user.c
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1 /*<html><pre> -<a href="qh-user.htm"
2  >-------------------------------</a><a name="TOP">-</a>
3 
4  user.c
5  user redefinable functions
6 
7  see user2.c for qh_fprintf, qh_malloc, qh_free
8 
9  see README.txt see COPYING.txt for copyright information.
10 
11  see libqhull.h for data structures, macros, and user-callable functions.
12 
13  see user_eg.c, user_eg2.c, and unix.c for examples.
14 
15  see user.h for user-definable constants
16 
17  use qh_NOmem in mem.h to turn off memory management
18  use qh_NOmerge in user.h to turn off facet merging
19  set qh_KEEPstatistics in user.h to 0 to turn off statistics
20 
21  This is unsupported software. You're welcome to make changes,
22  but you're on your own if something goes wrong. Use 'Tc' to
23  check frequently. Usually qhull will report an error if
24  a data structure becomes inconsistent. If so, it also reports
25  the last point added to the hull, e.g., 102. You can then trace
26  the execution of qhull with "T4P102".
27 
28  Please report any errors that you fix to qhull@qhull.org
29 
30  Qhull-template is a template for calling qhull from within your application
31 
32  if you recompile and load this module, then user.o will not be loaded
33  from qhull.a
34 
35  you can add additional quick allocation sizes in qh_user_memsizes
36 
37  if the other functions here are redefined to not use qh_print...,
38  then io.o will not be loaded from qhull.a. See user_eg.c for an
39  example. We recommend keeping io.o for the extra debugging
40  information it supplies.
41 */
42 
43 #include "qhull_a.h"
44 
45 #include <stdarg.h>
46 
47 /*-<a href="qh-user.htm#TOC"
48  >-------------------------------</a><a name="qhull_template">-</a>
49 
50  Qhull-template
51  Template for calling qhull from inside your program
52 
53  returns:
54  exit code(see qh_ERR... in libqhull.h)
55  all memory freed
56 
57  notes:
58  This can be called any number of times.
59 */
60 #if 0
61 {
62  int dim; /* dimension of points */
63  int numpoints; /* number of points */
64  coordT *points; /* array of coordinates for each point */
65  boolT ismalloc; /* True if qhull should free points in qh_freeqhull() or reallocation */
66  char flags[]= "qhull Tv"; /* option flags for qhull, see qh_opt.htm */
67  FILE *outfile= stdout; /* output from qh_produce_output()
68  use NULL to skip qh_produce_output() */
69  FILE *errfile= stderr; /* error messages from qhull code */
70  int exitcode; /* 0 if no error from qhull */
71  facetT *facet; /* set by FORALLfacets */
72  int curlong, totlong; /* memory remaining after qh_memfreeshort */
73 
74  QHULL_LIB_CHECK /* Check for compatible library */
75 
76 #if qh_QHpointer /* see user.h */
77  if (qh_qh){ /* should be NULL */
78  qh_printf_stderr(6238, "Qhull link error. The global variable qh_qh was not initialized\n\
79  to NULL by global.c. Please compile this program with -Dqh_QHpointer_dllimport\n\
80  as well as -Dqh_QHpointer, or use libqhullstatic, or use a different tool chain.\n\n");
81  exit(1);
82  }
83 #endif
84 
85  /* initialize dim, numpoints, points[], ismalloc here */
86  exitcode= qh_new_qhull(dim, numpoints, points, ismalloc,
87  flags, outfile, errfile);
88  if (!exitcode) { /* if no error */
89  /* 'qh facet_list' contains the convex hull */
90  FORALLfacets {
91  /* ... your code ... */
92  }
93  }
94  qh_freeqhull(!qh_ALL);
95  qh_memfreeshort(&curlong, &totlong);
96  if (curlong || totlong)
97  qh_fprintf(errfile, 7068, "qhull internal warning (main): did not free %d bytes of long memory(%d pieces)\n", totlong, curlong);
98 }
99 #endif
100 
101 /*-<a href="qh-user.htm#TOC"
102  >-------------------------------</a><a name="new_qhull">-</a>
103 
104  qh_new_qhull( dim, numpoints, points, ismalloc, qhull_cmd, outfile, errfile )
105  build new qhull data structure and return exitcode (0 if no errors)
106  if numpoints=0 and points=NULL, initializes qhull
107 
108  notes:
109  do not modify points until finished with results.
