
Convex hull of 8 points in 3-d:

  Number of vertices: 8
  Number of facets: 6
  Number of non-simplicial facets: 6

Statistics for:  | qhull s Tcv 

  Number of points processed: 8
  Number of hyperplanes created: 11
  Number of distance tests for qhull: 35
  Number of distance tests for merging: 142
  Number of distance tests for checking: 56
  Number of merged facets: 6
  CPU seconds to compute hull (after input):  0


Output completed.  Verifying that all points are below outer planes of
all facets.  Will make 48 distance computations.

Delaunay triangulation by the convex hull of 8 points in 4-d:

  Number of input sites: 8
  Number of Delaunay regions: 11

Statistics for:  | qhull s d Tcv 

  Number of points processed: 8
  Number of hyperplanes created: 28
  Number of facets in hull: 17
  Number of distance tests for qhull: 33
  CPU seconds to compute hull (after input):  0


Output completed.  Verifying that all points are below 9e-015 of
all facets.  Will make 88 distance computations.

Delaunay triangulation by the convex hull of 8 points in 4-d:

  Number of input sites: 8
  Number of Delaunay regions: 12

Statistics for:  | qhull s d Tcv 

  Number of points processed: 8
  Number of hyperplanes created: 24
  Number of facets in hull: 17
  Number of distance tests for qhull: 26
  CPU seconds to compute hull (after input):  0


Output completed.  Verifying that all points are below 8.9e-015 of
all facets.  Will make 96 distance computations.

Halfspace intersection by the convex hull of 8 points in 3-d:

  Number of halfspaces: 8
  Number of non-redundant halfspaces: 8
  Number of intersection points: 6
  Number of non-simplicial intersection points: 6

Statistics for:  | qhull H0 s Tcv Fp

  Number of points processed: 8
  Number of hyperplanes created: 11
  Number of distance tests for qhull: 35
  Number of distance tests for merging: 142
  Number of distance tests for checking: 56
  Number of merged facets: 6
  CPU seconds to compute hull (after input):  0


Output completed.  Verifying that all points are below outer planes of
all facets.  Will make 48 distance computations.
echo === errors if 'user_eg' and 'user_eg2' not found ===
=== errors if user_eg and user_eg2 not found ===
echo === check user_eg ${d:-`date`} =====================
date
=== check user_eg Wed Nov 23 22:04:40 EST 2011 =====================
user_eg "QR1 p n Qt" "v p" Fp
This is the output from user_eg.c

It shows how qhull() may be called from an application using the qhull
shared library.  It is not part of qhull itself.  If it appears accidently,
please remove user_eg.c from your project.


compute convex hull of cube after rotating input
input
    -1     -1     -1 
     1     -1     -1 
    -1      1     -1 
     1      1     -1 
    -1     -1      1 
     1     -1      1 
    -1      1      1 
     1      1      1 
3
8
0.9124763375291641 1.390833937612012 -0.4826674750136243 
-0.5764091147442416 0.565375258822974 -1.532352227508615 
-0.1847368907548403 1.250953879569599 1.183632828365285 
-1.673622343028246 0.4254952007805605 0.1339480758702939 
1.673622343028246 -0.4254952007805605 -0.1339480758702939 
0.1847368907548403 -1.250953879569599 -1.183632828365285 
0.5764091147442416 -0.565375258822974 1.532352227508615 
-0.9124763375291641 -1.390833937612012 0.4826674750136243 
4
12
-0.380573002749541 0.9081645691962865 -0.1743596995716652     -1 
-0.380573002749541 0.9081645691962865 -0.1743596995716652     -1 
0.744442726136703 0.4127293393945193 0.5248423762474955     -1 
0.744442726136703 0.4127293393945193 0.5248423762474955     -1 
-0.5486066141420022 -0.06994002902120672 0.8331501516894545     -1 
-0.5486066141420022 -0.06994002902120672 0.8331501516894545     -1 
0.5486066141420022 0.06994002902120679 -0.8331501516894545     -1 
0.5486066141420022 0.06994002902120679 -0.8331501516894545     -1 
0.380573002749541 -0.9081645691962865 0.1743596995716652     -1 
0.380573002749541 -0.9081645691962865 0.1743596995716652     -1 
-0.7444427261367029 -0.4127293393945193 -0.5248423762474955 -0.9999999999999997 
-0.7444427261367029 -0.4127293393945193 -0.5248423762474955 -0.9999999999999997 

8 vertices and 12 facets with normals:
 -0.38   0.91  -0.17 
 -0.38   0.91  -0.17 
  0.74   0.41   0.52 
  0.74   0.41   0.52 
 -0.55  -0.07   0.83 
 -0.55  -0.07   0.83 
  0.55   0.07  -0.83 
  0.55   0.07  -0.83 
  0.38  -0.91   0.17 
  0.38  -0.91   0.17 
 -0.74  -0.41  -0.52 
 -0.74  -0.41  -0.52 

compute 3-d Delaunay triangulation
seed: 1322103880
input
-0.454 -0.859 -0.555 
 0.957  0.439  0.612 
 0.261 -0.667  0.653 
-0.336  0.275  0.693 
-0.519 -0.0637 -0.753 
  0.69  0.496  0.717 
0.0315  0.142  0.352 
  0.38  0.501  0.547 
3
11
1.055978523384926 -0.3646158135533156 -0.7716526665441572 
0.8171638553283269 0.401264026995007 -0.7955601187068335 
0.6225513264063929 -0.08795222049685247 0.45766539798864 
-0.7237064719039652 -0.2923024472697715 0.1005311946780194 
-0.5023977118471468 1.432187098612461 -0.3310344440582338 
0.4834866854228005 -0.0895023344334871 0.7102055892762454 
0.2415029221447908 -0.0343530505125913 1.042775412059036 
0.2563338604002023 -0.02840369259448017 1.124392674378581 
0.62672652247501 -0.1303605771100868 0.491255687163586 
1.057033853658139 -0.4452463337370909 -0.6832436048735331 
-0.6119348841729465 -0.5449145513740108 0.3194934748845855 

8 vertices and 17 facets with normals:
  0.73  -0.25  -0.53  -0.35 
  0.62   0.31  -0.61  -0.38 
-0.076   0.93  -0.33   0.16 
  0.67 -0.095    0.5  -0.54 
 -0.52   0.37   0.07   0.76 
 -0.49   0.56 0.0095   0.67 
 -0.32   0.91  -0.14   0.24 
 -0.78  -0.31   0.11  -0.54 
 -0.31   0.88   -0.2  -0.31 
  0.48  -0.09   0.71   -0.5 
   0.2 -0.029   0.88  -0.42 
   0.2 -0.023   0.89   -0.4 
  0.66  -0.14   0.52  -0.53 
 -0.37  -0.07   0.65   0.66 
  0.74  -0.31  -0.48  -0.35 
 -0.44  -0.08   0.47   0.76 
  -0.6  -0.54   0.32  -0.49 

find 3-d Delaunay triangle closest to [0.5, 0.5, ...]
 0.38  0.50  0.55 
 0.03  0.14  0.35 
 0.96  0.44  0.61 
-0.52 -0.06 -0.75 

save first triangulation and compute a new triangulation
seed: 1322103881
input
-0.454 -0.596  0.957 
-0.126 -0.495  -0.95 
 0.355  0.691 0.0119 
-0.467 -0.958 -0.268 
-0.857 0.00544 -0.646 
 -0.25 -0.161  0.733 
 0.798  0.276 -0.813 
-0.247 -0.321  0.407 
3
12
-1.40139437602718 -0.4903363806146227 0.2697011389791514 
3.013225186878528 -2.643156209768993 0.8591003164615868 
-0.2996272421526284 -0.268084993544268 -0.2994456538467814 
0.8415134912005295 -0.8499506611228171 -0.1231572268155028 
-0.04889512719841398 0.3104517692237675 -0.6497668054180039 
-0.08467913030960342 0.05949850035514959 -0.3223353693093948 
0.2548447045276229 -0.1212603537693163 -0.2689044205597102 
3.032061213734874 -1.713747580399033 1.330294884509217 
25.54746834903245 -4.844012632488196 15.72237921155334 
1.734809709023358 -0.595660969003913 0.7665316845537354 
-1.489691016775167 -0.04413328784377588 0.4592197496989515 
-0.6425784076696081 0.5801147792597374 0.1622257215473305 

8 vertices and 17 facets with normals:
 -0.88  -0.31   0.17  -0.31 
  0.73  -0.64   0.21  -0.12 
 -0.19    0.7   0.17   0.67 
 -0.14 -0.065  -0.19   0.97 
 -0.15 -0.076  -0.19   0.97 
 -0.12 -0.092  -0.18   0.97 
 -0.42  -0.38  -0.42  -0.71 
  0.65  -0.65 -0.095  -0.38 
-0.056   0.35  -0.74  -0.57 
 -0.14  0.099  -0.53  -0.83 
   0.4  -0.19  -0.42  -0.79 
  0.81  -0.46   0.35  -0.13 
  0.84  -0.16   0.52 -0.016 
  0.85  -0.29   0.37  -0.24 
 -0.91 -0.027   0.28  -0.31 
 -0.56   0.72   0.38   0.14 
 -0.63   0.57   0.16  -0.49 

save second triangulation and restore first one

8 vertices and 17 facets with normals:
  0.73  -0.25  -0.53  -0.35 
  0.62   0.31  -0.61  -0.38 
-0.076   0.93  -0.33   0.16 
  0.67 -0.095    0.5  -0.54 
 -0.52   0.37   0.07   0.76 
 -0.49   0.56 0.0095   0.67 
 -0.32   0.91  -0.14   0.24 
 -0.78  -0.31   0.11  -0.54 
 -0.31   0.88   -0.2  -0.31 
  0.48  -0.09   0.71   -0.5 
   0.2 -0.029   0.88  -0.42 
   0.2 -0.023   0.89   -0.4 
  0.66  -0.14   0.52  -0.53 
 -0.37  -0.07   0.65   0.66 
  0.74  -0.31  -0.48  -0.35 
 -0.44  -0.08   0.47   0.76 
  -0.6  -0.54   0.32  -0.49 

free first triangulation and restore second one.

8 vertices and 17 facets with normals:
 -0.88  -0.31   0.17  -0.31 
  0.73  -0.64   0.21  -0.12 
 -0.19    0.7   0.17   0.67 
 -0.14 -0.065  -0.19   0.97 
 -0.15 -0.076  -0.19   0.97 
 -0.12 -0.092  -0.18   0.97 
 -0.42  -0.38  -0.42  -0.71 
  0.65  -0.65 -0.095  -0.38 
-0.056   0.35  -0.74  -0.57 
 -0.14  0.099  -0.53  -0.83 
   0.4  -0.19  -0.42  -0.79 
  0.81  -0.46   0.35  -0.13 
  0.84  -0.16   0.52 -0.016 
  0.85  -0.29   0.37  -0.24 
 -0.91 -0.027   0.28  -0.31 
 -0.56   0.72   0.38   0.14 
 -0.63   0.57   0.16  -0.49 

compute halfspace intersection about the origin for a diamond
input as halfspace coefficients + offsets
    -1     -1     -1     -1 
     1     -1     -1     -1 
    -1      1     -1     -1 
     1      1     -1     -1 
    -1     -1      1     -1 
     1     -1      1     -1 
    -1      1      1     -1 
     1      1      1     -1 
3
6
     0      0     -1 
     0     -1      0 
     1      0      0 
    -1      0      0 
     0      1      0 
     0      0      1 

8 vertices and 6 facets with normals:
    -0     -0     -1 
     0     -1      0 
     1     -0     -0 
    -1     -0     -0 
     0      1     -0 
    -0     -0      1 

Convex hull of 8 points in 3-d:

  Number of vertices: 8
  Number of facets: 12
  Number of triangulated facets: 6

Statistics for:  | qhull s Tcv QR1 p n Qt QR1

  Number of points processed: 8
  Number of hyperplanes created: 11
  Number of distance tests for qhull: 24
  Number of distance tests for merging: 186
  Number of distance tests for checking: 56
  Number of merged facets: 7
  CPU seconds to compute hull (after input):  0


Output completed.  Verifying that all points are below outer planes of
all facets.  Will make 96 distance computations.

Voronoi diagram by the convex hull of 8 points in 4-d:

  Number of Voronoi regions: 8
  Number of Voronoi vertices: 11

Statistics for:  | qhull s d Tcv v p

  Number of points processed: 8
  Number of hyperplanes created: 28
  Number of facets in hull: 17
  Number of distance tests for qhull: 33
  CPU seconds to compute hull (after input):  0


Output completed.  Verifying that all points are below 9e-015 of
all facets.  Will make 88 distance computations.

Voronoi diagram by the convex hull of 8 points in 4-d:

  Number of Voronoi regions: 8
  Number of Voronoi vertices: 12

Statistics for:  | qhull s d Tcv v p

  Number of points processed: 8
  Number of hyperplanes created: 24
  Number of facets in hull: 17
  Number of distance tests for qhull: 26
  CPU seconds to compute hull (after input):  0


Output completed.  Verifying that all points are below 8.9e-015 of
all facets.  Will make 96 distance computations.

Halfspace intersection by the convex hull of 8 points in 3-d:

  Number of halfspaces: 8
  Number of non-redundant halfspaces: 8
  Number of intersection points: 6
  Number of non-simplicial intersection points: 6

Statistics for:  | qhull H0 s Tcv Fp

  Number of points processed: 8
  Number of hyperplanes created: 11
  Number of distance tests for qhull: 35
  Number of distance tests for merging: 142
  Number of distance tests for checking: 56
  Number of merged facets: 6
  CPU seconds to compute hull (after input):  0


Output completed.  Verifying that all points are below outer planes of
all facets.  Will make 48 distance computations.
user_eg2 "QR1 p" "v p" Fp
This is the output from user_eg2.c

It shows how qhull() may be called from an application in the same way as
qconvex.  It is not part of qhull itself.  If it appears accidently,
please remove user_eg2.c from your project.


compute triangulated convex hull of cube after rotating input

8 vertices and 12 facets with normals:
 -0.38   0.91  -0.17 
 -0.38   0.91  -0.17 
  0.74   0.41   0.52 
  0.74   0.41   0.52 
 -0.55  -0.07   0.83 
 -0.55  -0.07   0.83 
  0.55   0.07  -0.83 
  0.55   0.07  -0.83 
  0.38  -0.91   0.17 
  0.38  -0.91   0.17 
 -0.74  -0.41  -0.52 
 -0.74  -0.41  -0.52 

add points in a diamond
9 vertices and 14 facets
10 vertices and 16 facets
11 vertices and 18 facets
12 vertices and 20 facets
13 vertices and 22 facets
14 vertices and 24 facets

14 vertices and 24 facets with normals:
 -0.67   0.63  -0.39 
  0.67  -0.63   0.39 
  -0.1  -0.71    0.7 
  0.65  -0.66   0.37 
  0.63   -0.7  -0.34 
 -0.65  -0.65  -0.41 
 -0.62  -0.62  -0.49 
 -0.65   0.66  -0.37 
 -0.63    0.7   0.34 
   0.1   0.71   -0.7 
  0.65   0.65   0.41 
  0.62   0.62   0.49 
  -0.7  -0.13   -0.7 
 -0.62  -0.49  -0.62 
  0.11    0.7  -0.71 
  0.68   0.29  -0.68 
  0.66  -0.34  -0.67 
  0.67   -0.3  -0.67 
 -0.66   0.34   0.67 
 -0.67    0.3   0.67 
   0.7   0.13    0.7 
  0.62   0.49   0.62 
 -0.11   -0.7   0.71 
 -0.68  -0.29   0.68 
3
14
0.9124763375291641 1.390833937612012 -0.4826674750136243 
-0.5764091147442416 0.565375258822974 -1.532352227508615 
-0.1847368907548403 1.250953879569599 1.183632828365285 
-1.673622343028246 0.4254952007805605 0.1339480758702939 
1.673622343028246 -0.4254952007805605 -0.1339480758702939 
0.1847368907548403 -1.250953879569599 -1.183632828365285 
0.5764091147442416 -0.565375258822974 1.532352227508615 
-0.9124763375291641 -1.390833937612012 0.4826674750136243 
    -2      0      0 
     2      0      0 
     0     -2      0 
     0      2      0 
     0      0     -2 
     0      0      2 

compute 2-d Delaunay triangulation
seed: 1322103880

8 vertices and 12 facets with normals:
 -0.98  -0.18  -0.12 
 -0.34   0.62  -0.71 
 -0.54  -0.16   0.83 
  0.39  -0.12   0.91 
 -0.32   0.68  -0.66 
  0.51  -0.66   0.56 
  0.19  -0.14  -0.97 
 -0.27  -0.63  -0.73 
  0.24  0.082  -0.97 
  0.16   0.19  -0.97 
  0.57   0.11   0.81 
  0.43   0.89   0.18 
2
7
-4.068833863142581 -0.7537848823147527 
-0.238392077671283 0.4368373501180812 
-0.2386041499061649 0.5148053313424543 
0.09684777917315318 -0.07324187811151195 
-0.1842213727247133 -0.4364965645295652 
0.1237542345879292 0.04239098724585988 
0.0836168728094453 0.09927194631096603 

add points to triangulation
added point p8:        -1  -0.7369    1.543
9 points, 0 extra points, 9 vertices, and 14 facets in total
added point p9:    0.5112  -0.0827   0.2682
10 points, 0 extra points, 10 vertices, and 16 facets in total
added point p10:   0.06553  -0.5621   0.3202
11 points, 0 extra points, 11 vertices, and 18 facets in total
added point p11:   -0.9059   0.3577   0.9486
12 points, 0 extra points, 12 vertices, and 20 facets in total
added point p12:    0.3586   0.8694   0.8844
13 points, 0 extra points, 13 vertices, and 22 facets in total
added point p13:    -0.233  0.03883   0.0558
14 points, 0 extra points, 14 vertices, and 24 facets in total
2
17
-0.6549264999832026 -0.4749429316689148 
1.646368036202902 0.3863693903043897 
0.330152747247965 -0.3507184263758297 
0.3091407581696226 -0.3416986517702098 
-0.343054904342996 -0.4494698081103229 
-0.03636301949253369 -0.9864828479239056 
-1.067780850794688 -0.179728823807914 
-0.09201707764374939 0.9755504574206503 
0.400741399693187 0.7413087671639999 
-0.4892143147831943 0.367596157169687 
-0.5023164831720048 0.3399488189748565 
-0.2728501304718273 0.5402820582546699 
-0.2251873791351292 0.5569762516361401 
-0.2556623749971488 -0.3470389778724706 
0.1161835094406034 -0.1623083804984112 
0.1614702095730213 0.2568904986180597 
0.1809173283824135 0.2340904095719151 

find Delaunay triangle closest to [0.5, 0.5, ...]
 0.51 -0.08 
 0.44  0.61 
 0.65 -0.34 

compute halfspace intersection about the origin for a diamond
3
6
     0      0     -1 
     0     -1      0 
     1      0      0 
    -1      0      0 
     0      1      0 
     0      0      1 

add halfspaces for cube to intersection
added offset -1 and normal    1.732        0        0
8 points, 1 extra points, 9 vertices, and 9 facets in total
added offset -1 and normal   -1.732        0        0
8 points, 2 extra points, 10 vertices, and 12 facets in total
added offset -1 and normal        0    1.732        0
8 points, 3 extra points, 11 vertices, and 15 facets in total
added offset -1 and normal        0   -1.732        0
8 points, 4 extra points, 12 vertices, and 18 facets in total
added offset -1 and normal        0        0    1.732
8 points, 5 extra points, 13 vertices, and 21 facets in total
added offset -1 and normal        0        0   -1.732
8 points, 6 extra points, 14 vertices, and 24 facets in total
3
24
0.5773502691896257      0 -0.4226497308103742 
0.5773502691896257 -0.4226497308103742      0 
0.5773502691896257 0.4226497308103742      0 
0.5773502691896257      0 0.4226497308103742 
-0.5773502691896257      0 -0.4226497308103742 
-0.5773502691896257 -0.4226497308103742      0 
-0.5773502691896257 0.4226497308103742      0 
-0.5773502691896257      0 0.4226497308103742 
     0 0.5773502691896257 -0.4226497308103742 
0.4226497308103742 0.5773502691896257      0 
-0.4226497308103742 0.5773502691896257      0 
     0 0.5773502691896257 0.4226497308103742 
     0 -0.5773502691896257 -0.4226497308103742 
-0.4226497308103742 -0.5773502691896257      0 
0.4226497308103742 -0.5773502691896257      0 
     0 -0.5773502691896257 0.4226497308103742 
     0 -0.4226497308103742 0.5773502691896257 
0.4226497308103742      0 0.5773502691896257 
     0 0.4226497308103742 0.5773502691896257 
-0.4226497308103742      0 0.5773502691896257 
-0.4226497308103742      0 -0.5773502691896257 
     0 -0.4226497308103742 -0.5773502691896257 
0.4226497308103742      0 -0.5773502691896257 
     0 0.4226497308103742 -0.5773502691896257 

Output completed.  Verifying that all points are below outer planes of
all facets.  Will make 96 distance computations.

