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
668
669
670
671
672
673
674
675
676
677
678
679
680
681
682
683
684
685
686
687
688
689
690
691
692
693
694
695
696
697
698
699
700
701
702
703
704
705
706
707
708
709
710
711
712
713
714
715
716
717
718
719
720
721
722
723
724
725
726
727
728
729
730
731
732
733
734
735
736
737
738
739
740
741
742
743
744
745
746
747
748
749
750
751
752
753
754
755
756
757
758
759
760
761
762
763
764
765
766
767
768
769
770
771
772
773
774
775
776
777
778
779
780
781
782
783
784
785
786
787
788
789
790
791
792
793
794
795
796
797
798
799
800
801
802
803
804
805
806
807
808
809
810
811
812
813
814
815
816
817
818
819
820
821
822
823
824
825
826
|
<!DOCTYPE HTML PUBLIC "-//IETF//DTD HTML//EN">
<html>
<head>
<title>Imprecision in Qhull</title>
</head>
<body>
<!-- Navigation links -->
<p><a name="TOP"><b>Up:</b></a> <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="#TOC">Qhull imprecision</a>: Table of Contents
(please wait while loading)
<hr>
<!-- Main text of document -->
<h1><a
href="http://www.geom.uiuc.edu/graphics/pix/Special_Topics/Computational_Geometry/4dcube.html"><img
src="qh--4d.gif" alt="[4-d cube]" align="middle" width="100"
height="100"></a> Imprecision in Qhull</h1>
<p>This section of the Qhull manual discusses the problems caused
by coplanar points and why Qhull uses options '<a
href="qh-optc.htm#C0">C-0</a>' or '<a href="qh-optq.htm#Qx">Qx</a>'
by default. If you ignore precision issues with option '<a
href="qh-optq.htm#Q0">Q0</a>', the output from Qhull can be
arbitrarily bad. Qhull avoids precision problems if you merge facets (the default) or joggle
the input ('<a
href="qh-optq.htm#QJn">QJ</a>'). </p>
<p>Use option '<a href="qh-optt.htm#Tv">Tv</a>' to verify the
output from Qhull. It verifies that adjacent facets are clearly
convex. It verifies that all points are on or below all facets. </p>
<p>Qhull automatically tests for convexity if it detects
precision errors while constructing the hull. </p>
<p><b>Copyright © 1995-2015 C.B. Barber</b></p>
<hr>
<h2><a href="#TOP">»</a><a name="TOC">Qhull
imprecision: Table of Contents </a></h2>
<ul>
<li><a href="#prec">Precision problems</a></li>
<li><a href="#joggle">Merged facets or joggled input</a></li>
<li><a href="#delaunay">Delaunay triangulations</a></li>
<li><a href="#halfspace">Halfspace intersection/a></li>
<li><a href="#imprecise">Merged facets</a></li>
<li><a href="#how">How Qhull merges facets</a></li>
<li><a href="#limit">Limitations of merged facets</a></li>
<li><a href="#injoggle">Joggled input</a></li>
<li><a href="#exact">Exact arithmetic</a></li>
<li><a href="#approximate">Approximating a convex hull</a></li>
</ul>
<hr>
<h2><a href="#TOC">»</a><a name="prec">Precision problems</a></h2>
<p>Since Qhull uses floating point arithmetic, roundoff error
occurs with each calculation. This causes problems for
geometric algorithms. Other floating point codes for convex
hulls, Delaunay triangulations, and Voronoi diagrams also suffer
from these problems. Qhull handles most of them.</p>
<p>There are several kinds of precision errors:</p>
<ul>
<li>Representation error occurs when there are not enough
digits to represent a number, e.g., 1/3.</li>
<li>Measurement error occurs when the input coordinates are
from measurements. </li>
<li>Roundoff error occurs when a calculation is rounded to a
fixed number of digits, e.g., a floating point
calculation.</li>
<li>Approximation error occurs when the user wants an
approximate result because the exact result contains too
much detail.</li>
</ul>
<p>Under imprecision, calculations may return erroneous results.
