1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
522
523
524
525
526
527
528
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
545
546
547
548
549
550
551
552
553
554
555
556
557
558
559
560
561
562
563
564
565
566
567
568
569
570
571
572
573
574
575
576
577
578
579
580
581
582
583
584
585
586
587
588
589
590
591
592
593
594
595
596
597
598
599
600
601
602
603
604
605
606
607
608
609
610
611
612
613
614
615
616
617
618
619
620
621
622
623
624
625
626
627
628
629
630
631
632
633
634
635
636
637
638
639
640
641
642
643
644
645
646
647
648
649
650
651
652
653
654
655
656
657
658
659
660
661
662
663
664
665
666
667
|
<!DOCTYPE HTML PUBLIC "-//IETF//DTD HTML//EN">
<html>
<head>
<title>qvoronoi -- Voronoi diagram</title>
</head>
<body>
<!-- Navigation links -->
<a name="TOP"><b>Up</b></a><b>:</b>
<a href="http://www.qhull.org">Home page</a> for Qhull<br>
<b>Up:</b> <a href="index.htm#TOC">Qhull manual</a>: Table of Contents<br>
<b>To:</b> <a href="qh-quick.htm#programs">Programs</a>
• <a href="qh-quick.htm#options">Options</a>
• <a href="qh-opto.htm#output">Output</a>
• <a href="qh-optf.htm#format">Formats</a>
• <a href="qh-optg.htm#geomview">Geomview</a>
• <a href="qh-optp.htm#print">Print</a>
• <a href="qh-optq.htm#qhull">Qhull</a>
• <a href="qh-optc.htm#prec">Precision</a>
• <a href="qh-optt.htm#trace">Trace</a>
• <a href="../src/libqhull_r/index.htm">Functions</a><br>
<b>To:</b> <a href="#synopsis">sy</a>nopsis
• <a href="#input">in</a>put • <a href="#outputs">ou</a>tputs
• <a href="#controls">co</a>ntrols • <a href="#graphics">gr</a>aphics
• <a href="#notes">no</a>tes • <a href="#conventions">co</a>nventions
• <a href="#options">op</a>tions
<hr>
<!-- Main text of document -->
<h1><a
href="http://www.archinect.com/gallery/displayimage.php?pos=-4658"><img
src="normal_voronoi_knauss_oesterle.jpg" alt="[voronoi]" align="middle"
height="100"></a>qvoronoi -- Voronoi diagram</h1>
<p>The Voronoi diagram is the nearest-neighbor map for a set of
points. Each region contains those points that are nearer
one input site than any other input site. It has many useful properties and applications. See the
survey article by Aurenhammer [<a href="index.htm#aure91">'91</a>]
and the detailed introduction by O'Rourke [<a
href="index.htm#orou94">'94</a>]. The Voronoi diagram is the
dual of the <a href=qdelaun.htm>Delaunay triangulation</a>. </p>
<blockquote>
<dl>
<dt><b>Example:</b> rbox 10 D3 | qvoronoi <a href="qh-opto.htm#s">s</a>
<a href="qh-opto.htm#o">o</a> <a href="qh-optt.htm#TO">TO
result</a></dt>
<dd>Compute the 3-d Voronoi diagram of 10 random points. Write a
summary to the console and the Voronoi vertices and
regions to 'result'. The first vertex of the result
indicates unbounded regions.</dd>
<dt> </dt>
<dt><b>Example:</b> rbox r y c G0.1 D2 | qvoronoi
<a href="qh-opto.htm#s">s</a>
<a href="qh-opto.htm#o">o</a> <a href="qh-optt.htm#TO">TO
result</a></dt>
<dd>Compute the 2-d Voronoi diagram of a triangle and a small
square. Write a
summary to the console and Voronoi vertices and regions
to 'result'. Report a single Voronoi vertex for
cocircular input sites. The first vertex of the result
indicates unbounded regions. The origin is the Voronoi
vertex for the square.</dd>
<dt> </dt>
<dt><b>Example:</b> rbox r y c G0.1 D2 | qvoronoi <a href="qh-optf.htm#Fv2">Fv</a>
<a href="qh-optt.htm#TO">TO result</a></dt>
<dd>Compute the 2-d Voronoi diagram of a triangle and a small
square. Write a
summary to the console and the Voronoi ridges to
'result'. Each ridge is the perpendicular bisector of a
pair of input sites. Vertex "0" indicates
unbounded ridges. Vertex "8" is the Voronoi
vertex for the square.</dd>
<dt> </dt>
<dt><b>Example:</b> rbox r y c G0.1 D2 | qvoronoi <a href="qh-optf.htm#Fi2">Fi</a></dt>
<dd>Print the bounded, separating hyperplanes for the 2-d Voronoi diagram of a
triangle and a small
square. Note the four hyperplanes (i.e., lines) for Voronoi vertex
"8". It is at the origin.