Convex hull of 14 points in 3-d:

  Number of vertices: 14
  Number of facets: 24

Statistics for: user_eg cube | qhull s Tcv Q11 QR1 p  QR1

  Number of points processed: 14
  Number of hyperplanes created: 45
  Number of distance tests for qhull: 68
  Number of distance tests for merging: 370
  Number of distance tests for checking: 214
  Number of merged facets: 7
  CPU seconds to compute hull (after input):  0


Output completed.  Verifying that all points are below outer planes of
all facets.  Will make 192 distance computations.

Voronoi diagram by the convex hull of 8 points in 3-d:

  Number of Voronoi regions: 8
  Number of Voronoi vertices: 7

Statistics for: user_eg Delaunay | qhull s d Tcv v p

  Number of points processed: 8
  Number of hyperplanes created: 19
  Number of facets in hull: 12
  Number of distance tests for qhull: 27
  CPU seconds to compute hull (after input):  0


Output completed.  Verifying that all points are below 5.1e-015 of
all facets.  Will make 56 distance computations.

Voronoi diagram by the convex hull of 14 points in 3-d:

  Number of Voronoi regions: 14
  Number of Voronoi vertices: 17

Statistics for: user_eg Delaunay | qhull s d Tcv v p

  Number of points processed: 14
  Number of hyperplanes created: 49
  Number of facets in hull: 24
  Number of distance tests for qhull: 69
  CPU seconds to compute hull (after input):  0


Output completed.  Verifying that all points are below 5.1e-015 of
all facets.  Will make 238 distance computations.

Halfspace intersection by the convex hull of 8 points in 3-d:

  Number of halfspaces: 8
  Number of non-redundant halfspaces: 8
  Number of intersection points: 6
  Number of non-simplicial intersection points: 6

Statistics for: user_eg halfspaces | qhull H0 s Tcv Fp

  Number of points processed: 8
  Number of hyperplanes created: 11
  Number of distance tests for qhull: 35
  Number of distance tests for merging: 142
  Number of distance tests for checking: 56
  Number of merged facets: 6
  CPU seconds to compute hull (after input):  0


Output completed.  Verifying that all points are below outer planes of
all facets.  Will make 48 distance computations.

Halfspace intersection by the convex hull of 14 points in 3-d:

  Number of halfspaces: 14
  Number of non-redundant halfspaces: 14
  Number of intersection points: 24

Statistics for: user_eg halfspaces | qhull H0 s Tcv Fp

  Number of points processed: 14
  Number of hyperplanes created: 35
  Number of distance tests for qhull: 59
  Number of distance tests for merging: 306
  Number of distance tests for checking: 214
  Number of merged facets: 6
  CPU seconds to compute hull (after input):  0


Output completed.  Verifying that all points are below outer planes of
all facets.  Will make 192 distance computations.
user_eg3 rbox "10 D2" "2 D2" qhull "p"
rbox 10 D2
rbox 2 D2

Results of p
2
5
-0.02222276248244826 -0.4979727817680433 
-0.4285431913366012 0.4745826469497594 
0.3790312361708201 0.3779794437605696 
0.3443122672329771 -0.1437312230875075 
-0.3674659290047977 0.0001301793382784133 
echo === check front ends ${d:-`date`} ==================
date
=== check front ends Wed Nov 23 22:04:40 EST 2011 ==================
qconvex -

qconvex- compute the convex hull
    http://www.qhull.org  2011.2 2011/11/21

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:
    Qt   - triangulated output
    QJ   - joggled input instead of merged facets
    Qc   - keep coplanar points with nearest facet
    Qi   - keep interior points with nearest facet

Qhull control options:
    Qbk:n   - scale coord k so that low bound is n
      QBk:n - scale coord k so that upper bound is n (QBk is 0.5)
    QbB  - scale input to unit cube centered at the origin
    Qbk:0Bk:0 - remove k-th coordinate from input
    QJn  - randomly joggle input in range [-n,n]
    QRn  - random rotation (n=seed, n=0 time, n=-1 time/no rotate)
    Qs   - search all points for the initial simplex
    QGn  - good facet if visible from point n, -n for not visible
    QVn  - good facet if it includes 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   - print statistics
    Tv   - verify result: structure, convexity, and point inclusion
    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]
    Un   - max distance below plane for a new, coplanar point
    Wn   - min facet width for outside point (before roundoff)

Output formats (may be combined; if none, produces a summary to stdout):
    f    - facet dump
    G    - Geomview output (see below)
    i    - vertices incident to each facet
    m    - Mathematica output (2-d and 3-d)
    n    - normals with offsets
    o    - OFF file format (dim, points and facets; Voronoi regions)
    p    - point coordinates 
    s    - summary (stderr)

More formats:
    Fa   - area for each facet
    FA   - compute total area and volume for option 's'
    Fc   - count plus coplanar points for each facet
           use 'Qc' (default) for coplanar and 'Qi' for interior
    FC   - centrum for each facet
    Fd   - use cdd format for input (homogeneous with offset first)
    FD   - use cdd format for numeric output (offset first)
    FF   - facet dump without ridges
    Fi   - inner plane for each facet
    FI   - ID for each facet
    Fm   - merge count for each facet (511 max)
    Fn   - count plus neighboring facets for each facet
    FN   - count plus neighboring facets for each point
    Fo   - outer plane (or max_outside) for each facet
    FO   - options and precision constants
    FP   - nearest vertex for each coplanar point
    FQ   - command used for qconvex
    Fs   - summary: #int (8), dimension, #points, tot vertices, tot facets,
                      for output: #vertices, #facets,
                                  #coplanar points, #non-simplicial facets
                    #real (2), max outer plane, min vertex
    FS   - sizes:   #int (0) 
                    #real (2) tot area, tot volume
    Ft   - triangulation with centrums for non-simplicial facets (OFF format)
    Fv   - count plus vertices for each facet
    FV   - average of vertices (a feasible point for 'H')
    Fx   - extreme points (in order for 2-d)

Geomview output (2-d, 3-d, and 4-d)
    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 facets by area
    Pdk:n - drop facet if normal[k] <= n (default 0.0)
    PDk:n - drop facet if normal[k] >= n
    Pg   - print good facets (needs 'QGn' or 'QVn')
    PFn  - keep facets whose area is at least n
    PG   - print neighbors of good facets
    PMn  - keep n facets 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
qconvex .

Qhull 2011.2 2011/11/21.
Except for 'F.' and 'PG', upper-case options take an argument.

 incidences     mathematica    normals        OFF_format     points
 summary        facet_dump

 Farea          FArea_total    Fcoplanars     FCentrums      Fd_cdd_in
 FD_cdd_out     FFacet_xridge  Finner         FIDs           Fmerges
 Fneighbors     FNeigh_vertex  Fouter         FOptions       FPoint_near
 FQhull         Fsummary       FSize          Fvertices      FVertex_ave
 Fxtremes       FMaple

 Gvertices      Gpoints        Gall_points    Gno_planes     Ginner
 Gcentrums      Ghyperplanes   Gridges        Gouter         GDrop_dim

 PArea_keep     Pdrop d0:0D0   PFacet_area_keep Pgood        PGood_neighbors
 PMerge_keep    Poutput_forced Pprecision_not

 QbBound 0:0.5  QbB_scale_box  Qcoplanar      QGood_point    Qinterior
 QJoggle        Qrandom        QRotate        Qsearch_1st    Qtriangulate
 QVertex_good

 T4_trace       Tcheck_often   Tstatistics    Tverify        Tz_stdout
 TFacet_log     TInput_file    TPoint_trace   TMerge_trace   TOutput_file
 TWide_trace    TVertex_stop   TCone_stop

 Angle_max      Centrum_size   Random_dist    Ucoplanar_max  Wide_outside
qconvex

qconvex- compute the convex hull.  Qhull 2011.2 2011/11/21
    input (stdin): dimension, number of points, point coordinates
    comments start with a non-numeric character

options (qconvex.htm):
    Qt   - triangulated output
    QJ   - joggled input instead of merged facets
    Tv   - verify result: structure, convexity, and point inclusion
    .    - concise list of all options
    -    - one-line description of all options

output options (subset):
    s    - summary of results (default)
    i    - vertices incident to each facet
    n    - normals with offsets
    p    - vertex coordinates (includes coplanar points if 'Qc')
    Fx   - extreme points (convex hull vertices)
    FA   - report total area and volume
    FS   - compute total area and volume
    o    - OFF format (dim, n, points, facets)
    G    - Geomview output (2-d, 3-d, and 4-d)
    m    - Mathematica output (2-d and 3-d)
    QVn  - print facets that include point n, -n if not
    TO file- output results to file, may be enclosed in single quotes

examples:
    rbox c D2 | qconvex s n                    rbox c D2 | qconvex i
    rbox c D2 | qconvex o                      rbox 1000 s | qconvex s Tv FA
    rbox c d D2 | qconvex s Qc Fx              rbox y 1000 W0 | qconvex s n
    rbox y 1000 W0 | qconvex s QJ              rbox d G1 D12 | qconvex QR0 FA Pp
    rbox c D7 | qconvex FA TF1000

rbox c D3 | qconvex s n Qt 
4
12
    -0     -0     -1   -0.5 
    -0     -0     -1   -0.5 
     0     -1      0   -0.5 
     0     -1      0   -0.5 
     1     -0     -0   -0.5 
     1     -0     -0   -0.5 
    -1     -0     -0   -0.5 
    -1     -0     -0   -0.5 
     0      1     -0   -0.5 
     0      1     -0   -0.5 
    -0     -0      1   -0.5 
    -0     -0      1   -0.5 

Convex hull of 8 points in 3-d:

  Number of vertices: 8
  Number of facets: 12
  Number of triangulated facets: 6

Statistics for: rbox c D3 | qconvex s n Qt

  Number of points processed: 8
  Number of hyperplanes created: 11
  Number of distance tests for qhull: 35
  Number of distance tests for merging: 108
  Number of distance tests for checking: 56
  Number of merged facets: 6
  CPU seconds to compute hull (after input):  0

rbox c D2 | qconvex s i 
4
0 2 
1 0 
2 3 
3 1 

Convex hull of 4 points in 2-d:

  Number of vertices: 4
  Number of facets: 4

Statistics for: rbox c D2 | qconvex s i

  Number of points processed: 4
  Number of hyperplanes created: 6
  Number of distance tests for qhull: 6
  CPU seconds to compute hull (after input):  0

rbox c D2 | qconvex o
2
4 4 4
  -0.5   -0.5 
  -0.5    0.5 
   0.5   -0.5 
   0.5    0.5 
2 0 2 
2 1 0 
2 2 3 
2 3 1 
rbox 1000 s | qconvex s Tv FA

Convex hull of 1000 points in 3-d:

  Number of vertices: 1000
  Number of facets: 1996

Statistics for: rbox 1000 s | qconvex s Tv FA

  Number of points processed: 1000
  Number of hyperplanes created: 5545
  Number of distance tests for qhull: 25488
  CPU seconds to compute hull (after input): 0.015
  Total facet area:   3.1201951
  Total volume:       0.51650274


qhull output completed.  Verifying that 1000 points are
below 2.1e-015 of the nearest facet.
rbox c d D2 | qconvex s Qc Fx
4
0
2
3
1

Convex hull of 8 points in 2-d:

  Number of vertices: 4
  Number of coplanar points: 4
  Number of facets: 4

Statistics for: rbox c d D2 | qconvex s Qc Fx

  Number of points processed: 4
  Number of hyperplanes created: 6
  Number of distance tests for qhull: 36
  CPU seconds to compute hull (after input):  0

rbox y 1000 W0 | qconvex s n 
4
4
0.2135848119563837 0.1168704340428167 0.9699086708284445 -0.09806074564261891 
0.08553875579270345 -0.8316435307282575 -0.5486822022403077 -0.1234144150397865 
-0.749541668500822 -0.02779445272382559 -0.6613733859014792 -0.2159556159595279 
0.3120013537900506 0.9498158823230658 -0.02247097950761911 -0.02453689035981485 

Convex hull of 1004 points in 3-d:

  Number of vertices: 4
  Number of facets: 4

Statistics for: rbox y 1000 W0 | qconvex s n

  Number of points processed: 4
  Number of hyperplanes created: 5
  Number of distance tests for qhull: 11001
  CPU seconds to compute hull (after input):  0

rbox y 1000 W0 | qconvex s QJ

Convex hull of 1004 points in 3-d:

  Number of vertices: 51
  Number of facets: 98

Statistics for: rbox y 1000 W0 | qconvex s QJ

  Number of points processed: 52
  Number of hyperplanes created: 181
  Number of distance tests for qhull: 6424
  CPU seconds to compute hull (after input):  0
  Input joggled by: 2.1e-011

rbox d G1 D12 | qconvex QR0 FA

Convex hull of 24 points in 12-d:

  Number of vertices: 24
  Number of facets: 4096

Statistics for: rbox d G1 D12 | qconvex QR0 FA QR1322103880

  Number of points processed: 24
  Number of hyperplanes created: 4108
  Number of distance tests for qhull: 4216
  Number of distance tests for merging: 121128
  Number of distance tests for checking: 196656
  Number of merged facets: 2036
  CPU seconds to compute hull (after input): 0.078
  Approximate facet area:   0.00035546337
  Approximate volume:       8.5511197e-006

rbox c D6 | qconvex FA TF500

Convex hull of 64 points in 6-d:

  Number of vertices: 64
  Number of facets: 12
  Number of non-simplicial facets: 12

Statistics for: rbox c D6 | qconvex FA TF500

  Number of points processed: 64
  Number of hyperplanes created: 359
  Number of distance tests for qhull: 2510
  Number of distance tests for merging: 14115
  Number of distance tests for checking: 832
  Number of merged facets: 1457
  CPU seconds to compute hull (after input): 0.015
  Approximate facet area:   12
  Approximate volume:        1


At 22:04:41 &  0 CPU secs, qhull has created 543 facets and merged 289.
 The current hull contains 91 facets and 23 vertices.  There are 41
 outside points.  Next is point p27(v24), 0.64 above f107.

At 22:04:41 &  0 CPU secs, qhull has created 1061 facets and merged 737.
 The current hull contains 69 facets and 33 vertices.  There are 31
 outside points.  Next is point p22(v34), 0.52 above f610.

At 22:04:41 & 0.015 CPU secs, qhull has created 1575 facets and merged 1230.
 The current hull contains 25 facets and 51 vertices.  There are 13
 outside points.  Next is point p45(v52), 0.45 above f1476.

At 22:04:41 & 0.015 CPU secs, qhull has created 1815 facets and merged 1457.
 The current hull contains 12 facets and 64 vertices.  Last point was p35

Convex hull of 64 points in 6-d:

  Number of vertices: 64
  Number of facets: 12
  Number of non-simplicial facets: 12

Statistics for: rbox c D6 | qconvex FA TF500

  Number of points processed: 64
  Number of hyperplanes created: 359
  Number of distance tests for qhull: 2510
  Number of distance tests for merging: 13995
  Number of distance tests for checking: 0
  Number of merged facets: 1457


First post-merge with 'C3.5e-015' and 'A1.8e+308'

At 22:04:41 & 0.015 CPU secs, qhull has created 1815 facets and merged 1457.
 The current hull contains 12 facets and 64 vertices.  Last point was p35

Testing all coplanar points.
computing area of each facet and volume of the convex hull
rbox c P0 d D2 | qconvex p Fa Fc FP FI Fn FN FS Fv Fx 
2
8
  -0.5   -0.5 
  -0.5    0.5 
   0.5   -0.5 
   0.5    0.5 
     0   -0.5 
     0    0.5 
  -0.5      0 
   0.5      0 
4
     1 
     1 
     1 
     1 
4
1 5
1 7
1 8
1 6
4
3 5 1    0.5 
2 7 2    0.5 
4 8 4    0.5 
4 6 5    0.5 
4
1
2
4
5
4
2 1 2
2 0 3
2 0 3
2 1 2
9
0
2 0 1
2 1 3
2 0 2
2 2 3
1 0
1 3
1 1
1 2
0
2      4      1 
4
2 3 1
2 2 1
2 4 3
2 4 2
4
1
3
4
2
rbox c d D2 | qconvex s i QV0
2
0 2 
1 0 

Convex hull of 8 points in 2-d:

  Number of vertices: 4
  Number of facets: 4
  Number of 'good' facets: 2

Statistics for: rbox c d D2 | qconvex s i QV0

  Number of points processed: 4
  Number of hyperplanes created: 6
  Number of distance tests for qhull: 36
  CPU seconds to compute hull (after input):  0

rbox c | qconvex Q0
QH6029 qhull error: option 'Q0' is not used with this program.
             It may be used with qhull.

While executing:  | qconvex Q0
Options selected for Qhull 2011.2 2011/11/21:
  run-id 586632459
qvoronoi -

qvoronoi- compute the Voronoi diagram
    http://www.qhull.org  2011.2 2011/11/21

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:
    Qz   - add point-at-infinity to Voronoi diagram
    Qs   - search all points for the initial simplex
    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
qvoronoi .

Qhull 2011.2 2011/11/21.
Except for 'F.' and 'PG', upper-case options take an argument.