For example, roundoff error can turn a small, positive number
into a small, negative number. See Milenkovic [<a
href="index.htm#mile93">'93</a>] for a discussion of <em>strict
robust geometry</em>. Qhull does not meet Milenkovic's criterion
for accuracy. Qhull's error bound is empirical instead of
theoretical.</p>
<p>Qhull 1.0 checked for precision errors but did not handle
them. The output could contain concave facets, facets with
inverted orientation ("flipped" facets), more than two
facets adjacent to a ridge, and two facets with exactly the same
set of vertices.</p>
<p>Qhull 2.4 and later automatically handles errors due to
machine round-off. Option '<a href="qh-optc.htm#C0">C-0</a>' or '<a
href="qh-optq.htm#Qx">Qx</a>' is set by default. In 5-d and
higher, the output is clearly convex but an input point could be
outside of the hull. This may be corrected by using option '<a
href="qh-optc.htm#C0">C-0</a>', but then the output may contain
wide facets.</p>
<p>Qhull 2.5 and later provides option '<a href="qh-optq.htm#QJn">QJ</a>'
to joggled input. Each input coordinate is modified by a
small, random quantity. If a precision error occurs, a larger
modification is tried. When no precision errors occur, Qhull is
done. </p>
<p>Qhull 3.1 and later provides option '<a href="qh-optq.htm#Qt">Qt</a>'
for triangulated output. This removes the need for
joggled input ('<a href="qh-optq.htm#QJn">QJ</a>').
Non-simplicial facets are triangulated.
The facets may have zero area.
Triangulated output is particularly useful for Delaunay triangulations.</p>
<p>By handling round-off errors, Qhull can provide a variety of
output formats. For example, it can return the halfspace that
defines each facet ('<a href="qh-opto.htm#n">n</a>'). The
halfspaces include roundoff error. If the halfspaces were exact,
their intersection would return the original extreme points. With
imprecise halfspaces and exact arithmetic, nearly incident points
may be returned for an original extreme point. By handling
roundoff error, Qhull returns one intersection point for each of
the original extreme points. Qhull may split or merge an extreme
point, but this appears to be unlikely.</p>
<p>The following pipe implements the identity function for
extreme points (with roundoff):
<blockquote>
qconvex <a href="qh-optf.htm#FV">FV</a> <a
href="qh-opto.htm#n">n</a> | qhalf <a href="qh-optf.htm#Fp">Fp</a>
</blockquote>
<p>Bernd Gartner published his
<a href=http://www.inf.ethz.ch/personal/gaertner/miniball.html>Miniball</a>
algorithm ["Fast and robust smallest enclosing balls", <i>Algorithms - ESA '99</i>, LNCS 1643].
It uses floating point arithmetic and a carefully designed primitive operation.
It is practical to 20-D or higher, and identifies at least two points on the
convex hull of the input set. Like Qhull, it is an incremental algorithm that
processes points furthest from the intermediate result and ignores
points that are close to the intermediate result.
<h2><a href="#TOC">»</a><a name="joggle">Merged facets or joggled input</a></h2>
<p>This section discusses the choice between merged facets and joggled input.
By default, Qhull uses merged facets to handle
precision problems. With option '<a href="qh-optq.htm#QJn">QJ</a>',
the input is joggled. See <a href="qh-eg.htm#joggle">examples</a>
of joggled input and triangulated output.
<ul>
<li>Use merged facets (the default)
when you want non-simplicial output (e.g., the faces of a cube).
<li>Use merged facets and triangulated output ('<a href="qh-optq.htm#Qt">Qt</a>') when
you want simplicial output and coplanar facets (e.g., triangles for a Delaunay triangulation).
<li>Use joggled input ('<a href="qh-optq.htm#QJn">QJ</a>') when you need clearly-convex,
simplicial output.
</ul>
<p>The choice between merged facets and joggled input depends on
the application. Both run about the same speed. Joggled input may
be faster if the initial joggle is sufficiently large to avoid
precision errors.
<p>Most applications should used merged facets
with triangulated output. </p>
<p>Use merged facets (the
default, '<a href="qh-optc.htm#C0">C-0</a>')
or triangulated output ('<a href="qh-optq.htm#Qt">Qt</a>') if </p>
<ul>
<li>Your application supports non-simplicial facets, or
it allows degenerate, simplicial facets (option '<a href="qh-optq.htm#Qt">Qt</a>'). </li>
<li>You do not want the input modified. </li>
<li>You want to set additional options for approximating the
hull. </li>
<li>You use single precision arithmetic (<a href="../src/libqhull/user.h#realT">realT</a>).