</dd>
</dl>
</blockquote>
<p>Qhull computes the Voronoi diagram via the <a href="qdelaun.htm">Delaunay
triangulation</a>. Each Voronoi
vertex is the circumcenter of a facet of the Delaunay
triangulation. Each Voronoi region corresponds to a vertex (i.e., input site) of the
Delaunay triangulation. </p>
<p>Qhull outputs the Voronoi vertices for each Voronoi region. With
option '<a href="qh-optf.htm#Fv2">Fv</a>',
it lists all ridges of the Voronoi diagram with the corresponding
pairs of input sites. With
options '<a href="qh-optf.htm#Fi2">Fi</a>' and '<a href="qh-optf.htm#Fo2">Fo</a>',
it lists the bounded and unbounded separating hyperplanes.
You can also output a single Voronoi region
for further processing [see <a href="#graphics">graphics</a>].</p>
<p>Use option '<a href="qh-optq.htm#Qz">Qz</a>' if the input is circular, cospherical, or
nearly so. It improves precision by adding a point "at infinity," above the corresponding paraboloid.
<p>See <a href="http://www.qhull.org/html/qh-faq.htm#TOC">Qhull FAQ</a> - Delaunay and
Voronoi diagram questions.</p>
<p>The 'qvonoroi' program is equivalent to
'<a href=qhull.htm#outputs>qhull v</a> <a href=qh-optq.htm#Qbb>Qbb</a>' in 2-d to 3-d, and
'<a href=qhull.htm#outputs>qhull v</a> <a href=qh-optq.htm#Qbb>Qbb</a> <a href=qh-optq.htm#Qx>Qx</a>'
in 4-d and higher. It disables the following Qhull
<a href=qh-quick.htm#options>options</a>: <i>d n v Qbb QbB Qf Qg Qm
Qr QR Qv Qx Qz TR E V Fa FA FC FD FS Ft FV Gt Q0,etc</i>.
<p><b>Copyright © 1995-2015 C.B. Barber</b></p>
<p>Voronoi image by KOOK Architecture, Silvan Oesterle and Michael Knauss.
<hr>
<h3><a href="#TOP">»</a><a name="synopsis">qvoronoi synopsis</a></h3>
<pre>
qvoronoi- compute the Voronoi diagram.