 OFF_format     p_vertices     i_delaunay     summary        facet_dump

 Fcoincident    Fd_cdd_in      FD_cdd_out     FF-dump-xridge Fi_bounded
 Fxtremes       Fmerges        Fneighbors     FNeigh_region  FOptions
 Fo_unbounded   FPoint_near    FQvoronoi      Fsummary       Fvoronoi
 FIDs

 Gvertices      Gpoints        Gall_points    Gno_planes     Ginner
 Gcentrums      Ghyperplanes   Gridges        Gouter         GDrop_dim

 PArea_keep     Pdrop d0:0D0   Pgood          PFacet_area_keep
 PGood_neighbors PMerge_keep   Poutput_forced Pprecision_not

 QG_vertex_good Qsearch_1st    Qupper_voronoi QV_point_good  Qzinfinite
 T4_trace       Tcheck_often   Tstatistics    Tverify        Tz_stdout
 TFacet_log     TInput_file    TPoint_trace   TMerge_trace   TOutput_file
 TWide_trace    TVertex_stop   TCone_stop

 Angle_max      Centrum_size   Random_dist    Wide_outside
qvoronoi

qvoronoi- compute the Voronoi diagram.  Qhull 2011.2 2011/11/21
    input (stdin): dimension, number of points, point coordinates
    comments start with a non-numeric character

options (qvoronoi.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

rbox c P0 D2 | qvoronoi s o
2
5 5 1
-10.101 -10.101 
     0   -0.5 
  -0.5      0 
   0.5      0 
     0    0.5 
4 4 2 1 3
3 2 0 1
3 4 0 2
3 3 0 1
3 4 0 3

Voronoi diagram by the convex hull of 5 points in 3-d:

  Number of Voronoi regions: 5
  Number of Voronoi vertices: 4

Statistics for: rbox c P0 D2 | qvoronoi s o

  Number of points processed: 5
  Number of hyperplanes created: 7
  Number of facets in hull: 5
  Number of distance tests for qhull: 8
  Number of distance tests for merging: 29
  Number of distance tests for checking: 30
  Number of merged facets: 1
  CPU seconds to compute hull (after input):  0

rbox c P0 D2 | qvoronoi Fi Tv
4
5 0 3 0.7071067811865475 -0.7071067811865475 -0.3535533905932737 
5 0 1 -0.7071067811865475 -0.7071067811865475 -0.3535533905932737 
5 0 2 -0.7071067811865475 0.7071067811865475 -0.3535533905932737 
5 0 4 0.7071067811865475 0.7071067811865475 -0.3535533905932737 

Voronoi ridge statistics
      4 bounded ridges
      0   max. distance of midpoint to ridge
      4 bounded ridges with ok normal
2.2e-016   ave. angle to ridge
2.2e-016   max. angle to ridge

Output completed.  Verifying that all points are below outer planes of
all facets.  Will make 20 distance computations.
rbox c P0 D2 | qvoronoi Fo
4
5 1 2     -1      0   -0.5 
5 1 3      1     -0     -0 
5 2 4      1     -0     -0 
5 3 4     -1      0    0.5 
rbox c P0 D2 | qvoronoi Fv
8
4 0 3 1 3
4 0 1 1 2
4 0 2 2 4
4 0 4 3 4
4 1 2 0 2
4 1 3 0 1
4 2 4 0 4
4 3 4 0 3
rbox c P0 D2 | qvoronoi s Qu Qt Fv
4
4 1 3 0 2
4 1 2 0 1
4 2 4 0 1
4 3 4 0 2

Furthest-site Voronoi vertices by the convex hull of 5 points in 3-d:

  Number of Voronoi regions: 5
  Number of Voronoi vertices: 2
  Number of triangulated facets: 1

Statistics for: rbox c P0 D2 | qvoronoi s Qu Qt Fv

  Number of points processed: 5
  Number of hyperplanes created: 7
  Number of facets in hull: 6
  Number of distance tests for qhull: 8
  Number of distance tests for merging: 33
  Number of distance tests for checking: 30
  Number of merged facets: 1
  CPU seconds to compute hull (after input):  0

rbox c P0 D2 | qvoronoi Qu Fo
4
5 1 2     -0      1      0 
5 1 3      1     -0     -0 
5 2 4      1     -0     -0 
5 3 4     -0      1     -0 
rbox c G1 d D2 | qvoronoi s p 
2
9
     0  -1.75 
 -1.75      0 
-0.5833333333333334 -0.5833333333333333 
     0   1.75 
-0.5833333333333333 0.5833333333333334 
  1.75      0 
0.5833333333333334 -0.5833333333333333 
0.5833333333333334 0.5833333333333333 
     0      0 

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 c G1 d D2 | qvoronoi s p

  Number of points processed: 8
  Number of hyperplanes created: 17
  Number of facets in hull: 10
  Number of distance tests for qhull: 42
  Number of distance tests for merging: 103
  Number of distance tests for checking: 80
  Number of merged facets: 3
  CPU seconds to compute hull (after input):  0

rbox c G1 d D2 | qvoronoi QJ p 
2
10
-2.57276422388486e-011 1.749999999802917 
-2.05463598231932e-011 -1.750000000331276 
0.5833333333083517 -0.5833333333052267 
1.862254794815499e-011 -7.815052443180038e-011 
1.750000000002628 3.498100747664324e-011 
0.5833333333132078 0.5833333333796981 
-0.5833333332888888 0.5833333333317835 
-7.592293460589872e-011 -7.815050692933324e-011 
-1.750000000049026 2.293196879294717e-011 
-0.5833333333977308 -0.5833333333229861 
rbox c P-0.1 P+0.1 P+0.1 D2 | qvoronoi s Fc FP FQ Fn FN 
4
0
1 1
0
0
1
2 1 8      0 
rbox c P-0.1 P+0.1 P+0.1 D2 | qvoronoi s Fc FP FQ Fn FN
4
3 -1 2 3
3 -1 2 3
4 -1 0 1 3
4 0 -1 2 1
7
3 3 2 0
1 1
3 3 1 2
3 0 -1 2
3 3 -1 0
3 1 -1 2
3 1 -1 3

Voronoi diagram by the convex hull of 7 points in 3-d:

  Number of Voronoi regions: 6
  Number of nearly incident points: 1
  Number of Voronoi vertices: 4
  Number of non-simplicial Voronoi vertices: 2

Statistics for: rbox c P-0.1 P+0.1 P+0.1 D2 | qvoronoi s Fc FP FQ Fn FN

  Number of points processed: 6
  Number of hyperplanes created: 8
  Number of facets in hull: 5
  Number of distance tests for qhull: 33
  Number of distance tests for merging: 49
  Number of distance tests for checking: 43
  Number of merged facets: 3
  CPU seconds to compute hull (after input):  0

rbox P0 c D2 | qvoronoi s Fv QV0
4
4 0 3 1 3
4 0 1 1 2
4 0 2 2 4
4 0 4 3 4

Voronoi diagram by the convex hull of 5 points in 3-d:

  Number of Voronoi regions: 5
  Number of 'good' Voronoi vertices: 4

Statistics for: rbox P0 c D2 | qvoronoi s Fv QV0

  Number of points processed: 5
  Number of hyperplanes created: 7
  Number of facets in hull: 5
  Number of distance tests for qhull: 8
  Number of distance tests for merging: 29
  Number of distance tests for checking: 30
  Number of merged facets: 1
  CPU seconds to compute hull (after input):  0

qdelaunay -

qdelaunay- compute the Delaunay triangulation
    http://www.qhull.org  2011.2 2011/11/21

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 Delaunay triangulation
    Qt   - triangulated output
    QJ   - joggled input instead of merged facets

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 Delaunay triangulation
    QGn  - print Delaunay region if visible from point n, -n if not
    QVn  - print Delaunay regions that include 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   - print 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 outside point (before roundoff)

Output formats (may be combined; if none, produces a summary to stdout):
    f    - facet dump
    G    - Geomview output (see below)
    i    - vertices incident to each Delaunay region
    m    - Mathematica output (2-d only, lifted to a paraboloid)
    o    - OFF format (dim, points, and facets as a paraboloid)
    p    - point coordinates (lifted to a paraboloid)
    s    - summary (stderr)

More formats:
    Fa   - area for each Delaunay region
    FA   - compute total area for option 's'
    Fc   - count plus coincident points for each Delaunay region
    Fd   - use cdd format for input (homogeneous with offset first)
    FD   - use cdd format for numeric output (offset first)
    FF   - facet dump without ridges
    FI   - ID of each Delaunay region
    Fm   - merge count for each Delaunay region (511 max)
    FM   - Maple output (2-d only, lifted to a paraboloid)
    Fn   - count plus neighboring region for each Delaunay region
    FN   - count plus neighboring region for each point
    FO   - options and precision constants
    FP   - nearest point and distance for each coincident point
    FQ   - command used for qdelaunay
    Fs   - summary: #int (8), dimension, #points, tot vertices, tot facets,
                    for output: #vertices, #Delaunay regions,
                                #coincident points, #non-simplicial regions
                    #real (2), max outer plane, min vertex
    FS   - sizes:   #int (0)
                    #real (2), tot area, 0
    Fv   - count plus vertices for each Delaunay region
    Fx   - extreme points of Delaunay triangulation (on convex hull)

Geomview options (2-d and 3-d)
    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
    Gt   - transparent outer ridges to view 3-d Delaunay

Print options:
    PAn  - keep n largest Delaunay regions by area
    Pdk:n - drop facet if normal[k] <= n (default 0.0)
    PDk:n - drop facet if normal[k] >= n
    Pg   - print good Delaunay regions (needs 'QGn' or 'QVn')
    PFn  - keep Delaunay regions whose area is at least n
    PG   - print neighbors of good regions (needs 'QGn' or 'QVn')
    PMn  - keep n Delaunay regions 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
qdelaunay .

Qhull 2011.2 2011/11/21.
Except for 'F.' and 'PG', upper-case options take an argument.

 incidences     mathematica    OFF_format     points_lifted  summary
 facet_dump

 Farea          FArea_total    Fcoincident    Fd_cdd_in      FD_cdd_out
 FF_dump_xridge FIDs           Fmerges        Fneighbors     FNeigh_vertex
 FOptions       FPoint_near    FQdelaun       Fsummary       FSize
 Fvertices      Fxtremes       FMaple

 Gvertices      Gpoints        Gall_points    Gno_planes     Ginner
 Gcentrums      Ghyperplanes   Gridges        Gouter         GDrop_dim
 Gtransparent

 PArea_keep     Pdrop d0:0D0   Pgood          PFacet_area_keep
 PGood_neighbors PMerge_keep   Poutput_forced Pprecision_not

 QGood_point    QJoggle        Qsearch_1st    Qtriangulate   QupperDelaunay
 QVertex_good   Qzinfinite

 T4_trace       Tcheck_often   Tstatistics    Tverify        Tz_stdout
 TFacet_log     TInput_file    TPoint_trace   TMerge_trace   TOutput_file
 TWide_trace    TVertex_stop   TCone_stop

 Angle_max      Centrum_size   Random_dist    Wide_outside
qdelaunay

qdelaunay- compute the Delaunay triangulation.  Qhull 2011.2 2011/11/21
    input (stdin): dimension, number of points, point coordinates
    comments start with a non-numeric character

options (qdelaun.htm):
    Qu   - furthest-site Delaunay triangulation
    Qt   - triangulated output
    QJ   - joggled input instead of merged facets
    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)
    i    - vertices incident to each Delaunay region
    Fx   - extreme points (vertices of the convex hull)
    o    - OFF format (shows the points lifted to a paraboloid)
    G    - Geomview output (2-d and 3-d points lifted to a paraboloid)
    m    - Mathematica output (2-d inputs lifted to a paraboloid)
    QVn  - print Delaunay regions that include point n, -n if not
    TO file- output results to file, may be enclosed in single quotes

examples:
    rbox c P0 D2 | qdelaunay s o          rbox c P0 D2 | qdelaunay i
    rbox c P0 D2 | qdelaunay Fv           rbox c P0 D2 | qdelaunay s Qu Fv
    rbox c G1 d D2 | qdelaunay s i        rbox c G1 d D2 | qdelaunay Qt
    rbox M3,4 z 100 D2 | qdelaunay s      rbox M3,4 z 100 D2 | qdelaunay s Qt

rbox c P0 D2 | qdelaunay s o
3
5 4 6
     0      0      0 
  -0.5   -0.5      1 
  -0.5    0.5      1 
   0.5   -0.5      1 
   0.5    0.5      1 
3 3 0 1 
3 0 2 1 
3 4 0 3 
3 0 4 2 

Delaunay triangulation by the convex hull of 5 points in 3-d:

  Number of input sites: 5
  Number of Delaunay regions: 4

Statistics for: rbox c P0 D2 | qdelaunay s o

  Number of points processed: 5
  Number of hyperplanes created: 7
  Number of facets in hull: 5
  Number of distance tests for qhull: 8
  Number of distance tests for merging: 29
  Number of distance tests for checking: 30
  Number of merged facets: 1
  CPU seconds to compute hull (after input):  0

rbox c P0 D2 | qdelaunay i
4
3 0 1 
0 2 1 
4 0 3 
0 4 2 
rbox c P0 D2 | qdelaunay Fv
4
3 0 3 1
3 0 2 1
3 4 0 3
3 4 0 2
rbox c P0 D2 | qdelaunay s Qu Qt Fv
2
3 4 2 1
3 4 3 1

Furthest-site Delaunay triangulation by the convex hull of 5 points in 3-d:

  Number of input sites: 5
  Number of Delaunay regions: 2
  Number of triangulated facets: 1

Statistics for: rbox c P0 D2 | qdelaunay s Qu Qt Fv

  Number of points processed: 5
  Number of hyperplanes created: 7
  Number of facets in hull: 6
  Number of distance tests for qhull: 8
  Number of distance tests for merging: 33
  Number of distance tests for checking: 30
  Number of merged facets: 1
  CPU seconds to compute hull (after input):  0

rbox c G1 d D2 | qdelaunay s i
9
2 4 0 
6 1 0 
4 6 0 
5 3 1 
6 5 1 
3 7 2 
7 4 2 
5 7 3 
4 7 5 6 

Delaunay triangulation by the convex hull of 8 points in 3-d:

  Number of input sites: 8
  Number of Delaunay regions: 9
  Number of non-simplicial Delaunay regions: 1

Statistics for: rbox c G1 d D2 | qdelaunay s i

  Number of points processed: 8
  Number of hyperplanes created: 17
  Number of facets in hull: 10
  Number of distance tests for qhull: 42
  Number of distance tests for merging: 103
  Number of distance tests for checking: 80
  Number of merged facets: 3
  CPU seconds to compute hull (after input):  0

rbox c G1 d D2 | qhull d Qbb Ft
2
9 12 14
    -1     -1 
    -1      1 
     1     -1 
     1      1 
     0   -0.5 
     0    0.5 
  -0.5      0 
   0.5      0 
     0      0 
3 2 4 0 
3 6 1 0 
3 4 6 0 
3 5 3 1 
3 6 5 1 
3 3 7 2 
3 7 4 2 
3 5 7 3 
3 8 5 6 
3 8 6 4 
3 8 4 7 
3 8 7 5 
rbox c G1 d D2 | qhull d Qbb QJ s Ft
2
8 10 15
-1.000000000061343 -0.999999999957446 
-0.9999999999945448 0.9999999999532114 
1.000000000029778 -0.9999999999701502 
0.9999999999806051 1.000000000003233 
-7.748592400792811e-011 -0.5000000000743411 
2.849349704487493e-011 0.49999999991804 
-0.5000000000721135 -1.37371989142159e-011 
0.5000000000148132 7.166044474310069e-011 
3 1 5 3 
3 2 4 0 
3 7 4 2 
3 4 7 5 
3 3 7 2 
3 5 7 3 
3 1 6 5 
3 6 4 5 
3 6 1 0 
3 4 6 0 

Delaunay triangulation by the convex hull of 8 points in 3-d:

  Number of input sites: 8
  Number of Delaunay regions: 10

Statistics for: rbox c G1 d D2 | qhull d Qbb QJ s Ft

  Number of points processed: 8
  Number of hyperplanes created: 20
  Number of facets in hull: 12
  Number of distance tests for qhull: 32
  CPU seconds to compute hull (after input):  0
  Input joggled by: 8.3e-011

rbox M3,4 z 100 D2 | qdelaunay s

Delaunay triangulation by the convex hull of 100 points in 3-d:

  Number of input sites: 100
  Number of Delaunay regions: 81
  Number of non-simplicial Delaunay regions: 81

Statistics for: rbox M3,4 z 100 D2 | qdelaunay s

  Number of points processed: 100
  Number of hyperplanes created: 262
  Number of facets in hull: 86
  Number of distance tests for qhull: 1652
  Number of distance tests for merging: 2778
  Number of distance tests for checking: 1348
  Number of merged facets: 152
  CPU seconds to compute hull (after input):  0

rbox c P-0.1 P+0.1 P+0.1 D2 | qdelaunay s Fx Fa Fc FP FQ Fn FN
4
4
3
6
5
4
   0.2 
   0.2 
   0.3 
   0.3 
4
0
1 1
0
0
1
2 1 8      0 
rbox c P-0.1 P+0.1 P+0.1 D2 | qdelaunay s Fx Fa Fc FP FQ Fn FN
4
3 -1 2 3
3 -1 2 3
4 -1 0 1 3
4 0 -1 2 1
7
3 3 2 0
1 1
3 3 1 2
3 0 -1 2
3 3 -1 0
3 1 -1 2
3 1 -1 3

Delaunay triangulation by the convex hull of 7 points in 3-d:

  Number of input sites: 6
  Number of nearly incident points: 1
  Number of Delaunay regions: 4
  Number of non-simplicial Delaunay regions: 2

Statistics for: rbox c P-0.1 P+0.1 P+0.1 D2 | qdelaunay s Fx Fa Fc FP FQ Fn FN

  Number of points processed: 6
  Number of hyperplanes created: 8
  Number of facets in hull: 5
  Number of distance tests for qhull: 33
  Number of distance tests for merging: 49
  Number of distance tests for checking: 42
  Number of merged facets: 3
  CPU seconds to compute hull (after input):  0
  Approximate facet area:    1

rbox P0 P0 c D2 | qdelaunay s FP QV0
1
0 1 2      0 

Delaunay triangulation by the convex hull of 6 points in 3-d:

  Number of input sites: 5
  Number of nearly incident points: 1
  Number of 'good' Delaunay regions: 4

Statistics for: rbox P0 P0 c D2 | qdelaunay s FP QV0

  Number of points processed: 5
  Number of hyperplanes created: 7
  Number of facets in hull: 5
  Number of distance tests for qhull: 20
  Number of distance tests for merging: 29
  Number of distance tests for checking: 36
  Number of merged facets: 1
  CPU seconds to compute hull (after input):  0

qhalf -

qhalf- compute the intersection of halfspaces about a point
    http://www.qhull.org  2011.2 2011/11/21

input (stdin):
    optional interior point: dimension, 1, coordinates
    first lines: dimension+1 and number of halfspaces
    other lines: halfspace coefficients followed by offset
    comments:    start with a non-numeric character

options:
    Hn,n - specify coordinates of interior point
    Qt   - triangulated output
    QJ   - joggled input instead of merged facets
    Qc   - keep coplanar halfspaces
    Qi   - keep other redundant halfspaces

Qhull control options:
    QJn  - randomly joggle input in range [-n,n]
    Qbk:0Bk:0 - remove k-th coordinate from input
    Qs   - search all halfspaces for the initial simplex
    QGn  - print intersection if visible to halfspace n, -n for not
    QVn  - print intersections for halfspace n, -n if not

Trace options:
    T4   - trace at level n, 4=all, 5=mem/gauss, -1= events
    Tc   - check frequently during execution
    Ts   - print statistics
    Tv   - verify result: structure, convexity, and redundancy
    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 halfspace n added to intersection
    TMn  - turn on tracing at merge n
    TWn  - trace merge facets when width > n
    TVn  - stop qhull after adding halfspace n, -n for before (see TCn)
    TCn  - stop qhull after building cone for halfspace 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]
    Un   - max distance below plane for a new, coplanar halfspace
    Wn   - min facet width for outside halfspace (before roundoff)

Output formats (may be combined; if none, produces a summary to stdout):
    f    - facet dump
    G    - Geomview output (dual convex hull)
    i    - non-redundant halfspaces incident to each intersection
    m    - Mathematica output (dual convex hull)
    o    - OFF format (dual convex hull: dimension, points, and facets)
    p    - vertex coordinates of dual convex hull (coplanars if 'Qc' or 'Qi')
    s    - summary (stderr)

More formats:
    Fc   - count plus redundant halfspaces for each intersection
         -   Qc (default) for coplanar and Qi for other redundant
    Fd   - use cdd format for input (homogeneous with offset first)
    FF   - facet dump without ridges
    FI   - ID of each intersection
    Fm   - merge count for each intersection (511 max)
    FM   - Maple output (dual convex hull)
    Fn   - count plus neighboring intersections for each intersection
    FN   - count plus intersections for each non-redundant halfspace
    FO   - options and precision constants
    Fp   - dim, count, and intersection coordinates
    FP   - nearest halfspace and distance for each redundant halfspace
    FQ   - command used for qhalf
    Fs   - summary: #int (8), dim, #halfspaces, #non-redundant, #intersections
                      for output: #non-redundant, #intersections, #coplanar
                                  halfspaces, #non-simplicial intersections
                    #real (2), max outer plane, min vertex
    Fv   - count plus non-redundant halfspaces for each intersection
    Fx   - non-redundant halfspaces

Geomview output (2-d, 3-d and 4-d; dual convex hull)
    Ga   - all points (i.e., transformed halfspaces) as dots
     Gp  -  coplanar points and vertices as radii
     Gv  -  vertices (i.e., non-redundant halfspaces) as spheres
    Gi   - inner planes (i.e., halfspace intersections) 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 facets (i.e., intersections) by area
    Pdk:n- drop facet if normal[k] <= n (default 0.0)
    PDk:n- drop facet if normal[k] >= n
    Pg   - print good facets (needs 'QGn' or 'QVn')
    PFn  - keep facets whose area is at least n
    PG   - print neighbors of good facets
    PMn  - keep n facets 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
qhalf .