</li>
</ul>
<p>Use joggled input ('<a href="qh-optq.htm#QJn">QJ</a>') if </p>
<ul>
<li>Your application needs clearly convex, simplicial output</li>
<li>Your application supports perturbed input points and narrow triangles.</li>
<li>Seven significant digits is sufficient accuracy.</li>
</ul>
<p>You may use both techniques or combine joggle with
post-merging ('<a href="qh-optc.htm#Cn2">Cn</a>'). </p>
<p>Other researchers have used techniques similar to joggled
input. Sullivan and Beichel [ref?] randomly perturb the input
before computing the Delaunay triangulation. Corkum and Wyllie
[news://comp.graphics, 1990] randomly rotate a polytope before
testing point inclusion. Edelsbrunner and Mucke [Symp. Comp.
Geo., 1988] and Yap [J. Comp. Sys. Sci., 1990] symbolically
perturb the input to remove singularities. </p>
<p>Merged facets ('<a href="qh-optc.htm#C0">C-0</a>') handles
precision problems directly. If a precision problem occurs, Qhull
merges one of the offending facets into one of its neighbors.
Since all precision problems in Qhull are associated with one or
more facets, Qhull will either fix the problem or attempt to merge the
last remaining facets. </p>
<h2><a href="#TOC">»</a><a name="delaunay">Delaunay
triangulations </a></h2>
<p>Programs that use Delaunay triangulations traditionally assume
a triangulated input. By default, <a href=qdelaun.htm>qdelaunay</a>
merges regions with cocircular or cospherical input sites.
If you want a simplicial triangulation
use triangulated output ('<a href="qh-optq.htm#Qt">Qt</a>') or joggled
input ('<a href="qh-optq.htm#QJn">QJ</a>').
<p>For Delaunay triangulations, triangulated
output should produce good results. All points are within roundoff error of
a paraboloid. If two points are nearly incident, one will be a
coplanar point. So all points are clearly separated and convex.
If qhull reports deleted vertices, the triangulation
may contain serious precision faults. Deleted vertices are reported
in the summary ('<a href="qh-opto.htm#s">s</a>', '<a href="qh-optf.htm#Fs">Fs</a>'</p>
<p>You should use option '<a href="qh-optq.htm#Qbb">Qbb</a>' with Delaunay
triangulations. It scales the last coordinate and may reduce
roundoff error. It is automatically set for <a href=qdelaun.htm>qdelaunay</a>,
<a href=qvoronoi.htm>qvoronoi</a>, and option '<a
href="qh-optq.htm#QJn">QJ</a>'.</p>
<p>Edelsbrunner, H, <i>Geometry and Topology for Mesh Generation</i>, Cambridge University Press, 2001.
Good mathematical treatise on Delaunay triangulation and mesh generation for 2-d
and 3-d surfaces. The chapter on surface simplification is
particularly interesting. It is similar to facet merging in Qhull.
<p>Veron and Leon published an algorithm for shape preserving polyhedral
simplification with bounded error [<i>Computers and Graphics</i>, 22.5:565-585, 1998].
It remove nodes using front propagation and multiple remeshing.
<h2><a href="#TOC">»</a><a name="halfspace">Halfspace intersection</a></h2>
<p>
The identity pipe for Qhull reveals some precision questions for
halfspace intersections. The identity pipe creates the convex hull of
a set of points and intersects the facets' hyperplanes. It should return the input
points, but narrow distributions may drop points while offset distributions may add
points. It may be better to normalize the input set about the origin.
For example, compare the first results with the later two results: [T. Abraham]
<blockquote>
rbox 100 s t | tee r | qconvex FV n | qhalf Fp | cat - r | /bin/sort -n | tail
<br>
rbox 100 L1e5 t | tee r | qconvex FV n | qhalf Fp | cat - r | /bin/sort -n | tail
<br>
rbox 100 s O10 t | tee r | qconvex FV n | qhalf Fp | cat - r | /bin/sort -n | tail
</blockquote>
<h2><a href="#TOC">»</a><a name="imprecise">Merged facets </a></h2>
<p>Qhull detects precision
problems when computing distances. A precision problem occurs if
the distance computation is less than the maximum roundoff error.
Qhull treats the result of a hyperplane computation as if it
were exact.</p>
<p>Qhull handles precision problems by merging non-convex facets.
The result of merging two facets is a thick facet defined by an <i>inner
plane</i> and an <i>outer plane</i>. The inner and outer planes
are offsets from the facet's hyperplane. The inner plane is
clearly below the facet's vertices. At the end of Qhull, the
outer planes are clearly above all input points. Any exact convex
hull must lie between the inner and outer planes.</p>
<p>Qhull tests for convexity by computing a point for each facet.
