middle of the inner square.\n");
2519 printf(" 1\n");
2520 printf(" 1 1.5 1.5\n");
2521 printf("\n");
2522 printf(
2523" Note that some segments are missing from the outer square, so you must\n");
2524 printf(
2525" use the `-c' switch. After `triangle -pqc box.poly', here is the output\n"
2526);
2527 printf(
2528" file `box.1.node', with twelve vertices. The last four vertices were\n");
2529 printf(
2530" added to meet the angle constraint. Vertices 1, 2, and 9 have markers\n");
2531 printf(
2532" from segment 1. Vertices 6 and 8 have markers from segment 4. All the\n");
2533 printf(
2534" other vertices but 4 have been marked to indicate that they lie on a\n");
2535 printf(" boundary.\n\n");
2536 printf(" 12 2 0 1\n");
2537 printf(" 1 0 0 5\n");
2538 printf(" 2 0 3 5\n");
2539 printf(" 3 3 0 1\n");
2540 printf(" 4 3 3 33\n");
2541 printf(" 5 1 1 1\n");
2542 printf(" 6 1 2 10\n");
2543 printf(" 7 2 1 1\n");
2544 printf(" 8 2 2 10\n");
2545 printf(" 9 0 1.5 5\n");
2546 printf(" 10 1.5 0 1\n");
2547 printf(" 11 3 1.5 1\n");
2548 printf(" 12 1.5 3 1\n");
2549 printf(" # Generated by triangle -pqc box.poly\n");
2550 printf("\n");
2551 printf(" Here is the output file `box.1.ele', with twelve triangles.\n");
2552 printf("\n");
2553 printf(" 12 3 0\n");
2554 printf(" 1 5 6 9\n");
2555 printf(" 2 10 3 7\n");
2556 printf(" 3 6 8 12\n");
2557 printf(" 4 9 1 5\n");
2558 printf(" 5 6 2 9\n");
2559 printf(" 6 7 3 11\n");
2560 printf(" 7 11 4 8\n");
2561 printf(" 8 7 5 10\n");
2562 printf(" 9 12 2 6\n");
2563 printf(" 10 8 7 11\n");
2564 printf(" 11 5 1 10\n");
2565 printf(" 12 8 4 12\n");
2566 printf(" # Generated by triangle -pqc box.poly\n\n");
2567 printf(
2568" Here is the output file `box.1.poly'. Note that segments have been added\n"
2569);
2570 printf(
2571" to represent the convex hull, and some segments have been subdivided by\n");
2572 printf(
2573" newly added vertices. Note also that <# of vertices> is set to zero to\n");
2574 printf(" indicate that the vertices should be read from the .node file.\n");
2575 printf("\n");
2576 printf(" 0 2 0 1\n");
2577 printf(" 12 1\n");
2578 printf(" 1 1 9 5\n");
2579 printf(" 2 5 7 1\n");
2580 printf(" 3 8 7 1\n");
2581 printf(" 4 6 8 10\n");
2582 printf(" 5 5 6 1\n");
2583 printf(" 6 3 10 1\n");
2584 printf(" 7 4 11 1\n");
2585 printf(" 8 2 12 1\n");
2586 printf(" 9 9 2 5\n");
2587 printf(" 10 10 1 1\n");
2588 printf(" 11 11 3 1\n");
2589 printf(" 12 12 4 1\n");
2590 printf(" 1\n");
2591 printf(" 1 1.5 1.5\n");
2592 printf(" # Generated by triangle -pqc box.poly\n");
2593 printf("\n");
2594 printf("Refinement and Area Constraints:\n");
2595 printf("\n");
2596 printf(
2597" The -r switch causes a mesh (.node and .ele files) to be read and\n");
2598 printf(
2599" refined. If the -p switch is also used, a .poly file is read and used to\n"
2600);
2601 printf(
2602" specify edges that are constrained and cannot be eliminated (although\n");
2603 printf(
2604" they can be subdivided into smaller edges) by the refinement process.\n");
2605 printf("\n");
2606 printf(
2607" When you refine a mesh, you generally want to impose tighter constraints.\n"
2608);
2609 printf(
2610" One way to accomplish this is to use -q with a larger angle, or -a\n");
2611 printf(
2612" followed by a smaller area than you used to generate the mesh you are\n");
2613 printf(
2614" refining. Another way to do this is to create an .area file, which\n");
2615 printf(
2616" specifies a maximum area for each triangle, and use the -a switch\n");
2617 printf(
2618" (without a number following). Each triangle's area constraint is applied\n"
2619);
2620 printf(
2621" to that triangle. Area constraints tend to diffuse as the mesh is\n");
2622 printf(
2623" refined, so if there are large variations in area constraint between\n");
2624 printf(
2625" adjacent triangles, you may not get the results you want. In that case,\n"
2626);
2627 printf(
2628" consider instead using the -u switch and writing a C procedure that\n");
2629 printf(" determines which triangles are too large.\n\n");
2630 printf(
2631" If you are refining a mesh composed of linear (three-node) elements, the\n"
2632);
2633 printf(
2634" output mesh contains all the nodes present in the input mesh, in the same\n"
2635);
2636 printf(
2637" order, with new nodes added at the end of the .node file. However, the\n");
2638 printf(
2639" refinement is not hierarchical: there is no guarantee that each output\n");
2640 printf(
2641" element is contained in a single input element. Often, an output element\n"
2642);
2643 printf(
2644" can overlap two or three input elements, and some input edges are not\n");
2645 printf(
2646" present in the output mesh. Hence, a sequence of refined meshes forms a\n"
2647);
2648 printf(
2649" hierarchy of nodes, but not a hierarchy of elements. If you refine a\n");
2650 printf(
2651" mesh of higher-order elements, the hierarchical property applies only to\n"
2652);
2653 printf(
2654" the nodes at the corners of an element; the midpoint nodes on each edge\n");
2655 printf(" are discarded before the mesh is refined.\n\n");
2656 printf(
2657" Maximum area constraints in .poly files operate differently from those in\n"
2658);
2659 printf(
2660" .area files. A maximum area in a .poly file applies to the whole\n");
2661 printf(
2662" (segment-bounded) region in which a point falls, whereas a maximum area\n");
2663 printf(
2664" in an .area file applies to only one triangle. Area constraints in .poly\n"
2665);
2666 printf(
2667" files are used only when a mesh is first generated, whereas area\n");
2668 printf(
2669" constraints in .area files are used only to refine an existing mesh, and\n"
2670);
2671 printf(
2672" are typically based on a posteriori error estimates resulting from a\n");
2673 printf(" finite element simulation on that mesh.\n\n");
2674 printf(
2675" `triangle -rq25 object.1' reads object.1.node and object.1.ele, then\n");
2676 printf(
2677" refines the triangulation to enforce a 25 degree minimum angle, and then\n"
2678);
2679 printf(
2680" writes the refined triangulation to object.2.node and object.2.ele.\n");
2681 printf("\n");
2682 printf(
2683" `triangle -rpaa6.2 z.3' reads z.3.node, z.3.ele, z.3.poly, and z.3.area.\n"
2684);
2685 printf(
2686" After reconstructing the mesh and its subsegments, Triangle refines the\n");
2687 printf(
2688" mesh so that no triangle has area greater than 6.2, and furthermore the\n");
2689 printf(
2690" triangles satisfy the maximum area constraints in z.3.area. No angle\n");
2691 printf(
2692" bound is imposed at all. The output is written to z.4.node, z.4.ele, and\n"
2693);
2694 printf(" z.4.poly.\n\n");
2695 printf(
2696" The sequence `triangle -qa1 x', `triangle -rqa.3 x.1', `triangle -rqa.1\n");
2697 printf(
2698" x.2' creates a sequence of successively finer meshes x.1, x.2, and x.3,\n");
2699 printf(" suitable for multigrid.\n\n");
2700 printf("Convex Hulls and Mesh Boundaries:\n\n");
2701 printf(
2702" If the input is a vertex set (not a PSLG), Triangle produces its convex\n");
2703 printf(
2704" hull as a by-product in the output .poly file if you use the -c switch.\n");
2705 printf(
2706" There are faster algorithms for finding a two-dimensional convex hull\n");
2707 printf(" than triangulation, of course, but this one comes for free.\n\n");
2708 printf(
2709" If the input is an unconstrained mesh (you are using the -r switch but\n");
2710 printf(
2711" not the -p switch), Triangle produces a list of its boundary edges\n");
2712 printf(
2713" (including hole boundaries) as a by-product when you use the -c switch.\n");
2714 printf(
2715" If you also use the -p switch, the output .poly file contains all the\n");
2716 printf(" segments from the input .poly file as well.\n\n");
2717 printf("Voronoi Diagrams:\n\n");
2718 printf(
2719" The -v switch produces a Voronoi diagram, in files suffixed .v.node and\n");
2720 printf(
2721" .v.edge. For example, `triangle -v points' reads points.node, produces\n");
2722 printf(
2723" its Delaunay triangulation in points.1.node and points.1.ele, and\n");
2724 printf(
2725" produces its Voronoi diagram in points.1.v.node and points.1.v.edge. The\n"
2726);
2727 printf(
2728" .v.node file contains a list of all Voronoi vertices, and the .v.edge\n");
2729 printf(
2730" file contains a list of all Voronoi edges, some of which may be infinite\n"
2731);
2732 printf(
2733" rays. (The choice of filenames makes it easy to run the set of Voronoi\n");
2734 printf(" vertices through Triangle, if so desired.)\n\n");
2735 printf(
2736" This implementation does not use exact arithmetic to compute the Voronoi\n"
2737);
2738 printf(
2739" vertices, and does not check whether neighboring vertices are identical.\n"
2740);
2741 printf(
2742" Be forewarned that if the Delaunay triangulation is degenerate or\n");
2743 printf(
2744" near-degenerate, the Voronoi diagram may have duplicate vertices or\n");
2745 printf(" crossing edges.\n\n");
2746 printf(
2747" The result is a valid Voronoi diagram only if Triangle's output is a true\n"
2748);
2749 printf(
2750" Delaunay triangulation. The Voronoi output is usually meaningless (and\n");
2751 printf(
2752" may contain crossing edges and other pathology) if the output is a CDT or\n"
2753);
2754 printf(
2755" CCDT, or if it has holes or concavities. If the triangulated domain is\n");
2756 printf(
2757" convex and has no holes, you can use -D switch to force Triangle to\n");
2758 printf(
2759" construct a conforming Delaunay triangulation instead of a CCDT, so the\n");
2760 printf(" Voronoi diagram will be valid.\n\n");
2761 printf("Mesh Topology:\n\n");
2762 printf(
2763" You may wish to know which triangles are adjacent to a certain Delaunay\n");
2764 printf(
2765" edge in an .edge file, which Voronoi cells are adjacent to a certain\n");
2766 printf(
2767" Voronoi edge in a .v.edge file, or which Voronoi cells are adjacent to\n");
2768 printf(
2769" each other. All of this information can be found by cross-referencing\n");
2770 printf(
2771" output files with the recollection that the Delaunay triangulation and\n");
2772 printf(" the Voronoi diagram are planar duals.\n\n");
2773 printf(
2774" Specifically, edge i of an .edge file is the dual of Voronoi edge i of\n");
2775 printf(
2776" the corresponding .v.edge file, and is rotated 90 degrees counterclock-\n");
2777 printf(
2778" wise from the Voronoi edge. Triangle j of an .ele file is the dual of\n");
2779 printf(
2780" vertex j of the corresponding .v.node file. Voronoi cell k is the dual\n");
2781 printf(" of vertex k of the corresponding .node file.\n\n");
2782 printf(
2783" Hence, to find the triangles adjacent to a Delaunay edge, look at the\n");
2784 printf(
2785" vertices of the corresponding Voronoi edge. If the endpoints of a\n");
2786 printf(
2787" Voronoi edge are Voronoi vertices 2 and 6 respectively, then triangles 2\n"
2788);
2789 printf(
2790" and 6 adjoin the left and right sides of the corresponding Delaunay edge,\n"
2791);
2792 printf(
2793" respectively. To find the Voronoi cells adjacent to a Voronoi edge, look\n"
2794);
2795 printf(
2796" at the endpoints of the corresponding Delaunay edge. If the endpoints of\n"
2797);
2798 printf(
2799" a Delaunay edge are input vertices 7 and 12, then Voronoi cells 7 and 12\n"
2800);
2801 printf(
2802" adjoin the right and left sides of the corresponding Voronoi edge,\n");
2803 printf(
2804" respectively. To find which Voronoi cells are adjacent to each other,\n");
2805 printf(" just read the list of Delaunay edges.\n\n");
2806 printf(
2807" Triangle does not write a list of the edges adjoining each Voronoi cell,\n"
2808);
2809 printf(
2810" but you can reconstructed it straightforwardly. For instance, to find\n");
2811 printf(
2812" all the edges of Voronoi cell 1, search the output .edge file for every\n");
2813 printf(
2814" edge that has input vertex 1 as an endpoint. The corresponding dual\n");
2815 printf(
2816" edges in the output .v.edge file form the boundary of Voronoi cell 1.\n");
2817 printf("\n");
2818 printf(
2819" For each Voronoi vertex, the .neigh file gives a list of the three\n");
2820 printf(
2821" Voronoi vertices attached to it. You might find this more convenient\n");
2822 printf(" than the .v.edge file.\n\n");
2823 printf("Quadratic Elements:\n\n");
2824 printf(
2825" Triangle generates meshes with subparametric quadratic elements if the\n");
2826 printf(
2827" -o2 switch is specified. Quadratic elements have six nodes per element,\n"
2828);
2829 printf(
2830" rather than three. `Subparametric' means that the edges of the triangles\n"
2831);
2832 printf(
2833" are always straight, so that subparametric quadratic elements are\n");
2834 printf(
2835" geometrically identical to linear elements, even though they can be used\n"
2836);
2837 printf(
2838" with quadratic interpolating functions. The three extra nodes of an\n");
2839 printf(
2840" element fall at the midpoints of the three edges, with the fourth, fifth,\n"
2841);
2842 printf(
2843" and sixth nodes appearing opposite the first, second, and third corners\n");
2844 printf(" respectively.\n\n");
2845 printf("Domains with Small Angles:\n\n");
2846 printf(
2847" If two input segments adjoin each other at a small angle, clearly the -q\n"
2848);
2849 printf(
2850" switch cannot remove the small angle. Moreover, Triangle may have no\n");
2851 printf(
2852" choice but to generate additional triangles whose smallest angles are\n");
2853 printf(
2854" smaller than the specified bound. However, these triangles only appear\n");