Qhull 2011.2 2011/11/21.
Except for 'F.' and 'PG', upper_case options take an argument.

 incidences     Geomview       mathematica    OFF_format     point_dual
 summary        facet_dump

 Fc_redundant   Fd_cdd_in      FF_dump_xridge FIDs           Fmerges
 Fneighbors     FN_intersect   FOptions       Fp_coordinates FP_nearest
 FQhalf         Fsummary       Fv_halfspace   FMaple         Fx_non_redundant

 Gvertices      Gpoints        Gall_points    Gno_planes     Ginner
 Gcentrums      Ghyperplanes   Gridges        Gouter         GDrop_dim

 PArea_keep     Pdrop d0:0D0   Pgood          PFacet_area_keep
 PGood_neighbors PMerge_keep   Poutput_forced Pprecision_not

 Qbk:0Bk:0_drop Qcoplanar      QG_half_good   Qi_redundant   QJoggle
 Qsearch_1st    Qtriangulate   QVertex_good

 T4_trace       Tcheck_often   Tstatistics    Tverify        Tz_stdout
 TFacet_log     TInput_file    TPoint_trace   TMerge_trace   TOutput_file
 TWide_trace    TVertex_stop   TCone_stop

 Angle_max      Centrum_size   Random_dist    Ucoplanar_max  Wide_outside
qhalf

qhalf- halfspace intersection about a point.  Qhull 2011.2 2011/11/21
    input (stdin): [dim, 1, interior point], dim+1, n, coefficients+offset
    comments start with a non-numeric character

options (qhalf.htm):
    Hn,n - specify coordinates of interior point
    Qt   - triangulated output
    QJ   - joggled input instead of merged facets
    Tv   - verify result: structure, convexity, and redundancy
    .    - concise list of all options
    -    - one-line description of all options

output options (subset):
    s    - summary of results (default)
    Fp   - intersection coordinates
    Fv   - non-redundant halfspaces incident to each intersection
    Fx   - non-redundant halfspaces
    o    - OFF file format (dual convex hull)
    G    - Geomview output (dual convex hull)
    m    - Mathematica output (dual convex hull)
    QVn  - print intersections for halfspace n, -n if not
    TO file - output results to file, may be enclosed in single quotes

examples:
    rbox d | qconvex FQ n | qhalf s H0,0,0 Fp
    rbox c | qconvex FQ FV n | qhalf s i
    rbox c | qconvex FQ FV n | qhalf s o

rbox d | qhull FQ n | qhalf s Qt H0,0,0 Fp
3
12
     0      0    0.5 
     0      0    0.5 
     0    0.5      0 
     0    0.5      0 
  -0.5      0      0 
  -0.5      0      0 
   0.5      0      0 
   0.5      0      0 
     0   -0.5      0 
     0   -0.5      0 
     0      0   -0.5 
     0      0   -0.5 

Halfspace intersection by the convex hull of 8 points in 3-d:

  Number of halfspaces: 8
  Number of non-redundant halfspaces: 8
  Number of intersection points: 12
  Number of triangulated facets: 6

Statistics for: rbox d | qhull FQ n | qhalf s Qt H0,0,0 Fp

  Number of points processed: 8
  Number of hyperplanes created: 11
  Number of distance tests for qhull: 33
  Number of distance tests for merging: 119
  Number of distance tests for checking: 56
  Number of merged facets: 7
  CPU seconds to compute hull (after input):  0

rbox c | qhull FQ FV n | qhalf s i
8
4 5 3 
5 4 2 
0 4 3 
4 0 2 
1 5 2 
5 1 3 
1 0 3 
0 1 2 

Halfspace intersection by the convex hull of 6 points in 3-d:

  Number of halfspaces: 6
  Number of non-redundant halfspaces: 6
  Number of intersection points: 8

Statistics for: rbox c | qhull FQ FV n | qhalf s i

  Number of points processed: 6
  Number of hyperplanes created: 11
  Number of distance tests for qhull: 11
  Number of distance tests for merging: 45
  Number of distance tests for checking: 54
  Number of merged facets: 1
  CPU seconds to compute hull (after input):  0

rbox c | qhull FQ FV n | qhalf o
3
6 8 12
    -0     -0     -2 
     0     -2      0 
     2     -0     -0 
    -2     -0     -0 
     0      2     -0 
    -0     -0      2 
3 4 5 3 
3 5 4 2 
3 0 4 3 
3 4 0 2 
3 1 5 2 
3 5 1 3 
3 1 0 3 
3 0 1 2 
rbox d D2 | qhull FQ n | qhalf s H0 Fc FP Fn FN FQ Fv Fx
4
0
0
0
0
0
4
2 1 2
2 0 3
2 0 3
2 1 2
4
2 0 1
2 0 2
2 1 3
2 2 3
rbox d D2 | qhull FQ n | qhalf s H0 Fc FP Fn FN FQ Fv Fx
4
2 1 0
2 2 0
2 3 1
2 3 2
4
1
0
2
3

Halfspace intersection by the convex hull of 4 points in 2-d:

  Number of halfspaces: 4
  Number of non-redundant halfspaces: 4
  Number of intersection points: 4

Statistics for: rbox d D2 | qhull FQ n | qhalf s H0 Fc FP Fn FN FQ Fv Fx

  Number of points processed: 4
  Number of hyperplanes created: 6
  Number of distance tests for qhull: 6
  CPU seconds to compute hull (after input):  0

echo === check quality of Qhull for ${d:-`hostname`} ${d:-`date`}
hostname
date
=== check quality of Qhull for ispy Wed Nov 23 22:04:42 EST 2011
rbox 1000 W0 | qhull QR2 QJ s Fs Tv
10 3 1000 144 284 144 284 0 0 33 0
2 5.045982607091333e-011 -5.045982607091333e-011

Convex hull of 1000 points in 3-d:

  Number of vertices: 144
  Number of facets: 284

Statistics for: rbox 1000 W0 | qhull QR2 QJ s Fs Tv QR2

  Number of points processed: 171
  Number of hyperplanes created: 1233
  Number of distance tests for qhull: 33562
  CPU seconds to compute hull (after input):  0
  After 1 retries, input joggled by: 2.9e-011


Output completed.  Verifying that all points are below 2.9e-015 of
all facets.  Will make 284000 distance computations.
rbox 1000 W0 | qhull QR2 s Fs Tv
10 3 1000 84 98 84 98 0 6 13 0
2 1.942124858599573e-015 -1.241679291552168e-015

Convex hull of 1000 points in 3-d:

  Number of vertices: 84
  Number of facets: 98
  Number of non-simplicial facets: 6

Statistics for: rbox 1000 W0 | qhull QR2 s Fs Tv QR2

  Number of points processed: 97
  Number of hyperplanes created: 299
  Number of distance tests for qhull: 18051
  Number of distance tests for merging: 3325
  Number of distance tests for checking: 15397
  Number of merged facets: 92
  CPU seconds to compute hull (after input): 0.015


Output completed.  Verifying that all points are below outer planes of
all facets.  Will make 98000 distance computations.
rbox 1000 s | qhull C0.02 Qc Tv

Convex hull of 1000 points in 3-d:

  Number of vertices: 114
  Number of coplanar points: 886
  Number of facets: 60
  Number of non-simplicial facets: 60

Statistics for: rbox 1000 s | qhull C0.02 Qc Tv

  Number of points processed: 1000
  Number of hyperplanes created: 5545
  Number of distance tests for qhull: 83858
  Number of distance tests for merging: 82226
  Number of distance tests for checking: 7929
  Number of merged facets: 1936
  CPU seconds to compute hull (after input): 0.015
  Maximum distance of vertex below facet: -0.072 (1.2x)


Output completed.  Verifying that all points are below outer planes of
all facets.  Will make 60000 distance computations.
rbox 500 s D4 | qhull C0.01 Qc Tv

Convex hull of 500 points in 4-d:

  Number of vertices: 467
  Number of coplanar points: 33
  Number of facets: 357
  Number of non-simplicial facets: 343

Statistics for: rbox 500 s D4 | qhull C0.01 Qc Tv

  Number of points processed: 500
  Number of hyperplanes created: 11284
  Number of distance tests for qhull: 41395
  Number of distance tests for merging: 162691
  Number of distance tests for checking: 20813
  Number of merged facets: 2809
  CPU seconds to compute hull (after input): 0.046
  Maximum distance of vertex below facet: -0.073 (1.8x)


Output completed.  Verifying that all points are below outer planes of
all facets.  Will make 178500 distance computations.
rbox 1000 s | qhull C-0.02 Qc Tv

Convex hull of 1000 points in 3-d:

  Number of vertices: 47
  Number of coplanar points: 953
  Number of facets: 47
  Number of non-simplicial facets: 35

Statistics for: rbox 1000 s | qhull C-0.02 Qc Tv

  Number of points processed: 50
  Number of hyperplanes created: 199
  Number of distance tests for qhull: 52519
  Number of distance tests for merging: 2966
  Number of distance tests for checking: 24231
  Number of merged facets: 98
  CPU seconds to compute hull (after input): 0.015
  Maximum distance of point above facet: 0.065 (1.1x)
  Maximum distance of vertex below facet: -0.062 (1.0x)


Output completed.  Verifying that all points are below outer planes of
all facets.  Will make 47000 distance computations.
rbox 1000 s | qhull C-0.02 Qc Tv

Convex hull of 1000 points in 3-d:

  Number of vertices: 47
  Number of coplanar points: 953
  Number of facets: 47
  Number of non-simplicial facets: 35

Statistics for: rbox 1000 s | qhull C-0.02 Qc Tv

  Number of points processed: 50
  Number of hyperplanes created: 199
  Number of distance tests for qhull: 52519
  Number of distance tests for merging: 2966
  Number of distance tests for checking: 24231
  Number of merged facets: 98
  CPU seconds to compute hull (after input):  0
  Maximum distance of point above facet: 0.065 (1.1x)
  Maximum distance of vertex below facet: -0.062 (1.0x)


Output completed.  Verifying that all points are below outer planes of
all facets.  Will make 47000 distance computations.
rbox 1000 s D4 | qhull C-0.01 Qc Tv

Convex hull of 1000 points in 4-d:

  Number of vertices: 117
  Number of coplanar points: 883
  Number of facets: 232
  Number of non-simplicial facets: 149

Statistics for: rbox 1000 s D4 | qhull C-0.01 Qc Tv

  Number of points processed: 117
  Number of hyperplanes created: 1232
  Number of distance tests for qhull: 372627
  Number of distance tests for merging: 31334
  Number of distance tests for checking: 122166
  Number of merged facets: 1167
  CPU seconds to compute hull (after input): 0.031
  Maximum distance of point above facet: 0.062 (1.5x)
  Maximum distance of vertex below facet: -0.064 (1.6x)


Output completed.  Verifying that all points are below outer planes of
all facets.  Will make 232000 distance computations.
rbox 200 s D5 | qhull C-0.01 Qx Qc Tv 

Convex hull of 200 points in 5-d:

  Number of vertices: 160
  Number of coplanar points: 40
  Number of facets: 388
  Number of non-simplicial facets: 342

Statistics for: rbox 200 s D5 | qhull C-0.01 Qx Qc Tv

  Number of points processed: 160
  Number of hyperplanes created: 4522
  Number of distance tests for qhull: 235312
  Number of distance tests for merging: 143132
  Number of distance tests for checking: 46266
  Number of merged facets: 5940
  CPU seconds to compute hull (after input): 0.046
  Maximum distance of point above facet: 0.059 (1.2x)
  Maximum distance of vertex below facet: -0.11 (2.1x)

QH7076 qhull input warning: exact merge ('Qx').  Verify may report that a point
is outside of a facet.  See qh-optq.htm#Qx

Output completed.  Verifying that all points are below outer planes of
all facets.  Will make 77600 distance computations.
rbox 100 s D6 | qhull C-0.001 Qx Qc Tv

Convex hull of 100 points in 6-d:

  Number of vertices: 100
  Number of facets: 4465
  Number of non-simplicial facets: 809

Statistics for: rbox 100 s D6 | qhull C-0.001 Qx Qc Tv

  Number of points processed: 100
  Number of hyperplanes created: 20175
  Number of distance tests for qhull: 33383
  Number of distance tests for merging: 299744
  Number of distance tests for checking: 66106
  Number of merged facets: 3948
  CPU seconds to compute hull (after input): 0.109
  Maximum distance of point above facet: 0.0029 (0.5x)
  Maximum distance of vertex below facet: -0.014 (2.4x)

QH7076 qhull input warning: exact merge ('Qx').  Verify may report that a point
is outside of a facet.  See qh-optq.htm#Qx

Output completed.  Verifying that all points are below outer planes of
all facets.  Will make 446500 distance computations.
rbox 1000 W1e-4 | qhull C-1e-6 Qc Tv

Convex hull of 1000 points in 3-d:

  Number of vertices: 144
  Number of coplanar points: 75
  Number of facets: 243
  Number of non-simplicial facets: 26

Statistics for: rbox 1000 W1e-4 | qhull C-1e-6 Qc Tv

  Number of points processed: 185
  Number of hyperplanes created: 939
  Number of distance tests for qhull: 42000
  Number of distance tests for merging: 8652
  Number of distance tests for checking: 5684
  Number of merged facets: 145
  CPU seconds to compute hull (after input): 0.015
  Maximum distance of point above facet: 1.3e-006 (0.4x)
  Maximum distance of vertex below facet: -3.2e-006 (1.1x)


Output completed.  Verifying that all points are below outer planes of
all facets.  Will make 243000 distance computations.
rbox 1000 W5e-4 D4 | qhull C-1e-5 Qc Tv

Convex hull of 1000 points in 4-d:

  Number of vertices: 290
  Number of coplanar points: 433
  Number of facets: 1296
  Number of non-simplicial facets: 154

Statistics for: rbox 1000 W5e-4 D4 | qhull C-1e-5 Qc Tv

  Number of points processed: 368
  Number of hyperplanes created: 6238
  Number of distance tests for qhull: 182095
  Number of distance tests for merging: 75580
  Number of distance tests for checking: 52937
  Number of merged facets: 1353
  CPU seconds to compute hull (after input): 0.031
  Maximum distance of point above facet: 9.1e-005 (2.3x)
  Maximum distance of vertex below facet: -0.00012 (2.9x)


Output completed.  Verifying that all points are below outer planes of
all facets.  Will make 1296000 distance computations.
rbox 400 W1e-3 D5 | qhull C-1e-5 Qx Qc Tv

Convex hull of 400 points in 5-d:

  Number of vertices: 305
  Number of coplanar points: 35
  Number of facets: 5552
  Number of non-simplicial facets: 281

Statistics for: rbox 400 W1e-3 D5 | qhull C-1e-5 Qx Qc Tv

  Number of points processed: 336
  Number of hyperplanes created: 24527
  Number of distance tests for qhull: 251847
  Number of distance tests for merging: 277585
  Number of distance tests for checking: 61531
  Number of merged facets: 1482
  CPU seconds to compute hull (after input): 0.093
  Maximum distance of point above facet: 5.5e-005 (1.1x)
  Maximum distance of vertex below facet: -0.0001 (2.1x)

QH7076 qhull input warning: exact merge ('Qx').  Verify may report that a point
is outside of a facet.  See qh-optq.htm#Qx

Output completed.  Verifying that all points are below outer planes of
all facets.  Will make 2220800 distance computations.
echo === check input format etc. ${d:-`date`}
date
=== check input format etc. Wed Nov 23 22:04:43 EST 2011
qhull <<EOF

Convex hull of 4 points in 2-d:

  Number of vertices: 3
  Number of facets: 3

Statistics for: #;laskdjf | qhull

  Number of points processed: 3
  Number of hyperplanes created: 4
  Number of distance tests for qhull: 9
  CPU seconds to compute hull (after input):  0

qhull <<EOF

Convex hull of 4 points in 2-d:

  Number of vertices: 3
  Number of facets: 3

Statistics for: #;laskdjf | qhull

  Number of points processed: 3
  Number of hyperplanes created: 4
  Number of distance tests for qhull: 9
  CPU seconds to compute hull (after input):  0

QH7073 qhull warning: instead of 4 2-dimensional points, input contains
4 points and 1 extra coordinates.  Line 2 is the first
point, line 1 is the first comment, line 2 is the first long line.  Continue with 4 points.
qhull <<EOF
QH7073 qhull warning: instead of 4 2-dimensional points, input contains
3 points and 1 extra coordinates.  Line 2 is the first
point, line 1 is the first comment, line 2 is the first long line.  Continue with 3 points.
QH6013 qhull input error: input is less than 2-dimensional since it has the same x coordinate