This point is called the facet's <i>centrum</i>. It is the
arithmetic center of the facet's vertices projected to the
facet's hyperplane. For simplicial facets with <em>d</em>
vertices, the centrum is equivalent to the centroid or center of
gravity. </p>
<p>Two neighboring facets are convex if each centrum is clearly
below the other hyperplane. The '<a href="qh-optc.htm#Cn2">Cn</a>'
or '<a href="qh-optc.htm#Cn">C-n</a>' options sets the centrum's
radius to <i>n </i>. A centrum is clearly below a hyperplane if
the computed distance from the centrum to the hyperplane is
greater than the centrum's radius plus two maximum roundoff
errors. Two are required because the centrum can be the maximum
roundoff error above its hyperplane and the distance computation
can be high by the maximum roundoff error.</p>
<p>With the '<a href="qh-optc.htm#Cn">C-n</a>' or '<a
href="qh-optc.htm#An">A-n </a>' options, Qhull merges non-convex
facets while constructing the hull. The remaining facets are
clearly convex. With the '<a href="qh-optq.htm#Qx">Qx </a>'
option, Qhull merges coplanar facets after constructing the hull.
While constructing the hull, it merges coplanar horizon facets,
flipped facets, concave facets and duplicated ridges. With '<a
href="qh-optq.htm#Qx">Qx</a>', coplanar points may be missed, but
it appears to be unlikely.</p>
<p>If the user sets the '<a href="qh-optc.htm#An2">An</a>' or '<a
href="qh-optc.htm#An">A-n</a>' option, then all neighboring
facets are clearly convex and the maximum (exact) cosine of an
angle is <i>n</i>.</p>
<p>If '<a href="qh-optc.htm#C0">C-0</a>' or '<a
href="qh-optq.htm#Qx">Qx</a>' is used without other precision
options (default), Qhull tests vertices instead of centrums for
adjacent simplices. In 3-d, if simplex <i>abc</i> is adjacent to
simplex <i>bcd</i>, Qhull tests that vertex <i>a</i> is clearly
below simplex <i>bcd </i>, and vertex <i>d</i> is clearly below
simplex <i>abc</i>. When building the hull, Qhull tests vertices
if the horizon is simplicial and no merges occur. </p>
<h2><a href="#TOC">»</a><a name="how">How Qhull merges facets</a></h2>
<p>If two facets are not clearly convex, then Qhull removes one
or the other facet by merging the facet into a neighbor. It
selects the merge which minimizes the distance from the
neighboring hyperplane to the facet's vertices. Qhull also
performs merges when a facet has fewer than <i>d</i> neighbors (called a
degenerate facet), when a facet's vertices are included in a
neighboring facet's vertices (called a redundant facet), when a
facet's orientation is flipped, or when a ridge occurs between
more than two facets.</p>
<p>Qhull performs merges in a series of passes sorted by merge
angle. Each pass merges those facets which haven't already been
merged in that pass. After a pass, Qhull checks for redundant
vertices. For example, if a vertex has only two neighbors in 3-d,
the vertex is redundant and Qhull merges it into an adjacent
vertex.</p>
<p>Merging two simplicial facets creates a non-simplicial facet
of <em>d+1</em> vertices. Additional merges create larger facets.
When merging facet A into facet B, Qhull retains facet B's
hyperplane. It merges the vertices, neighbors, and ridges of both
facets. It recomputes the centrum if a wide merge has not
occurred (qh_WIDEcoplanar) and the number of extra vertices is
smaller than a constant (qh_MAXnewcentrum).</p>
<h2><a href="#TOC">»</a><a name="limit">Limitations of merged
facets</a></h2>
<ul>
<li><b>Uneven dimensions</b> --
If one coordinate has a larger absolute value than other
coordinates, it may dominate the effect of roundoff errors on
distance computations. You may use option '<a
href="qh-optq.htm#QbB">QbB</a>' to scale points to the unit cube.
For Delaunay triangulations and Voronoi diagrams, <a href=qdelaun.htm>qdelaunay</a>
and <a href=qvoronoi.htm>qvoronoi</a> always set
option '<a href="qh-optq.htm#Qbb">Qbb</a>'. It scales the last
coordinate to [0,m] where <i>m</i> is the maximum width of the
other coordinates. Option '<a href="qh-optq.htm#Qbb">Qbb</a>' is
needed for Delaunay triangulations of integer coordinates
and nearly cocircular points.