input (stdin): dimension, number of points, point coordinates
comments start with a non-numeric character
options (qh-voron.htm):
Qu - compute furthest-site Voronoi diagram
Tv - verify result: structure, convexity, and in-circle test
. - concise list of all options
- - one-line description of all options
output options (subset):
s - summary of results (default)
p - Voronoi vertices
o - OFF file format (dim, Voronoi vertices, and Voronoi regions)
FN - count and Voronoi vertices for each Voronoi region
Fv - Voronoi diagram as Voronoi vertices between adjacent input sites
Fi - separating hyperplanes for bounded regions, 'Fo' for unbounded
G - Geomview output (2-d only)
QVn - Voronoi vertices for input point n, -n if not
TO file- output results to file, may be enclosed in single quotes
examples:
rbox c P0 D2 | qvoronoi s o rbox c P0 D2 | qvoronoi Fi
rbox c P0 D2 | qvoronoi Fo rbox c P0 D2 | qvoronoi Fv
rbox c P0 D2 | qvoronoi s Qu Fv rbox c P0 D2 | qvoronoi Qu Fo
rbox c G1 d D2 | qvoronoi s p rbox c P0 D2 | qvoronoi s Fv QV0
</pre>
<h3><a href="#TOP">»</a><a name="input">qvoronoi input</a></h3>
<blockquote>
The input data on <tt>stdin</tt> consists of:
<ul>
<li>dimension
<li>number of points</li>
<li>point coordinates</li>
</ul>
<p>Use I/O redirection (e.g., qvoronoi < data.txt), a pipe (e.g., rbox 10 | qvoronoi),
or the '<a href=qh-optt.htm#TI>TI</a>' option (e.g., qvoronoi TI data.txt).
<p>For example, this is four cocircular points inside a square. Their Voronoi
diagram has nine vertices and eight regions. Notice the Voronoi vertex
at the origin, and the Voronoi vertices (on each axis) for the four
sides of the square.
<p>
<blockquote>
<tt>rbox s 4 W0 c G1 D2 > data</tt>
<blockquote><pre>
2 RBOX s 4 W0 c D2
8
-0.4941988586954018 -0.07594397977563715
-0.06448037284989526 0.4958248496365813
0.4911154367094632 0.09383830681375946
-0.348353580869097 -0.3586778257652367
-1 -1
-1 1
1 -1
1 1
</pre></blockquote>
<p><tt>qvoronoi s p < data</tt>
<blockquote><pre>
Voronoi diagram by the convex hull of 8 points in 3-d:
Number of Voronoi regions: 8
Number of Voronoi vertices: 9
Number of non-simplicial Voronoi vertices: 1
Statistics for: RBOX s 4 W0 c D2 | QVORONOI s p
Number of points processed: 8
Number of hyperplanes created: 18
Number of facets in hull: 10
Number of distance tests for qhull: 33
Number of merged facets: 2
Number of distance tests for merging: 102
CPU seconds to compute hull (after input): 0.094
2
9
4.217546450968612e-17 1.735507986399734
-8.402566836762659e-17 -1.364368854147395
0.3447488772716865 -0.6395484723719818
1.719446929853986 2.136555906154247e-17
0.4967882915039657 0.68662371396699
-1.729928876283549 1.343733067524222e-17
-0.8906163241424728 -0.4594150543829102
-0.6656840313875723 0.5003013793414868
-7.318364664277155e-19 -1.188217818408333e-16
</pre></blockquote>
</blockquote>
</blockquote>
<h3><a href="#TOP">»</a> <a name="outputs">qvoronoi
outputs</a></h3>
<blockquote>
<p>These options control the output of Voronoi diagrams.</p>
<blockquote>
<dl compact>
<dt> </dt>
<dd><b>Voronoi vertices</b></dd>
<dt><a href="qh-opto.htm#p">p</a></dt>
<dd>print the coordinates of the Voronoi vertices. The first line
is the dimension. The second line is the number of vertices. Each
remaining line is a Voronoi vertex.</dd>
<dt><a href="qh-optf.htm#Fn">Fn</a></dt>
<dd>list the neighboring Voronoi vertices for each Voronoi
vertex. The first line is the number of Voronoi vertices. Each
remaining line starts with the number of neighboring vertices.
Negative vertices (e.g., <em>-1</em>) indicate vertices
outside of the Voronoi diagram.