110  The qhull data structure contains pointers into the points array.
111  do not call qhull functions before qh_new_qhull().
112  The qhull data structure is not initialized until qh_new_qhull().
113 
114  Default errfile is stderr, outfile may be null
115  qhull_cmd must start with "qhull "
116  projects points to a new point array for Delaunay triangulations ('d' and 'v')
117  transforms points into a new point array for halfspace intersection ('H')
118 
119 
120  To allow multiple, concurrent calls to qhull()
121  - set qh_QHpointer in user.h
122  - use qh_save_qhull and qh_restore_qhull to swap the global data structure between calls.
123  - use qh_freeqhull(qh_ALL) to free intermediate convex hulls
124 
125  see:
126  Qhull-template at the beginning of this file.
127  An example of using qh_new_qhull is user_eg.c
128 */
129 int qh_new_qhull(int dim, int numpoints, coordT *points, boolT ismalloc,
130  char *qhull_cmd, FILE *outfile, FILE *errfile) {
131  int exitcode, hulldim;
132  boolT new_ismalloc;
133  static boolT firstcall = True;
134  coordT *new_points;
135 
136  if(!errfile){
137  errfile= stderr;
138  }
139  if (firstcall) {
140  qh_meminit(errfile);
141  firstcall= False;
142  } else {
143  qh_memcheck();
144  }
145  if (strncmp(qhull_cmd, "qhull ", (size_t)6)) {
146  qh_fprintf(errfile, 6186, "qhull error (qh_new_qhull): start qhull_cmd argument with \"qhull \"\n");
147  return qh_ERRinput;
148  }
149  qh_initqhull_start(NULL, outfile, errfile);
150  if(numpoints==0 && points==NULL){
151  trace1((qh ferr, 1047, "qh_new_qhull: initialize Qhull\n"));
152  return 0;
153  }
154  trace1((qh ferr, 1044, "qh_new_qhull: build new Qhull for %d %d-d points with %s\n", numpoints, dim, qhull_cmd));
155  exitcode = setjmp(qh errexit);
156  if (!exitcode)
157  {
158  qh NOerrexit = False;
159  qh_initflags(qhull_cmd);
160  if (qh DELAUNAY)
161  qh PROJECTdelaunay= True;
162  if (qh HALFspace) {
163  /* points is an array of halfspaces,
164  the last coordinate of each halfspace is its offset */
165  hulldim= dim-1;
166  qh_setfeasible(hulldim);
167  new_points= qh_sethalfspace_all(dim, numpoints, points, qh feasible_point);
168  new_ismalloc= True;
169  if (ismalloc)
170  qh_free(points);
171  }else {
172  hulldim= dim;
173  new_points= points;
174  new_ismalloc= ismalloc;
175  }
176  qh_init_B(new_points, numpoints, hulldim, new_ismalloc);
177  qh_qhull();
178  qh_check_output();
179  if (outfile) {
180  qh_produce_output();
181  }else {
182  qh_prepare_output();
183  }
184  if (qh VERIFYoutput && !qh STOPpoint && !qh STOPcone)
185  qh_check_points();
186  }
187  qh NOerrexit = True;
188  return exitcode;
189 } /* new_qhull */
190 
191 /*-<a href="qh-user.htm#TOC"
192  >-------------------------------</a><a name="errexit">-</a>
193 
194  qh_errexit( exitcode, facet, ridge )
195  report and exit from an error
196  report facet and ridge if non-NULL
197  reports useful information such as last point processed
198  set qh.FORCEoutput to print neighborhood of facet
199 
200  see:
201  qh_errexit2() in libqhull.c for printing 2 facets
202 
203  design:
204  check for error within error processing
205  compute qh.hulltime
206  print facet and ridge (if any)
207  report commandString, options, qh.furthest_id
208  print summary and statistics (including precision statistics)
209  if qh_ERRsingular
210  print help text for singular data set
211  exit program via long jump (if defined) or exit()
212 */
213 void qh_errexit(int exitcode, facetT *facet, ridgeT *ridge) {
214 
215  if (qh ERREXITcalled) {
216  qh_fprintf(qh ferr, 8126, "\nqhull error while processing previous error. Exit program\n");
217  qh_exit(qh_ERRqhull);
218  }
219  qh ERREXITcalled= True;
220  if (!qh QHULLfinished)
221  qh hulltime= qh_CPUclock - qh hulltime;
222  qh_errprint("ERRONEOUS", facet, NULL, ridge, NULL);
223  qh_fprintf(qh ferr, 8127, "\nWhile executing: %s | %s\n", qh rbox_command, qh qhull_command);