2855 printf(
2856" between input segments separated by small angles. Moreover, if you\n");
2857 printf(
2858" request a minimum angle of theta degrees, Triangle will generally produce\n"
2859);
2860 printf(
2861" no angle larger than 180 - 2 theta, even if it is forced to compromise on\n"
2862);
2863 printf(" the minimum angle.\n\n");
2864 printf("Statistics:\n\n");
2865 printf(
2866" After generating a mesh, Triangle prints a count of entities in the\n");
2867 printf(
2868" output mesh, including the number of vertices, triangles, edges, exterior\n"
2869);
2870 printf(
2871" boundary edges (i.e. subsegments on the boundary of the triangulation,\n");
2872 printf(
2873" including hole boundaries), interior boundary edges (i.e. subsegments of\n"
2874);
2875 printf(
2876" input segments not on the boundary), and total subsegments. If you've\n");
2877 printf(
2878" forgotten the statistics for an existing mesh, run Triangle on that mesh\n"
2879);
2880 printf(
2881" with the -rNEP switches to read the mesh and print the statistics without\n"
2882);
2883 printf(
2884" writing any files. Use -rpNEP if you've got a .poly file for the mesh.\n");
2885 printf("\n");
2886 printf(
2887" The -V switch produces extended statistics, including a rough estimate\n");
2888 printf(
2889" of memory use, the number of calls to geometric predicates, and\n");
2890 printf(
2891" histograms of the angles and the aspect ratios of the triangles in the\n");
2892 printf(" mesh.\n\n");
2893 printf("Exact Arithmetic:\n\n");
2894 printf(
2895" Triangle uses adaptive exact arithmetic to perform what computational\n");
2896 printf(
2897" geometers call the `orientation' and `incircle' tests. If the floating-\n"
2898);
2899 printf(
2900" point arithmetic of your machine conforms to the IEEE 754 standard (as\n");
2901 printf(
2902" most workstations do), and does not use extended precision internal\n");
2903 printf(
2904" floating-point registers, then your output is guaranteed to be an\n");
2905 printf(
2906" absolutely true Delaunay or constrained Delaunay triangulation, roundoff\n"
2907);
2908 printf(
2909" error notwithstanding. The word `adaptive' implies that these arithmetic\n"
2910);
2911 printf(
2912" routines compute the result only to the precision necessary to guarantee\n"
2913);
2914 printf(
2915" correctness, so they are usually nearly as fast as their approximate\n");
2916 printf(" counterparts.\n\n");
2917 printf(
2918" May CPUs, including Intel x86 processors, have extended precision\n");
2919 printf(
2920" floating-point registers. These must be reconfigured so their precision\n"
2921);
2922 printf(
2923" is reduced to memory precision. Triangle does this if it is compiled\n");
2924 printf(" correctly. See the makefile for details.\n\n");
2925 printf(
2926" The exact tests can be disabled with the -X switch. On most inputs, this\n"
2927);
2928 printf(
2929" switch reduces the computation time by about eight percent--it's not\n");
2930 printf(
2931" worth the risk. There are rare difficult inputs (having many collinear\n");
2932 printf(
2933" and cocircular vertices), however, for which the difference in speed\n");
2934 printf(
2935" could be a factor of two. Be forewarned that these are precisely the\n");
2936 printf(
2937" inputs most likely to cause errors if you use the -X switch. Hence, the\n"
2938);
2939 printf(" -X switch is not recommended.\n\n");
2940 printf(
2941" Unfortunately, the exact tests don't solve every numerical problem.\n");
2942 printf(
2943" Exact arithmetic is not used to compute the positions of new vertices,\n");
2944 printf(
2945" because the bit complexity of vertex coordinates would grow without\n");
2946 printf(
2947" bound. Hence, segment intersections aren't computed exactly; in very\n");
2948 printf(
2949" unusual cases, roundoff error in computing an intersection point might\n");
2950 printf(
2951" actually lead to an inverted triangle and an invalid triangulation.\n");
2952 printf(
2953" (This is one reason to specify your own intersection points in your .poly\n"
2954);
2955 printf(
2956" files.) Similarly, exact arithmetic is not used to compute the vertices\n"
2957);
2958 printf(" of the Voronoi diagram.\n\n");
2959 printf(
2960" Another pair of problems not solved by the exact arithmetic routines is\n");
2961 printf(
2962" underflow and overflow. If Triangle is compiled for double precision\n");
2963 printf(
2964" arithmetic, I believe that Triangle's geometric predicates work correctly\n"
2965);
2966 printf(
2967" if the exponent of every input coordinate falls in the range [-148, 201].\n"
2968);
2969 printf(
2970" Underflow can silently prevent the orientation and incircle tests from\n");
2971 printf(
2972" being performed exactly, while overflow typically causes a floating\n");
2973 printf(" exception.\n\n");
2974 printf("Calling Triangle from Another Program:\n\n");
2975 printf(" Read the file triangle.h for details.\n\n");
2976 printf("Troubleshooting:\n\n");
2977 printf(" Please read this section before mailing me bugs.\n\n");
2978 printf(" `My output mesh has no triangles!'\n\n");
2979 printf(
2980" If you're using a PSLG, you've probably failed to specify a proper set\n"
2981);
2982 printf(
2983" of bounding segments, or forgotten to use the -c switch. Or you may\n");
2984 printf(
2985" have placed a hole badly, thereby eating all your triangles. To test\n");
2986 printf(" these possibilities, try again with the -c and -O switches.\n");
2987 printf(
2988" Alternatively, all your input vertices may be collinear, in which case\n"
2989);
2990 printf(" you can hardly expect to triangulate them.\n\n");
2991 printf(" `Triangle doesn't terminate, or just crashes.'\n\n");
2992 printf(
2993" Bad things can happen when triangles get so small that the distance\n");
2994 printf(
2995" between their vertices isn't much larger than the precision of your\n");
2996 printf(
2997" machine's arithmetic. If you've compiled Triangle for single-precision\n"
2998);
2999 printf(
3000" arithmetic, you might do better by recompiling it for double-precision.\n"
3001);
3002 printf(
3003" Then again, you might just have to settle for more lenient constraints\n"
3004);
3005 printf(
3006" on the minimum angle and the maximum area than you had planned.\n");
3007 printf("\n");
3008 printf(
3009" You can minimize precision problems by ensuring that the origin lies\n");
3010 printf(
3011" inside your vertex set, or even inside the densest part of your\n");
3012 printf(
3013" mesh. If you're triangulating an object whose x-coordinates all fall\n");
3014 printf(
3015" between 6247133 and 6247134, you're not leaving much floating-point\n");
3016 printf(" precision for Triangle to work with.\n\n");
3017 printf(
3018" Precision problems can occur covertly if the input PSLG contains two\n");
3019 printf(
3020" segments that meet (or intersect) at an extremely small angle, or if\n");
3021 printf(
3022" such an angle is introduced by the -c switch. If you don't realize\n");
3023 printf(
3024" that a tiny angle is being formed, you might never discover why\n");
3025 printf(
3026" Triangle is crashing. To check for this possibility, use the -S switch\n"
3027);
3028 printf(
3029" (with an appropriate limit on the number of Steiner points, found by\n");
3030 printf(
3031" trial-and-error) to stop Triangle early, and view the output .poly file\n"
3032);
3033 printf(
3034" with Show Me (described below). Look carefully for regions where dense\n"
3035);
3036 printf(
3037" clusters of vertices are forming and for small angles between segments.\n"
3038);
3039 printf(
3040" Zoom in closely, as such segments might look like a single segment from\n"
3041);
3042 printf(" a distance.\n\n");
3043 printf(
3044" If some of the input values are too large, Triangle may suffer a\n");
3045 printf(
3046" floating exception due to overflow when attempting to perform an\n");
3047 printf(
3048" orientation or incircle test. (Read the section on exact arithmetic\n");
3049 printf(
3050" above.) Again, I recommend compiling Triangle for double (rather\n");
3051 printf(" than single) precision arithmetic.\n\n");
3052 printf(
3053" Unexpected problems can arise if you use quality meshing (-q, -a, or\n");
3054 printf(
3055" -u) with an input that is not segment-bounded--that is, if your input\n");
3056 printf(
3057" is a vertex set, or you're using the -c switch. If the convex hull of\n"
3058);
3059 printf(
3060" your input vertices has collinear vertices on its boundary, an input\n");
3061 printf(
3062" vertex that you think lies on the convex hull might actually lie just\n");
3063 printf(
3064" inside the convex hull. If so, the vertex and the nearby convex hull\n");
3065 printf(
3066" edge form an extremely thin triangle. When Triangle tries to refine\n");
3067 printf(
3068" the mesh to enforce angle and area constraints, Triangle might generate\n"
3069);
3070 printf(
3071" extremely tiny triangles, or it might fail because of insufficient\n");
3072 printf(" floating-point precision.\n\n");
3073 printf(
3074" `The numbering of the output vertices doesn't match the input vertices.'\n"
3075);
3076 printf("\n");
3077 printf(
3078" You may have had duplicate input vertices, or you may have eaten some\n");
3079 printf(
3080" of your input vertices with a hole, or by placing them outside the area\n"
3081);
3082 printf(
3083" enclosed by segments. In any case, you can solve the problem by not\n");
3084 printf(" using the -j switch.\n\n");
3085 printf(
3086" `Triangle executes without incident, but when I look at the resulting\n");
3087 printf(
3088" mesh, it has overlapping triangles or other geometric inconsistencies.'\n");
3089 printf("\n");
3090 printf(
3091" If you select the -X switch, Triangle occasionally makes mistakes due\n");
3092 printf(
3093" to floating-point roundoff error. Although these errors are rare,\n");
3094 printf(
3095" don't use the -X switch. If you still have problems, please report the\n"
3096);
3097 printf(" bug.\n\n");
3098 printf(
3099" `Triangle executes without incident, but when I look at the resulting\n");
3100 printf(" Voronoi diagram, it has overlapping edges or other geometric\n");
3101 printf(" inconsistencies.'\n");
3102 printf("\n");
3103 printf(
3104" If your input is a PSLG (-p), you can only expect a meaningful Voronoi\n"
3105);
3106 printf(
3107" diagram if the domain you are triangulating is convex and free of\n");
3108 printf(
3109" holes, and you use the -D switch to construct a conforming Delaunay\n");
3110 printf(" triangulation (instead of a CDT or CCDT).\n\n");
3111 printf(
3112" Strange things can happen if you've taken liberties with your PSLG. Do\n");
3113 printf(
3114" you have a vertex lying in the middle of a segment? Triangle sometimes\n");
3115 printf(
3116" copes poorly with that sort of thing. Do you want to lay out a collinear\n"
3117);
3118 printf(
3119" row of evenly spaced, segment-connected vertices? Have you simply\n");
3120 printf(
3121" defined one long segment connecting the leftmost vertex to the rightmost\n"
3122);
3123 printf(
3124" vertex, and a bunch of vertices lying along it? This method occasionally\n"
3125);
3126 printf(
3127" works, especially with horizontal and vertical lines, but often it\n");
3128 printf(
3129" doesn't, and you'll have to connect each adjacent pair of vertices with a\n"
3130);
3131 printf(" separate segment. If you don't like it, tough.\n\n");
3132 printf(
3133" Furthermore, if you have segments that intersect other than at their\n");
3134 printf(
3135" endpoints, try not to let the intersections fall extremely close to PSLG\n"
3136);
3137 printf(" vertices or each other.\n\n");
3138 printf(
3139" If you have problems refining a triangulation not produced by Triangle:\n");
3140 printf(
3141" Are you sure the triangulation is geometrically valid? Is it formatted\n");
3142 printf(
3143" correctly for Triangle? Are the triangles all listed so the first three\n"
3144);
3145 printf(
3146" vertices are their corners in counterclockwise order? Are all of the\n");
3147 printf(
3148" triangles constrained Delaunay? Triangle's Delaunay refinement algorithm\n"
3149);
3150 printf(" assumes that it starts with a CDT.\n\n");
3151 printf("Show Me:\n\n");
3152 printf(
3153" Triangle comes with a separate program named `Show Me', whose primary\n");
3154 printf(
3155" purpose is to draw meshes on your screen or in PostScript. Its secondary\n"
3156);
3157 printf(
3158" purpose is to check the validity of your input files, and do so more\n");
3159 printf(
3160" thoroughly than Triangle does. Unlike Triangle, Show Me requires that\n");
3161 printf(
3162" you have the X Windows system. Sorry, Microsoft Windows users.\n");
3163 printf("\n");
3164 printf("Triangle on the Web:\n");
3165 printf("\n");
3166 printf(" To see an illustrated version of these instructions, check out\n");
3167 printf("\n");
3168 printf(" http://www.cs.cmu.edu/~quake/triangle.html\n");
3169 printf("\n");
3170 printf("A Brief Plea:\n");
3171 printf("\n");
3172 printf(
3173" If you use Triangle, and especially if you use it to accomplish real\n");
3174 printf(
3175" work, I would like very much to hear from you. A short letter or email\n");
3176 printf(
3177" (to jrs@cs.berkeley.edu) describing how you use Triangle will mean a lot\n"
3178);
3179 printf(
3180" to me. The more people I know are using this program, the more easily I\n"
3181);
3182 printf(
3183" can justify spending time on improvements, which in turn will benefit\n");
3184 printf(