While executing: #;laskdjf | qhull
Options selected for Qhull 2011.2 2011/11/21:
  run-id 586666073  _pre-merge  _zero-centrum  _max-width  2
  Error-roundoff 1.7e-015  _one-merge 8.6e-015  _near-inside 4.3e-014
  Visible-distance 3.4e-015  U-coplanar-distance 3.4e-015
  Width-outside 6.9e-015  _wide-facet 2.1e-014
qhull <<EOF
QH7073 qhull warning: instead of 4 2-dimensional points, input contains
3 points and 1 extra coordinates.  Line 3 is the first
point, line 2 is the first comment, line 3 is the first short
line, line 4 is the first long line.  Continue with 3 points.
QH6013 qhull input error: input is less than 2-dimensional since it has the same x coordinate

While executing: #;laskdjf | qhull
Options selected for Qhull 2011.2 2011/11/21:
  run-id 586666073  _pre-merge  _zero-centrum  _max-width  2
  Error-roundoff 1.7e-015  _one-merge 8.6e-015  _near-inside 4.3e-014
  Visible-distance 3.4e-015  U-coplanar-distance 3.4e-015
  Width-outside 6.9e-015  _wide-facet 2.1e-014
qhull <<EOF

Convex hull of 4 points in 2-d:

  Number of vertices: 3
  Number of facets: 3

Statistics for:  | qhull

  Number of points processed: 3
  Number of hyperplanes created: 4
  Number of distance tests for qhull: 9
  CPU seconds to compute hull (after input):  0

qhull d Qz <<EOF

Delaunay triangulation by the convex hull of 5 points in 3-d:

  Number of input sites and at-infinity: 5
  Number of Delaunay regions: 1
  Number of non-simplicial Delaunay regions: 1

Statistics for:  | qhull d Qz

  Number of points processed: 5
  Number of hyperplanes created: 7
  Number of facets in hull: 5
  Number of distance tests for qhull: 6
  Number of distance tests for merging: 29
  Number of distance tests for checking: 30
  Number of merged facets: 1
  CPU seconds to compute hull (after input):  0

QH7073 qhull warning: instead of 5 2-dimensional points, input contains
4 points and 1 extra coordinates.  Line 1 is the first
point, line 1 is the first long line.  Continue with 4 points.
qhull d Q8 Qz <<EOF

Delaunay triangulation by the convex hull of 5 points in 3-d:

  Number of input sites and at-infinity: 5
  Number of Delaunay regions: 1
  Number of non-simplicial Delaunay regions: 1

Statistics for:  | qhull d Q8 Qz

  Number of points processed: 5
  Number of hyperplanes created: 7
  Number of facets in hull: 5
  Number of distance tests for qhull: 6
  Number of distance tests for merging: 29
  Number of distance tests for checking: 30
  Number of merged facets: 1
  CPU seconds to compute hull (after input):  0

QH7073 qhull warning: instead of 5 2-dimensional points, input contains
4 points and 1 extra coordinates.  Line 1 is the first
point, line 1 is the first long line.  Continue with 4 points.
rbox d h | qhull Fd FV n FD Tcv | qhull Fd H Fp Tcv

Output completed.  Verifying that all points are below outer planes of
all facets.  Will make 48 distance computations.
3
6
     0      0    0.5 
     0    0.5      0 
  -0.5      0      0 
   0.5      0      0 
     0   -0.5      0 
     0      0   -0.5 

Output completed.  Verifying that all points are below outer planes of
all facets.  Will make 48 distance computations.
rbox 10 h | qhull Fd FD p Tcv | qhull Fd d Tcv 

Output completed.  Verifying that all points are below 1.7e-015 of
all facets.  Will make 140 distance computations.

Delaunay triangulation by the convex hull of 9 points in 4-d:

  Number of input sites: 9
  Number of Delaunay regions: 11

Statistics for: rbox 10 h | qhull Fd FD p Tcv | qhull Fd d Tcv

  Number of points processed: 9
  Number of hyperplanes created: 34
  Number of facets in hull: 22
  Number of distance tests for qhull: 46
  CPU seconds to compute hull (after input):  0


Output completed.  Verifying that all points are below 2.5e-015 of
all facets.  Will make 99 distance computations.
echo === check rbox ${d:-`date`}
date
=== check rbox Wed Nov 23 22:04:43 EST 2011
rbox 3 n D2
2
3
0.2234321234034673 0.2236923805658635 
-0.4021654933711192 -0.1954478539484068 
0.1079163058734651 -0.2506519423338137 
rbox 3 D2
2 rbox 3 D2
3
-0.3570490315156514 0.07692619792830779 
-0.1013959339832849 -0.1614655765345931 
0.2480525339469802 0.01893219260399437 
rbox 3 h D2
rbox 3 h D2
begin
        3 3 real
     1 0.16093073846859 -0.2370837309743126 
     1 0.3337314574362071 0.02459860548805315 
     1 0.4287583324394733 0.141286041719174 
end
hull
rbox 3 z D2
2 rbox 3 z D2
3
696870 290489 
252081 728918 
932465 -69056 
rbox 3 z h D2
rbox 3 z h D2
begin
        3 3 integer
1 -951655 -464136 
1 -741320 630989 
1 -964370 -171508 
end
hull
rbox 3 B10 D2
2 rbox 3 B10 D2
3
6.07777520648928 9.167769643634342 
2.704250554278727 -9.661033665445665 
7.007182209759188 9.71126632738045 
rbox 3 z B10 D2
2 rbox 3 z B10 D2
3
5 2 
5 -1 
3 9 
rbox 4 L2 r D2
2 rbox 4 L2 r D2
6
   0.5      0
0.1736481776669304 0.1187823492277694
0.1736481776669304 -0.1187823492277694
-0.1736481776669304 0.1187823492277694
-0.1736481776669304 -0.1187823492277694
  -0.5      0
rbox 8 L2 D2
2 rbox 8 L2 D2
8
-9.037112801233254e-006 -0.1339745961747267 
-0.4680200990905804 0.01769241704257374 
-0.05424246984329754 -0.1325023897522757 
0.07519664081423347 -0.1311433207603374 
0.1915694083864337 0.1154536646477271 
0.4772264523261766 0.01275495170842944 
-0.4048291575354194 0.04836693717712182 
0.3401207718037282 -0.07435635943071228 
rbox 4 L4 r D3
3 rbox 4 L4 r D3
12
   0.5      0      0
  0.25      0 0.04782181019473453
  0.25      0 -0.04782181019473453
3.061616997868383e-017    0.5      0
1.530808498934192e-017   0.25 0.04782181019473453
1.530808498934192e-017   0.25 -0.04782181019473453
  -0.5 6.123233995736766e-017      0
 -0.25 3.061616997868383e-017 0.04782181019473453
 -0.25 3.061616997868383e-017 -0.04782181019473453
-9.184850993605148e-017   -0.5      0
-4.592425496802574e-017  -0.25 0.04782181019473453
-4.592425496802574e-017  -0.25 -0.04782181019473453
rbox 4 L4 s D5 W1e-3 
5 rbox 4 L4 s D5 W1e-3
4
-0.1779876601833928 -0.1311655069863711 0.09099061083332423 -0.0147197628642461 0.0491007664448838 
0.08530477699633816 0.08551065958609008 0.2073152333839907 -0.05556059038062115 0.04824638578429845 
-0.2043906348252541 0.01359444539903207 0.07833891569611255 -0.2253375927975145 0.03863248544102459 
0.07678497944972421 0.03657959246981071 0.1769587053753909 0.1423142450665639 -0.04874898497507386 
rbox y
3 rbox y
4
-0.3062920647731908 0.1492658412542808 -0.2890111205997049 
-0.4099055704752966 -0.2829236833219637 -0.09834729097629646 
0.07707741817187275 0.4401626986825491 -0.1855306003107974 
-0.2128018845923262 0.4387234094913336 -0.3756640258949846 
rbox 10 M3,4 
3 rbox 10 M3,4
10
     0      0      0 
     3      4      0 
     6      8      0 
    -4      3      0 
    -1      7      0 
     2     11      0 
    -8      6      0 
    -5     10      0 
    -2     14      0 
     0      0      5 
rbox 10 L2 s D3 | qhull Tcv

Convex hull of 10 points in 3-d:

  Number of vertices: 10
  Number of facets: 16

Statistics for: rbox 10 L2 s D3 | qhull Tcv

  Number of points processed: 10
  Number of hyperplanes created: 29
  Number of distance tests for qhull: 50
  CPU seconds to compute hull (after input):  0


Output completed.  Verifying that all points are below 1.5e-015 of
all facets.  Will make 160 distance computations.
rbox 10 L4 s W1e-3 D3 | qhull Tcv

Convex hull of 10 points in 3-d:

  Number of vertices: 10
  Number of facets: 16

Statistics for: rbox 10 L4 s W1e-3 D3 | qhull Tcv

  Number of points processed: 10
  Number of hyperplanes created: 27
  Number of distance tests for qhull: 43
  CPU seconds to compute hull (after input):  0


Output completed.  Verifying that all points are below 1.9e-015 of
all facets.  Will make 160 distance computations.
rbox 10 L6 D3 | qhull Tcv

Convex hull of 10 points in 3-d:

  Number of vertices: 10
  Number of facets: 16

Statistics for: rbox 10 L6 D3 | qhull Tcv

  Number of points processed: 10
  Number of hyperplanes created: 29
  Number of distance tests for qhull: 48
  CPU seconds to compute hull (after input):  0


Output completed.  Verifying that all points are below 1.7e-015 of
all facets.  Will make 160 distance computations.
rbox 10 L1.1 s D4 | qhull Tcv

Convex hull of 10 points in 4-d:

  Number of vertices: 10
  Number of facets: 26

Statistics for: rbox 10 L1.1 s D4 | qhull Tcv

  Number of points processed: 10
  Number of hyperplanes created: 44
  Number of distance tests for qhull: 53
  CPU seconds to compute hull (after input):  0


Output completed.  Verifying that all points are below 2.4e-015 of
all facets.  Will make 260 distance computations.
rbox y r 100 W0 O0.5 | qhull s p Tcv
3
4
     1    0.5    0.5 
   0.5      1    0.5 
   0.5    0.5      1 
     0      0      0 

Convex hull of 104 points in 3-d:

  Number of vertices: 4
  Number of facets: 4

Statistics for: rbox y r 100 W0 O0.5 | qhull s p Tcv

  Number of points processed: 4
  Number of hyperplanes created: 5
  Number of distance tests for qhull: 1101
  CPU seconds to compute hull (after input):  0


Output completed.  Verifying that all points are below outer planes of
all facets.  Will make 416 distance computations.
rbox x r 100 W0 O0.5 | qhull s Tcv

Convex hull of 100 points in 3-d:

  Number of vertices: 27
  Number of facets: 35
  Number of non-simplicial facets: 4

Statistics for: rbox x r 100 W0 O0.5 | qhull s Tcv

  Number of points processed: 27
  Number of hyperplanes created: 76
  Number of distance tests for qhull: 1285
  Number of distance tests for merging: 774
  Number of distance tests for checking: 905
  Number of merged facets: 15
  CPU seconds to compute hull (after input):  0


Output completed.  Verifying that all points are below outer planes of
all facets.  Will make 3500 distance computations.
rbox 12 D8 | qhull Tcv

Convex hull of 12 points in 8-d:

  Number of vertices: 12
  Number of facets: 92

Statistics for: rbox 12 D8 | qhull Tcv

  Number of points processed: 12
  Number of hyperplanes created: 132
  Number of distance tests for qhull: 91
  CPU seconds to compute hull (after input):  0

QH7076 qhull input warning: exact merge ('Qx').  Verify may report that a point
is outside of a facet.  See qh-optq.htm#Qx

Output completed.  Verifying that all points are below 7.9e-015 of
all facets.  Will make 1104 distance computations.
rbox 12 D9 | qhull Tcv

Convex hull of 12 points in 9-d:

  Number of vertices: 12
  Number of facets: 70

Statistics for: rbox 12 D9 | qhull Tcv

  Number of points processed: 12
  Number of hyperplanes created: 89
  Number of distance tests for qhull: 47
  CPU seconds to compute hull (after input):  0

QH7076 qhull input warning: exact merge ('Qx').  Verify may report that a point
is outside of a facet.  See qh-optq.htm#Qx

Output completed.  Verifying that all points are below 9e-015 of
all facets.  Will make 840 distance computations.
rbox 1000 D4 | qhull s i A-0.97 C0.2 A0.7 Tcv
48
117 111 263 638 
117 263 111 146 
117 263 146 638 
117 146 111 638 
118 111 771 456 
118 111 146 638 
118 146 111 456 
118 771 146 456 
118 771 430 465 
118 771 465 638 
118 771 111 638 
118 146 771 638 
118 465 430 638 
118 430 771 638 
119 771 263 299 
119 430 465 638 
119 111 263 771 
119 771 430 638 
119 465 299 638 
119 263 111 638 
119 430 771 299 
119 111 771 638 
119 299 263 638 
119 430 299 465 
120 430 771 465 
120 299 465 638 
120 263 771 299 
120 146 263 638 
120 299 430 465 
120 465 771 638 
120 263 146 299 
120 771 263 456 
120 299 771 456 
120 771 430 299 
120 771 146 638 
120 146 771 299 
120 263 299 456 
120 263 299 638 
121 111 146 456 
121 146 263 299 
121 771 299 456 
121 771 146 299 
121 263 111 771 
121 146 771 456 
121 299 263 456 
121 111 263 146 
121 771 111 456 
121 263 771 456 

Convex hull of 1000 points in 4-d:

  Number of vertices: 9
  Number of facets: 5
  Number of non-simplicial facets: 4

Statistics for: rbox 1000 D4 | qhull s i A-0.97 C0.2 A0.7 Tcv

  Number of points processed: 116
  Number of hyperplanes created: 2284
  Number of distance tests for qhull: 518926
  Number of distance tests for merging: 76090
  Number of distance tests for checking: 5989
  Number of merged facets: 1889
  CPU seconds to compute hull (after input): 0.062
  Maximum distance of point above facet: 0.07
  Maximum distance of vertex below facet: -0.92 (0.6x)


Output completed.  Verifying that all points are below outer planes of
all facets.  Will make 5000 distance computations.
rbox 3 D2 | qhull Qb0B1:-2 p 
2
3
  -0.5     -2 
-0.1839506649197531 -0.1614655765345934 
0.2480525339469802 -1.552736350137872 
rbox 100 r D2 | qhull Pd0:0.7 PD0:0.8 n Tcv
3
6
0.7501110696304588 0.6613118653236527 -0.4997532801828658 
0.7071067811865485 0.7071067811865467 -0.4997532801828658 
0.7501110696304543 -0.6613118653236579 -0.4997532801828658 
0.7901550123756914 0.6129070536529753 -0.4997532801828658 
0.7901550123756885 -0.6129070536529789 -0.4997532801828659 
0.707106781186544 -0.7071067811865511 -0.4997532801828658 

Output completed.  Verifying that all points are below 1.3e-015 of
all facets.  Will make 600 distance computations.
rbox 1000 s | qhull  C0.05 Tcv

Convex hull of 1000 points in 3-d:

  Number of vertices: 48
  Number of facets: 26
  Number of non-simplicial facets: 26

Statistics for: rbox 1000 s | qhull C0.05 Tcv

  Number of points processed: 1000
  Number of hyperplanes created: 5545
  Number of distance tests for qhull: 55658
  Number of distance tests for merging: 100391
  Number of distance tests for checking: 6790
  Number of merged facets: 1970
  CPU seconds to compute hull (after input): 0.016
  Maximum distance of vertex below facet: -0.21 (1.4x)


Output completed.  Verifying that all points are below outer planes of
all facets.  Will make 26000 distance computations.
rbox 1000 s t | qhull Qm C0.05 Tcv

Convex hull of 1000 points in 3-d:

  Number of vertices: 48
  Number of facets: 26
  Number of non-simplicial facets: 26

Statistics for: rbox 1000 s t1322103884 | qhull Qm C0.05 Tcv

  Number of points processed: 1000
  Number of hyperplanes created: 5578
  Number of distance tests for qhull: 57511
  Number of distance tests for merging: 100160
  Number of distance tests for checking: 6806
  Number of merged facets: 1970
  CPU seconds to compute hull (after input): 0.016
  Maximum distance of vertex below facet: -0.14 (0.9x)


Output completed.  Verifying that all points are below outer planes of
all facets.  Will make 26000 distance computations.
rbox 500 D2 | qhull n A-0.95 C0.1 Tcv
3
4
0.004814116480453948 0.9999884120741163 -0.4852084337990873 
-0.9999815737204624 -0.006070602898184517 -0.4913197370926178 
-0.006423098021811921 -0.9999793716931376 -0.4958250456626817 
0.9999972817907392 0.002331611273993484 -0.4974344499395051 

Output completed.  Verifying that all points are below outer planes of
all facets.  Will make 2000 distance computations.
rbox 500 s P1,1,1 | qhull QgG0 Pp Tcv

Convex hull of 500 points in 3-d:

  Number of vertices: 226
  Number of facets: 448
  Number of 'good' facets: 340

Statistics for: rbox 500 s P1,1,1 | qhull QgG0 Pp Tcv

  Number of points processed: 294
  Number of hyperplanes created: 1612
  Number of distance tests for qhull: 9968
  CPU seconds to compute hull (after input):  0


rbox d | qhull m
{
Polygon[{{      0.00000000,       0.50000000,       0.00000000},
{      0.00000000,       0.00000000,       0.50000000},
{     -0.50000000,       0.00000000,       0.00000000}}],
Polygon[{{      0.00000000,       0.00000000,       0.50000000},
{      0.00000000,       0.50000000,       0.00000000},
{      0.50000000,       0.00000000,       0.00000000}}],
Polygon[{{      0.00000000,       0.00000000,      -0.50000000},
{      0.00000000,       0.50000000,       0.00000000},
{     -0.50000000,       0.00000000,       0.00000000}}],
Polygon[{{      0.00000000,       0.50000000,       0.00000000},
{      0.00000000,       0.00000000,      -0.50000000},
{      0.50000000,       0.00000000,       0.00000000}}],
Polygon[{{      0.00000000,      -0.50000000,       0.00000000},
{      0.00000000,       0.00000000,       0.50000000},
{      0.50000000,       0.00000000,       0.00000000}}],
Polygon[{{      0.00000000,       0.00000000,       0.50000000},
{      0.00000000,      -0.50000000,       0.00000000},
{     -0.50000000,       0.00000000,       0.00000000}}],
Polygon[{{      0.00000000,      -0.50000000,       0.00000000},
{      0.00000000,       0.00000000,      -0.50000000},
{     -0.50000000,       0.00000000,       0.00000000}}],
Polygon[{{      0.00000000,       0.00000000,      -0.50000000},
{      0.00000000,      -0.50000000,       0.00000000},
{      0.50000000,       0.00000000,       0.00000000}}]}
rbox d | qhull FM
PLOT3D(POLYGONS(
[[      0.00000000,       0.50000000,       0.00000000],
[      0.00000000,       0.00000000,       0.50000000],
[     -0.50000000,       0.00000000,       0.00000000]],
[[      0.00000000,       0.00000000,       0.50000000],
[      0.00000000,       0.50000000,       0.00000000],
[      0.50000000,       0.00000000,       0.00000000]],
[[      0.00000000,       0.00000000,      -0.50000000],
[      0.00000000,       0.50000000,       0.00000000],
[     -0.50000000,       0.00000000,       0.00000000]],
[[      0.00000000,       0.50000000,       0.00000000],
[      0.00000000,       0.00000000,      -0.50000000],
[      0.50000000,       0.00000000,       0.00000000]],
[[      0.00000000,      -0.50000000,       0.00000000],
[      0.00000000,       0.00000000,       0.50000000],
[      0.50000000,       0.00000000,       0.00000000]],
[[      0.00000000,       0.00000000,       0.50000000],
[      0.00000000,      -0.50000000,       0.00000000],
[     -0.50000000,       0.00000000,       0.00000000]],
[[      0.00000000,      -0.50000000,       0.00000000],
[      0.00000000,       0.00000000,      -0.50000000],
[     -0.50000000,       0.00000000,       0.00000000]],
[[      0.00000000,       0.00000000,      -0.50000000],
[      0.00000000,      -0.50000000,       0.00000000],
[      0.50000000,       0.00000000,       0.00000000]]));
rbox c D2 | qhull Tcv Q0