224  qh_fprintf(qh ferr, 8128, "Options selected for Qhull %s:\n%s\n", qh_version, qh qhull_options);
225  if (qh furthest_id >= 0) {
226  qh_fprintf(qh ferr, 8129, "Last point added to hull was p%d.", qh furthest_id);
227  if (zzval_(Ztotmerge))
228  qh_fprintf(qh ferr, 8130, " Last merge was #%d.", zzval_(Ztotmerge));
229  if (qh QHULLfinished)
230  qh_fprintf(qh ferr, 8131, "\nQhull has finished constructing the hull.");
231  else if (qh POSTmerging)
232  qh_fprintf(qh ferr, 8132, "\nQhull has started post-merging.");
233  qh_fprintf(qh ferr, 8133, "\n");
234  }
235  if (qh FORCEoutput && (qh QHULLfinished || (!facet && !ridge)))
236  qh_produce_output();
237  else if (exitcode != qh_ERRinput) {
238  if (exitcode != qh_ERRsingular && zzval_(Zsetplane) > qh hull_dim+1) {
239  qh_fprintf(qh ferr, 8134, "\nAt error exit:\n");
240  qh_printsummary(qh ferr);
241  if (qh PRINTstatistics) {
242  qh_collectstatistics();
243  qh_printstatistics(qh ferr, "at error exit");
244  qh_memstatistics(qh ferr);
245  }
246  }
247  if (qh PRINTprecision)
248  qh_printstats(qh ferr, qhstat precision, NULL);
249  }
250  if (!exitcode)
251  exitcode= qh_ERRqhull;
252  else if (exitcode == qh_ERRsingular)
253  qh_printhelp_singular(qh ferr);
254  else if (exitcode == qh_ERRprec && !qh PREmerge)
255  qh_printhelp_degenerate(qh ferr);
256  if (qh NOerrexit) {
257  qh_fprintf(qh ferr, 6187, "qhull error while ending program, or qh->NOerrexit not cleared after setjmp(). Exit program with error.\n");
258  qh_exit(qh_ERRqhull);
259  }
260  qh ERREXITcalled= False;
261  qh NOerrexit= True;
262  qh ALLOWrestart= False; /* longjmp will undo qh_build_withrestart */
263  longjmp(qh errexit, exitcode);
264 } /* errexit */
265 
266 
267 /*-<a href="qh-user.htm#TOC"
268  >-------------------------------</a><a name="errprint">-</a>
269 
270  qh_errprint( fp, string, atfacet, otherfacet, atridge, atvertex )
271  prints out the information of facets and ridges to fp
272  also prints neighbors and geomview output
273 
274  notes:
275  except for string, any parameter may be NULL
276 */
277 void qh_errprint(const char *string, facetT *atfacet, facetT *otherfacet, ridgeT *atridge, vertexT *atvertex) {
278  int i;
279 
280  if (atfacet) {
281  qh_fprintf(qh ferr, 8135, "%s FACET:\n", string);
282  qh_printfacet(qh ferr, atfacet);
283  }
284  if (otherfacet) {
285  qh_fprintf(qh ferr, 8136, "%s OTHER FACET:\n", string);
286  qh_printfacet(qh ferr, otherfacet);
287  }
288  if (atridge) {
289  qh_fprintf(qh ferr, 8137, "%s RIDGE:\n", string);
290  qh_printridge(qh ferr, atridge);
291  if (atridge->top && atridge->top != atfacet && atridge->top != otherfacet)
292  qh_printfacet(qh ferr, atridge->top);
293  if (atridge->bottom
294  && atridge->bottom != atfacet && atridge->bottom != otherfacet)
295  qh_printfacet(qh ferr, atridge->bottom);
296  if (!atfacet)
297  atfacet= atridge->top;
298  if (!otherfacet)
299  otherfacet= otherfacet_(atridge, atfacet);
300  }
301  if (atvertex) {
302  qh_fprintf(qh ferr, 8138, "%s VERTEX:\n", string);
303  qh_printvertex(qh ferr, atvertex);
304  }
305  if (qh fout && qh FORCEoutput && atfacet && !qh QHULLfinished && !qh IStracing) {
306  qh_fprintf(qh ferr, 8139, "ERRONEOUS and NEIGHBORING FACETS to output\n");
307  for (i=0; i < qh_PRINTEND; i++) /* use fout for geomview output */
308  qh_printneighborhood(qh fout, qh PRINTout[i], atfacet, otherfacet,
309  !qh_ALL);
310  }
311 } /* errprint */
312 
313 
314 /*-<a href="qh-user.htm#TOC"
315  >-------------------------------</a><a name="printfacetlist">-</a>
316 
317  qh_printfacetlist( fp, facetlist, facets, printall )
318  print all fields for a facet list and/or set of facets to fp
319  if !printall,
320  only prints good facets
321 
322  notes:
323  also prints all vertices
324 */
325 void qh_printfacetlist(facetT *facetlist, setT *facets, boolT printall) {
326  facetT *facet, **facetp;
327 
328  qh_printbegin(qh ferr, qh_PRINTfacets, facetlist, facets, printall);