3185" you. Also, I can put you on a list to receive email whenever a new\n");
3186 printf(" version of Triangle is available.\n\n");
3187 printf(
3188" If you use a mesh generated by Triangle in a publication, please include\n"
3189);
3190 printf(
3191" an acknowledgment as well. And please spell Triangle with a capital `T'!\n"
3192);
3193 printf(
3194" If you want to include a citation, use `Jonathan Richard Shewchuk,\n");
3195 printf(
3196" ``Triangle: Engineering a 2D Quality Mesh Generator and Delaunay\n");
3197 printf(
3198" Triangulator,'' in Applied Computational Geometry: Towards Geometric\n");
3199 printf(
3200" Engineering (Ming C. Lin and Dinesh Manocha, editors), volume 1148 of\n");
3201 printf(
3202" Lecture Notes in Computer Science, pages 203-222, Springer-Verlag,\n");
3203 printf(
3204" Berlin, May 1996. (From the First ACM Workshop on Applied Computational\n"
3205);
3206 printf(" Geometry.)'\n\n");
3207 printf("Research credit:\n\n");
3208 printf(
3209" Of course, I can take credit for only a fraction of the ideas that made\n");
3210 printf(
3211" this mesh generator possible. Triangle owes its existence to the efforts\n"
3212);
3213 printf(
3214" of many fine computational geometers and other researchers, including\n");
3215 printf(
3216" Marshall Bern, L. Paul Chew, Kenneth L. Clarkson, Boris Delaunay, Rex A.\n"
3217);
3218 printf(
3219" Dwyer, David Eppstein, Steven Fortune, Leonidas J. Guibas, Donald E.\n");
3220 printf(
3221" Knuth, Charles L. Lawson, Der-Tsai Lee, Gary L. Miller, Ernst P. Mucke,\n");
3222 printf(
3223" Steven E. Pav, Douglas M. Priest, Jim Ruppert, Isaac Saias, Bruce J.\n");
3224 printf(
3225" Schachter, Micha Sharir, Peter W. Shor, Daniel D. Sleator, Jorge Stolfi,\n"
3226);
3227 printf(" Robert E. Tarjan, Alper Ungor, Christopher J. Van Wyk, Noel J.\n");
3228 printf(
3229" Walkington, and Binhai Zhu. See the comments at the beginning of the\n");
3230 printf(" source code for references.\n\n");
3231 triexit(0);
3232}
3233
3234#endif /* not TRILIBRARY */
3235
3236/*****************************************************************************/
3237/* */
3238/* internalerror() Ask the user to send me the defective product. Exit. */
3239/* */
3240/*****************************************************************************/
3241
3243{
3244 printf(" Please report this bug to jrs@cs.berkeley.edu\n");
3245 printf(" Include the message above, your input data set, and the exact\n");
3246 printf(" command line you used to run Triangle.\n");
3247 triexit(1);
3248}
3249
3250/*****************************************************************************/
3251/* */
3252/* parsecommandline() Read the command line, identify switches, and set */
3253/* up options and file names. */
3254/* */
3255/*****************************************************************************/
3256
3257#ifdef ANSI_DECLARATORS
3258void parsecommandline(int argc, char **argv, struct behavior *b)
3259#else /* not ANSI_DECLARATORS */
3260void parsecommandline(argc, argv, b)
3261int argc;
3262char **argv;
3263struct behavior *b;
3264#endif /* not ANSI_DECLARATORS */
3265
3266{
3267#ifdef TRILIBRARY
3268#define STARTINDEX 0
3269#else /* not TRILIBRARY */
3270#define STARTINDEX 1
3271 int increment;
3272 int meshnumber;
3273#endif /* not TRILIBRARY */
3274 /* int i, j, k; */
3275 int i, j;
3276 /* char workstring[FILENAMESIZE]; */
3277
3278 b->poly = b->refine = b->quality = 0;
3279 b->vararea = b->fixedarea = b->usertest = 0;
3280 b->regionattrib = b->convex = b->weighted = b->jettison = 0;
3281 b->firstnumber = 1;
3282 b->edgesout = b->voronoi = b->neighbors = b->geomview = 0;
3283 b->nobound = b->nopolywritten = b->nonodewritten = b->noelewritten = 0;
3284 b->noiterationnum = 0;
3285 b->noholes = b->noexact = 0;
3286 b->incremental = b->sweepline = 0;
3287 b->dwyer = 1;
3288 b->splitseg = 0;
3289 b->docheck = 0;
3290 b->nobisect = 0;
3291 b->conformdel = 0;
3292 b->steiner = -1;
3293 b->order = 1;
3294 b->minangle = 0.0;
3295 b->maxarea = -1.0;
3296 b->quiet = b->verbose = 0;
3297#ifndef TRILIBRARY
3298 b->innodefilename[0] = '\0';
3299#endif /* not TRILIBRARY */
3300
3301 for (i = STARTINDEX; i < argc; i++) {
3302#ifndef TRILIBRARY
3303 if (argv[i][0] == '-') {
3304#endif /* not TRILIBRARY */
3305 for (j = STARTINDEX; argv[i][j] != '\0'; j++) {
3306 if (argv[i][j] == 'p') {
3307 b->poly = 1;
3308 }
3309#ifndef CDT_ONLY
3310 if (argv[i][j] == 'r') {
3311 b->refine = 1;
3312 }
3313 if (argv[i][j] == 'q') {
3314 b->quality = 1;
3315 if (((argv[i][j + 1] >= '0') && (argv[i][j + 1] <= '9')) ||
3316 (argv[i][j + 1] == '.')) {
3317 k = 0;
3318 while (((argv[i][j + 1] >= '0') && (argv[i][j + 1] <= '9')) ||
3319 (argv[i][j + 1] == '.')) {
3320 j++;
3321 workstring[k] = argv[i][j];
3322 k++;
3323 }
3324 workstring[k] = '\0';
3325 b->minangle = (REAL) strtod(workstring, (char **) NULL);
3326 } else {
3327 b->minangle = 20.0;
3328 }
3329 }
3330 if (argv[i][j] == 'a') {
3331 b->quality = 1;
3332 if (((argv[i][j + 1] >= '0') && (argv[i][j + 1] <= '9')) ||
3333 (argv[i][j + 1] == '.')) {
3334 b->fixedarea = 1;
3335 k = 0;
3336 while (((argv[i][j + 1] >= '0') && (argv[i][j + 1] <= '9')) ||
3337 (argv[i][j + 1] == '.')) {
3338 j++;
3339 workstring[k] = argv[i][j];
3340 k++;
3341 }
3342 workstring[k] = '\0';
3343 b->maxarea = (REAL) strtod(workstring, (char **) NULL);
3344 if (b->maxarea <= 0.0) {
3345 printf("Error: Maximum area must be greater than zero.\n");
3346 triexit(1);
3347 }
3348 } else {
3349 b->vararea = 1;
3350 }
3351 }
3352 if (argv[i][j] == 'u') {
3353 b->quality = 1;
3354 b->usertest = 1;
3355 }
3356#endif /* not CDT_ONLY */
3357 if (argv[i][j] == 'A') {
3358 b->regionattrib = 1;
3359 }
3360 if (argv[i][j] == 'c') {
3361 b->convex = 1;
3362 }
3363 if (argv[i][j] == 'w') {
3364 b->weighted = 1;
3365 }
3366 if (argv[i][j] == 'W') {
3367 b->weighted = 2;
3368 }
3369 if (argv[i][j] == 'j') {
3370 b->jettison = 1;
3371 }
3372 if (argv[i][j] == 'z') {
3373 b->firstnumber = 0;
3374 }
3375 if (argv[i][j] == 'e') {
3376 b->edgesout = 1;
3377 }
3378 if (argv[i][j] == 'v') {
3379 b->voronoi = 1;
3380 }
3381 if (argv[i][j] == 'n') {
3382 b->neighbors = 1;
3383 }
3384 if (argv[i][j] == 'g') {
3385 b->geomview = 1;
3386 }
3387 if (argv[i][j] == 'B') {
3388 b->nobound = 1;
3389 }
3390 if (argv[i][j] == 'P') {
3391 b->nopolywritten = 1;
3392 }
3393 if (argv[i][j] == 'N') {
3394 b->nonodewritten = 1;
3395 }
3396 if (argv[i][j] == 'E') {
3397 b->noelewritten = 1;
3398 }
3399#ifndef TRILIBRARY
3400 if (argv[i][j] == 'I') {
3401 b->noiterationnum = 1;
3402 }
3403#endif /* not TRILIBRARY */
3404 if (argv[i][j] == 'O') {
3405 b->noholes = 1;
3406 }
3407 if (argv[i][j] == 'X') {
3408 b->noexact = 1;
3409 }
3410 if (argv[i][j] == 'o') {
3411 if (argv[i][j + 1] == '2') {
3412 j++;
3413 b->order = 2;
3414 }
3415 }
3416#ifndef CDT_ONLY
3417 if (argv[i][j] == 'Y') {
3418 b->nobisect++;
3419 }
3420 if (argv[i][j] == 'S') {
3421 b->steiner = 0;
3422 while ((argv[i][j + 1] >= '0') && (argv[i][j + 1] <= '9')) {
3423 j++;
3424 b->steiner = b->steiner * 10 + (int) (argv[i][j] - '0');
3425 }
3426 }
3427#endif /* not CDT_ONLY */
3428#ifndef REDUCED
3429 if (argv[i][j] == 'i') {
3430 b->incremental = 1;
3431 }
3432 if (argv[i][j] == 'F') {
3433 b->sweepline = 1;
3434 }
3435#endif /* not REDUCED */
3436 if (argv[i][j] == 'l') {
3437 b->dwyer = 0;
3438 }
3439#ifndef REDUCED
3440#ifndef CDT_ONLY
3441 if (argv[i][j] == 's') {
3442 b->splitseg = 1;
3443 }
3444 if ((argv[i][j] == 'D') || (argv[i][j] == 'L')) {
3445 b->quality = 1;
3446 b->conformdel = 1;
3447 }
3448#endif /* not CDT_ONLY */
3449 if (argv[i][j] == 'C') {
3450 b->docheck = 1;
3451 }
3452#endif /* not REDUCED */
3453 if (argv[i][j] == 'Q') {
3454 b->quiet = 1;
3455 }
3456 if (argv[i][j] == 'V') {
3457 b->verbose++;
3458 }
3459#ifndef TRILIBRARY
3460 if ((argv[i][j] == 'h') || (argv[i][j] == 'H') ||
3461 (argv[i][j] == '?')) {
3462 info();
3463 }
3464#endif /* not TRILIBRARY */
3465 }
3466#ifndef TRILIBRARY
3467 } else {
3468 strncpy(b->innodefilename, argv[i], FILENAMESIZE - 1);
3469 b->innodefilename[FILENAMESIZE - 1] = '\0';
3470 }
3471#endif /* not TRILIBRARY */
3472 }
3473#ifndef TRILIBRARY
3474 if (b->innodefilename[0] == '\0') {
3475 syntax();
3476 }
3477 if (!strcmp(&b->innodefilename[strlen(b->innodefilename) - 5], ".node")) {
3478 b->innodefilename[strlen(b->innodefilename) - 5] = '\0';
3479 }
3480 if (!strcmp(&b->innodefilename[strlen(b->innodefilename) - 5], ".poly")) {
3481 b->innodefilename[strlen(b->innodefilename) - 5] = '\0';
3482 b->poly = 1;
3483 }
3484#ifndef CDT_ONLY
3485 if (!strcmp(&b->innodefilename[strlen(b->innodefilename) - 4], ".ele")) {
3486 b->innodefilename[strlen(b->innodefilename) - 4] = '\0';
3487 b->refine = 1;
3488 }
3489 if (!strcmp(&b->innodefilename[strlen(b->innodefilename) - 5], ".area")) {
3490 b->innodefilename[strlen(b->innodefilename) - 5] = '\0';
3491 b->refine = 1;
3492 b->quality = 1;
3493 b->vararea = 1;
3494 }
3495#endif /* not CDT_ONLY */
3496#endif /* not TRILIBRARY */
3497 b->usesegments = b->poly || b->refine || b->quality || b->convex;
3498 b->goodangle = cos(b->minangle * PI / 180.0);
3499 if (b->goodangle == 1.0) {
3500 b->offconstant = 0.0;
3501 } else {
3502 b->offconstant = 0.475 * sqrt((1.0 + b->goodangle) / (1.0 - b->goodangle));
3503 }
3504 b->goodangle *= b->goodangle;
3505 if (b->refine && b->noiterationnum) {
3506 printf(
3507 "Error: You cannot use the -I switch when refining a triangulation.\n");
3508 triexit(1);
3509 }
3510 /* Be careful not to allocate space for element area constraints that */
3511 /* will never be assigned any value (other than the default -1.0). */
3512 if (!b->refine && !b->poly) {
3513 b->vararea = 0;
3514 }
3515 /* Be careful not to add an extra attribute to each element unless the */
3516 /* input supports it (PSLG in, but not refining a preexisting mesh). */
3517 if (b->refine || !b->poly) {
3518 b->regionattrib = 0;
3519 }
3520 /* Regular/weighted triangulations are incompatible with PSLGs */
3521 /* and meshing. */
3522 if (b->weighted && (b->poly || b->quality)) {
3523 b->weighted = 0;
3524 if (!b->quiet) {
3525 printf("Warning: weighted triangulations (-w, -W) are incompatible\n");
3526 printf(" with PSLGs (-p) and meshing (-q, -a, -u). Weights ignored.\n"
3527 );
3528 }
3529 }
3530 if (b->jettison && b->nonodewritten && !b->quiet) {
3531 printf("Warning: -j and -N switches are somewhat incompatible.\n");
3532 printf(" If any vertices are jettisoned, you will need the output\n");
3533 printf(" .node file to reconstruct the new node indices.");
3534 }
3535
3536#ifndef TRILIBRARY
3537 strcpy(b->inpolyfilename, b->innodefilename);
3538 strcpy(b->inelefilename, b->innodefilename);
3539 strcpy(b->areafilename, b->innodefilename);
3540 increment = 0;
3541 strcpy(workstring, b->innodefilename);
3542 j = 1;
3543 while (workstring[j] != '\0') {
3544 if ((workstring[j] == '.') && (workstring[j + 1] != '\0')) {
3545 increment = j + 1;
3546 }
3547 j++;
3548 }
3549 meshnumber = 0;
3550 if (increment > 0) {
3551 j = increment;
3552 do {
3553 if ((workstring[j] >= '0') && (workstring[j] <= '9')) {
3554 meshnumber = meshnumber * 10 + (int) (workstring[j] - '0');
3555 } else {
3556 increment = 0;
3557 }
3558 j++;
3559 } while (workstring[j] != '\0');
3560 }
3561 if (b->noiterationnum) {
3562 strcpy(b->outnodefilename, b->innodefilename);
3563 strcpy(b->outelefilename, b->innodefilename);
3564 strcpy(b->edgefilename, b->innodefilename);
3565 strcpy(b->vnodefilename, b->innodefilename);
3566 strcpy(b->vedgefilename, b->innodefilename);
3567 strcpy(b->neighborfilename, b->innodefilename);
3568 strcpy(b->offfilename, b->innodefilename);
3569 strcat(b->outnodefilename, ".node");
3570 strcat(b->outelefilename, ".ele");
3571 strcat(b->edgefilename, ".edge");
3572 strcat(b->vnodefilename, ".v.node");
3573 strcat(b->vedgefilename, ".v.edge");
3574 strcat(b->neighborfilename, ".neigh");
3575 strcat(b->offfilename, ".off");
3576 } else if (increment == 0) {
3577 strcpy(b->outnodefilename, b->innodefilename);
3578 strcpy(b->outpolyfilename, b->innodefilename);
3579 strcpy(b->outelefilename, b->innodefilename);
3580 strcpy(b->edgefilename, b->innodefilename);
3581 strcpy(b->vnodefilename, b->innodefilename);
3582 strcpy(b->vedgefilename, b->innodefilename);
3583 strcpy(b->neighborfilename, b->innodefilename);
3584 strcpy(b->offfilename, b->innodefilename);
3585 strcat(b->outnodefilename, ".1.node");
3586 strcat(b->outpolyfilename, ".1.poly");
3587 strcat(b->outelefilename, ".1.ele");
3588 strcat(b->edgefilename, ".1.edge");
3589 strcat(b->vnodefilename, ".1.v.node");
3590 strcat(b->vedgefilename, ".1.v.edge");
3591 strcat(b->neighborfilename, ".1.neigh");
3592 strcat(b->offfilename, ".1.off");
3593 } else {
3594 workstring[increment] = '%';
3595 workstring[increment + 1] = 'd';
3596 workstring[increment + 2] = '\0';
3597 sprintf(b->outnodefilename, workstring, meshnumber + 1);
3598 strcpy(b->outpolyfilename, b->outnodefilename);
3599 strcpy(b->outelefilename, b->outnodefilename);
3600 strcpy(b->edgefilename, b->outnodefilename);
3601 strcpy(b->vnodefilename, b->outnodefilename);
3602 strcpy(b->vedgefilename, b->outnodefilename);
3603 strcpy(b->neighborfilename, b->outnodefilename);
3604 strcpy(b->offfilename, b->outnodefilename);
3605 strcat(b->outnodefilename, ".node");
3606 strcat(b->outpolyfilename, ".poly");
3607 strcat(b->outelefilename, ".ele");
3608 strcat(b->edgefilename, ".edge");
3609 strcat(b->vnodefilename, ".v.node");
3610 strcat(b->vedgefilename, ".v.edge");
3611 strcat(b->neighborfilename, ".neigh");