<p>For example, compare
<pre>
rbox 1000 W0 t | qconvex Qb2:-1e-14B2:1e-14
</pre>
with
<pre>
rbox 1000 W0 t | qconvex
</pre>
The distributions are the same but the first is compressed to a 2e-14 slab.
<p>
<li><b>Post-merging of coplanar facets</b> -- In 5-d and higher, option '<a href="qh-optq.htm#Qx">Qx</a>'
(default) delays merging of coplanar facets until post-merging.
This may allow "dents" to occur in the intermediate
convex hulls. A point may be poorly partitioned and force a poor
approximation. See option '<a href="qh-optq.htm#Qx">Qx</a>' for
further discussion.</p>
<p>This is difficult to produce in 5-d and higher. Option '<a href="qh-optq.htm#Q6">Q6</a>' turns off merging of concave
facets. This is similar to 'Qx'. It may lead to serious precision errors,
for example,
<pre>
rbox 10000 W1e-13 | qhull Q6 Tv
</pre>
<p>
<li><b>Maximum facet width</b> --
Qhull reports the maximum outer plane and inner planes (if
more than roundoff error apart). There is no upper bound
for either figure. This is an area for further research. Qhull
does a good job of post-merging in all dimensions. Qhull does a
good job of pre-merging in 2-d, 3-d, and 4-d. With the '<a
href="qh-optq.htm#Qx">Qx</a>' option, it does a good job in
higher dimensions. In 5-d and higher, Qhull does poorly at
detecting redundant vertices. </p>
<p>In the summary ('<a href="qh-opto.htm#s">s</a>'), look at the
ratio between the maximum facet width and the maximum width of a
single merge, e.g., "(3.4x)". Qhull usually reports a
ratio of four or lower in 3-d and six or lower in 4-d. If it
reports a ratio greater than 10, this may indicate an
implementation error. Narrow distributions (see following) may
produce wide facets.
<p>For example, if special processing for narrow distributions is
turned off ('<a href="qh-optq.htm#Q10">Q10</a>'), qhull may produce
a wide facet:</p>
<pre>
rbox 1000 L100000 s G1e-16 t1002074964 | qhull Tv Q10
</pre>
<p>
<li><b>Narrow distribution</b> -- In 3-d, a narrow distribution may result in a poor
approximation. For example, if you do not use qdelaunay nor option
'<a href="qh-optq.htm#Qbb">Qbb</a>', the furthest-site
Delaunay triangulation of nearly cocircular points may produce a poor
approximation:
<pre>
rbox s 5000 W1e-13 D2 t1002151341 | qhull d Qt
rbox 1000 s W1e-13 t1002231672 | qhull d Tv
</pre>
<p>During
construction of the hull, a point may be above two
facets with opposite orientations that span the input
set. Even though the point may be nearly coplanar with both
facets, and can be distant from the precise convex
hull of the input sites. Additional facets leave the point distant from
a facet. To fix this problem, add option '<a href="qh-optq.htm#Qbb">Qbb</a>'
(it scales the last coordinate). Option '<a href="qh-optq.htm#Qbb">Qbb</a>'
is automatically set for <a href=qdelaun.htm>qdelaunay</a> and <a href=qvoronoi.htm>qvoronoi</a>.
<p>Qhull generates a warning if the initial simplex is narrow.
For narrow distributions, Qhull changes how it processes coplanar
points -- it does not make a point coplanar until the hull is
finished.
Use option '<a href="qh-optq.htm#Q10">Q10</a>' to try Qhull without
special processing for narrow distributions.
For example, special processing is needed for:
<pre>
rbox 1000 L100000 s G1e-16 t1002074964 | qhull Tv Q10
</pre>
<p>You may turn off the warning message by reducing
qh_WARNnarrow in <tt>user.h</tt> or by setting option
'<a href="qh-optp.htm#Pp">Pp</a>'. </p>
<p>Similar problems occur for distributions with a large flat facet surrounded
with many small facet at a sharp angle to the large facet.
Qhull 3.1 fixes most of these problems, but a poor approximation can occur.
A point may be left outside of the convex hull ('<a href="qh-optt.htm#Tv">Tv</a>').
Examples include
the furthest-site Delaunay triangulation of nearly cocircular points plus the origin, and the convex hull of a cone of nearly cocircular points. The width of the band is 10^-13.