329  FORALLfacet_(facetlist)
330  qh_printafacet(qh ferr, qh_PRINTfacets, facet, printall);
331  FOREACHfacet_(facets)
332  qh_printafacet(qh ferr, qh_PRINTfacets, facet, printall);
333  qh_printend(qh ferr, qh_PRINTfacets, facetlist, facets, printall);
334 } /* printfacetlist */
335 
336 
337 /*-<a href="qh-io.htm#TOC"
338  >-------------------------------</a><a name="printhelp_degenerate">-</a>
339 
340  qh_printhelp_degenerate( fp )
341  prints descriptive message for precision error
342 
343  notes:
344  no message if qh_QUICKhelp
345 */
346 void qh_printhelp_degenerate(FILE *fp) {
347 
348  if (qh MERGEexact || qh PREmerge || qh JOGGLEmax < REALmax/2)
349  qh_fprintf(fp, 9368, "\n\
350 A Qhull error has occurred. Qhull should have corrected the above\n\
351 precision error. Please send the input and all of the output to\n\
352 qhull_bug@qhull.org\n");
353  else if (!qh_QUICKhelp) {
354  qh_fprintf(fp, 9369, "\n\
355 Precision problems were detected during construction of the convex hull.\n\
356 This occurs because convex hull algorithms assume that calculations are\n\
357 exact, but floating-point arithmetic has roundoff errors.\n\
358 \n\
359 To correct for precision problems, do not use 'Q0'. By default, Qhull\n\
360 selects 'C-0' or 'Qx' and merges non-convex facets. With option 'QJ',\n\
361 Qhull joggles the input to prevent precision problems. See \"Imprecision\n\
362 in Qhull\" (qh-impre.htm).\n\
363 \n\
364 If you use 'Q0', the output may include\n\
365 coplanar ridges, concave ridges, and flipped facets. In 4-d and higher,\n\
366 Qhull may produce a ridge with four neighbors or two facets with the same \n\
367 vertices. Qhull reports these events when they occur. It stops when a\n\
368 concave ridge, flipped facet, or duplicate facet occurs.\n");
369 #if REALfloat
370  qh_fprintf(fp, 9370, "\
371 \n\
372 Qhull is currently using single precision arithmetic. The following\n\
373 will probably remove the precision problems:\n\
374  - recompile qhull for realT precision(#define REALfloat 0 in user.h).\n");
375 #endif
376  if (qh DELAUNAY && !qh SCALElast && qh MAXabs_coord > 1e4)
377  qh_fprintf(fp, 9371, "\
378 \n\
379 When computing the Delaunay triangulation of coordinates > 1.0,\n\
380  - use 'Qbb' to scale the last coordinate to [0,m] (max previous coordinate)\n");
381  if (qh DELAUNAY && !qh ATinfinity)
382  qh_fprintf(fp, 9372, "\
383 When computing the Delaunay triangulation:\n\
384  - use 'Qz' to add a point at-infinity. This reduces precision problems.\n");
385 
386  qh_fprintf(fp, 9373, "\
387 \n\
388 If you need triangular output:\n\
389  - use option 'Qt' to triangulate the output\n\
390  - use option 'QJ' to joggle the input points and remove precision errors\n\
391  - use option 'Ft'. It triangulates non-simplicial facets with added points.\n\
392 \n\
393 If you must use 'Q0',\n\
394 try one or more of the following options. They can not guarantee an output.\n\
395  - use 'QbB' to scale the input to a cube.\n\
396  - use 'Po' to produce output and prevent partitioning for flipped facets\n\
397  - use 'V0' to set min. distance to visible facet as 0 instead of roundoff\n\
398  - use 'En' to specify a maximum roundoff error less than %2.2g.\n\
399  - options 'Qf', 'Qbb', and 'QR0' may also help\n",
400  qh DISTround);
401  qh_fprintf(fp, 9374, "\
402 \n\
403 To guarantee simplicial output:\n\
404  - use option 'Qt' to triangulate the output\n\
405  - use option 'QJ' to joggle the input points and remove precision errors\n\
406  - use option 'Ft' to triangulate the output by adding points\n\
407  - use exact arithmetic (see \"Imprecision in Qhull\", qh-impre.htm)\n\
408 ");
409  }
410 } /* printhelp_degenerate */
411 
412 
413 /*-<a href="qh-globa.htm#TOC"
414  >-------------------------------</a><a name="printhelp_narrowhull">-</a>
415 
416  qh_printhelp_narrowhull( minangle )
417  Warn about a narrow hull
418 
419  notes:
420  Alternatively, reduce qh_WARNnarrow in user.h
421 