3612 strcat(b->offfilename, ".off");
3613 }
3614 strcat(b->innodefilename, ".node");
3615 strcat(b->inpolyfilename, ".poly");
3616 strcat(b->inelefilename, ".ele");
3617 strcat(b->areafilename, ".area");
3618#endif /* not TRILIBRARY */
3619}
3620
3621/** **/
3622/** **/
3623/********* User interaction routines begin here *********/
3624
3625/********* Debugging routines begin here *********/
3626/** **/
3627/** **/
3628
3629/*****************************************************************************/
3630/* */
3631/* printtriangle() Print out the details of an oriented triangle. */
3632/* */
3633/* I originally wrote this procedure to simplify debugging; it can be */
3634/* called directly from the debugger, and presents information about an */
3635/* oriented triangle in digestible form. It's also used when the */
3636/* highest level of verbosity (`-VVV') is specified. */
3637/* */
3638/*****************************************************************************/
3639
3640#ifdef ANSI_DECLARATORS
3641void printtriangle(struct mesh *m, struct behavior *b, struct otri *t)
3642#else /* not ANSI_DECLARATORS */
3643void printtriangle(m, b, t)
3644struct mesh *m;
3645struct behavior *b;
3646struct otri *t;
3647#endif /* not ANSI_DECLARATORS */
3648
3649{
3650 struct otri printtri;
3651 struct osub printsh;
3652 vertex printvertex;
3653
3654 printf("triangle x%zx with orientation %d:\n", (size_t) t->tri,
3655 t->orient);
3656 decode(t->tri[0], printtri);
3657 if (printtri.tri == m->dummytri) {
3658 printf(" [0] = Outer space\n");
3659 } else {
3660 printf(" [0] = x%zx %d\n", (size_t) printtri.tri,
3661 printtri.orient);
3662 }
3663 decode(t->tri[1], printtri);
3664 if (printtri.tri == m->dummytri) {
3665 printf(" [1] = Outer space\n");
3666 } else {
3667 printf(" [1] = x%zx %d\n", (size_t) printtri.tri,
3668 printtri.orient);
3669 }
3670 decode(t->tri[2], printtri);
3671 if (printtri.tri == m->dummytri) {
3672 printf(" [2] = Outer space\n");
3673 } else {
3674 printf(" [2] = x%zx %d\n", (size_t) printtri.tri,
3675 printtri.orient);
3676 }
3677
3678 org(*t, printvertex);
3679 if (printvertex == (vertex) NULL)
3680 printf(" Origin[%d] = NULL\n", (t->orient + 1) % 3 + 3);
3681 else
3682 printf(" Origin[%d] = x%zx (%.12g, %.12g)\n",
3683 (t->orient + 1) % 3 + 3, (size_t) printvertex,
3684 printvertex[0], printvertex[1]);
3685 dest(*t, printvertex);
3686 if (printvertex == (vertex) NULL)
3687 printf(" Dest [%d] = NULL\n", (t->orient + 2) % 3 + 3);
3688 else
3689 printf(" Dest [%d] = x%zx (%.12g, %.12g)\n",
3690 (t->orient + 2) % 3 + 3, (size_t) printvertex,
3691 printvertex[0], printvertex[1]);
3692 apex(*t, printvertex);
3693 if (printvertex == (vertex) NULL)
3694 printf(" Apex [%d] = NULL\n", t->orient + 3);
3695 else
3696 printf(" Apex [%d] = x%zx (%.12g, %.12g)\n",
3697 t->orient + 3, (size_t) printvertex,
3698 printvertex[0], printvertex[1]);
3699
3700 if (b->usesegments) {
3701 sdecode(t->tri[6], printsh);
3702 if (printsh.ss != m->dummysub) {
3703 printf(" [6] = x%zx %d\n", (size_t) printsh.ss,
3704 printsh.ssorient);
3705 }
3706 sdecode(t->tri[7], printsh);
3707 if (printsh.ss != m->dummysub) {
3708 printf(" [7] = x%zx %d\n", (size_t) printsh.ss,
3709 printsh.ssorient);
3710 }
3711 sdecode(t->tri[8], printsh);
3712 if (printsh.ss != m->dummysub) {
3713 printf(" [8] = x%zx %d\n", (size_t) printsh.ss,
3714 printsh.ssorient);
3715 }
3716 }
3717
3718 if (b->vararea) {
3719 printf(" Area constraint: %.4g\n", areabound(*t));
3720 }
3721}
3722
3723/*****************************************************************************/
3724/* */
3725/* printsubseg() Print out the details of an oriented subsegment. */
3726/* */
3727/* I originally wrote this procedure to simplify debugging; it can be */
3728/* called directly from the debugger, and presents information about an */
3729/* oriented subsegment in digestible form. It's also used when the highest */
3730/* level of verbosity (`-VVV') is specified. */
3731/* */
3732/*****************************************************************************/
3733
3734#ifdef ANSI_DECLARATORS
3735void printsubseg(struct mesh *m, struct behavior *b, struct osub *s)
3736#else /* not ANSI_DECLARATORS */
3737void printsubseg(m, b, s)
3738struct mesh *m;
3739struct behavior *b;
3740struct osub *s;
3741#endif /* not ANSI_DECLARATORS */
3742
3743{
3744 struct osub printsh;
3745 struct otri printtri;
3746 vertex printvertex;
3747
3748 (void)b; /*LM: added to suppress warning */
3749
3750 printf("subsegment x%zx with orientation %d and mark %d:\n",
3751 (size_t) s->ss, s->ssorient, mark(*s));
3752 sdecode(s->ss[0], printsh);
3753 if (printsh.ss == m->dummysub) {
3754 printf(" [0] = No subsegment\n");
3755 } else {
3756 printf(" [0] = x%zx %d\n", (size_t) printsh.ss,
3757 printsh.ssorient);
3758 }
3759 sdecode(s->ss[1], printsh);
3760 if (printsh.ss == m->dummysub) {
3761 printf(" [1] = No subsegment\n");
3762 } else {
3763 printf(" [1] = x%zx %d\n", (size_t) printsh.ss,
3764 printsh.ssorient);
3765 }
3766
3767 sorg(*s, printvertex);
3768 if (printvertex == (vertex) NULL)
3769 printf(" Origin[%d] = NULL\n", 2 + s->ssorient);
3770 else
3771 printf(" Origin[%d] = x%zx (%.12g, %.12g)\n",
3772 2 + s->ssorient, (size_t) printvertex,
3773 printvertex[0], printvertex[1]);
3774 sdest(*s, printvertex);
3775 if (printvertex == (vertex) NULL)
3776 printf(" Dest [%d] = NULL\n", 3 - s->ssorient);
3777 else
3778 printf(" Dest [%d] = x%zx (%.12g, %.12g)\n",
3779 3 - s->ssorient, (size_t) printvertex,
3780 printvertex[0], printvertex[1]);
3781
3782 decode(s->ss[6], printtri);
3783 if (printtri.tri == m->dummytri) {
3784 printf(" [6] = Outer space\n");
3785 } else {
3786 printf(" [6] = x%zx %d\n", (size_t) printtri.tri,
3787 printtri.orient);
3788 }
3789 decode(s->ss[7], printtri);
3790 if (printtri.tri == m->dummytri) {
3791 printf(" [7] = Outer space\n");
3792 } else {
3793 printf(" [7] = x%zx %d\n", (size_t) printtri.tri,
3794 printtri.orient);
3795 }
3796
3797 segorg(*s, printvertex);
3798 if (printvertex == (vertex) NULL)
3799 printf(" Segment origin[%d] = NULL\n", 4 + s->ssorient);
3800 else
3801 printf(" Segment origin[%d] = x%zx (%.12g, %.12g)\n",
3802 4 + s->ssorient, (size_t) printvertex,
3803 printvertex[0], printvertex[1]);
3804 segdest(*s, printvertex);
3805 if (printvertex == (vertex) NULL)
3806 printf(" Segment dest [%d] = NULL\n", 5 - s->ssorient);
3807 else
3808 printf(" Segment dest [%d] = x%zx (%.12g, %.12g)\n",
3809 5 - s->ssorient, (size_t) printvertex,
3810 printvertex[0], printvertex[1]);
3811}
3812
3813/** **/
3814/** **/
3815/********* Debugging routines end here *********/
3816
3817/********* Memory management routines begin here *********/
3818/** **/
3819/** **/
3820
3821/*****************************************************************************/
3822/* */
3823/* poolzero() Set all of a pool's fields to zero. */
3824/* */
3825/* This procedure should never be called on a pool that has any memory */
3826/* allocated to it, as that memory would leak. */
3827/* */
3828/*****************************************************************************/
3829
3830#ifdef ANSI_DECLARATORS
3831void poolzero(struct memorypool *pool)
3832#else /* not ANSI_DECLARATORS */
3833void poolzero(pool)
3834struct memorypool *pool;
3835#endif /* not ANSI_DECLARATORS */
3836
3837{
3838 pool->firstblock = (VOID **) NULL;
3839 pool->nowblock = (VOID **) NULL;
3840 pool->nextitem = (VOID *) NULL;
3841 pool->deaditemstack = (VOID *) NULL;
3842 pool->pathblock = (VOID **) NULL;
3843 pool->pathitem = (VOID *) NULL;
3844 pool->alignbytes = 0;
3845 pool->itembytes = 0;
3846 pool->itemsperblock = 0;
3847 pool->itemsfirstblock = 0;
3848 pool->items = 0;
3849 pool->maxitems = 0;
3850 pool->unallocateditems = 0;
3851 pool->pathitemsleft = 0;
3852}
3853
3854/*****************************************************************************/
3855/* */
3856/* poolrestart() Deallocate all items in a pool. */
3857/* */
3858/* The pool is returned to its starting state, except that no memory is */
3859/* freed to the operating system. Rather, the previously allocated blocks */
3860/* are ready to be reused. */
3861/* */
3862/*****************************************************************************/
3863
3864#ifdef ANSI_DECLARATORS
3865void poolrestart(struct memorypool *pool)
3866#else /* not ANSI_DECLARATORS */
3867void poolrestart(pool)
3868struct memorypool *pool;
3869#endif /* not ANSI_DECLARATORS */
3870
3871{
3872 uintptr_t alignptr;
3873
3874 pool->items = 0;
3875 pool->maxitems = 0;
3876
3877 /* Set the currently active block. */
3878 pool->nowblock = pool->firstblock;
3879 /* Find the first item in the pool. Increment by the size of (VOID *). */
3880 alignptr = (uintptr_t) (pool->nowblock + 1);
3881 /* Align the item on an `alignbytes'-byte boundary. */
3882 pool->nextitem = (VOID *)
3883 (alignptr + (uintptr_t) pool->alignbytes -
3884 (alignptr % (uintptr_t) pool->alignbytes));
3885 /* There are lots of unallocated items left in this block. */
3886 pool->unallocateditems = pool->itemsfirstblock;
3887 /* The stack of deallocated items is empty. */
3888 pool->deaditemstack = (VOID *) NULL;
3889}
3890
3891/*****************************************************************************/
3892/* */
3893/* poolinit() Initialize a pool of memory for allocation of items. */
3894/* */
3895/* This routine initializes the machinery for allocating items. A `pool' */
3896/* is created whose records have size at least `bytecount'. Items will be */
3897/* allocated in `itemcount'-item blocks. Each item is assumed to be a */
3898/* collection of words, and either pointers or floating-point values are */
3899/* assumed to be the "primary" word type. (The "primary" word type is used */
3900/* to determine alignment of items.) If `alignment' isn't zero, all items */
3901/* will be `alignment'-byte aligned in memory. `alignment' must be either */
3902/* a multiple or a factor of the primary word size; powers of two are safe. */
3903/* `alignment' is normally used to create a few unused bits at the bottom */
3904/* of each item's pointer, in which information may be stored. */
3905/* */
3906/* Don't change this routine unless you understand it. */
3907/* */
3908/*****************************************************************************/
3909
3910#ifdef ANSI_DECLARATORS
3911void poolinit(struct memorypool *pool, int bytecount, int itemcount,
3912 int firstitemcount, int alignment)
3913#else /* not ANSI_DECLARATORS */
3914void poolinit(pool, bytecount, itemcount, firstitemcount, alignment)
3915struct memorypool *pool;
3916int bytecount;
3917int itemcount;
3918int firstitemcount;
3919int alignment;
3920#endif /* not ANSI_DECLARATORS */
3921
3922{
3923 /* Find the proper alignment, which must be at least as large as: */
3924 /* - The parameter `alignment'. */
3925 /* - sizeof(VOID *), so the stack of dead items can be maintained */
3926 /* without unaligned accesses. */
3927 if (alignment > (int) sizeof(VOID *)) {
3928 pool->alignbytes = alignment;
3929 } else {
3930 pool->alignbytes = sizeof(VOID *);
3931 }
3932 pool->itembytes = ((bytecount - 1) / pool->alignbytes + 1) *
3933 pool->alignbytes;
3934 pool->itemsperblock = itemcount;
3935 if (firstitemcount == 0) {
3936 pool->itemsfirstblock = itemcount;
3937 } else {
3938 pool->itemsfirstblock = firstitemcount;
3939 }
3940
3941 /* Allocate a block of items. Space for `itemsfirstblock' items and one */
3942 /* pointer (to point to the next block) are allocated, as well as space */
3943 /* to ensure alignment of the items. */
3944 pool->firstblock = (VOID **)
3945 trimalloc(pool->itemsfirstblock * pool->itembytes + (int) sizeof(VOID *) +
3946 pool->alignbytes);
3947 /* Set the next block pointer to NULL. */
3948 *(pool->firstblock) = (VOID *) NULL;
3949 poolrestart(pool);
3950}
3951
3952/*****************************************************************************/
3953/* */
3954/* pooldeinit() Free to the operating system all memory taken by a pool. */
3955/* */
3956/*****************************************************************************/
3957
3958#ifdef ANSI_DECLARATORS
3959void pooldeinit(struct memorypool *pool)
3960#else /* not ANSI_DECLARATORS */
3961void pooldeinit(pool)
3962struct memorypool *pool;
3963#endif /* not ANSI_DECLARATORS */
3964
3965{
3966 while (pool->firstblock != (VOID **) NULL) {
3967 pool->nowblock = (VOID **) *(pool->firstblock);
3968 trifree((VOID *) pool->firstblock);
3969 pool->firstblock = pool->nowblock;
3970 }
3971}
3972
3973/*****************************************************************************/
3974/* */
3975/* poolalloc() Allocate space for an item. */
3976/* */
3977/*****************************************************************************/
3978
3979#ifdef ANSI_DECLARATORS
3981#else /* not ANSI_DECLARATORS */
3982VOID *poolalloc(pool)
3983struct memorypool *pool;
3984#endif /* not ANSI_DECLARATORS */
3985
3986{
3987 VOID *newitem;
3988 VOID **newblock;
3989 uintptr_t alignptr;
3990
3991 /* First check the linked list of dead items. If the list is not */
3992 /* empty, allocate an item from the list rather than a fresh one. */
3993 if (pool->deaditemstack != (VOID *) NULL) {
3994 newitem = pool->deaditemstack; /* Take first item in list. */
3995 pool->deaditemstack = * (VOID **) pool->deaditemstack;
3996 } else {
3997 /* Check if there are any free items left in the current block. */
3998 if (pool->unallocateditems == 0) {
3999 /* Check if another block must be allocated. */
4000 if (*(pool->nowblock) == (VOID *) NULL) {