Convex hull of 4 points in 2-d:

  Number of vertices: 4
  Number of facets: 4

Statistics for: rbox c D2 | qhull Tcv Q0

  Number of points processed: 4
  Number of hyperplanes created: 6
  Number of distance tests for qhull: 6
  CPU seconds to compute hull (after input):  0


Output completed.  Verifying that all points are below 1.3e-015 of
all facets.  Will make 16 distance computations.
rbox d D2 | qhull Tcv

Convex hull of 4 points in 2-d:

  Number of vertices: 4
  Number of facets: 4

Statistics for: rbox d D2 | qhull Tcv

  Number of points processed: 4
  Number of hyperplanes created: 6
  Number of distance tests for qhull: 4
  CPU seconds to compute hull (after input):  0


Output completed.  Verifying that all points are below 1.3e-015 of
all facets.  Will make 16 distance computations.
rbox c D3 | qhull Tcv Q0

Convex hull of 8 points in 3-d:

  Number of vertices: 8
  Number of facets: 12

Statistics for: rbox c D3 | qhull Tcv Q0

  Number of points processed: 8
  Number of hyperplanes created: 17
  Number of distance tests for qhull: 35
  CPU seconds to compute hull (after input):  0


precision problems (corrected unless 'Q0' or an error)
     12 coplanar half ridges in output
      6 coplanar horizon facets for new vertices

Output completed.  Verifying that all points are below 2.1e-015 of
all facets.  Will make 96 distance computations.
rbox d D3 | qhull Tcv

Convex hull of 6 points in 3-d:

  Number of vertices: 6
  Number of facets: 8

Statistics for: rbox d D3 | qhull Tcv

  Number of points processed: 6
  Number of hyperplanes created: 11
  Number of distance tests for qhull: 11
  Number of distance tests for merging: 59
  Number of distance tests for checking: 54
  Number of merged facets: 1
  CPU seconds to compute hull (after input):  0


Output completed.  Verifying that all points are below outer planes of
all facets.  Will make 48 distance computations.
rbox c D4 | qhull Tcv Q0

Convex hull of 16 points in 4-d:

  Number of vertices: 16
  Number of facets: 44

Statistics for: rbox c D4 | qhull Tcv Q0

  Number of points processed: 16
  Number of hyperplanes created: 62
  Number of distance tests for qhull: 163
  CPU seconds to compute hull (after input):  0


precision problems (corrected unless 'Q0' or an error)
     80 coplanar half ridges in output
     32 coplanar horizon facets for new vertices

Output completed.  Verifying that all points are below 3e-015 of
all facets.  Will make 704 distance computations.
rbox d D4 | qhull Tcv

Convex hull of 8 points in 4-d:

  Number of vertices: 8
  Number of facets: 16

Statistics for: rbox d D4 | qhull Tcv

  Number of points processed: 8
  Number of hyperplanes created: 20
  Number of distance tests for qhull: 24
  Number of distance tests for merging: 165
  Number of distance tests for checking: 136
  Number of merged facets: 4
  CPU seconds to compute hull (after input):  0


Output completed.  Verifying that all points are below outer planes of
all facets.  Will make 128 distance computations.
rbox c D5 | qhull Tcv Q0

Convex hull of 32 points in 5-d:

  Number of vertices: 32
  Number of facets: 210

Statistics for: rbox c D5 | qhull Tcv Q0

  Number of points processed: 32
  Number of hyperplanes created: 297
  Number of distance tests for qhull: 856
  CPU seconds to compute hull (after input):  0


precision problems (corrected unless 'Q0' or an error)
    590 coplanar half ridges in output
    200 coplanar horizon facets for new vertices
    190 nearly singular or axis-parallel hyperplanes
    190 zero divisors during back substitute
    246 zero divisors during gaussian elimination

Output completed.  Verifying that all points are below 4.1e-015 of
all facets.  Will make 6720 distance computations.
rbox d D5 | qhull Tcv

Convex hull of 10 points in 5-d:

  Number of vertices: 10
  Number of facets: 32

Statistics for: rbox d D5 | qhull Tcv

  Number of points processed: 10
  Number of hyperplanes created: 37
  Number of distance tests for qhull: 47
  Number of distance tests for merging: 510
  Number of distance tests for checking: 330
  Number of merged facets: 11
  CPU seconds to compute hull (after input):  0

QH7076 qhull input warning: exact merge ('Qx').  Verify may report that a point
is outside of a facet.  See qh-optq.htm#Qx

Output completed.  Verifying that all points are below outer planes of
all facets.  Will make 320 distance computations.
rbox c D6 | qhull Tcv Q0

Convex hull of 64 points in 6-d:

  Number of vertices: 64
  Number of facets: 1242

Statistics for: rbox c D6 | qhull Tcv Q0

  Number of points processed: 64
  Number of hyperplanes created: 1760
  Number of distance tests for qhull: 5986
  CPU seconds to compute hull (after input): 0.015


precision problems (corrected unless 'Q0' or an error)
   4752 coplanar half ridges in output
   1382 coplanar horizon facets for new vertices
   1180 nearly singular or axis-parallel hyperplanes
   1180 zero divisors during back substitute
   1593 zero divisors during gaussian elimination

Output completed.  Verifying that all points are below 5.3e-015 of
all facets.  Will make 79488 distance computations.
rbox d D6 | qhull Tcv

Convex hull of 12 points in 6-d:

  Number of vertices: 12
  Number of facets: 64

Statistics for: rbox d D6 | qhull Tcv

  Number of points processed: 12
  Number of hyperplanes created: 70
  Number of distance tests for qhull: 88
  Number of distance tests for merging: 1176
  Number of distance tests for checking: 780
  Number of merged facets: 26
  CPU seconds to compute hull (after input):  0

QH7076 qhull input warning: exact merge ('Qx').  Verify may report that a point
is outside of a facet.  See qh-optq.htm#Qx

Output completed.  Verifying that all points are below outer planes of
all facets.  Will make 768 distance computations.
rbox d D7 | qhull Tcv

Convex hull of 14 points in 7-d:

  Number of vertices: 14
  Number of facets: 128

Statistics for: rbox d D7 | qhull Tcv

  Number of points processed: 14
  Number of hyperplanes created: 135
  Number of distance tests for qhull: 163
  Number of distance tests for merging: 2635
  Number of distance tests for checking: 1806
  Number of merged facets: 57
  CPU seconds to compute hull (after input):  0

QH7076 qhull input warning: exact merge ('Qx').  Verify may report that a point
is outside of a facet.  See qh-optq.htm#Qx

Output completed.  Verifying that all points are below outer planes of
all facets.  Will make 1792 distance computations.
rbox c D2 | qhull Tcv C-0

Convex hull of 4 points in 2-d:

  Number of vertices: 4
  Number of facets: 4

Statistics for: rbox c D2 | qhull Tcv C-0

  Number of points processed: 4
  Number of hyperplanes created: 6
  Number of distance tests for qhull: 6
  CPU seconds to compute hull (after input):  0


Output completed.  Verifying that all points are below 1.3e-015 of
all facets.  Will make 16 distance computations.
rbox c D3 | qhull Tcv C-0

Convex hull of 8 points in 3-d:

  Number of vertices: 8
  Number of facets: 6
  Number of non-simplicial facets: 6

Statistics for: rbox c D3 | qhull Tcv C-0

  Number of points processed: 8
  Number of hyperplanes created: 11
  Number of distance tests for qhull: 35
  Number of distance tests for merging: 142
  Number of distance tests for checking: 56
  Number of merged facets: 6
  CPU seconds to compute hull (after input):  0


Output completed.  Verifying that all points are below outer planes of
all facets.  Will make 48 distance computations.
rbox c D4 | qhull Tcv C-0

Convex hull of 16 points in 4-d:

  Number of vertices: 16
  Number of facets: 8
  Number of non-simplicial facets: 8

Statistics for: rbox c D4 | qhull Tcv C-0

  Number of points processed: 16
  Number of hyperplanes created: 26
  Number of distance tests for qhull: 168
  Number of distance tests for merging: 788
  Number of distance tests for checking: 144
  Number of merged facets: 36
  CPU seconds to compute hull (after input):  0


Output completed.  Verifying that all points are below outer planes of
all facets.  Will make 128 distance computations.
rbox c D5 | qhull Tcv C-0

Convex hull of 32 points in 5-d:

  Number of vertices: 32
  Number of facets: 10
  Number of non-simplicial facets: 10

Statistics for: rbox c D5 | qhull Tcv C-0

  Number of points processed: 32
  Number of hyperplanes created: 87
  Number of distance tests for qhull: 722
  Number of distance tests for merging: 4130
  Number of distance tests for checking: 352
  Number of merged facets: 210
  CPU seconds to compute hull (after input):  0


Output completed.  Verifying that all points are below outer planes of
all facets.  Will make 320 distance computations.
rbox c D6 | qhull Tcv C-0

Convex hull of 64 points in 6-d:

  Number of vertices: 64
  Number of facets: 12
  Number of non-simplicial facets: 12

Statistics for: rbox c D6 | qhull Tcv C-0

  Number of points processed: 64
  Number of hyperplanes created: 354
  Number of distance tests for qhull: 2474
  Number of distance tests for merging: 22065
  Number of distance tests for checking: 832
  Number of merged facets: 1488
  CPU seconds to compute hull (after input): 0.046


Output completed.  Verifying that all points are below outer planes of
all facets.  Will make 768 distance computations.
rbox c D7 | qhull Tv C-0

Convex hull of 128 points in 7-d:

  Number of vertices: 128
  Number of facets: 14
  Number of non-simplicial facets: 14

Statistics for: rbox c D7 | qhull Tv C-0

  Number of points processed: 128
  Number of hyperplanes created: 1786
  Number of distance tests for qhull: 9309
  Number of distance tests for merging: 88772
  Number of distance tests for checking: 1920
  Number of merged facets: 11556
  CPU seconds to compute hull (after input): 0.062


Output completed.  Verifying that all points are below outer planes of
all facets.  Will make 1792 distance computations.
rbox 20 l D3 | qhull Tcv

Convex hull of 20 points in 3-d:

  Number of vertices: 20
  Number of facets: 36

Statistics for: rbox 20 l D3 | qhull Tcv

  Number of points processed: 20
  Number of hyperplanes created: 63
  Number of distance tests for qhull: 139
  CPU seconds to compute hull (after input):  0


Output completed.  Verifying that all points are below 2.1e-015 of
all facets.  Will make 720 distance computations.
rbox 100 s D2 | qhull Tcv

Convex hull of 100 points in 2-d:

  Number of vertices: 100
  Number of facets: 100

Statistics for: rbox 100 s D2 | qhull Tcv

  Number of points processed: 100
  Number of hyperplanes created: 198
  Number of distance tests for qhull: 898
  CPU seconds to compute hull (after input):  0


Output completed.  Verifying that all points are below 1.3e-015 of
all facets.  Will make 10000 distance computations.
rbox 100 s D3 | qhull Tcv

Convex hull of 100 points in 3-d:

  Number of vertices: 100
  Number of facets: 196

Statistics for: rbox 100 s D3 | qhull Tcv

  Number of points processed: 100
  Number of hyperplanes created: 514
  Number of distance tests for qhull: 1691
  CPU seconds to compute hull (after input):  0


Output completed.  Verifying that all points are below 2e-015 of
all facets.  Will make 19600 distance computations.
rbox 100 s D4 | qhull Tcv

Convex hull of 100 points in 4-d:

  Number of vertices: 100
  Number of facets: 584

Statistics for: rbox 100 s D4 | qhull Tcv

  Number of points processed: 100
  Number of hyperplanes created: 1718
  Number of distance tests for qhull: 4091
  CPU seconds to compute hull (after input):  0


Output completed.  Verifying that all points are below 2.8e-015 of
all facets.  Will make 58400 distance computations.
rbox 100 s c D4 | qhull Tcv

Convex hull of 116 points in 4-d:

  Number of vertices: 16
  Number of facets: 8
  Number of non-simplicial facets: 8

Statistics for: rbox 100 s c D4 | qhull Tcv

  Number of points processed: 16
  Number of hyperplanes created: 26
  Number of distance tests for qhull: 2028
  Number of distance tests for merging: 788
  Number of distance tests for checking: 144
  Number of merged facets: 36
  CPU seconds to compute hull (after input):  0


Output completed.  Verifying that all points are below outer planes of
all facets.  Will make 928 distance computations.
rbox 100 s d G1.5 D4 | qhull Tcv

Convex hull of 108 points in 4-d:

  Number of vertices: 8
  Number of facets: 16

Statistics for: rbox 100 s d G1.5 D4 | qhull Tcv

  Number of points processed: 8
  Number of hyperplanes created: 20
  Number of distance tests for qhull: 1457
  Number of distance tests for merging: 165
  Number of distance tests for checking: 136
  Number of merged facets: 4
  CPU seconds to compute hull (after input):  0


Output completed.  Verifying that all points are below outer planes of
all facets.  Will make 1728 distance computations.
rbox 100 s W1e-2 | qhull Tcv

Convex hull of 100 points in 3-d:

  Number of vertices: 100
  Number of facets: 196

Statistics for: rbox 100 s W1e-2 | qhull Tcv

  Number of points processed: 100
  Number of hyperplanes created: 493
  Number of distance tests for qhull: 1582
  CPU seconds to compute hull (after input):  0


Output completed.  Verifying that all points are below 2.1e-015 of
all facets.  Will make 19600 distance computations.
rbox 100 | qhull Tcv

Convex hull of 100 points in 3-d:

  Number of vertices: 30
  Number of facets: 56

Statistics for: rbox 100 | qhull Tcv

  Number of points processed: 31
  Number of hyperplanes created: 117
  Number of distance tests for qhull: 1211
  CPU seconds to compute hull (after input):  0


Output completed.  Verifying that all points are below outer planes of
all facets.  Will make 5600 distance computations.
rbox 100 W1e-3 | qhull Tcv

Convex hull of 100 points in 3-d:

  Number of vertices: 60
  Number of facets: 116

Statistics for: rbox 100 W1e-3 | qhull Tcv

  Number of points processed: 65
  Number of hyperplanes created: 298
  Number of distance tests for qhull: 1753
  CPU seconds to compute hull (after input):  0


Output completed.  Verifying that all points are below outer planes of
all facets.  Will make 11600 distance computations.
rbox 100 r D2 | qhull Tcv

Convex hull of 100 points in 2-d:

  Number of vertices: 100
  Number of facets: 100

Statistics for: rbox 100 r D2 | qhull Tcv

  Number of points processed: 100
  Number of hyperplanes created: 198
  Number of distance tests for qhull: 853
  CPU seconds to compute hull (after input):  0


Output completed.  Verifying that all points are below 1.3e-015 of
all facets.  Will make 10000 distance computations.
rbox 100 r s Z1 | qhull Tcv

Convex hull of 102 points in 3-d:

  Number of vertices: 102
  Number of facets: 200

Statistics for: rbox 100 r s Z1 | qhull Tcv

  Number of points processed: 102
  Number of hyperplanes created: 299
  Number of distance tests for qhull: 1457
  Number of distance tests for merging: 16912
  Number of distance tests for checking: 2604
  Number of merged facets: 97
  CPU seconds to compute hull (after input): 0.015


Output completed.  Verifying that all points are below outer planes of
all facets.  Will make 20400 distance computations.
rbox 100 r s Z1 G0.1 | qhull Tcv C-0

Convex hull of 201 points in 3-d:

  Number of vertices: 201
  Number of facets: 201
  Number of non-simplicial facets: 101

Statistics for: rbox 100 r s Z1 G0.1 | qhull Tcv C-0

  Number of points processed: 201
  Number of hyperplanes created: 994
  Number of distance tests for qhull: 6676
  Number of distance tests for merging: 38219
  Number of distance tests for checking: 11601
  Number of merged facets: 323
  CPU seconds to compute hull (after input): 0.046


Output completed.  Verifying that all points are below outer planes of
all facets.  Will make 40401 distance computations.
rbox 100 s Z1 G0.1 | qhull Tcv 

Convex hull of 101 points in 3-d:

  Number of vertices: 101
  Number of facets: 198

Statistics for: rbox 100 s Z1 G0.1 | qhull Tcv

  Number of points processed: 101
  Number of hyperplanes created: 460
  Number of distance tests for qhull: 1622
  CPU seconds to compute hull (after input):  0


Output completed.  Verifying that all points are below 2.1e-015 of
all facets.  Will make 19998 distance computations.
rbox 100 s Z1e-5 G0.1 | qhull Tc Pp

Convex hull of 101 points in 3-d:

  Number of vertices: 101
  Number of facets: 196
  Number of non-simplicial facets: 2

Statistics for: rbox 100 s Z1e-5 G0.1 | qhull Tc Pp

  Number of points processed: 101
  Number of hyperplanes created: 453
  Number of distance tests for qhull: 1848
  Number of distance tests for merging: 2108
  Number of distance tests for checking: 1274
  Number of merged facets: 3
  CPU seconds to compute hull (after input):  0
  Maximum distance of vertex below facet: -1.1e-015 (0.3x)


echo === check qhull output formats ${d:-`date`}
date
=== check qhull output formats Wed Nov 23 22:04:45 EST 2011
rbox 5 r s D2 | qhull Tcv

Convex hull of 5 points in 2-d:

  Number of vertices: 5
  Number of facets: 5

Statistics for: rbox 5 r s D2 | qhull Tcv

  Number of points processed: 5
  Number of hyperplanes created: 8
  Number of distance tests for qhull: 8
  CPU seconds to compute hull (after input):  0


Output completed.  Verifying that all points are below 1.3e-015 of
all facets.  Will make 25 distance computations.
rbox 5 r s D2 | qhull s 

Convex hull of 5 points in 2-d:

  Number of vertices: 5
  Number of facets: 5

Statistics for: rbox 5 r s D2 | qhull s

  Number of points processed: 5
  Number of hyperplanes created: 8
  Number of distance tests for qhull: 8
  CPU seconds to compute hull (after input):  0

rbox 5 r s D2 | qhull s o 
2
5 5 5
0.1545084971874737 0.4755282581475768 
-0.4045084971874737 0.2938926261462366 
-0.4045084971874738 -0.2938926261462365 
0.1545084971874736 -0.4755282581475768 
   0.5 -1.224646799147353e-016 
2 4 0 
2 2 3 
2 3 4 
2 1 2 
2 0 1 