422 */
423 void qh_printhelp_narrowhull(FILE *fp, realT minangle) {
424 
425  qh_fprintf(fp, 9375, "qhull precision warning: \n\
426 The initial hull is narrow (cosine of min. angle is %.16f).\n\
427 Is the input lower dimensional (e.g., on a plane in 3-d)? Qhull may\n\
428 produce a wide facet. Options 'QbB' (scale to unit box) or 'Qbb' (scale\n\
429 last coordinate) may remove this warning. Use 'Pp' to skip this warning.\n\
430 See 'Limitations' in qh-impre.htm.\n",
431  -minangle); /* convert from angle between normals to angle between facets */
432 } /* printhelp_narrowhull */
433 
434 /*-<a href="qh-io.htm#TOC"
435  >-------------------------------</a><a name="printhelp_singular">-</a>
436 
437  qh_printhelp_singular( fp )
438  prints descriptive message for singular input
439 */
440 void qh_printhelp_singular(FILE *fp) {
441  facetT *facet;
442  vertexT *vertex, **vertexp;
443  realT min, max, *coord, dist;
444  int i,k;
445 
446  qh_fprintf(fp, 9376, "\n\
447 The input to qhull appears to be less than %d dimensional, or a\n\
448 computation has overflowed.\n\n\
449 Qhull could not construct a clearly convex simplex from points:\n",
450  qh hull_dim);
451  qh_printvertexlist(fp, "", qh facet_list, NULL, qh_ALL);
452  if (!qh_QUICKhelp)
453  qh_fprintf(fp, 9377, "\n\
454 The center point is coplanar with a facet, or a vertex is coplanar\n\
455 with a neighboring facet. The maximum round off error for\n\
456 computing distances is %2.2g. The center point, facets and distances\n\
457 to the center point are as follows:\n\n", qh DISTround);
458  qh_printpointid(fp, "center point", qh hull_dim, qh interior_point, qh_IDunknown);
459  qh_fprintf(fp, 9378, "\n");
460  FORALLfacets {
461  qh_fprintf(fp, 9379, "facet");
462  FOREACHvertex_(facet->vertices)
463  qh_fprintf(fp, 9380, " p%d", qh_pointid(vertex->point));
464  zinc_(Zdistio);
465  qh_distplane(qh interior_point, facet, &dist);
466  qh_fprintf(fp, 9381, " distance= %4.2g\n", dist);
467  }
468  if (!qh_QUICKhelp) {
469  if (qh HALFspace)
470  qh_fprintf(fp, 9382, "\n\
471 These points are the dual of the given halfspaces. They indicate that\n\
472 the intersection is degenerate.\n");
473  qh_fprintf(fp, 9383,"\n\
474 These points either have a maximum or minimum x-coordinate, or\n\
475 they maximize the determinant for k coordinates. Trial points\n\
476 are first selected from points that maximize a coordinate.\n");
477  if (qh hull_dim >= qh_INITIALmax)
478  qh_fprintf(fp, 9384, "\n\
479 Because of the high dimension, the min x-coordinate and max-coordinate\n\
480 points are used if the determinant is non-zero. Option 'Qs' will\n\
481 do a better, though much slower, job. Instead of 'Qs', you can change\n\
482 the points by randomly rotating the input with 'QR0'.\n");
483  }
484  qh_fprintf(fp, 9385, "\nThe min and max coordinates for each dimension are:\n");
485  for (k=0; k < qh hull_dim; k++) {
486  min= REALmax;
487  max= -REALmin;
488  for (i=qh num_points, coord= qh first_point+k; i--; coord += qh hull_dim) {
489  maximize_(max, *coord);
490  minimize_(min, *coord);
491  }
492  qh_fprintf(fp, 9386, " %d: %8.4g %8.4g difference= %4.4g\n", k, min, max, max-min);
493  }
494  if (!qh_QUICKhelp) {
495  qh_fprintf(fp, 9387, "\n\
496 If the input should be full dimensional, you have several options that\n\
497 may determine an initial simplex:\n\
498  - use 'QJ' to joggle the input and make it full dimensional\n\
499  - use 'QbB' to scale the points to the unit cube\n\
500  - use 'QR0' to randomly rotate the input for different maximum points\n\
501  - use 'Qs' to search all points for the initial simplex\n\
502  - use 'En' to specify a maximum roundoff error less than %2.2g.\n\
503  - trace execution with 'T3' to see the determinant for each point.\n",
504  qh DISTround);
505 #if REALfloat
506  qh_fprintf(fp, 9388, "\
507  - recompile qhull for realT precision(#define REALfloat 0 in libqhull.h).\n");
508 #endif
509  qh_fprintf(fp, 9389, "\n\