4001 /* Allocate a new block of items, pointed to by the previous block. */
4002 newblock = (VOID **) trimalloc(pool->itemsperblock * pool->itembytes +
4003 (int) sizeof(VOID *) +
4004 pool->alignbytes);
4005 *(pool->nowblock) = (VOID *) newblock;
4006 /* The next block pointer is NULL. */
4007 *newblock = (VOID *) NULL;
4008 }
4009
4010 /* Move to the new block. */
4011 pool->nowblock = (VOID **) *(pool->nowblock);
4012 /* Find the first item in the block. */
4013 /* Increment by the size of (VOID *). */
4014 alignptr = (uintptr_t) (pool->nowblock + 1);
4015 /* Align the item on an `alignbytes'-byte boundary. */
4016 pool->nextitem = (VOID *)
4017 (alignptr + (uintptr_t) pool->alignbytes -
4018 (alignptr % (uintptr_t) pool->alignbytes));
4019 /* There are lots of unallocated items left in this block. */
4020 pool->unallocateditems = pool->itemsperblock;
4021 }
4022
4023 /* Allocate a new item. */
4024 newitem = pool->nextitem;
4025 /* Advance `nextitem' pointer to next free item in block. */
4026 pool->nextitem = (VOID *) ((char *) pool->nextitem + pool->itembytes);
4027 pool->unallocateditems--;
4028 pool->maxitems++;
4029 }
4030 pool->items++;
4031 return newitem;
4032}
4033
4034/*****************************************************************************/
4035/* */
4036/* pooldealloc() Deallocate space for an item. */
4037/* */
4038/* The deallocated space is stored in a queue for later reuse. */
4039/* */
4040/*****************************************************************************/
4041
4042#ifdef ANSI_DECLARATORS
4043void pooldealloc(struct memorypool *pool, VOID *dyingitem)
4044#else /* not ANSI_DECLARATORS */
4045void pooldealloc(pool, dyingitem)
4046struct memorypool *pool;
4047VOID *dyingitem;
4048#endif /* not ANSI_DECLARATORS */
4049
4050{
4051 /* Push freshly killed item onto stack. */
4052 *((VOID **) dyingitem) = pool->deaditemstack;
4053 pool->deaditemstack = dyingitem;
4054 pool->items--;
4055}
4056
4057/*****************************************************************************/
4058/* */
4059/* traversalinit() Prepare to traverse the entire list of items. */
4060/* */
4061/* This routine is used in conjunction with traverse(). */
4062/* */
4063/*****************************************************************************/
4064
4065#ifdef ANSI_DECLARATORS
4066void traversalinit(struct memorypool *pool)
4067#else /* not ANSI_DECLARATORS */
4068void traversalinit(pool)
4069struct memorypool *pool;
4070#endif /* not ANSI_DECLARATORS */
4071
4072{
4073 uintptr_t alignptr;
4074
4075 /* Begin the traversal in the first block. */
4076 pool->pathblock = pool->firstblock;
4077 /* Find the first item in the block. Increment by the size of (VOID *). */
4078 alignptr = (uintptr_t) (pool->pathblock + 1);
4079 /* Align with item on an `alignbytes'-byte boundary. */
4080 pool->pathitem = (VOID *)
4081 (alignptr + (uintptr_t) pool->alignbytes -
4082 (alignptr % (uintptr_t) pool->alignbytes));
4083 /* Set the number of items left in the current block. */
4084 pool->pathitemsleft = pool->itemsfirstblock;
4085}
4086
4087/*****************************************************************************/
4088/* */
4089/* traverse() Find the next item in the list. */
4090/* */
4091/* This routine is used in conjunction with traversalinit(). Be forewarned */
4092/* that this routine successively returns all items in the list, including */
4093/* deallocated ones on the deaditemqueue. It's up to you to figure out */
4094/* which ones are actually dead. Why? I don't want to allocate extra */
4095/* space just to demarcate dead items. It can usually be done more */
4096/* space-efficiently by a routine that knows something about the structure */
4097/* of the item. */
4098/* */
4099/*****************************************************************************/
4100
4101#ifdef ANSI_DECLARATORS
4103#else /* not ANSI_DECLARATORS */
4104VOID *traverse(pool)
4105struct memorypool *pool;
4106#endif /* not ANSI_DECLARATORS */
4107
4108{
4109 VOID *newitem;
4110 uintptr_t alignptr;
4111
4112 /* Stop upon exhausting the list of items. */
4113 if (pool->pathitem == pool->nextitem) {
4114 return (VOID *) NULL;
4115 }
4116
4117 /* Check whether any untraversed items remain in the current block. */
4118 if (pool->pathitemsleft == 0) {
4119 /* Find the next block. */
4120 pool->pathblock = (VOID **) *(pool->pathblock);
4121 /* Find the first item in the block. Increment by the size of (VOID *). */
4122 alignptr = (uintptr_t) (pool->pathblock + 1);
4123 /* Align with item on an `alignbytes'-byte boundary. */
4124 pool->pathitem = (VOID *)
4125 (alignptr + (uintptr_t) pool->alignbytes -
4126 (alignptr % (uintptr_t) pool->alignbytes));
4127 /* Set the number of items left in the current block. */
4128 pool->pathitemsleft = pool->itemsperblock;
4129 }
4130
4131 newitem = pool->pathitem;
4132 /* Find the next item in the block. */
4133 pool->pathitem = (VOID *) ((char *) pool->pathitem + pool->itembytes);
4134 pool->pathitemsleft--;
4135 return newitem;
4136}
4137
4138/*****************************************************************************/
4139/* */
4140/* dummyinit() Initialize the triangle that fills "outer space" and the */
4141/* omnipresent subsegment. */
4142/* */
4143/* The triangle that fills "outer space," called `dummytri', is pointed to */
4144/* by every triangle and subsegment on a boundary (be it outer or inner) of */
4145/* the triangulation. Also, `dummytri' points to one of the triangles on */
4146/* the convex hull (until the holes and concavities are carved), making it */
4147/* possible to find a starting triangle for point location. */
4148/* */
4149/* The omnipresent subsegment, `dummysub', is pointed to by every triangle */
4150/* or subsegment that doesn't have a full complement of real subsegments */
4151/* to point to. */
4152/* */
4153/* `dummytri' and `dummysub' are generally required to fulfill only a few */
4154/* invariants: their vertices must remain NULL and `dummytri' must always */
4155/* be bonded (at offset zero) to some triangle on the convex hull of the */
4156/* mesh, via a boundary edge. Otherwise, the connections of `dummytri' and */
4157/* `dummysub' may change willy-nilly. This makes it possible to avoid */
4158/* writing a good deal of special-case code (in the edge flip, for example) */
4159/* for dealing with the boundary of the mesh, places where no subsegment is */
4160/* present, and so forth. Other entities are frequently bonded to */
4161/* `dummytri' and `dummysub' as if they were real mesh entities, with no */
4162/* harm done. */
4163/* */
4164/*****************************************************************************/
4165
4166#ifdef ANSI_DECLARATORS
4167void dummyinit(struct mesh *m, struct behavior *b, int trianglebytes,
4168 int subsegbytes)
4169#else /* not ANSI_DECLARATORS */
4170void dummyinit(m, b, trianglebytes, subsegbytes)
4171struct mesh *m;
4172struct behavior *b;
4173int trianglebytes;
4174int subsegbytes;
4175#endif /* not ANSI_DECLARATORS */
4176
4177{
4178 uintptr_t alignptr;
4179
4180 /* Set up `dummytri', the `triangle' that occupies "outer space." */
4181 m->dummytribase = (triangle *) trimalloc(trianglebytes +
4182 m->triangles.alignbytes);
4183 /* Align `dummytri' on a `triangles.alignbytes'-byte boundary. */
4184 alignptr = (uintptr_t) m->dummytribase;
4185 m->dummytri = (triangle *)
4186 (alignptr + (uintptr_t) m->triangles.alignbytes -
4187 (alignptr % (uintptr_t) m->triangles.alignbytes));
4188 /* Initialize the three adjoining triangles to be "outer space." These */
4189 /* will eventually be changed by various bonding operations, but their */
4190 /* values don't really matter, as long as they can legally be */
4191 /* dereferenced. */
4192 m->dummytri[0] = (triangle) m->dummytri;
4193 m->dummytri[1] = (triangle) m->dummytri;
4194 m->dummytri[2] = (triangle) m->dummytri;
4195 /* Three NULL vertices. */
4196 m->dummytri[3] = (triangle) NULL;
4197 m->dummytri[4] = (triangle) NULL;
4198 m->dummytri[5] = (triangle) NULL;
4199
4200 if (b->usesegments) {
4201 /* Set up `dummysub', the omnipresent subsegment pointed to by any */
4202 /* triangle side or subsegment end that isn't attached to a real */
4203 /* subsegment. */
4204 m->dummysubbase = (subseg *) trimalloc(subsegbytes +
4205 m->subsegs.alignbytes);
4206 /* Align `dummysub' on a `subsegs.alignbytes'-byte boundary. */
4207 alignptr = (uintptr_t) m->dummysubbase;
4208 m->dummysub = (subseg *)
4209 (alignptr + (uintptr_t) m->subsegs.alignbytes -
4210 (alignptr % (uintptr_t) m->subsegs.alignbytes));
4211 /* Initialize the two adjoining subsegments to be the omnipresent */
4212 /* subsegment. These will eventually be changed by various bonding */
4213 /* operations, but their values don't really matter, as long as they */
4214 /* can legally be dereferenced. */
4215 m->dummysub[0] = (subseg) m->dummysub;
4216 m->dummysub[1] = (subseg) m->dummysub;
4217 /* Four NULL vertices. */
4218 m->dummysub[2] = (subseg) NULL;
4219 m->dummysub[3] = (subseg) NULL;
4220 m->dummysub[4] = (subseg) NULL;
4221 m->dummysub[5] = (subseg) NULL;
4222 /* Initialize the two adjoining triangles to be "outer space." */
4223 m->dummysub[6] = (subseg) m->dummytri;
4224 m->dummysub[7] = (subseg) m->dummytri;
4225 /* Set the boundary marker to zero. */
4226 * (int *) (m->dummysub + 8) = 0;
4227
4228 /* Initialize the three adjoining subsegments of `dummytri' to be */
4229 /* the omnipresent subsegment. */
4230 m->dummytri[6] = (triangle) m->dummysub;
4231 m->dummytri[7] = (triangle) m->dummysub;
4232 m->dummytri[8] = (triangle) m->dummysub;
4233 }
4234}
4235
4236/*****************************************************************************/
4237/* */
4238/* initializevertexpool() Calculate the size of the vertex data structure */
4239/* and initialize its memory pool. */
4240/* */
4241/* This routine also computes the `vertexmarkindex' and `vertex2triindex' */
4242/* indices used to find values within each vertex. */
4243/* */
4244/*****************************************************************************/
4245
4246#ifdef ANSI_DECLARATORS
4247void initializevertexpool(struct mesh *m, struct behavior *b)
4248#else /* not ANSI_DECLARATORS */
4250struct mesh *m;
4251struct behavior *b;
4252#endif /* not ANSI_DECLARATORS */
4253
4254{
4255 int vertexsize;
4256
4257 /* The index within each vertex at which the boundary marker is found, */
4258 /* followed by the vertex type. Ensure the vertex marker is aligned to */
4259 /* a sizeof(int)-byte address. */
4260 m->vertexmarkindex = ((m->mesh_dim + m->nextras) * sizeof(REAL) +
4261 sizeof(int) - 1) /
4262 sizeof(int);
4263 vertexsize = (m->vertexmarkindex + 2) * sizeof(int);
4264 if (b->poly) {
4265 /* The index within each vertex at which a triangle pointer is found. */
4266 /* Ensure the pointer is aligned to a sizeof(triangle)-byte address. */
4267 m->vertex2triindex = (vertexsize + sizeof(triangle) - 1) /
4268 sizeof(triangle);
4269 vertexsize = (m->vertex2triindex + 1) * sizeof(triangle);
4270 }
4271
4272 /* Initialize the pool of vertices. */
4273 poolinit(&m->vertices, vertexsize, VERTEXPERBLOCK,
4274 m->invertices > VERTEXPERBLOCK ? m->invertices : VERTEXPERBLOCK,
4275 sizeof(REAL));
4276}
4277
4278/*****************************************************************************/
4279/* */
4280/* initializetrisubpools() Calculate the sizes of the triangle and */
4281/* subsegment data structures and initialize */
4282/* their memory pools. */
4283/* */
4284/* This routine also computes the `highorderindex', `elemattribindex', and */
4285/* `areaboundindex' indices used to find values within each triangle. */
4286/* */
4287/*****************************************************************************/
4288
4289#ifdef ANSI_DECLARATORS
4290void initializetrisubpools(struct mesh *m, struct behavior *b)
4291#else /* not ANSI_DECLARATORS */
4293struct mesh *m;
4294struct behavior *b;
4295#endif /* not ANSI_DECLARATORS */
4296
4297{
4298 int trisize;
4299
4300 /* The index within each triangle at which the extra nodes (above three) */
4301 /* associated with high order elements are found. There are three */
4302 /* pointers to other triangles, three pointers to corners, and possibly */
4303 /* three pointers to subsegments before the extra nodes. */
4304 m->highorderindex = 6 + (b->usesegments * 3);
4305 /* The number of bytes occupied by a triangle. */
4306 trisize = ((b->order + 1) * (b->order + 2) / 2 + (m->highorderindex - 3)) *
4307 sizeof(triangle);
4308 /* The index within each triangle at which its attributes are found, */
4309 /* where the index is measured in REALs. */
4310 m->elemattribindex = (trisize + sizeof(REAL) - 1) / sizeof(REAL);
4311 /* The index within each triangle at which the maximum area constraint */
4312 /* is found, where the index is measured in REALs. Note that if the */
4313 /* `regionattrib' flag is set, an additional attribute will be added. */
4314 m->areaboundindex = m->elemattribindex + m->eextras + b->regionattrib;
4315 /* If triangle attributes or an area bound are needed, increase the number */
4316 /* of bytes occupied by a triangle. */
4317 if (b->vararea) {
4318 trisize = (m->areaboundindex + 1) * sizeof(REAL);
4319 } else if (m->eextras + b->regionattrib > 0) {
4320 trisize = m->areaboundindex * sizeof(REAL);
4321 }
4322 /* If a Voronoi diagram or triangle neighbor graph is requested, make */
4323 /* sure there's room to store an integer index in each triangle. This */