<pre>
rbox s 1000 W1e-13 P0 D2 t996799242 | qhull d Tv
rbox 1000 s Z1 G1e-13 t1002152123 | qhull Tv
rbox 1000 s Z1 G1e-13 t1002231668 | qhull Tv
</pre>
<p>
<li><b>Quadratic running time</b> -- If the output contains large, non-simplicial
facets, the running time for Qhull may be quadratic in the size of the triangulated
output. For example, <tt>rbox 1000 s W1e-13 c G2 | qhull d</tt> is 4 times
faster for 500 points. The convex hull contains two large nearly spherical facets and
many nearly coplanar facets. Each new point retriangulates the spherical facet and repartitions the remaining points into all of the nearly coplanar facets.
In this case, quadratic running time is avoided if you use qdelaunay,
add option '<a href="qh-optq.htm#Qbb">Qbb</a>',
or add the origin ('P0') to the input.
<p>
<li><b>Nearly coincident points within 1e-13</b> --
Multiple, nearly coincident points within a 1e-13 ball of points in the unit cube
may lead to wide facets or quadratic running time.
For example, the convex hull a 1000 coincident, cospherical points in 4-D,
or the 3-D Delaunay triangulation of nearly coincident points, may lead to very
wide facets (e.g., 2267021951.3x).
<p>For Delaunay triangulations, the problem typically occurs for extreme points of the input
set (i.e., on the edge between the upper and lower convex hull). After multiple facet merges, four
facets may share the same, duplicate ridge and must be merged.
Some of these facets may be long and narrow, leading to a very wide merged facet.
If so, error QH6271 is reported. It may be overriden with option '<a href="qh-optq.htm#Q12">Q12</a>'.
<p>Duplicate ridges occur when the horizon facets for a new point is "pinched".
In a duplicate ridge, a subridge (e.g., a line segment in 3-d) is shared by two horizon facets.
At least two of its vertices are nearly coincident. It is easy to generate coincident points with
option 'Cn,r,m' of rbox. It generates n points within an r ball for each of m input sites. For example,
every point of the following distributions has a nearly coincident point within a 1e-13 ball.
Substantially smaller or larger balls do not lead to pinched horizons.
<pre>
rbox 1000 C1,1e-13 D4 s t | qhull
rbox 75 C1,1e-13 t | qhull d
</pre>
For Delaunay triangulations, a bounding box may alleviate this error (e.g., <tt>rbox 500 C1,1E-13 t c G1 | qhull d</tt>).
A later release of qhull will avoid pinched horizons by merging duplicate subridges. A subridge is
merged by merging adjacent vertices.
<p>
<li><b>Facet with zero-area</b> --
It is possible for a zero-area facet to be convex with its
neighbors. This can occur if the hyperplanes of neighboring
facets are above the facet's centrum, and the facet's hyperplane
is above the neighboring centrums. Qhull computes the facet's
hyperplane so that it passes through the facet's vertices. The
vertices can be collinear. </p>
<p>
<li><b>No more facets</b> -- Qhull reports an error if there are <em>d+1</em> facets left
and two of the facets are not clearly convex. This typically
occurs when the convexity constraints are too strong or the input
points are degenerate. The former is more likely in 5-d and
higher -- especially with option '<a href="qh-optc.htm#Cn">C-n</a>'.</p>
<p>
<li><b>Deleted cone</b> -- Lots of merging can end up deleting all
of the new facets for a point. This is a rare event that has
only been seen while debugging the code.
<p>
<li><b>Triangulated output leads to precision problems</b> -- With sufficient
merging, the ridges of a non-simplicial facet may have serious topological
and geometric problems. A ridge may be between more than two
neighboring facets. If so, their triangulation ('<a href="qh-optq.htm#Qt">Qt</a>')
will fail since two facets have the same vertex set. Furthermore,
a triangulated facet may have flipped orientation compared to its
neighbors.</li>
<p>The triangulation process detects degenerate facets with
only two neighbors. These are marked degenerate. They have
zero area.
<p>
<li><b>Coplanar points</b> --
Option '<a href="qh-optq.htm#Qc">Qc</a>' is determined by
qh_check_maxout() after constructing the hull. Qhull needs to
retain all possible coplanar points in the facets' coplanar sets.