In the circle-in-box example, the
Voronoi vertex at the origin has four neighbors.</dd>
<dt><a href="qh-optf.htm#FN">FN</a></dt>
<dd>list the Voronoi vertices for each Voronoi region. The first line is
the number of Voronoi regions. Each remaining line starts with the
number of Voronoi vertices. Negative indices (e.g., <em>-1</em>) indicate vertices
outside of the Voronoi diagram.
In the circle-in-box example, the four bounded regions are defined by four
Voronoi vertices.</dd>
<dt> </dt>
<dt> </dt>
<dd><b>Voronoi regions</b></dd>
<dt><a href="qh-opto.htm#o">o</a></dt>
<dd>print the Voronoi regions in OFF format. The first line is the
dimension. The second line is the number of vertices, the number
of input sites, and "1". The third line represents the vertex-at-infinity.
Its coordinates are "-10.101". The next lines are the coordinates
of the Voronoi vertices. Each remaining line starts with the number
of Voronoi vertices in a Voronoi region. In 2-d, the vertices are
listed in adjacency order (unoriented). In 3-d and higher, the
vertices are listed in numeric order. In the circle-in-square
example, each bounded region includes the Voronoi vertex at
the origin. Lines consisting of <em>0</em> indicate
coplanar input sites or '<a href="qh-optq.htm#Qz">Qz</a>'. </dd>
<dt><a href="qh-optf.htm#Fi2">Fi</a></dt>
<dd>print separating hyperplanes for inner, bounded Voronoi
regions. The first number is the number of separating
hyperplanes. Each remaining line starts with <i>3+dim</i>. The
next two numbers are adjacent input sites. The next <i>dim</i>
numbers are the coefficients of the separating hyperplane. The
last number is its offset. Use '<a href="qh-optt.htm#Tv">Tv</a>' to verify that the
hyperplanes are perpendicular bisectors. It will list relevant
statistics to stderr. </dd>
<dt><a href="qh-optf.htm#Fo2">Fo</a></dt>
<dd>print separating hyperplanes for outer, unbounded Voronoi
regions. The first number is the number of separating
hyperplanes. Each remaining line starts with <i>3+dim</i>. The
next two numbers are adjacent input sites on the convex hull. The
next <i>dim</i>
numbers are the coefficients of the separating hyperplane. The
last number is its offset. Use '<a href="qh-optt.htm#Tv">Tv</a>' to verify that the
hyperplanes are perpendicular bisectors. It will list relevant
statistics to stderr,</dd>
<dt> </dt>
<dt> </dt>
<dd><b>Input sites</b></dd>
<dt><a href="qh-optf.htm#Fv2">Fv</a></dt>
<dd>list ridges of Voronoi vertices for pairs of input sites. The
first line is the number of ridges. Each remaining line starts with
two plus the number of Voronoi vertices in the ridge. The next
two numbers are two adjacent input sites. The remaining numbers list
the Voronoi vertices. As with option 'o', a <em>0</em> indicates
the vertex-at-infinity
and an unbounded, separating hyperplane.
The perpendicular bisector (separating hyperplane)
of the input sites is a flat through these vertices.
In the circle-in-square example, the ridge for each edge of the square
is unbounded.</dd>
<dt><a href="qh-optf.htm#Fc">Fc</a></dt>
<dd>list coincident input sites for each Voronoi vertex.
The first line is the number of vertices. The remaining lines start with
the number of coincident sites and deleted vertices. Deleted vertices
indicate highly degenerate input (see'<a href="qh-optf.htm#Fs">Fs</a>').
A coincident site is assigned to one Voronoi
vertex. Do not use '<a href="qh-optq.htm#QJn">QJ</a>' with 'Fc'; the joggle will separate
coincident sites.</dd>
<dt><a href="qh-optf.htm#FP">FP</a></dt>
<dd>print coincident input sites with distance to
nearest site (i.e., vertex). The first line is the
number of coincident sites. Each remaining line starts with the point ID of
an input site, followed by the point ID of a coincident point, its vertex, and distance.