4324 /* integer index can occupy the same space as the subsegment pointers */
4325 /* or attributes or area constraint or extra nodes. */
4326 if ((b->voronoi || b->neighbors) &&
4327 (trisize < (int) ( 6 * sizeof(triangle) + sizeof(int)))) {
4328 trisize = 6 * sizeof(triangle) + sizeof(int);
4329 }
4330
4331 /* Having determined the memory size of a triangle, initialize the pool. */
4332 poolinit(&m->triangles, trisize, TRIPERBLOCK,
4333 (2 * m->invertices - 2) > TRIPERBLOCK ? (2 * m->invertices - 2) :
4334 TRIPERBLOCK, 4);
4335
4336 if (b->usesegments) {
4337 /* Initialize the pool of subsegments. Take into account all eight */
4338 /* pointers and one boundary marker. */
4339 poolinit(&m->subsegs, 8 * sizeof(triangle) + sizeof(int),
4341
4342 /* Initialize the "outer space" triangle and omnipresent subsegment. */
4343 dummyinit(m, b, m->triangles.itembytes, m->subsegs.itembytes);
4344 } else {
4345 /* Initialize the "outer space" triangle. */
4346 dummyinit(m, b, m->triangles.itembytes, 0);
4347 }
4348}
4349
4350/*****************************************************************************/
4351/* */
4352/* triangledealloc() Deallocate space for a triangle, marking it dead. */
4353/* */
4354/*****************************************************************************/
4355
4356#ifdef ANSI_DECLARATORS
4357void triangledealloc(struct mesh *m, triangle *dyingtriangle)
4358#else /* not ANSI_DECLARATORS */
4359void triangledealloc(m, dyingtriangle)
4360struct mesh *m;
4361triangle *dyingtriangle;
4362#endif /* not ANSI_DECLARATORS */
4363
4364{
4365 /* Mark the triangle as dead. This makes it possible to detect dead */
4366 /* triangles when traversing the list of all triangles. */
4367 killtri(dyingtriangle);
4368 pooldealloc(&m->triangles, (VOID *) dyingtriangle);
4369}
4370
4371/*****************************************************************************/
4372/* */
4373/* triangletraverse() Traverse the triangles, skipping dead ones. */
4374/* */
4375/*****************************************************************************/
4376
4377#ifdef ANSI_DECLARATORS
4379#else /* not ANSI_DECLARATORS */
4381struct mesh *m;
4382#endif /* not ANSI_DECLARATORS */
4383
4384{
4385 triangle *newtriangle;
4386
4387 do {
4388 newtriangle = (triangle *) traverse(&m->triangles);
4389 if (newtriangle == (triangle *) NULL) {
4390 return (triangle *) NULL;
4391 }
4392 } while (deadtri(newtriangle)); /* Skip dead ones. */
4393 return newtriangle;
4394}
4395
4396/*****************************************************************************/
4397/* */
4398/* subsegdealloc() Deallocate space for a subsegment, marking it dead. */
4399/* */
4400/*****************************************************************************/
4401
4402#ifdef ANSI_DECLARATORS
4403void subsegdealloc(struct mesh *m, subseg *dyingsubseg)
4404#else /* not ANSI_DECLARATORS */
4405void subsegdealloc(m, dyingsubseg)
4406struct mesh *m;
4407subseg *dyingsubseg;
4408#endif /* not ANSI_DECLARATORS */
4409
4410{
4411 /* Mark the subsegment as dead. This makes it possible to detect dead */
4412 /* subsegments when traversing the list of all subsegments. */
4413 killsubseg(dyingsubseg);
4414 pooldealloc(&m->subsegs, (VOID *) dyingsubseg);
4415}
4416
4417/*****************************************************************************/
4418/* */
4419/* subsegtraverse() Traverse the subsegments, skipping dead ones. */
4420/* */
4421/*****************************************************************************/
4422
4423#ifdef ANSI_DECLARATORS
4425#else /* not ANSI_DECLARATORS */
4427struct mesh *m;
4428#endif /* not ANSI_DECLARATORS */
4429
4430{
4431 subseg *newsubseg;
4432
4433 do {
4434 newsubseg = (subseg *) traverse(&m->subsegs);
4435 if (newsubseg == (subseg *) NULL) {
4436 return (subseg *) NULL;
4437 }
4438 } while (deadsubseg(newsubseg)); /* Skip dead ones. */
4439 return newsubseg;
4440}
4441
4442/*****************************************************************************/
4443/* */
4444/* vertexdealloc() Deallocate space for a vertex, marking it dead. */
4445/* */
4446/*****************************************************************************/
4447
4448#ifdef ANSI_DECLARATORS
4449void vertexdealloc(struct mesh *m, vertex dyingvertex)
4450#else /* not ANSI_DECLARATORS */
4451void vertexdealloc(m, dyingvertex)
4452struct mesh *m;
4453vertex dyingvertex;
4454#endif /* not ANSI_DECLARATORS */
4455
4456{
4457 /* Mark the vertex as dead. This makes it possible to detect dead */
4458 /* vertices when traversing the list of all vertices. */
4459 setvertextype(dyingvertex, DEADVERTEX);
4460 pooldealloc(&m->vertices, (VOID *) dyingvertex);
4461}
4462
4463/*****************************************************************************/
4464/* */
4465/* vertextraverse() Traverse the vertices, skipping dead ones. */
4466/* */
4467/*****************************************************************************/
4468
4469#ifdef ANSI_DECLARATORS
4471#else /* not ANSI_DECLARATORS */
4473struct mesh *m;
4474#endif /* not ANSI_DECLARATORS */
4475
4476{
4477 vertex newvertex;
4478
4479 do {
4480 newvertex = (vertex) traverse(&m->vertices);
4481 if (newvertex == (vertex) NULL) {
4482 return (vertex) NULL;
4483 }
4484 } while (vertextype(newvertex) == DEADVERTEX); /* Skip dead ones. */
4485 return newvertex;
4486}
4487
4488/*****************************************************************************/
4489/* */
4490/* badsubsegdealloc() Deallocate space for a bad subsegment, marking it */
4491/* dead. */
4492/* */
4493/*****************************************************************************/
4494
4495#ifndef CDT_ONLY
4496
4497#ifdef ANSI_DECLARATORS
4498void badsubsegdealloc(struct mesh *m, struct badsubseg *dyingseg)
4499#else /* not ANSI_DECLARATORS */
4500void badsubsegdealloc(m, dyingseg)
4501struct mesh *m;
4502struct badsubseg *dyingseg;
4503#endif /* not ANSI_DECLARATORS */
4504
4505{
4506 /* Set subsegment's origin to NULL. This makes it possible to detect dead */
4507 /* badsubsegs when traversing the list of all badsubsegs . */
4508 dyingseg->subsegorg = (vertex) NULL;
4509 pooldealloc(&m->badsubsegs, (VOID *) dyingseg);
4510}
4511
4512#endif /* not CDT_ONLY */
4513
4514/*****************************************************************************/
4515/* */
4516/* badsubsegtraverse() Traverse the bad subsegments, skipping dead ones. */
4517/* */
4518/*****************************************************************************/
4519
4520#ifndef CDT_ONLY
4521
4522#ifdef ANSI_DECLARATORS
4523struct badsubseg *badsubsegtraverse(struct mesh *m)
4524#else /* not ANSI_DECLARATORS */
4525struct badsubseg *badsubsegtraverse(m)
4526struct mesh *m;
4527#endif /* not ANSI_DECLARATORS */
4528
4529{
4530 struct badsubseg *newseg;
4531
4532 do {
4533 newseg = (struct badsubseg *) traverse(&m->badsubsegs);
4534 if (newseg == (struct badsubseg *) NULL) {
4535 return (struct badsubseg *) NULL;
4536 }
4537 } while (newseg->subsegorg == (vertex) NULL); /* Skip dead ones. */
4538 return newseg;
4539}
4540
4541#endif /* not CDT_ONLY */
4542
4543/*****************************************************************************/
4544/* */
4545/* getvertex() Get a specific vertex, by number, from the list. */
4546/* */
4547/* The first vertex is number 'firstnumber'. */
4548/* */
4549/* Note that this takes O(n) time (with a small constant, if VERTEXPERBLOCK */
4550/* is large). I don't care to take the trouble to make it work in constant */
4551/* time. */
4552/* */
4553/*****************************************************************************/
4554
4555#ifdef ANSI_DECLARATORS
4556vertex getvertex(struct mesh *m, struct behavior *b, int number)
4557#else /* not ANSI_DECLARATORS */
4558vertex getvertex(m, b, number)
4559struct mesh *m;
4560struct behavior *b;
4561int number;
4562#endif /* not ANSI_DECLARATORS */
4563
4564{
4565 VOID **getblock;
4566 char *foundvertex;
4567 uintptr_t alignptr;
4568 int current;
4569
4570 getblock = m->vertices.firstblock;
4571 current = b->firstnumber;
4572
4573 /* Find the right block. */
4574 if (current + m->vertices.itemsfirstblock <= number) {
4575 getblock = (VOID **) *getblock;
4576 current += m->vertices.itemsfirstblock;
4577 while (current + m->vertices.itemsperblock <= number) {
4578 getblock = (VOID **) *getblock;
4579 current += m->vertices.itemsperblock;
4580 }
4581 }
4582
4583 /* Now find the right vertex. */
4584 alignptr = (uintptr_t) (getblock + 1);
4585 foundvertex = (char *) (alignptr + (uintptr_t) m->vertices.alignbytes -
4586 (alignptr % (uintptr_t) m->vertices.alignbytes));
4587 return (vertex) (foundvertex + m->vertices.itembytes * (number - current));
4588}
4589
4590/*****************************************************************************/
4591/* */
4592/* triangledeinit() Free all remaining allocated memory. */
4593/* */
4594/*****************************************************************************/
4595
4596#ifdef ANSI_DECLARATORS
4597void triangledeinit(struct mesh *m, struct behavior *b)
4598#else /* not ANSI_DECLARATORS */
4599void triangledeinit(m, b)
4600struct mesh *m;
4601struct behavior *b;
4602#endif /* not ANSI_DECLARATORS */
4603
4604{
4605 pooldeinit(&m->triangles);
4606 trifree((VOID *) m->dummytribase);
4607 if (b->usesegments) {
4608 pooldeinit(&m->subsegs);
4609 trifree((VOID *) m->dummysubbase);
4610 }
4611 pooldeinit(&m->vertices);
4612#ifndef CDT_ONLY
4613 if (b->quality) {
4614 pooldeinit(&m->badsubsegs);
4615 if ((b->minangle > 0.0) || b->vararea || b->fixedarea || b->usertest) {
4616 pooldeinit(&m->badtriangles);
4617 pooldeinit(&m->flipstackers);
4618 }
4619 }
4620#endif /* not CDT_ONLY */
4621}
4622
4623/** **/
4624/** **/
4625/********* Memory management routines end here *********/
4626
4627/********* Constructors begin here *********/
4628/** **/
4629/** **/
4630
4631/*****************************************************************************/
4632/* */
4633/* maketriangle() Create a new triangle with orientation zero. */
4634/* */
4635/*****************************************************************************/
4636
4637#ifdef ANSI_DECLARATORS
4638void maketriangle(struct mesh *m, struct behavior *b, struct otri *newotri)
4639#else /* not ANSI_DECLARATORS */
4640void maketriangle(m, b, newotri)
4641struct mesh *m;
4642struct behavior *b;
4643struct otri *newotri;
4644#endif /* not ANSI_DECLARATORS */
4645
4646{
4647 int i;
4648
4649 newotri->tri = (triangle *) poolalloc(&m->triangles);
4650 /* Initialize the three adjoining triangles to be "outer space". */
4651 newotri->tri[0] = (triangle) m->dummytri;
4652 newotri->tri[1] = (triangle) m->dummytri;
4653 newotri->tri[2] = (triangle) m->dummytri;
4654 /* Three NULL vertices. */
4655 newotri->tri[3] = (triangle) NULL;
4656 newotri->tri[4] = (triangle) NULL;
4657 newotri->tri[5] = (triangle) NULL;
4658 if (b->usesegments) {
4659 /* Initialize the three adjoining subsegments to be the omnipresent */
4660 /* subsegment. */
4661 newotri->tri[6] = (triangle) m->dummysub;
4662 newotri->tri[7] = (triangle) m->dummysub;
4663 newotri->tri[8] = (triangle) m->dummysub;
4664 }
4665 for (i = 0; i < m->eextras; i++) {
4666 setelemattribute(*newotri, i, 0.0);
4667 }
4668 if (b->vararea) {
4669 setareabound(*newotri, -1.0);
4670 }
4671
4672 newotri->orient = 0;
4673}
4674
4675/*****************************************************************************/
4676/* */
4677/* makesubseg() Create a new subsegment with orientation zero. */
4678/* */
4679/*****************************************************************************/
4680
4681#ifdef ANSI_DECLARATORS
4682void makesubseg(struct mesh *m, struct osub *newsubseg)
4683#else /* not ANSI_DECLARATORS */
4684void makesubseg(m, newsubseg)
4685struct mesh *m;
4686struct osub *newsubseg;
4687#endif /* not ANSI_DECLARATORS */
4688
4689{
4690 newsubseg->ss = (subseg *) poolalloc(&m->subsegs);
4691 /* Initialize the two adjoining subsegments to be the omnipresent */
4692 /* subsegment. */
4693 newsubseg->ss[0] = (subseg) m->dummysub;
4694 newsubseg->ss[1] = (subseg) m->dummysub;
4695 /* Four NULL vertices. */
4696 newsubseg->ss[2] = (subseg) NULL;
4697 newsubseg->ss[3] = (subseg) NULL;
4698 newsubseg->ss[4] = (subseg) NULL;
4699 newsubseg->ss[5] = (subseg) NULL;
4700 /* Initialize the two adjoining triangles to be "outer space." */
4701 newsubseg->ss[6] = (subseg) m->dummytri;
4702 newsubseg->ss[7] = (subseg) m->dummytri;
4703 /* Set the boundary marker to zero. */
4704 setmark(*newsubseg, 0);
4705
4706 newsubseg->ssorient = 0;
4707}
4708
4709/** **/
4710/** **/
4711/********* Constructors end here *********/
4712
4713/********* Geometric primitives begin here *********/
4714/** **/
4715/** **/
4716
4717/* The adaptive exact arithmetic geometric predicates implemented herein are */
4718/* described in detail in my paper, "Adaptive Precision Floating-Point */
4719/* Arithmetic and Fast Robust Geometric Predicates." See the header for a */
4720/* full citation. */
4721
4722/* Which of the following two methods of finding the absolute values is */
4723/* fastest is compiler-dependent. A few compilers can inline and optimize */
4724/* the fabs() call; but most will incur the overhead of a function call, */
4725/* which is disastrously slow. A faster way on IEEE machines might be to */
4726/* mask the appropriate bit, but that's difficult to do in C without */
4727/* forcing the value to be stored to memory (rather than be kept in the */
4728/* register to which the optimizer assigned it). */
4729