Convex hull of 5 points in 2-d:

  Number of vertices: 5
  Number of facets: 5

Statistics for: rbox 5 r s D2 | qhull s o

  Number of points processed: 5
  Number of hyperplanes created: 8
  Number of distance tests for qhull: 8
  CPU seconds to compute hull (after input):  0

rbox 5 r s D2 | qhull f
Vertices and facets:
- p0(v2):  0.15  0.48
- p4(v1):   0.5 -1.2e-016
- p3(v4):  0.15 -0.48
- p2(v0):  -0.4 -0.29
- p1(v5):  -0.4  0.29
- f3
    - flags: bottom simplicial
    - normal:     0.809   0.5878
    - offset: -0.4045085
    - vertices: p0(v2) p4(v1)
    - neighboring facets: f5 f7
- f4
    - flags: bottom simplicial
    - normal:    -0.309  -0.9511
    - offset: -0.4045085
    - vertices: p3(v4) p2(v0)
    - neighboring facets: f6 f5
- f5
    - flags: top simplicial
    - normal:     0.809  -0.5878
    - offset: -0.4045085
    - vertices: p3(v4) p4(v1)
    - neighboring facets: f3 f4
- f6
    - flags: top simplicial
    - normal:        -1 9.444e-017
    - offset: -0.4045085
    - vertices: p1(v5) p2(v0)
    - neighboring facets: f4 f7
- f7
    - flags: bottom simplicial
    - normal:    -0.309   0.9511
    - offset: -0.4045085
    - vertices: p1(v5) p0(v2)
    - neighboring facets: f3 f6
rbox 5 r s D2 | qhull i 
5
4 0 
2 3 
3 4 
1 2 
0 1 
rbox 5 r s D2 | qhull m 
{
Line[{{      0.50000000,      -0.00000000}, {      0.15450850,       0.47552826}}]
,Line[{{     -0.40450850,      -0.29389263}, {      0.15450850,      -0.47552826}}]
,Line[{{      0.15450850,      -0.47552826}, {      0.50000000,      -0.00000000}}]
,Line[{{     -0.40450850,       0.29389263}, {     -0.40450850,      -0.29389263}}]
,Line[{{      0.15450850,       0.47552826}, {     -0.40450850,       0.29389263}}]
}
rbox 5 r s D2 | qhull FM 
PLOT(CURVES(
[[      0.50000000,      -0.00000000], [      0.15450850,       0.47552826]]
,[[     -0.40450850,      -0.29389263], [      0.15450850,      -0.47552826]]
,[[      0.15450850,      -0.47552826], [      0.50000000,      -0.00000000]]
,[[     -0.40450850,       0.29389263], [     -0.40450850,      -0.29389263]]
,[[      0.15450850,       0.47552826], [     -0.40450850,       0.29389263]]
));
rbox 5 r s D2 | qhull n 
3
5
0.8090169943749476 0.587785252292473 -0.4045084971874737 
-0.3090169943749476 -0.9510565162951536 -0.4045084971874738 
0.8090169943749472 -0.5877852522924733 -0.4045084971874737 
    -1 9.444121133484361e-017 -0.4045084971874738 
-0.3090169943749474 0.9510565162951536 -0.4045084971874738 
rbox 5 r s D2 | qhull p 
2
5
0.1545084971874737 0.4755282581475768 
-0.4045084971874737 0.2938926261462366 
-0.4045084971874738 -0.2938926261462365 
0.1545084971874736 -0.4755282581475768 
   0.5 -1.224646799147353e-016 
rbox 5 r s D2 | qhull o 
2
5 5 5
0.1545084971874737 0.4755282581475768 
-0.4045084971874737 0.2938926261462366 
-0.4045084971874738 -0.2938926261462365 
0.1545084971874736 -0.4755282581475768 
   0.5 -1.224646799147353e-016 
2 4 0 
2 2 3 
2 3 4 
2 1 2 
2 0 1 
rbox 5 r s D2 | qhull Ft
2
5 5 5
0.1545084971874737 0.4755282581475768 
-0.4045084971874737 0.2938926261462366 
-0.4045084971874738 -0.2938926261462365 
0.1545084971874736 -0.4755282581475768 
   0.5 -1.224646799147353e-016 
2 4 0 
2 2 3 
2 3 4 
2 1 2 
2 0 1 
rbox 5 r s D2 | qhull Fx
5
4
0
1
2
3
rbox 5 r s D2 | qhull p n i p p
2
5
0.1545084971874737 0.4755282581475768 
-0.4045084971874737 0.2938926261462366 
-0.4045084971874738 -0.2938926261462365 
0.1545084971874736 -0.4755282581475768 
   0.5 -1.224646799147353e-016 
3
5
0.8090169943749476 0.587785252292473 -0.4045084971874737 
-0.3090169943749476 -0.9510565162951536 -0.4045084971874738 
0.8090169943749472 -0.5877852522924733 -0.4045084971874737 
    -1 9.444121133484361e-017 -0.4045084971874738 
-0.3090169943749474 0.9510565162951536 -0.4045084971874738 
5
4 0 
2 3 
3 4 
1 2 
0 1 
rbox 10 D3 | qhull f Tcv
Vertices and facets:
- p0(v5): -0.022 -0.37  0.33
- p9(v2):  0.38 -0.47 -0.22
- p6(v0): -0.31 -0.011 -0.49
- p1(v3): -0.067 -0.16  0.46
- p7(v1):  0.39 0.045  0.12
- p3(v6):  0.31 0.084  -0.1
- p4(v7):  0.18  0.12 0.049
- p5(v8): -0.12 0.015 -0.14
- p2(v9): 0.028 0.042 0.058
- f5
    - flags: top simplicial
    - normal:   -0.4934  -0.8471  -0.1973
    - offset: -0.2568231
    - vertices: p0(v5) p9(v2) p6(v0)
    - neighboring facets: f9 f6 f7
- f6
    - flags: bottom simplicial
    - normal:   -0.9301  -0.3157   0.1877
    - offset: -0.1977325
    - vertices: p0(v5) p1(v3) p6(v0)
    - neighboring facets: f17 f5 f8
- f7
    - flags: bottom simplicial
    - normal:    0.7008  -0.3956   0.5936
    - offset:  -0.323519
    - vertices: p0(v5) p9(v2) p7(v1)
    - neighboring facets: f10 f8 f5
- f8
    - flags: top simplicial
    - normal:    0.6544   -0.298   0.6949
    - offset: -0.3219447
    - vertices: p0(v5) p1(v3) p7(v1)
    - neighboring facets: f13 f7 f6
- f9
    - flags: bottom simplicial
    - normal:    0.4866   0.2316  -0.8424
    - offset: -0.2582907
    - vertices: p3(v6) p9(v2) p6(v0)
    - neighboring facets: f5 f16 f10
- f10
    - flags: top simplicial
    - normal:     0.943   0.1711  -0.2855
    - offset: -0.3386013
    - vertices: p3(v6) p9(v2) p7(v1)
    - neighboring facets: f7 f14 f9
- f13
    - flags: bottom simplicial
    - normal:   0.08536   0.8035   0.5892
    - offset: -0.1388354
    - vertices: p4(v7) p1(v3) p7(v1)
    - neighboring facets: f8 f14 f20
- f14
    - flags: top simplicial
    - normal:    0.3121    0.948  0.06295
    - offset: -0.1707289
    - vertices: p4(v7) p3(v6) p7(v1)
    - neighboring facets: f10 f13 f16
- f16
    - flags: bottom simplicial
    - normal:  -0.008104   0.9738  -0.2271
    - offset: -0.1026508
    - vertices: p4(v7) p3(v6) p6(v0)
    - neighboring facets: f9 f18 f14
- f17
    - flags: top simplicial
    - normal:   -0.6179   0.7381   0.2709
    - offset: -0.04995414
    - vertices: p5(v8) p1(v3) p6(v0)
    - neighboring facets: f6 f18 f21
- f18
    - flags: bottom simplicial
    - normal:   -0.3978   0.9064   0.1423
    - offset: -0.04320607
    - vertices: p5(v8) p4(v7) p6(v0)
    - neighboring facets: f16 f17 f22
- f20
    - flags: top simplicial
    - normal:   -0.4124   0.8514   0.3242
    - offset: -0.04294297
    - vertices: p2(v9) p4(v7) p1(v3)
    - neighboring facets: f13 f21 f22
- f21
    - flags: bottom simplicial
    - normal:   -0.5086   0.8133   0.2826
    - offset: -0.03620543
    - vertices: p2(v9) p5(v8) p1(v3)
    - neighboring facets: f17 f20 f22
- f22
    - flags: top simplicial
    - normal:    -0.432   0.8759   0.2148
    - offset: -0.03703814
    - vertices: p2(v9) p5(v8) p4(v7)
    - neighboring facets: f18 f20 f21

Output completed.  Verifying that all points are below 2e-015 of
all facets.  Will make 140 distance computations.
rbox 10 D3 | qhull i
14
0 9 6 
1 0 6 
9 0 7 
0 1 7 
9 3 6 
3 9 7 
1 4 7 
4 3 7 
3 4 6 
5 1 6 
4 5 6 
2 4 1 
5 2 1 
2 5 4 
rbox 10 D3 | qhull p 
3
9
-0.0222149361131852 -0.366434993563625 0.3270621312102882 
-0.06676722137887703 -0.1566931052661437 0.4589771055234383 
0.02820502736438535 0.04189077954915421 0.05832764185809314 
0.3126723396709863 0.08400649026409401 -0.1029227018383543 
0.1781470954214661 0.1182274414396169 0.04860343742054274 
-0.1220315663349177 0.01546165115708642 -0.1360330368727754 
-0.3072535691850387 -0.01073880122111998 -0.4870359524963758 
0.3867462923626847 0.04492879989084675 0.118335500935405 
0.3789805913148268 -0.4732086509216658 -0.2177962499836425 
rbox 10 D3 | qhull o 
3
10 14 21
-0.0222149361131852 -0.366434993563625 0.3270621312102882 
-0.06676722137887703 -0.1566931052661437 0.4589771055234383 
0.02820502736438535 0.04189077954915421 0.05832764185809314 
0.3126723396709863 0.08400649026409401 -0.1029227018383543 
0.1781470954214661 0.1182274414396169 0.04860343742054274 
-0.1220315663349177 0.01546165115708642 -0.1360330368727754 
-0.3072535691850387 -0.01073880122111998 -0.4870359524963758 
0.3867462923626847 0.04492879989084675 0.118335500935405 
-0.1352406177997967 0.01093378431250691 -0.2358910583293913 
0.3789805913148268 -0.4732086509216658 -0.2177962499836425 
3 0 9 6 
3 1 0 6 
3 9 0 7 
3 0 1 7 
3 9 3 6 
3 3 9 7 
3 1 4 7 
3 4 3 7 
3 3 4 6 
3 5 1 6 
3 4 5 6 
3 2 4 1 
3 5 2 1 
3 2 5 4 
rbox 10 D3 | qhull Fx
9
0
1
2
3
4
5
6
7
9
rbox 27 M1,0,1 | qhull Qc

Convex hull of 27 points in 3-d:

  Number of vertices: 8
  Number of coplanar points: 18
  Number of facets: 6
  Number of non-simplicial facets: 6

Statistics for: rbox 27 M1,0,1 | qhull Qc

  Number of points processed: 8
  Number of hyperplanes created: 11
  Number of distance tests for qhull: 320
  Number of distance tests for merging: 84
  Number of distance tests for checking: 188
  Number of merged facets: 6
  CPU seconds to compute hull (after input):  0

rbox 50 D3 s | qhull C0.1 Qc Pd0d1d2 s p Tcv
3
7
0.1955434141942995 0.09315491718098735 0.4506494586374647 
0.3786694231678469 0.2419725568326128 -0.2192230592287637 
0.1164290160865212 0.2250749615572271 0.4310284745734756 
0.3010852516984067 0.3425921496145067 0.2048860420629442 
0.4995640899182717 0.005583735684776723 0.02011322848106417 
0.4035338761538298 0.1419567271683243 0.2588603840063664 
0.3642936952899509 -0.2163723184172344 -0.2654677445467591 

Convex hull of 50 points in 3-d:

  Number of vertices: 19
  Number of coplanar points: 31
  Number of facets: 12
  Number of 'good' facets: 1
  Number of 'good' non-simplicial facets: 1

Statistics for: rbox 50 D3 s | qhull C0.1 Qc Pd0d1d2 s p Tcv

  Number of points processed: 50
  Number of hyperplanes created: 237
  Number of distance tests for qhull: 1177
  Number of distance tests for merging: 3999
  Number of distance tests for checking: 367
  Number of merged facets: 84
  CPU seconds to compute hull (after input):  0
  Maximum distance of vertex below facet: -0.24 (0.8x)


Output completed.  Verifying that all points are below outer planes of
all facets.  Will make 50 distance computations.
rbox 10 D2 P0 P1e-15 | qhull d Qc FP s Tcv
1
0 1 12 1e-015 

Delaunay triangulation by the convex hull of 12 points in 3-d:

  Number of input sites: 11
  Number of nearly incident points: 1
  Number of Delaunay regions: 16

Statistics for: rbox 10 D2 P0 P1e-15 | qhull d Qc FP s Tcv

  Number of points processed: 11
  Number of hyperplanes created: 33
  Number of facets in hull: 18
  Number of distance tests for qhull: 111
  CPU seconds to compute hull (after input):  0


Output completed.  Verifying that all points are below outer planes of
all facets.  Will make 192 distance computations.
rbox 100 s | qhull C-0.003 Qc FP s
10
29 92 222 0.05221620842441997 
49 2 292 0.03790045221711885 
99 9 338 0.02993864679572162 
54 84 341 0.06479664045626413 
54 72 226 0.03827167959574595 
96 6 368 0.02282529175708785 
19 1 372 0.01048228658130246 
69 26 409 0.04998878658421259 
19 66 371 0.008281093788857875 
19 45 371 0.04608027722786196 

Convex hull of 100 points in 3-d:

  Number of vertices: 90
  Number of coplanar points: 10
  Number of facets: 118
  Number of non-simplicial facets: 46

Statistics for: rbox 100 s | qhull C-0.003 Qc FP s

  Number of points processed: 90
  Number of hyperplanes created: 400
  Number of distance tests for qhull: 2016
  Number of distance tests for merging: 3695
  Number of distance tests for checking: 1469
  Number of merged facets: 102
  CPU seconds to compute hull (after input):  0
  Maximum distance of point above facet: 0.0057 (0.6x)
  Maximum distance of vertex below facet: -0.0087 (1.0x)

rbox 100 s D2 | qhull C0.1 i Fx Tcv
6
4 28 
70 27 
77 70 
68 4 
27 68 
28 77 
6
4
28
77
70
27
68

Output completed.  Verifying that all points are below outer planes of
all facets.  Will make 600 distance computations.
rbox 4 s D3 | qhull Qc Ghipv Tcv 
{appearance {+edge -evert linewidth 2} LIST # rbox 4 s D3 | qhull Qc Ghipv Tcv
{appearance {-edge -normal normscale 0} {
INST geom {define vsphere OFF
18 32 48

0 0 1
1 0 0
0 1 0
-1 0 0
0 -1 0
0 0 -1
0.707107 0 0.707107
0 -0.707107 0.707107
0.707107 -0.707107 0
-0.707107 0 0.707107
-0.707107 -0.707107 0
0 0.707107 0.707107
-0.707107 0.707107 0
0.707107 0.707107 0
0.707107 0 -0.707107
0 0.707107 -0.707107
-0.707107 0 -0.707107
0 -0.707107 -0.707107

3 0 6 11
3 0 7 6 
3 0 9 7 
3 0 11 9
3 1 6 8 
3 1 8 14
3 1 13 6
3 1 14 13
3 2 11 13
3 2 12 11
3 2 13 15
3 2 15 12
3 3 9 12
3 3 10 9
3 3 12 16
3 3 16 10
3 4 7 10
3 4 8 7
3 4 10 17
3 4 17 8
3 5 14 17
3 5 15 14
3 5 16 15
3 5 17 16
3 6 13 11
3 7 8 6
3 9 10 7
3 11 12 9
3 14 8 17
3 15 13 14
3 16 12 15
3 17 10 16
} transforms { TLIST
0.009951 0 0 0 # v2
 0 0.009951 0 0
0 0 0.009951 0
-0.04234  -0.2077  -0.4529  # p3
1
0.009951 0 0 0 # v1
 0 0.009951 0 0
0 0 0.009951 0
0.004313  -0.4976  0.04904  # p2
1
0.009951 0 0 0 # v0
 0 0.009951 0 0
0 0 0.009951 0
 -0.4876  0.03313  -0.1055  # p0
1
0.009951 0 0 0 # v3
 0 0.009951 0 0
0 0 0.009951 0
 -0.3939  -0.1068  -0.2889  # p1
1
}}}
{ OFF 3 1 1 # f1
-0.04933  -0.2152  -0.4566 
-0.002675  -0.5051  0.04532 
 -0.4946  0.02557  -0.1092 
3 0 1 2   0.1808   0.1547   0.3302 1.0 }
VECT 1 2 1 2 1 # intersect f1 f2
0.004313  -0.4976  0.04904 # projected p2
 -0.4876  0.03313  -0.1055 # projected p0
       0        0        0 1.0
VECT 1 2 1 2 1 # intersect f1 f3
-0.04234  -0.2077  -0.4529 # projected p3
 -0.4876  0.03313  -0.1055 # projected p0
       0        0        0 1.0
VECT 1 2 1 2 1 # intersect f1 f4
-0.04234  -0.2077  -0.4529 # projected p3
0.004313  -0.4976  0.04904 # projected p2
       0        0        0 1.0
{ OFF 3 1 1 # f2
 0.01247  -0.4904  0.04773 
 -0.3857 -0.09961  -0.2902 
 -0.4795  0.04031  -0.1068 
3 0 1 2   0.8726    0.828   0.4402 1.0 }
VECT 1 2 1 2 1 # intersect f2 f3
 -0.3939  -0.1068  -0.2889 # projected p1
 -0.4876  0.03313  -0.1055 # projected p0
       0        0        0 1.0
VECT 1 2 1 2 1 # intersect f2 f4
 -0.3939  -0.1068  -0.2889 # projected p1
0.004313  -0.4976  0.04904 # projected p2
       0        0        0 1.0
{ OFF 3 1 1 # f3
 -0.3931  -0.1153   -0.282 
-0.04157  -0.2162   -0.446 
 -0.4869  0.02464 -0.09863 
3 0 1 2   0.5351   0.1123   0.8138 1.0 }
VECT 1 2 1 2 1 # intersect f3 f4
 -0.3939  -0.1068  -0.2889 # projected p1
-0.04234  -0.2077  -0.4529 # projected p3
       0        0        0 1.0
{ OFF 3 1 1 # f4
-0.03767  -0.1989  -0.4482 
 -0.3892 -0.09803  -0.2843 
0.008983  -0.4888  0.05366 
3 0 1 2   0.7133   0.8999   0.7112 1.0 }
}