Includes deleted vertices which
indicate highly degenerate input (see'<a href="qh-optf.htm#Fs">Fs</a>').
Do not use '<a href="qh-optq.htm#QJn">QJ</a>' with 'FP'; the joggle will separate
coincident sites.</dd>
<dt> </dt>
<dt> </dt>
<dd><b>General</b></dd>
<dt><a href="qh-opto.htm#s">s</a></dt>
<dd>print summary of the Voronoi diagram. Use '<a
href="qh-optf.htm#Fs">Fs</a>' for numeric data.</dd>
<dt><a href="qh-opto.htm#i">i</a></dt>
<dd>list input sites for each <a href=qdelaun.htm>Delaunay region</a>. Use option '<a href="qh-optp.htm#Pp">Pp</a>'
to avoid the warning. The first line is the number of regions. The
remaining lines list the input sites for each region. The regions are
oriented. In the circle-in-square example, the cocircular region has four
edges. In 3-d and higher, report cospherical sites by adding extra points.
</dd>
<dt><a href="qh-optg.htm#G">G</a></dt>
<dd>Geomview output for 2-d Voronoi diagrams.</dd>
</dl>
</blockquote>
</blockquote>
<h3><a href="#TOP">»</a> <a name="controls">qvoronoi
controls</a></h3>
<blockquote>
<p>These options provide additional control:</p>
<blockquote>
<dl compact>
<dt><a href="qh-optq.htm#Qu">Qu</a></dt>
<dd>compute the <a href="qvoron_f.htm">furthest-site Voronoi diagram</a>.</dd>
<dt><a href="qh-optq.htm#QVn">QVn</a></dt>
<dd>select Voronoi vertices for input site <em>n</em> </dd>
<dt><a href="qh-optq.htm#Qz">Qz</a></dt>
<dd>add a point above the paraboloid to reduce precision
errors. Use it for nearly cocircular/cospherical input
(e.g., 'rbox c | qvoronoi Qz').</dd>
<dt><a href="qh-optt.htm#Tv">Tv</a></dt>
<dd>verify result</dd>
<dt><a href="qh-optt.htm#TO">TI file</a></dt>
<dd>input data from file. The filename may not use spaces or quotes.</dd>
<dt><a href="qh-optt.htm#TO">TO file</a></dt>
<dd>output results to file. Use single quotes if the filename
contains spaces (e.g., <tt>TO 'file with spaces.txt'</tt></dd>
<dt><a href="qh-optt.htm#TFn">TFn</a></dt>
<dd>report progress after constructing <em>n</em> facets</dd>
<dt><a href="qh-optp.htm#PDk">PDk:1</a></dt>
<dd>include upper and lower facets in the output. Set <em>k</em>
to the last dimension (e.g., 'PD2:1' for 2-d inputs). </dd>
<dt><a href="qh-opto.htm#f">f </a></dt>
<dd>facet dump. Print the data structure for each facet (i.e.,
Voronoi vertex).</dd>
</dl>
</blockquote>
</blockquote>
<h3><a href="#TOP">»</a> <a name="graphics">qvoronoi
graphics</a></h3>
<blockquote>
<p>In 2-d, Geomview output ('<a href="qh-optg.htm#G">G</a>')
displays a Voronoi diagram with extra edges to close the
unbounded Voronoi regions. To view the unbounded rays, enclose
the input points in a square.</p>
<p>You can also view <i>individual</i> Voronoi regions in 3-d. To
view the Voronoi region for site 3 in Geomview, execute</p>
<blockquote>
<p>qvoronoi <data <a href="qh-optq.htm#QVn">QV3</a> <a
href="qh-opto.htm#p">p</a> | qconvex s G >output</p>
</blockquote>
<p>The <tt>qvoronoi</tt> command returns the Voronoi vertices
for input site 3. The <tt>qconvex</tt> command computes their convex hull.