4730#define Absolute(a) ((a) >= 0.0 ? (a) : -(a))
4731/* #define Absolute(a) fabs(a) */
4732
4733/* Many of the operations are broken up into two pieces, a main part that */
4734/* performs an approximate operation, and a "tail" that computes the */
4735/* roundoff error of that operation. */
4736/* */
4737/* The operations Fast_Two_Sum(), Fast_Two_Diff(), Two_Sum(), Two_Diff(), */
4738/* Split(), and Two_Product() are all implemented as described in the */
4739/* reference. Each of these macros requires certain variables to be */
4740/* defined in the calling routine. The variables `bvirt', `c', `abig', */
4741/* `_i', `_j', `_k', `_l', `_m', and `_n' are declared `INEXACT' because */
4742/* they store the result of an operation that may incur roundoff error. */
4743/* The input parameter `x' (or the highest numbered `x_' parameter) must */
4744/* also be declared `INEXACT'. */
4745
4746#define Fast_Two_Sum_Tail(a, b, x, y) \
4747 bvirt = x - a; \
4748 y = b - bvirt
4749
4750#define Fast_Two_Sum(a, b, x, y) \
4751 x = (REAL) (a + b); \
4752 Fast_Two_Sum_Tail(a, b, x, y)
4753
4754#define Two_Sum_Tail(a, b, x, y) \
4755 bvirt = (REAL) (x - a); \
4756 avirt = x - bvirt; \
4757 bround = b - bvirt; \
4758 around = a - avirt; \
4759 y = around + bround
4760
4761#define Two_Sum(a, b, x, y) \
4762 x = (REAL) (a + b); \
4763 Two_Sum_Tail(a, b, x, y)
4764
4765#define Two_Diff_Tail(a, b, x, y) \
4766 bvirt = (REAL) (a - x); \
4767 avirt = x + bvirt; \
4768 bround = bvirt - b; \
4769 around = a - avirt; \
4770 y = around + bround
4771
4772#define Two_Diff(a, b, x, y) \
4773 x = (REAL) (a - b); \
4774 Two_Diff_Tail(a, b, x, y)
4775
4776#define Split(a, ahi, alo) \
4777 c = (REAL) (splitter * a); \
4778 abig = (REAL) (c - a); \
4779 ahi = c - abig; \
4780 alo = a - ahi
4781
4782#define Two_Product_Tail(a, b, x, y) \
4783 Split(a, ahi, alo); \
4784 Split(b, bhi, blo); \
4785 err1 = x - (ahi * bhi); \
4786 err2 = err1 - (alo * bhi); \
4787 err3 = err2 - (ahi * blo); \
4788 y = (alo * blo) - err3
4789
4790#define Two_Product(a, b, x, y) \
4791 x = (REAL) (a * b); \
4792 Two_Product_Tail(a, b, x, y)
4793
4794/* Two_Product_Presplit() is Two_Product() where one of the inputs has */
4795/* already been split. Avoids redundant splitting. */
4796
4797#define Two_Product_Presplit(a, b, bhi, blo, x, y) \
4798 x = (REAL) (a * b); \
4799 Split(a, ahi, alo); \
4800 err1 = x - (ahi * bhi); \
4801 err2 = err1 - (alo * bhi); \
4802 err3 = err2 - (ahi * blo); \
4803 y = (alo * blo) - err3
4804
4805/* Square() can be done more quickly than Two_Product(). */
4806
4807#define Square_Tail(a, x, y) \
4808 Split(a, ahi, alo); \
4809 err1 = x - (ahi * ahi); \
4810 err3 = err1 - ((ahi + ahi) * alo); \
4811 y = (alo * alo) - err3
4812
4813#define Square(a, x, y) \
4814 x = (REAL) (a * a); \
4815 Square_Tail(a, x, y)
4816
4817/* Macros for summing expansions of various fixed lengths. These are all */
4818/* unrolled versions of Expansion_Sum(). */
4819
4820#define Two_One_Sum(a1, a0, b, x2, x1, x0) \
4821 Two_Sum(a0, b , _i, x0); \
4822 Two_Sum(a1, _i, x2, x1)
4823
4824#define Two_One_Diff(a1, a0, b, x2, x1, x0) \
4825 Two_Diff(a0, b , _i, x0); \
4826 Two_Sum( a1, _i, x2, x1)
4827
4828#define Two_Two_Sum(a1, a0, b1, b0, x3, x2, x1, x0) \
4829 Two_One_Sum(a1, a0, b0, _j, _0, x0); \
4830 Two_One_Sum(_j, _0, b1, x3, x2, x1)
4831
4832#define Two_Two_Diff(a1, a0, b1, b0, x3, x2, x1, x0) \
4833 Two_One_Diff(a1, a0, b0, _j, _0, x0); \
4834 Two_One_Diff(_j, _0, b1, x3, x2, x1)
4835
4836/* Macro for multiplying a two-component expansion by a single component. */
4837
4838#define Two_One_Product(a1, a0, b, x3, x2, x1, x0) \
4839 Split(b, bhi, blo); \
4840 Two_Product_Presplit(a0, b, bhi, blo, _i, x0); \
4841 Two_Product_Presplit(a1, b, bhi, blo, _j, _0); \
4842 Two_Sum(_i, _0, _k, x1); \
4843 Fast_Two_Sum(_j, _k, x3, x2)
4844
4845/*****************************************************************************/
4846/* */
4847/* exactinit() Initialize the variables used for exact arithmetic. */
4848/* */
4849/* `epsilon' is the largest power of two such that 1.0 + epsilon = 1.0 in */
4850/* floating-point arithmetic. `epsilon' bounds the relative roundoff */
4851/* error. It is used for floating-point error analysis. */
4852/* */
4853/* `splitter' is used to split floating-point numbers into two half- */
4854/* length significands for exact multiplication. */
4855/* */
4856/* I imagine that a highly optimizing compiler might be too smart for its */
4857/* own good, and somehow cause this routine to fail, if it pretends that */
4858/* floating-point arithmetic is too much like real arithmetic. */
4859/* */
4860/* Don't change this routine unless you fully understand it. */
4861/* */
4862/*****************************************************************************/
4863
4865{
4866 REAL half;
4867 REAL check, lastcheck;
4868 int every_other;
4869#ifdef LINUX
4870 fenv_t fenv;
4871#endif /* LINUX */
4872
4873#ifdef CPU86
4874#ifdef SINGLE
4875 _control87(_PC_24, _MCW_PC); /* Set FPU control word for single precision. */
4876#else /* not SINGLE */
4877 _control87(_PC_53, _MCW_PC); /* Set FPU control word for double precision. */
4878#endif /* not SINGLE */
4879#endif /* CPU86 */
4880#ifdef LINUX
4881#ifdef SINGLE
4882 /* fenv.__control_word = 4223; */
4883 fenv.__control_word = 4210; /* set FPU control word for single precision */
4884#else /* not SINGLE */
4885 /* fenv.__control_word = 4735; */
4886 fenv.__control_word = 4722; /* set FPU control word for double precision */
4887#endif /* not SINGLE */
4888 fesetenv(&fenv);
4889#endif /* LINUX */
4890
4891 every_other = 1;
4892 half = 0.5;
4893 epsilon = 1.0;
4894 splitter = 1.0;
4895 check = 1.0;
4896 /* Repeatedly divide `epsilon' by two until it is too small to add to */
4897 /* one without causing roundoff. (Also check if the sum is equal to */
4898 /* the previous sum, for machines that round up instead of using exact */
4899 /* rounding. Not that these routines will work on such machines.) */
4900 do {
4901 lastcheck = check;
4902 epsilon *= half;
4903 if (every_other) {
4904 splitter *= 2.0;
4905 }
4906 every_other = !every_other;
4907 check = 1.0 + epsilon;
4908 } while ((check != 1.0) && (check != lastcheck));
4909 splitter += 1.0;
4910 /* Error bounds for orientation and incircle tests. */
4911 resulterrbound = (3.0 + 8.0 * epsilon) * epsilon;
4912 ccwerrboundA = (3.0 + 16.0 * epsilon) * epsilon;
4913 ccwerrboundB = (2.0 + 12.0 * epsilon) * epsilon;
4914 ccwerrboundC = (9.0 + 64.0 * epsilon) * epsilon * epsilon;
4915 iccerrboundA = (10.0 + 96.0 * epsilon) * epsilon;
4916 iccerrboundB = (4.0 + 48.0 * epsilon) * epsilon;
4917 iccerrboundC = (44.0 + 576.0 * epsilon) * epsilon * epsilon;
4918 o3derrboundA = (7.0 + 56.0 * epsilon) * epsilon;
4919 o3derrboundB = (3.0 + 28.0 * epsilon) * epsilon;
4920 o3derrboundC = (26.0 + 288.0 * epsilon) * epsilon * epsilon;
4921}
4922
4923/*****************************************************************************/
4924/* */
4925/* fast_expansion_sum_zeroelim() Sum two expansions, eliminating zero */
4926/* components from the output expansion. */
4927/* */
4928/* Sets h = e + f. See my Robust Predicates paper for details. */
4929/* */
4930/* If round-to-even is used (as with IEEE 754), maintains the strongly */
4931/* nonoverlapping property. (That is, if e is strongly nonoverlapping, h */
4932/* will be also.) Does NOT maintain the nonoverlapping or nonadjacent */
4933/* properties. */
4934/* */
4935/*****************************************************************************/
4936
4937#ifdef ANSI_DECLARATORS
4938int fast_expansion_sum_zeroelim(int elen, REAL *e, int flen, REAL *f, REAL *h)
4939#else /* not ANSI_DECLARATORS */
4940int fast_expansion_sum_zeroelim(elen, e, flen, f, h) /* h cannot be e or f. */
4941int elen;
4942REAL *e;
4943int flen;
4944REAL *f;
4945REAL *h;
4946#endif /* not ANSI_DECLARATORS */
4947
4948{
4949 REAL Q;
4950 INEXACT REAL Qnew;
4951 INEXACT REAL hh;
4952 INEXACT REAL bvirt;
4953 REAL avirt, bround, around;
4954 int eindex, findex, hindex;
4955 REAL enow, fnow;
4956
4957 enow = e[0];
4958 fnow = f[0];
4959 eindex = findex = 0;
4960 if ((fnow > enow) == (fnow > -enow)) {
4961 Q = enow;
4962 enow = e[++eindex];
4963 } else {
4964 Q = fnow;
4965 fnow = f[++findex];
4966 }
4967 hindex = 0;
4968 if ((eindex < elen) && (findex < flen)) {
4969 if ((fnow > enow) == (fnow > -enow)) {
4970 Fast_Two_Sum(enow, Q, Qnew, hh);
4971 enow = e[++eindex];
4972 } else {
4973 Fast_Two_Sum(fnow, Q, Qnew, hh);
4974 fnow = f[++findex];
4975 }
4976 Q = Qnew;
4977 if (hh != 0.0) {
4978 h[hindex++] = hh;
4979 }
4980 while ((eindex < elen) && (findex < flen)) {
4981 if ((fnow > enow) == (fnow > -enow)) {
4982 Two_Sum(Q, enow, Qnew, hh);
4983 enow = e[++eindex];
4984 } else {
4985 Two_Sum(Q, fnow, Qnew, hh);
4986 fnow = f[++findex];
4987 }
4988 Q = Qnew;
4989 if (hh != 0.0) {
4990 h[hindex++] = hh;
4991 }
4992 }
4993 }
4994 while (eindex < elen) {
4995 Two_Sum(Q, enow, Qnew, hh);
4996 enow = e[++eindex];
4997 Q = Qnew;
4998 if (hh != 0.0) {
4999 h[hindex++] = hh;
5000 }
5001 }
5002 while (findex < flen) {
5003 Two_Sum(Q, fnow, Qnew, hh);
5004 fnow = f[++findex];
5005 Q = Qnew;
5006 if (hh != 0.0) {
5007 h[hindex++] = hh;
5008 }
5009 }
5010 if ((Q != 0.0) || (hindex == 0)) {
5011 h[hindex++] = Q;
5012 }
5013 return hindex;
5014}
5015
5016/*****************************************************************************/
5017/* */
5018/* scale_expansion_zeroelim() Multiply an expansion by a scalar, */
5019/* eliminating zero components from the */
5020/* output expansion. */
5021/* */
5022/* Sets h = be. See my Robust Predicates paper for details. */
5023/* */
5024/* Maintains the nonoverlapping property. If round-to-even is used (as */
5025/* with IEEE 754), maintains the strongly nonoverlapping and nonadjacent */
5026/* properties as well. (That is, if e has one of these properties, so */
5027/* will h.) */
5028/* */
5029/*****************************************************************************/
5030
5031#ifdef ANSI_DECLARATORS
5033#else /* not ANSI_DECLARATORS */
5034int scale_expansion_zeroelim(elen, e, b, h) /* e and h cannot be the same. */
5035int elen;
5036REAL *e;
5037REAL b;
5038REAL *h;
5039#endif /* not ANSI_DECLARATORS */
5040
5041{
5042 INEXACT REAL Q, sum;
5043 REAL hh;
5044 INEXACT REAL product1;
5045 REAL product0;
5046 int eindex, hindex;
5047 REAL enow;
5048 INEXACT REAL bvirt;
5049 REAL avirt, bround, around;
5050 INEXACT REAL c;
5051 INEXACT REAL abig;
5052 REAL ahi, alo, bhi, blo;
5053 REAL err1, err2, err3;
5054
5055 Split(b, bhi, blo);
5056 Two_Product_Presplit(e[0], b, bhi, blo, Q, hh);
5057 hindex = 0;
5058 if (hh != 0) {
5059 h[hindex++] = hh;
5060 }
5061 for (eindex = 1; eindex < elen; eindex++) {
5062 enow = e[eindex];
5063 Two_Product_Presplit(enow, b, bhi, blo, product1, product0);
5064 Two_Sum(Q, product0, sum, hh);
5065 if (hh != 0) {
5066 h[hindex++] = hh;
5067 }
5068 Fast_Two_Sum(product1, sum, Q, hh);
5069 if (hh != 0) {
5070 h[hindex++] = hh;
5071 }
5072 }
5073 if ((Q != 0.0) || (hindex == 0)) {
5074 h[hindex++] = Q;
5075 }
5076 return hindex;
5077}
5078
5079/*****************************************************************************/
5080/* */
5081/* estimate() Produce a one-word estimate of an expansion's value. */
5082/* */
5083/* See my Robust Predicates paper for details. */
5084/* */
5085/*****************************************************************************/
5086
5087#ifdef ANSI_DECLARATORS
5088REAL estimate(int elen, REAL *e)
5089#else /* not ANSI_DECLARATORS */
5090REAL estimate(elen, e)
5091int elen;
5092REAL *e;
5093#endif /* not ANSI_DECLARATORS */
5094
5095{
5096 REAL Q;
5097 int eindex;
5098
5099 Q = e[0];
5100 for (eindex = 1; eindex < elen; eindex++) {
5101 Q += e[eindex];
5102 }
5103 return Q;
5104}
5105
5106/*****************************************************************************/
5107/* */
5108/* counterclockwise() Return a positive value if the points pa, pb, and */
5109/* pc occur in counterclockwise order; a negative */
5110/* value if they occur in clockwise order; and zero */
5111/* if they are collinear. The result is also a rough */
5112/* approximation of twice the signed area of the */
5113/* triangle defined by the three points. */
5114/* */
5115/* Uses exact arithmetic if necessary to ensure a correct answer. The */
5116/* result returned is the determinant of a matrix. This determinant is */
5117/* computed adaptively, in the sense that exact arithmetic is used only to */
5118/* the degree it is needed to ensure that the returned value has the */
5119/* correct sign. Hence, this function is usually quite fast, but will run */
5120/* more slowly when the input points are collinear or nearly so. */
5121/* */
5122/* See my Robust Predicates paper for details. */
5123/* */
5124/*****************************************************************************/
5125
5126#ifdef ANSI_DECLARATORS
5128#else /* not ANSI_DECLARATORS */
5129REAL counterclockwiseadapt(pa, pb, pc, detsum)
5130vertex pa;
5131vertex pb;
5132vertex pc;
5133REAL detsum;
5134#endif /* not ANSI_DECLARATORS */
5135
5136{
5137 INEXACT REAL acx, acy, bcx, bcy;
5138 REAL acxtail, acytail, bcxtail, bcytail;
5139 INEXACT REAL detleft, detright;
5140 REAL detlefttail, detrighttail;
5141 REAL det, errbound;
5142 REAL B[4], C1[8], C2[12], D[16];
5143 INEXACT REAL B3;
5144 int C1length, C2length, Dlength;
5145 REAL u[4];
5146 INEXACT REAL u3;
5147 INEXACT REAL s1, t1;
5148 REAL s0, t0;
5149
5150 INEXACT REAL bvirt;