This depends on qh_RATIOnearInside in <tt>user.h.</tt>
Furthermore, the cutoff for a coplanar point is arbitrarily set
at the minimum vertex. If coplanar points are important to your
application, remove the interior points by hand (set '<a
href="qh-optq.htm#Qc">Qc</a> <a href="qh-optq.htm#Qi">Qi</a>') or
make qh_RATIOnearInside sufficiently large.</p>
<p>
<li><b>Maximum roundoff error</b> -- Qhull computes the maximum roundoff error from the maximum
coordinates of the point set. Usually the maximum roundoff error
is a reasonable choice for all distance computations. The maximum
roundoff error could be computed separately for each point or for
each distance computation. This is expensive and it conflicts
with option '<a href="qh-optc.htm#Cn">C-n</a>'.
<p>
<li><b>All flipped or upper Delaunay</b> -- When a lot of merging occurs for
Delaunay triangulations, a new point may lead to no good facets. For example,
try a strong convexity constraint:
<pre>
rbox 1000 s t993602376 | qdelaunay C-1e-3
</pre>
</ul>
<h2><a href="#TOC">»</a><a name="injoggle">Joggled input</a></h2>
<p>Joggled input is a simple work-around for precision problems
in computational geometry ["joggle: to shake or jar
slightly," Amer. Heritage Dictionary]. Other names are
<i>jostled input</i> or <i>random perturbation</i>.
Qhull joggles the
input by modifying each coordinate by a small random quantity. If
a precision problem occurs, Qhull joggles the input with a larger
quantity and the algorithm is restarted. This process continues
until no precision problems occur. Unless all inputs incur
precision problems, Qhull will terminate. Qhull adjusts the inner
and outer planes to account for the joggled input. </p>
<p>Neither joggle nor merged facets has an upper bound for the width of the output
facets, but both methods work well in practice. Joggled input is
easier to justify. Precision errors occur when the points are
nearly singular. For example, four points may be coplanar or
three points may be collinear. Consider a line and an incident
point. A precision error occurs if the point is within some
epsilon of the line. Now joggle the point away from the line by a
small, uniformly distributed, random quantity. If the point is
changed by more than epsilon, the precision error is avoided. The
probability of this event depends on the maximum joggle. Once the
maximum joggle is larger than epsilon, doubling the maximum
joggle will halve the probability of a precision error. </p>
<p>With actual data, an analysis would need to account for each
point changing independently and other computations. It is easier
to determine the probabilities empirically ('<a href="qh-optt.htm#TRn">TRn</a>') . For example, consider
computing the convex hull of the unit cube centered on the
origin. The arithmetic has 16 significant decimal digits. </p>
<blockquote>
<p><b>Convex hull of unit cube</b> </p>
<table border="1">
<tr>
<th align="left">joggle</th>
<th align="right">error prob. </th>
</tr>
<tr>
<td align="right">1.0e-15</td>
<td align="center">0.983 </td>
</tr>
<tr>
<td align="right">2.0e-15</td>
<td align="center">0.830 </td>
</tr>
<tr>
<td align="right">4.0e-15</td>
<td align="center">0.561 </td>
</tr>
<tr>
<td align="right">8.0e-15</td>
<td align="center">0.325 </td>
</tr>
<tr>
<td align="right">1.6e-14</td>
<td align="center">0.185 </td>
</tr>
<tr>
<td align="right">3.2e-14</td>
<td align="center">0.099 </td>
</tr>
<tr>
<td align="right">6.4e-14</td>
<td align="center">0.051 </td>
</tr>
<tr>
<td align="right">1.3e-13</td>
<td align="center">0.025 </td>
</tr>
<tr>
<td align="right">2.6e-13</td>
<td align="center">0.010 </td>
</tr>
<tr>
<td align="right">5.1e-13</td>
<td align="center">0.004 </td>
</tr>
<tr>
<td align="right">1.0e-12</td>
<td align="center">0.002 </td>
</tr>
<tr>
<td align="right">2.0e-12</td>
<td align="center">0.001 </td>
</tr>
</table>
</blockquote>
<p>A larger joggle is needed for multiple points. Since the
number of potential singularities increases, the probability of
one or more precision errors increases. Here is an example. </p>
<blockquote>
<p><b>Convex hull of 1000 points on unit cube</b> </p>
<table border="1">
<tr>
<th align="left">joggle</th>
<th align="right">error prob. </th>
</tr>
<tr>
<td align="right">1.0e-12</td>
<td align="center">0.870 </td>
</tr>
<tr>
<td align="right">2.0e-12</td>
<td align="center">0.700 </td>
</tr>
<tr>
<td align="right">4.0e-12</td>
<td align="center">0.450 </td>
</tr>
<tr>
<td align="right">8.0e-12</td>
<td align="center">0.250 </td>
</tr>
<tr>
<td align="right">1.6e-11</td>
<td align="center">0.110 </td>
</tr>
<tr>
<td align="right">3.2e-11</td>
<td align="center">0.065 </td>
</tr>
<tr>
<td align="right">6.4e-11</td>
<td align="center">0.030 </td>
</tr>
<tr>
<td align="right">1.3e-10</td>
<td align="center">0.010 </td>
</tr>
<tr>
<td align="right">2.6e-10</td>
<td align="center">0.008 </td>
</tr>
<tr>
<td align="right">5.1e-09</td>
<td align="center">0.003 </td>
</tr>
</table>
</blockquote>
<p>Other distributions behave similarly. No distribution should
behave significantly worse. In Euclidean space, the probability
measure of all singularities is zero. With floating point
numbers, the probability of a singularity is non-zero. With
sufficient digits, the probability of a singularity is extremely
small for random data. For a sufficiently large joggle, all data
is nearly random data. </p>
<p>Qhull uses an initial joggle of 30,000 times the maximum
roundoff error for a distance computation. This avoids most
potential singularities. If a failure occurs, Qhull retries at
the initial joggle (in case bad luck occurred). If it occurs
again, Qhull increases the joggle by ten-fold and tries again.