This is the Voronoi region for input site 3. Its
hyperplane normals (qconvex 'n') are the same as the separating hyperplanes
from options '<a href="qh-optf.htm#Fi">Fi</a>'
and '<a href="qh-optf.htm#Fo">Fo</a>' (up to roundoff error).
<p>See the <a href="qh-eg.htm#delaunay">Delaunay and Voronoi
examples</a> for 2-d and 3-d examples. Turn off normalization (on
Geomview's 'obscure' menu) when comparing the Voronoi diagram
with the corresponding Delaunay triangulation. </p>
</blockquote>
<h3><a href="#TOP">»</a><a name="notes">qvoronoi
notes</a></h3>
<blockquote>
<p>You can simplify the Voronoi diagram by enclosing the input
sites in a large square or cube. This is particularly recommended
for cocircular or cospherical input data.</p>
<p>See <a href="#graphics">Voronoi graphics</a> for computing
the convex hull of a Voronoi region. </p>
<p>Voronoi diagrams do not include facets that are
coplanar with the convex hull of the input sites. A facet is
coplanar if the last coefficient of its normal is
nearly zero (see <a href="../src/user.h#ZEROdelaunay">qh_ZEROdelaunay</a>).
<p>Unbounded regions can be confusing. For example, '<tt>rbox c |
qvoronoi Qz o</tt>' produces the Voronoi regions for the vertices
of a cube centered at the origin. All regions are unbounded. The
output is </p>
<blockquote>
<pre>3
2 9 1
-10.101 -10.101 -10.101
0 0 0
2 0 1
2 0 1
2 0 1
2 0 1
2 0 1
2 0 1
2 0 1
2 0 1
0
</pre>
</blockquote>
<p>The first line is the dimension. The second line is the number
of vertices and the number of regions. There is one region per
input point plus a region for the point-at-infinity added by
option '<a href="qh-optq.htm#Qz">Qz</a>'. The next two lines
lists the Voronoi vertices. The first vertex is the infinity
vertex. It is indicate by the coordinates <em>-10.101</em>. The
second vertex is the origin. The next nine lines list the
regions. Each region lists two vertices -- the infinity vertex
and the origin. The last line is "0" because no region
is associated with the point-at-infinity. A "0" would
also be listed for nearly incident input sites. </p>
<p>To use option '<a href="qh-optf.htm#Fv">Fv</a>', add an
interior point. For example, </p>
<blockquote>
<pre>
rbox c P0 | qvoronoi Fv
20
5 0 7 1 3 5
5 0 3 1 4 5
5 0 5 1 2 3
5 0 1 1 2 4
5 0 6 2 3 6
5 0 2 2 4 6
5 0 4 4 5 6
5 0 8 5 3 6
5 1 2 0 2 4
5 1 3 0 1 4
5 1 5 0 1 2
5 2 4 0 4 6
5 2 6 0 2 6
5 3 4 0 4 5
5 3 7 0 1 5
5 4 8 0 6 5
5 5 6 0 2 3
5 5 7 0 1 3
5 6 8 0 6 3
5 7 8 0 3 5
</pre>
</blockquote>
<p>The output consists of 20 ridges and each ridge lists a pair
of input sites and a triplet of Voronoi vertices. The first eight
ridges connect the origin ('P0'). The remainder list the edges of
the cube. Each edge generates an unbounded ray through the
midpoint. The corresponding separating planes ('Fo') follow each
pair of coordinate axes. </p>
<p>Options '<a href="qh-optq.htm#Qt">Qt</a>' (triangulated output)
and '<a href="qh-optq.htm#QJn">QJ</a>' (joggled input) are deprecated. They may produce
unexpected results. If you use these options, cocircular and cospherical input sites will
produce duplicate or nearly duplicate Voronoi vertices. See also <a
href="qh-impre.htm#joggle">Merged facets or joggled input</a>. </p>
</blockquote>
<h3><a href="#TOP">»</a><a name="conventions">qvoronoi conventions</a></h3>
<blockquote>
<p>The following terminology is used for Voronoi diagrams in
Qhull. The underlying structure is a Delaunay triangulation from
a convex hull in one higher dimension. Facets of the Delaunay
triangulation correspond to vertices of the Voronoi diagram.