510 If the input is lower dimensional:\n\
511  - use 'QJ' to joggle the input and make it full dimensional\n\
512  - use 'Qbk:0Bk:0' to delete coordinate k from the input. You should\n\
513  pick the coordinate with the least range. The hull will have the\n\
514  correct topology.\n\
515  - determine the flat containing the points, rotate the points\n\
516  into a coordinate plane, and delete the other coordinates.\n\
517  - add one or more points to make the input full dimensional.\n\
518 ");
519  }
520 } /* printhelp_singular */
521 
522 /*-<a href="qh-globa.htm#TOC"
523  >-------------------------------</a><a name="user_memsizes">-</a>
524 
525  qh_user_memsizes()
526  allocate up to 10 additional, quick allocation sizes
527 
528  notes:
529  increase maximum number of allocations in qh_initqhull_mem()
530 */
531 void qh_user_memsizes(void) {
532 
533  /* qh_memsize(size); */
534 } /* user_memsizes */
535 
536 
qh_ERRinput
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#define qh_CPUclock
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Definition: user.c:325
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#define FORALLfacet_(facetlist)
Definition: poly.h:77
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#define qh_ALL
Definition: libqhull.h:180
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#define FOREACHfacet_(facets)
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Definition: stat.h:413
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qh_check_output
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Definition: poly2.c:302
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#define realT
Definition: user.h:154
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@ qh_IDunknown
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Definition: libqhull.h:375
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@ qh_PRINTfacets
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@ Zsetplane
Definition: stat.h:246
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Definition: io.c:1071
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Definition: user.h:420
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Definition: libqhull.c:60
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@ Zdistio
Definition: stat.h:106
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Definition: mem.c:288
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Definition: io.c:1690
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Definition: libqhull.h:396
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Definition: libqhull.h:950
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Definition: io.c:2809
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@ qh_PRINTEND
Definition: libqhull.h:172
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Definition: stat.c:674
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Definition: stat.h:250
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Definition: userprintf.c:42
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Definition: geom.h:51
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Definition: user.c:277
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Definition: mem.c:317
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Definition: global.c:534
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Definition: global.c:1827
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Definition: stat.c:316
facetT::vertices
setT * vertices
Definition: libqhull.h:295
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Definition: libqhull.h:195
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Definition: mem.c:431
True
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Definition: libqhull.h:129
qh_printbegin
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Definition: io.c:1309

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autogenerated on Fri Aug 2 2024 02:45:15