5151 REAL avirt, bround, around;
5152 INEXACT REAL c;
5153 INEXACT REAL abig;
5154 REAL ahi, alo, bhi, blo;
5155 REAL err1, err2, err3;
5156 INEXACT REAL _i, _j;
5157 REAL _0;
5158
5159 acx = (REAL) (pa[0] - pc[0]);
5160 bcx = (REAL) (pb[0] - pc[0]);
5161 acy = (REAL) (pa[1] - pc[1]);
5162 bcy = (REAL) (pb[1] - pc[1]);
5163
5164 Two_Product(acx, bcy, detleft, detlefttail);
5165 Two_Product(acy, bcx, detright, detrighttail);
5166
5167 Two_Two_Diff(detleft, detlefttail, detright, detrighttail,
5168 B3, B[2], B[1], B[0]);
5169 B[3] = B3;
5170
5171 det = estimate(4, B);
5172 errbound = ccwerrboundB * detsum;
5173 if ((det >= errbound) || (-det >= errbound)) {
5174 return det;
5175 }
5176
5177 Two_Diff_Tail(pa[0], pc[0], acx, acxtail);
5178 Two_Diff_Tail(pb[0], pc[0], bcx, bcxtail);
5179 Two_Diff_Tail(pa[1], pc[1], acy, acytail);
5180 Two_Diff_Tail(pb[1], pc[1], bcy, bcytail);
5181
5182 if ((acxtail == 0.0) && (acytail == 0.0)
5183 && (bcxtail == 0.0) && (bcytail == 0.0)) {
5184 return det;
5185 }
5186
5187 errbound = ccwerrboundC * detsum + resulterrbound * Absolute(det);
5188 det += (acx * bcytail + bcy * acxtail)
5189 - (acy * bcxtail + bcx * acytail);
5190 if ((det >= errbound) || (-det >= errbound)) {
5191 return det;
5192 }
5193
5194 Two_Product(acxtail, bcy, s1, s0);
5195 Two_Product(acytail, bcx, t1, t0);
5196 Two_Two_Diff(s1, s0, t1, t0, u3, u[2], u[1], u[0]);
5197 u[3] = u3;
5198 C1length = fast_expansion_sum_zeroelim(4, B, 4, u, C1);
5199
5200 Two_Product(acx, bcytail, s1, s0);
5201 Two_Product(acy, bcxtail, t1, t0);
5202 Two_Two_Diff(s1, s0, t1, t0, u3, u[2], u[1], u[0]);
5203 u[3] = u3;
5204 C2length = fast_expansion_sum_zeroelim(C1length, C1, 4, u, C2);
5205
5206 Two_Product(acxtail, bcytail, s1, s0);
5207 Two_Product(acytail, bcxtail, t1, t0);
5208 Two_Two_Diff(s1, s0, t1, t0, u3, u[2], u[1], u[0]);
5209 u[3] = u3;
5210 Dlength = fast_expansion_sum_zeroelim(C2length, C2, 4, u, D);
5211
5212 return(D[Dlength - 1]);
5213}
5214
5215#ifdef ANSI_DECLARATORS
5217 vertex pa, vertex pb, vertex pc)
5218#else /* not ANSI_DECLARATORS */
5219REAL counterclockwise(m, b, pa, pb, pc)
5220struct mesh *m;
5221struct behavior *b;
5222vertex pa;
5223vertex pb;
5224vertex pc;
5225#endif /* not ANSI_DECLARATORS */
5226
5227{
5228 REAL detleft, detright, det;
5229 REAL detsum, errbound;
5230
5231 m->counterclockcount++;
5232
5233 detleft = (pa[0] - pc[0]) * (pb[1] - pc[1]);
5234 detright = (pa[1] - pc[1]) * (pb[0] - pc[0]);
5235 det = detleft - detright;
5236
5237 if (b->noexact) {
5238 return det;
5239 }
5240
5241 if (detleft > 0.0) {
5242 if (detright <= 0.0) {
5243 return det;
5244 } else {
5245 detsum = detleft + detright;
5246 }
5247 } else if (detleft < 0.0) {
5248 if (detright >= 0.0) {
5249 return det;
5250 } else {
5251 detsum = -detleft - detright;
5252 }
5253 } else {
5254 return det;
5255 }
5256
5257 errbound = ccwerrboundA * detsum;
5258 if ((det >= errbound) || (-det >= errbound)) {
5259 return det;
5260 }
5261
5262 return counterclockwiseadapt(pa, pb, pc, detsum);
5263}
5264
5265/*****************************************************************************/
5266/* */
5267/* incircle() Return a positive value if the point pd lies inside the */
5268/* circle passing through pa, pb, and pc; a negative value if */
5269/* it lies outside; and zero if the four points are cocircular.*/
5270/* The points pa, pb, and pc must be in counterclockwise */
5271/* order, or the sign of the result will be reversed. */
5272/* */
5273/* Uses exact arithmetic if necessary to ensure a correct answer. The */
5274/* result returned is the determinant of a matrix. This determinant is */
5275/* computed adaptively, in the sense that exact arithmetic is used only to */
5276/* the degree it is needed to ensure that the returned value has the */
5277/* correct sign. Hence, this function is usually quite fast, but will run */
5278/* more slowly when the input points are cocircular or nearly so. */
5279/* */
5280/* See my Robust Predicates paper for details. */
5281/* */
5282/*****************************************************************************/
5283
5284#ifdef ANSI_DECLARATORS
5286#else /* not ANSI_DECLARATORS */
5287REAL incircleadapt(pa, pb, pc, pd, permanent)
5288vertex pa;
5289vertex pb;
5290vertex pc;
5291vertex pd;
5292REAL permanent;
5293#endif /* not ANSI_DECLARATORS */
5294
5295{
5296 INEXACT REAL adx, bdx, cdx, ady, bdy, cdy;
5297 REAL det, errbound;
5298
5299 INEXACT REAL bdxcdy1, cdxbdy1, cdxady1, adxcdy1, adxbdy1, bdxady1;
5300 REAL bdxcdy0, cdxbdy0, cdxady0, adxcdy0, adxbdy0, bdxady0;
5301 REAL bc[4], ca[4], ab[4];
5302 INEXACT REAL bc3, ca3, ab3;
5303 REAL axbc[8], axxbc[16], aybc[8], ayybc[16], adet[32];
5304 int axbclen, axxbclen, aybclen, ayybclen, alen;
5305 REAL bxca[8], bxxca[16], byca[8], byyca[16], bdet[32];
5306 int bxcalen, bxxcalen, bycalen, byycalen, blen;
5307 REAL cxab[8], cxxab[16], cyab[8], cyyab[16], cdet[32];
5308 int cxablen, cxxablen, cyablen, cyyablen, clen;
5309 REAL abdet[64];
5310 int ablen;
5311 REAL fin1[1152], fin2[1152];
5312 REAL *finnow, *finother, *finswap;
5313 int finlength;
5314
5315 REAL adxtail, bdxtail, cdxtail, adytail, bdytail, cdytail;
5316 INEXACT REAL adxadx1, adyady1, bdxbdx1, bdybdy1, cdxcdx1, cdycdy1;
5317 REAL adxadx0, adyady0, bdxbdx0, bdybdy0, cdxcdx0, cdycdy0;
5318 REAL aa[4], bb[4], cc[4];
5319 INEXACT REAL aa3, bb3, cc3;
5320 INEXACT REAL ti1, tj1;
5321 REAL ti0, tj0;
5322 REAL u[4], v[4];
5323 INEXACT REAL u3, v3;
5324 REAL temp8[8], temp16a[16], temp16b[16], temp16c[16];
5325 REAL temp32a[32], temp32b[32], temp48[48], temp64[64];
5326 int temp8len, temp16alen, temp16blen, temp16clen;
5327 int temp32alen, temp32blen, temp48len, temp64len;
5328 REAL axtbb[8], axtcc[8], aytbb[8], aytcc[8];
5329 int axtbblen, axtcclen, aytbblen, aytcclen;
5330 REAL bxtaa[8], bxtcc[8], bytaa[8], bytcc[8];
5331 int bxtaalen, bxtcclen, bytaalen, bytcclen;
5332 REAL cxtaa[8], cxtbb[8], cytaa[8], cytbb[8];
5333 int cxtaalen, cxtbblen, cytaalen, cytbblen;
5334 REAL axtbc[8], aytbc[8], bxtca[8], bytca[8], cxtab[8], cytab[8];
5335 int axtbclen, aytbclen, bxtcalen, bytcalen, cxtablen, cytablen;
5336 REAL axtbct[16], aytbct[16], bxtcat[16], bytcat[16], cxtabt[16], cytabt[16];
5337 int axtbctlen, aytbctlen, bxtcatlen, bytcatlen, cxtabtlen, cytabtlen;
5338 REAL axtbctt[8], aytbctt[8], bxtcatt[8];
5339 REAL bytcatt[8], cxtabtt[8], cytabtt[8];
5340 int axtbcttlen, aytbcttlen, bxtcattlen, bytcattlen, cxtabttlen, cytabttlen;
5341 REAL abt[8], bct[8], cat[8];
5342 int abtlen, bctlen, catlen;
5343 REAL abtt[4], bctt[4], catt[4];
5344 int abttlen, bcttlen, cattlen;
5345 INEXACT REAL abtt3, bctt3, catt3;
5346 REAL negate;
5347
5348 INEXACT REAL bvirt;
5349 REAL avirt, bround, around;
5350 INEXACT REAL c;
5351 INEXACT REAL abig;
5352 REAL ahi, alo, bhi, blo;
5353 REAL err1, err2, err3;
5354 INEXACT REAL _i, _j;
5355 REAL _0;
5356
5357 adx = (REAL) (pa[0] - pd[0]);
5358 bdx = (REAL) (pb[0] - pd[0]);
5359 cdx = (REAL) (pc[0] - pd[0]);
5360 ady = (REAL) (pa[1] - pd[1]);
5361 bdy = (REAL) (pb[1] - pd[1]);
5362 cdy = (REAL) (pc[1] - pd[1]);
5363
5364 Two_Product(bdx, cdy, bdxcdy1, bdxcdy0);
5365 Two_Product(cdx, bdy, cdxbdy1, cdxbdy0);
5366 Two_Two_Diff(bdxcdy1, bdxcdy0, cdxbdy1, cdxbdy0, bc3, bc[2], bc[1], bc[0]);
5367 bc[3] = bc3;
5368 axbclen = scale_expansion_zeroelim(4, bc, adx, axbc);
5369 axxbclen = scale_expansion_zeroelim(axbclen, axbc, adx, axxbc);
5370 aybclen = scale_expansion_zeroelim(4, bc, ady, aybc);
5371 ayybclen = scale_expansion_zeroelim(aybclen, aybc, ady, ayybc);
5372 alen = fast_expansion_sum_zeroelim(axxbclen, axxbc, ayybclen, ayybc, adet);
5373
5374 Two_Product(cdx, ady, cdxady1, cdxady0);
5375 Two_Product(adx, cdy, adxcdy1, adxcdy0);
5376 Two_Two_Diff(cdxady1, cdxady0, adxcdy1, adxcdy0, ca3, ca[2], ca[1], ca[0]);
5377 ca[3] = ca3;
5378 bxcalen = scale_expansion_zeroelim(4, ca, bdx, bxca);
5379 bxxcalen = scale_expansion_zeroelim(bxcalen, bxca, bdx, bxxca);
5380 bycalen = scale_expansion_zeroelim(4, ca, bdy, byca);
5381 byycalen = scale_expansion_zeroelim(bycalen, byca, bdy, byyca);
5382 blen = fast_expansion_sum_zeroelim(bxxcalen, bxxca, byycalen, byyca, bdet);
5383
5384 Two_Product(adx, bdy, adxbdy1, adxbdy0);
5385 Two_Product(bdx, ady, bdxady1, bdxady0);
5386 Two_Two_Diff(adxbdy1, adxbdy0, bdxady1, bdxady0, ab3, ab[2], ab[1], ab[0]);
5387 ab[3] = ab3;
5388 cxablen = scale_expansion_zeroelim(4, ab, cdx, cxab);
5389 cxxablen = scale_expansion_zeroelim(cxablen, cxab, cdx, cxxab);
5390 cyablen = scale_expansion_zeroelim(4, ab, cdy, cyab);
5391 cyyablen = scale_expansion_zeroelim(cyablen, cyab, cdy, cyyab);
5392 clen = fast_expansion_sum_zeroelim(cxxablen, cxxab, cyyablen, cyyab, cdet);
5393
5394 ablen = fast_expansion_sum_zeroelim(alen, adet, blen, bdet, abdet);
5395 finlength = fast_expansion_sum_zeroelim(ablen, abdet, clen, cdet, fin1);
5396
5397 det = estimate(finlength, fin1);
5398 errbound = iccerrboundB * permanent;
5399 if ((det >= errbound) || (-det >= errbound)) {
5400 return det;
5401 }
5402
5403 Two_Diff_Tail(pa[0], pd[0], adx, adxtail);
5404 Two_Diff_Tail(pa[1], pd[1], ady, adytail);
5405 Two_Diff_Tail(pb[0], pd[0], bdx, bdxtail);
5406 Two_Diff_Tail(pb[1], pd[1], bdy, bdytail);
5407 Two_Diff_Tail(pc[0], pd[0], cdx, cdxtail);
5408 Two_Diff_Tail(pc[1], pd[1], cdy, cdytail);
5409 if ((adxtail == 0.0) && (bdxtail == 0.0) && (cdxtail == 0.0)
5410 && (adytail == 0.0) && (bdytail == 0.0) && (cdytail == 0.0)) {
5411 return det;
5412 }
5413
5414 errbound = iccerrboundC * permanent + resulterrbound * Absolute(det);
5415 det += ((adx * adx + ady * ady) * ((bdx * cdytail + cdy * bdxtail)
5416 - (bdy * cdxtail + cdx * bdytail))
5417 + 2.0 * (adx * adxtail + ady * adytail) * (bdx * cdy - bdy * cdx))
5418 + ((bdx * bdx + bdy * bdy) * ((cdx * adytail + ady * cdxtail)
5419 - (cdy * adxtail + adx * cdytail))
5420 + 2.0 * (bdx * bdxtail + bdy * bdytail) * (cdx * ady - cdy * adx))
5421 + ((cdx * cdx + cdy * cdy) * ((adx * bdytail + bdy * adxtail)
5422 - (ady * bdxtail + bdx * adytail))
5423 + 2.0 * (cdx * cdxtail + cdy * cdytail) * (adx * bdy - ady * bdx));
5424 if ((det >= errbound) || (-det >= errbound)) {
5425 return det;
5426 }
5427
5428 finnow = fin1;
5429 finother = fin2;
5430
5431 if ((bdxtail != 0.0) || (bdytail != 0.0)
5432 || (cdxtail != 0.0) || (cdytail != 0.0)) {
5433 Square(adx, adxadx1, adxadx0);
5434 Square(ady, adyady1, adyady0);
5435 Two_Two_Sum(adxadx1, adxadx0, adyady1, adyady0, aa3, aa[2], aa[1], aa[0]);
5436 aa[3] = aa3;
5437 }
5438 if ((cdxtail != 0.0) || (cdytail != 0.0)
5439 || (adxtail != 0.0) || (adytail != 0.0)) {
5440 Square(bdx, bdxbdx1, bdxbdx0);
5441 Square(bdy, bdybdy1, bdybdy0);
5442 Two_Two_Sum(bdxbdx1, bdxbdx0, bdybdy1, bdybdy0, bb3, bb[2], bb[1], bb[0]);
5443 bb[3] = bb3;
5444 }
5445 if ((adxtail != 0.0) || (adytail != 0.0)
5446 || (bdxtail != 0.0) || (bdytail != 0.0)) {
5447 Square(cdx, cdxcdx1, cdxcdx0);
5448 Square(cdy, cdycdy1, cdycdy0);
5449 Two_Two_Sum(cdxcdx1, cdxcdx0, cdycdy1, cdycdy0, cc3, cc[2], cc[1], cc[0]);
5450 cc[3] = cc3;
5451 }
5452
5453 if (adxtail != 0.0) {
5454 axtbclen = scale_expansion_zeroelim(4, bc, adxtail, axtbc);
5455 temp16alen = scale_expansion_zeroelim(axtbclen, axtbc, 2.0 * adx,
5456 temp16a);
5457
5458 axtcclen = scale_expansion_zeroelim(4, cc, adxtail, axtcc);
5459 temp16blen = scale_expansion_zeroelim(axtcclen, axtcc, bdy, temp16b);
5460
5461 axtbblen = scale_expansion_zeroelim(4, bb, adxtail, axtbb);
5462 temp16clen = scale_expansion_zeroelim(axtbblen, axtbb, -cdy, temp16c);
5463
5464 temp32alen = fast_expansion_sum_zeroelim(temp16alen, temp16a,
5465 temp16blen, temp16b, temp32a);
5466 temp48len = fast_expansion_sum_zeroelim(temp16clen, temp16c,
5467 temp32alen, temp32a, temp48);
5468 finlength = fast_expansion_sum_zeroelim(finlength, finnow, temp48len,
5469 temp48, finother);
5470 finswap = finnow; finnow = finother; finother = finswap;
5471 }
5472 if (adytail != 0.0) {
5473 aytbclen = scale_expansion_zeroelim(4, bc, adytail, aytbc);
5474 temp16alen = scale_expansion_zeroelim(aytbclen, aytbc, 2.0 * ady,
5475 temp16a);
5476
5477 aytbblen = scale_expansion_zeroelim(4, bb, adytail, aytbb);
5478 temp16blen = scale_expansion_zeroelim(aytbblen, aytbb, cdx, temp16b);
5479
5480 aytcclen = scale_expansion_zeroelim(4, cc, adytail, aytcc);
5481 temp16clen = scale_expansion_zeroelim(aytcclen, aytcc, -bdx, temp16c);
5482
5483 temp32alen = fast_expansion_sum_zeroelim(temp16alen, temp16a,
5484 temp16blen, temp16b, temp32a);
5485 temp48len = fast_expansion_sum_zeroelim(temp16clen, temp16c,
5486 temp32alen, temp32a, temp48);
5487 finlength = fast_expansion_sum_zeroelim(finlength, finnow, temp48len,
5488 temp48, finother);
5489 finswap = finnow; finnow = finother; finother = finswap;
5490 }
5491 if (bdxtail != 0.0) {
5492 bxtcalen = scale_expansion_zeroelim(4, ca, bdxtail, bxtca);
5493 temp16alen = scale_expansion_zeroelim(bxtcalen, bxtca, 2.0 * bdx,
5494 temp16a);
5495
5496 bxtaalen = scale_expansion_zeroelim(4, aa, bdxtail, bxtaa);
5497 temp16blen = scale_expansion_zeroelim(bxtaalen, bxtaa, cdy, temp16b);
5498
5499 bxtcclen = scale_expansion_zeroelim(4, cc, bdxtail, bxtcc);
5500 temp16clen = scale_expansion_zeroelim(bxtcclen, bxtcc, -ady, temp16c);
5501
5502 temp32alen = fast_expansion_sum_zeroelim(temp16alen, temp16a,
5503 temp16blen, temp16b, temp32a);
5504 temp48len = fast_expansion_sum_zeroelim(temp16clen, temp16c,
5505 temp32alen, temp32a, temp48);
5506 finlength = fast_expansion_sum_zeroelim(finlength, finnow, temp48len,
5507 temp48, finother);
5508 finswap = finnow; finnow = finother; finother = finswap;