Vertices of the Delaunay triangulation correspond to input sites.
They also correspond to regions of the Voronoi diagram. See <a
href="qconvex.htm#conventions">convex hull conventions</a>, <a
href="qdelaun.htm#conventions">Delaunay conventions</a>, and
<a href="index.htm#structure">Qhull's data structures</a>.</p>
<blockquote>
<ul>
<li><em>input site</em> - a point in the input (one dimension
lower than a point on the convex hull)</li>
<li><em>point</em> - a point has <i>d+1</i> coordinates. The
last coordinate is the sum of the squares of the input
site's coordinates</li>
<li><em>coplanar point</em> - a <em>nearly incident</em>
input site</li>
<li><em>vertex</em> - a point on the paraboloid. It
corresponds to a unique input site. </li>
<li><em>point-at-infinity</em> - a point added above the
paraboloid by option '<a href="qh-optq.htm#Qz">Qz</a>'</li>
<li><em>Delaunay facet</em> - a lower facet of the
paraboloid. The last coefficient of its normal is
clearly negative.</li>
<li><em>Voronoi vertex</em> - the circumcenter of a Delaunay
facet</li>
<li><em>Voronoi region</em> - the Voronoi vertices for an
input site. The region of Euclidean space nearest to an
input site.</li>
<li><em>Voronoi diagram</em> - the graph of the Voronoi
regions. It includes the ridges (i.e., edges) between the
regions.</li>
<li><em>vertex-at-infinity</em> - the Voronoi vertex that
indicates unbounded Voronoi regions in '<a
href="qh-opto.htm#o">o</a>' output format. Its
coordinates are <em>-10.101</em>.</li>
<li><em>good facet</em> - a Voronoi vertex with optional
restrictions by '<a href="qh-optq.htm#QVn">QVn</a>', etc.</li>
</ul>
</blockquote>
</blockquote>
<h3><a href="#TOP">»</a><a name="options">qvoronoi options</a></h3>
<pre>
qvoronoi- compute the Voronoi diagram
http://www.qhull.org
input (stdin):
first lines: dimension and number of points (or vice-versa).
other lines: point coordinates, best if one point per line
comments: start with a non-numeric character
options:
Qu - compute furthest-site Voronoi diagram
Qhull control options:
QJn - randomly joggle input in range [-n,n]
Qs - search all points for the initial simplex
Qz - add point-at-infinity to Voronoi diagram
QGn - Voronoi vertices if visible from point n, -n if not
QVn - Voronoi vertices for input point n, -n if not
Trace options:
T4 - trace at level n, 4=all, 5=mem/gauss, -1= events
Tc - check frequently during execution
Ts - statistics
Tv - verify result: structure, convexity, and in-circle test
Tz - send all output to stdout
TFn - report summary when n or more facets created
TI file - input data from file, no spaces or single quotes
TO file - output results to file, may be enclosed in single quotes
TPn - turn on tracing when point n added to hull
TMn - turn on tracing at merge n
TWn - trace merge facets when width > n
TVn - stop qhull after adding point n, -n for before (see TCn)
TCn - stop qhull after building cone for point n (see TVn)
Precision options:
Cn - radius of centrum (roundoff added). Merge facets if non-convex
An - cosine of maximum angle. Merge facets if cosine > n or non-convex
C-0 roundoff, A-0.99/C-0.01 pre-merge, A0.99/C0.01 post-merge
Rn - randomly perturb computations by a factor of [1-n,1+n]
Wn - min facet width for non-coincident point (before roundoff)