This process repeats until the joggle is a hundredth of the width
of the input points. Qhull reports an error after 100 attempts.
This should never happen with double-precision arithmetic. Once
the probability of success is non-zero, the probability of
success increases about ten-fold at each iteration. The
probability of repeated failures becomes extremely small. </p>
<p>Merged facets produces a significantly better approximation.
Empirically, the maximum separation between inner and outer
facets is about 30 times the maximum roundoff error for a
distance computation. This is about 2,000 times better than
joggled input. Most applications though will not notice the
difference. </p>
<h2><a href="#TOC">»</a><a name="exact">Exact arithmetic</a></h2>
<p>Exact arithmetic may be used instead of floating point.
Singularities such as coplanar points can either be handled
directly or the input can be symbolically perturbed. Using exact
arithmetic is slower than using floating point arithmetic and the
output may take more space. Chaining a sequence of operations
increases the time and space required. Some operations are
difficult to do.</p>
<p>Clarkson's <a
href="http://www.netlib.org/voronoi/hull.html">hull
program</a> and Shewchuk's <a
href="http://www.cs.cmu.edu/~quake/triangle.html">triangle
program</a> are practical implementations of exact arithmetic.</p>
<p>Clarkson limits the input precision to about fifteen digits.
This reduces the number of nearly singular computations. When a
determinant is nearly singular, he uses exact arithmetic to
compute a precise result.</p>
<h2><a href="#TOC">»</a><a name="approximate">Approximating a
convex hull</a></h2>
<p>Qhull may be used for approximating a convex hull. This is
particularly valuable in 5-d and higher where hulls can be
immense. You can use '<a href="qh-optq.htm#Qx">Qx</a> <a
href="qh-optc.htm#Cn">C-n</a>' to merge facets as the hull is
being constructed. Then use '<a href="qh-optc.htm#Cn2">Cn</a>'
and/or '<a href="qh-optc.htm#An2">An</a>' to merge small facets
during post-processing. You can print the <i>n</i> largest facets
with option '<a href="qh-optp.htm#PAn">PAn</a>'. You can print
facets whose area is at least <i>n</i> with option '<a
href="qh-optp.htm#PFn">PFn</a>'. You can output the outer planes
and an interior point with '<a href="qh-optf.htm#FV">FV</a> <a
href="qh-optf.htm#Fo">Fo</a>' and then compute their intersection
with 'qhalf'. </p>
<p>To approximate a convex hull in 6-d and higher, use
post-merging with '<a href="qh-optc.htm#Wn">Wn</a>' (e.g., qhull
W1e-1 C1e-2 TF2000). Pre-merging with a convexity constraint
(e.g., qhull Qx C-1e-2) often produces a poor approximation or
terminates with a simplex. Option '<a href="qh-optq.htm#QbB">QbB</a>'
may help to spread out the data.</p>
<p>You will need to experiment to determine a satisfactory set of
options. Use <a href="rbox.htm">rbox</a> to generate test sets
quickly and <a href="index.htm#geomview">Geomview</a> to view
the results. You will probably want to write your own driver for
Qhull using the Qhull library. For example, you could select the
largest facet in each quadrant.</p>
<!-- 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="#TOC">Qhull imprecision: Table of Contents</a>
<!-- 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>
|