Output formats (may be combined; if none, produces a summary to stdout):
s - summary to stderr
p - Voronoi vertices
o - OFF format (dim, Voronoi vertices, and Voronoi regions)
i - Delaunay regions (use 'Pp' to avoid warning)
f - facet dump
More formats:
Fc - count plus coincident points (by Voronoi vertex)
Fd - use cdd format for input (homogeneous with offset first)
FD - use cdd format for output (offset first)
FF - facet dump without ridges
Fi - separating hyperplanes for bounded Voronoi regions
FI - ID for each Voronoi vertex
Fm - merge count for each Voronoi vertex (511 max)
Fn - count plus neighboring Voronoi vertices for each Voronoi vertex
FN - count and Voronoi vertices for each Voronoi region
Fo - separating hyperplanes for unbounded Voronoi regions
FO - options and precision constants
FP - nearest point and distance for each coincident point
FQ - command used for qvoronoi
Fs - summary: #int (8), dimension, #points, tot vertices, tot facets,
for output: #Voronoi regions, #Voronoi vertices,
#coincident points, #non-simplicial regions
#real (2), max outer plane and min vertex
Fv - Voronoi diagram as Voronoi vertices between adjacent input sites
Fx - extreme points of Delaunay triangulation (on convex hull)
Geomview options (2-d only)
Ga - all points as dots
Gp - coplanar points and vertices as radii
Gv - vertices as spheres
Gi - inner planes only
Gn - no planes
Go - outer planes only
Gc - centrums
Gh - hyperplane intersections
Gr - ridges
GDn - drop dimension n in 3-d and 4-d output
Print options:
PAn - keep n largest Voronoi vertices by 'area'
Pdk:n - drop facet if normal[k] <= n (default 0.0)
PDk:n - drop facet if normal[k] >= n
Pg - print good Voronoi vertices (needs 'QGn' or 'QVn')
PFn - keep Voronoi vertices whose 'area' is at least n
PG - print neighbors of good Voronoi vertices
PMn - keep n Voronoi vertices with most merges
Po - force output. If error, output neighborhood of facet
Pp - do not report precision problems
. - list of all options
- - one line descriptions of all options
</pre>
<!-- Navigation links -->
<hr>
<p><b>Up:</b> <a href="http://www.qhull.org">Home page</a> for Qhull<br>
<b>Up:</b> <a href="index.htm#TOC">Qhull manual</a>: Table of Contents<br>
<b>To:</b> <a href="qh-quick.htm#programs">Programs</a>
• <a href="qh-quick.htm#options">Options</a>
• <a href="qh-opto.htm#output">Output</a>
• <a href="qh-optf.htm#format">Formats</a>
• <a href="qh-optg.htm#geomview">Geomview</a>
• <a href="qh-optp.htm#print">Print</a>
• <a href="qh-optq.htm#qhull">Qhull</a>
• <a href="qh-optc.htm#prec">Precision</a>
• <a href="qh-optt.htm#trace">Trace</a>
• <a href="../src/libqhull_r/index.htm">Functions</a><br>
<b>To:</b> <a href="#synopsis">sy</a>nopsis
• <a href="#input">in</a>put • <a href="#outputs">ou</a>tputs
• <a href="#controls">co</a>ntrols • <a href="#graphics">gr</a>aphics
• <a href="#notes">no</a>tes • <a href="#conventions">co</a>nventions
• <a href="#options">op</a>tions
<!-- GC common information -->
<hr>
<p><a href="http://www.geom.uiuc.edu/"><img src="qh--geom.gif"
align="middle" width="40" height="40"></a><i>The Geometry Center
Home Page </i></p>
<p>Comments to: <a href=mailto:qhull@qhull.org>qhull@qhull.org</a>
</a><br>
Created: Sept. 25, 1995 --- <!-- hhmts start --> Last modified: see top <!-- hhmts end --> </p>
</body>
</html>
|