Output completed.  Verifying that all points are below 2.1e-015 of
all facets.  Will make 16 distance computations.
rbox 6 D4 | qhull f Tcv
Vertices and facets:
- p3(v4):   0.1 -0.13 -0.17 -0.42
- p0(v3): -0.33 -0.43 -0.46  0.15
- p2(v2):  0.11   0.4 -0.063 -0.18
- p1(v0): -0.35 -0.24  0.14 -0.18
- p4(v1):  0.14 -0.34 -0.084 0.087
- p5(v6): -0.25  -0.5  0.22 -0.033
- f4
    - flags: bottom simplicial
    - normal:   -0.6536   0.3379   -0.424  -0.5281
    - offset: -0.1850304
    - vertices: p3(v4) p0(v3) p2(v2) p1(v0)
    - neighboring facets: f6 f8 f10 f5
- f5
    - flags: top simplicial
    - normal:    0.6142  0.09488  -0.7743   0.1197
    - offset: -0.1313181
    - vertices: p3(v4) p0(v3) p2(v2) p4(v1)
    - neighboring facets: f7 f9 f11 f4
- f6
    - flags: top simplicial
    - normal:   -0.4209   0.3853   0.2879   0.7691
    - offset: 0.04819078
    - vertices: p5(v6) p0(v3) p2(v2) p1(v0)
    - neighboring facets: f4 f8 f10 f7
- f7
    - flags: bottom simplicial
    - normal:   -0.1759   0.3066   0.2842   0.8912
    - offset: 0.07466537
    - vertices: p5(v6) p0(v3) p2(v2) p4(v1)
    - neighboring facets: f5 f9 f11 f6
- f8
    - flags: bottom simplicial
    - normal:    0.2584  0.07433    0.814   -0.515
    - offset: -0.09844114
    - vertices: p5(v6) p3(v4) p2(v2) p1(v0)
    - neighboring facets: f4 f6 f10 f9
- f9
    - flags: top simplicial
    - normal:    0.6576 -0.06072   0.7252   -0.195
    - offset: -0.03612816
    - vertices: p5(v6) p3(v4) p2(v2) p4(v1)
    - neighboring facets: f5 f7 f11 f8
- f10
    - flags: top simplicial
    - normal:   -0.3537  -0.6003  -0.2017  -0.6883
    - offset: -0.3651888
    - vertices: p5(v6) p3(v4) p0(v3) p1(v0)
    - neighboring facets: f4 f6 f8 f11
- f11
    - flags: bottom simplicial
    - normal:    0.2962   -0.865  -0.2123  -0.3449
    - offset: -0.3220787
    - vertices: p5(v6) p3(v4) p0(v3) p4(v1)
    - neighboring facets: f5 f7 f9 f10

Output completed.  Verifying that all points are below 3e-015 of
all facets.  Will make 48 distance computations.
rbox 6 D4 | qhull i
8
0 3 2 1 
3 0 2 4 
5 0 2 1 
0 5 2 4 
3 5 2 1 
5 3 2 4 
5 3 0 1 
3 5 0 4 
rbox 6 D4 | qhull p 
4
6
-0.3257826863096865 -0.4296101703584289 -0.4581337649916613 0.1458114577883961 
-0.3468340047139991 -0.2391184267952278 0.1365988111464296 -0.1837860440684352 
0.1079548672847029 0.397449696341017 -0.062960620096848 -0.1791453880063681 
0.1034612661259819 -0.1265049433582509 -0.1685859450777862 -0.4239815160855479 
0.1426585555473888 -0.3376619446442108 -0.08430490557505277 0.08744874697872318 
-0.2489141260729303 -0.4997188728300099 0.224904344160952 -0.03269336003129686 
rbox 6 D4 | qhull o
4
6 8 16
-0.3257826863096865 -0.4296101703584289 -0.4581337649916613 0.1458114577883961 
-0.3468340047139991 -0.2391184267952278 0.1365988111464296 -0.1837860440684352 
0.1079548672847029 0.397449696341017 -0.062960620096848 -0.1791453880063681 
0.1034612661259819 -0.1265049433582509 -0.1685859450777862 -0.4239815160855479 
0.1426585555473888 -0.3376619446442108 -0.08430490557505277 0.08744874697872318 
-0.2489141260729303 -0.4997188728300099 0.224904344160952 -0.03269336003129686 
4 0 3 2 1 
4 3 0 2 4 
4 5 0 2 1 
4 0 5 2 4 
4 3 5 2 1 
4 5 3 2 4 
4 5 3 0 1 
4 3 5 0 4 
rbox 1000 s D2 | qhull FA Tcv

Convex hull of 1000 points in 2-d:

  Number of vertices: 1000
  Number of facets: 1000

Statistics for: rbox 1000 s D2 | qhull FA Tcv

  Number of points processed: 1000
  Number of hyperplanes created: 1998
  Number of distance tests for qhull: 13978
  CPU seconds to compute hull (after input): 0.015
  Total facet area:   3.1415592
  Total volume:       0.7853647


qhull output completed.  Verifying that 1000 points are
below 1.3e-015 of the nearest facet.
rbox 1000 s | qhull FA Tcv

Convex hull of 1000 points in 3-d:

  Number of vertices: 1000
  Number of facets: 1996

Statistics for: rbox 1000 s | qhull FA Tcv

  Number of points processed: 1000
  Number of hyperplanes created: 5545
  Number of distance tests for qhull: 25488
  CPU seconds to compute hull (after input):  0
  Total facet area:   3.1201951
  Total volume:       0.51650274


qhull output completed.  Verifying that 1000 points are
below 2.1e-015 of the nearest facet.
rbox c D4 | qhull FA Tcv

Convex hull of 16 points in 4-d:

  Number of vertices: 16
  Number of facets: 8
  Number of non-simplicial facets: 8

Statistics for: rbox c D4 | qhull FA Tcv

  Number of points processed: 16
  Number of hyperplanes created: 26
  Number of distance tests for qhull: 168
  Number of distance tests for merging: 788
  Number of distance tests for checking: 144
  Number of merged facets: 36
  CPU seconds to compute hull (after input):  0
  Approximate facet area:    8
  Approximate volume:        1


Output completed.  Verifying that all points are below outer planes of
all facets.  Will make 128 distance computations.
rbox c D5 | qhull FA Tcv

Convex hull of 32 points in 5-d:

  Number of vertices: 32
  Number of facets: 10
  Number of non-simplicial facets: 10

Statistics for: rbox c D5 | qhull FA Tcv

  Number of points processed: 32
  Number of hyperplanes created: 87
  Number of distance tests for qhull: 722
  Number of distance tests for merging: 2775
  Number of distance tests for checking: 352
  Number of merged facets: 210
  CPU seconds to compute hull (after input): 0.015
  Approximate facet area:   10
  Approximate volume:        1

QH7076 qhull input warning: exact merge ('Qx').  Verify may report that a point
is outside of a facet.  See qh-optq.htm#Qx

Output completed.  Verifying that all points are below outer planes of
all facets.  Will make 320 distance computations.
rbox c D5 | qhull FA Qt Tcv

Convex hull of 32 points in 5-d:

  Number of vertices: 32
  Number of facets: 316
  Number of triangulated facets: 10

Statistics for: rbox c D5 | qhull FA Qt Tcv

  Number of points processed: 32
  Number of hyperplanes created: 87
  Number of distance tests for qhull: 722
  Number of distance tests for merging: 3235
  Number of distance tests for checking: 352
  Number of merged facets: 210
  CPU seconds to compute hull (after input):  0
  Approximate facet area:   10
  Approximate volume:        1

QH7076 qhull input warning: exact merge ('Qx').  Verify may report that a point
is outside of a facet.  See qh-optq.htm#Qx

Output completed.  Verifying that all points are below outer planes of
all facets.  Will make 10112 distance computations.
rbox 10 D2 | qhull d FA Tcv

Delaunay triangulation by the convex hull of 10 points in 3-d:

  Number of input sites: 10
  Number of Delaunay regions: 14

Statistics for: rbox 10 D2 | qhull d FA Tcv

  Number of points processed: 10
  Number of hyperplanes created: 28
  Number of facets in hull: 16
  Number of distance tests for qhull: 48
  CPU seconds to compute hull (after input):  0
  Total facet area:   0.46254269


Output completed.  Verifying that all points are below 2.1e-015 of
all facets.  Will make 140 distance computations.
rbox 10 D2 | qhull d Qu FA Tcv

Furthest-site Delaunay triangulation by the convex hull of 10 points in 3-d:

  Number of input sites: 10
  Number of Delaunay regions: 2

Statistics for: rbox 10 D2 | qhull d Qu FA Tcv

  Number of points processed: 10
  Number of hyperplanes created: 28
  Number of facets in hull: 16
  Number of distance tests for qhull: 48
  CPU seconds to compute hull (after input):  0
  Total facet area:   0.46254269


Output completed.  Verifying that all points are below 2.1e-015 of
all facets.  Will make 20 distance computations.
rbox 10 D2 | qhull FA Tcv

Convex hull of 10 points in 2-d:

  Number of vertices: 4
  Number of facets: 4

Statistics for: rbox 10 D2 | qhull FA Tcv

  Number of points processed: 4
  Number of hyperplanes created: 6
  Number of distance tests for qhull: 52
  CPU seconds to compute hull (after input):  0
  Total facet area:   2.8999576
  Total volume:       0.46254269


Output completed.  Verifying that all points are below 1.3e-015 of
all facets.  Will make 40 distance computations.
rbox 10 c D2 | qhull Fx Tcv
4
10
12
13
11

Output completed.  Verifying that all points are below 1.3e-015 of
all facets.  Will make 56 distance computations.
rbox 1000 s | qhull FS Tcv
0
2 3.12019513507746 0.516502736886338 

qhull output completed.  Verifying that 1000 points are
below 2.1e-015 of the nearest facet.
rbox 10 W0 D2 | qhull p Qc FcC Tcv
2
10
-0.3863076515368257    0.5 
-0.05259535978789942    0.5 
0.2626424396993987    0.5 
0.3748708151978168    0.5 
-0.01474997775140219    0.5 
0.1582814042067914    0.5 
  -0.5 0.45089279622854 
  -0.5 -0.003085198815060031 
-0.3688616616361418   -0.5 
-0.2798755283280048    0.5 
5
0
0
0
0
5 2 4 5 9 1
2
5
0.003004576780837498 -8.023127287351645e-018 
  -0.5 0.22390379870674 
-0.4344308308180709 -0.25154259940753 
-0.4431538257684129 0.47544639811427 
-0.005718418169504469    0.5 

Output completed.  Verifying that all points are below outer planes of
all facets.  Will make 50 distance computations.
rbox 4 z h s D2 | qhull Fd s n FD Tcv
rbox 4 z h s D2 | qhull Fd s n FD Tcv
begin
    4 3 real
107643.7343385944 -0.2650304141580589 -0.964240052876465 
797791.7732245289 -0.6912673062074456 0.7225991360143615 
874890.2679778723 0.9916265954087946 0.1291382796771071 
864878.4439653738 0.3984446264475182 0.9171923896626583 
end

Convex hull of 4 points in 2-d:

  Number of vertices: 4
  Number of facets: 4

Statistics for: rbox 4 z h s D2 | qhull Fd s n FD Tcv

  Number of points processed: 4
  Number of hyperplanes created: 6
  Number of distance tests for qhull: 5
  CPU seconds to compute hull (after input):  0


Output completed.  Verifying that all points are below 2.6e-009 of
all facets.  Will make 16 distance computations.
rbox 6 s D3 | qhull C-0.1 Qc FF s FQ Fi n Fo FQ FI Fm Fn FN FO FO FQ Fs FS FV Fv Tcv
Vertices and facets:
- p1(v2):  -0.3  0.39 -0.092
- p5(v1):   0.4 -0.22  -0.2
- p4(v0): -0.46 -0.19 -0.018
- p2(v3):  0.25  0.28 -0.33
- f1
    - flags: top simplicial
    - area: 0.26
    - normal:    0.2054  0.06987   0.9762
    - offset:   0.125398
    - center: -0.1205015151724569 -0.008790385787885074 -0.1024776532548504 
    - vertices: p1(v2) p5(v1) p4(v0)
    - neighboring facets: f2 f3 f4
- f2
    - flags: bottom simplicial
    - area: 0.23
    - normal:   -0.2032  -0.3001   -0.932
    - offset: -0.1680571
    - center: 0.06430144219370747 -0.04270356053094847 -0.1805812707066012 
    - maxoutside:  0.1132672
    - coplanar set(furthest p3):
      p0:   0.07025  -0.4742   -0.142
      p3:    0.2976  -0.2965  -0.2712
      furthest distance= 0.11
    - vertices: p2(v3) p5(v1) p4(v0)
    - neighboring facets: f1 f3 f4
- f3
    - flags: top simplicial
    - area: 0.18
    - normal:   -0.3912 -0.01238  -0.9202
    - offset: -0.1989549
    - center: -0.1715856748140029 0.1590825675964338 -0.1454013928109104 
    - vertices: p2(v3) p1(v2) p4(v0)
    - neighboring facets: f1 f2 f4
- f4
    - flags: bottom simplicial
    - area: 0.16
    - normal:    0.4184   0.3424   0.8413
    - offset: 0.07254957
    - center: 0.116549362001881 0.1503252206347392 -0.2053893818610149 
    - vertices: p2(v3) p1(v2) p5(v1)
    - neighboring facets: f1 f2 f3
rbox 6 s D3 | qhull C-0.1 Qc FF s FQ Fi n Fo FQ FI Fm Fn FN FO FO FQ Fs FS FV Fv Tcv
4
4
0.205360378770873 0.06987429109559466 0.976188864039624 0.1253980026939646 
-0.2031613339148238 -0.3001304507855299 -0.9320124381751694 -0.1680570625036091 
-0.3911895500493898 -0.01238286325891905 -0.9202268202077502 -0.1989548866041478 
0.4183781509471984 0.3424157187037644 0.8412557271095321 0.07254956874669725 
4
4
0.205360378770873 0.06987429109559466 0.976188864039624 0.125398002693964 
-0.2031613339148238 -0.3001304507855299 -0.9320124381751694 -0.1680570625036098 
-0.3911895500493898 -0.01238286325891905 -0.9202268202077502 -0.1989548866041485 
0.4183781509471984 0.3424157187037644 0.8412557271095321 0.0725495687466966 
4
4
0.205360378770873 0.06987429109559466 0.976188864039624 0.1253980026939627 
-0.2031613339148238 -0.3001304507855299 -0.9320124381751694 -0.2813242952138536 
-0.3911895500493898 -0.01238286325891905 -0.9202268202077502 -0.1989548866041498 
0.4183781509471984 0.3424157187037644 0.8412557271095321 0.07254956874669528 
rbox 6 s D3 | qhull C-0.1 Qc FF s FQ Fi n Fo FQ FI Fm Fn FN FO FO FQ Fs FS FV Fv Tcv
4
1
2
3
4
4
0
0
0
0
4
3 1 2 3
3 0 2 3
3 0 1 3
3 0 1 2
6
1 1
3 3 0 2
3 3 1 2
1 1
3 2 0 1
3 3 0 1
Options selected for Qhull 2011.2 2011/11/21:
  run-id 586733301  Centrum-premerge- 0.1  Qcoplanar-keep  FFacets-xridge
  summary  FQhull  Finner  normals  Fouter  FQhull  FIDs  Fmerges  Fneighbors
  FNeighbors-vertex  FOptions  FQhull  Fsummary  FSize  FVertex-average
  Tcheck-frequently  Tverify  Fvertices  _max-width 0.86
  Error-roundoff 6.6e-016  _one-merge 0.3  _near-inside 1.5
  Visible-distance 0.1  U-coplanar-distance 0.1  Width-outside 0.2
  _wide-facet 0.6
rbox 6 s D3 | qhull C-0.1 Qc FF s FQ Fi n Fo FQ FI Fm Fn FN FO FO FQ Fs FS FV Fv Tcv
10 3 6 4 4 4 4 2 0 0 0
2 0.1132672327102437 -1.315898280434284e-015
0
2 0.8296676646636063 0.01051107893821417 
3 1
-0.02780909644771785 0.06447846047808484 -0.1584624246583442 
4
3 1 5 4
3 2 5 4
3 2 1 4
3 2 1 5
Options selected for Qhull 2011.2 2011/11/21:
  run-id 586733301  Centrum-premerge- 0.1  Qcoplanar-keep  FFacets-xridge
  summary  FQhull  Finner  normals  Fouter  FQhull  FIDs  Fmerges  Fneighbors
  FNeighbors-vertex  FOptions  FQhull  Fsummary  FSize  FVertex-average
  Tcheck-frequently  Tverify  Fvertices  _max-width 0.86
  Error-roundoff 6.6e-016  _one-merge 0.3  _near-inside 1.5
  Visible-distance 0.1  U-coplanar-distance 0.1  Width-outside 0.2
  _wide-facet 0.6

Convex hull of 6 points in 3-d:

  Number of vertices: 4
  Number of coplanar points: 2
  Number of facets: 4

Statistics for: rbox 6 s D3 | qhull C-0.1 Qc FF s FQ Fi n Fo FQ FI Fm Fn FN FO FO FQ Fs FS FV Fv Tcv

  Number of points processed: 4
  Number of hyperplanes created: 5
  Number of distance tests for qhull: 24
  CPU seconds to compute hull (after input):  0
  Total facet area:   0.82966766
  Total volume:       0.010511079
  Maximum distance of point above facet: 0.11 (0.4x)


Output completed.  Verifying that all points are below outer planes of
all facets.  Will make 24 distance computations.
rbox P0.5,0.5 P0.5,0.5 W0 5 D2 | qhull d FN Qc
7
5 3 0 2 -7 -8
1 0
5 -10 1 -6 -7 -8
4 -6 1 0 2
3 -7 2 -6
3 -10 -8 3
4 -10 1 0 3
rbox 10 D3 | qhull Fa PA5
5
0.2844546842397038 
0.1170134053930415 
0.1758021655943633 
0.07490600482805046 
0.2087585943133527 
rbox 10 D3 | qhull Fa PF0.4
0
QH7055 qhull warning: no facets printed

echo === test Qt ${d:-`date`}
date
=== test Qt Wed Nov 23 22:04:47 EST 2011
rbox c | qhull Qt s o Tcv
3
8 12 18
  -0.5   -0.5   -0.5 
  -0.5   -0.5    0.5 
  -0.5    0.5   -0.5 
  -0.5    0.5    0.5 
   0.5   -0.5   -0.5 
   0.5   -0.5    0.5 
   0.5    0.5   -0.5 
   0.5    0.5    0.5 
3 6 2 0 
3 4 6 0 
3 5 4 0 
3 1 5 0 
3 5 6 4 
3 6 5 7 
3 2 3 0 
3 3 1 0 
3 6 3 2 
3 3 6 7 
3 3 5 1 
3 5 3 7 

Convex hull of 8 points in 3-d:

  Number of vertices: 8
  Number of facets: 12
  Number of triangulated facets: 6

Statistics for: rbox c | qhull Qt s o Tcv

  Number of points processed: 8
  Number of hyperplanes created: 11
  Number of distance tests for qhull: 35
  Number of distance tests for merging: 166
  Number of distance tests for checking: 56
  Number of merged facets: 6
  CPU seconds to compute hull (after input):  0


Output completed.  Verifying that all points are below outer planes of
all facets.  Will make 96 distance computations.
rbox c | qhull Qt f i
Vertices and facets:
- p6(v6):   0.5   0.5  -0.5
  neighbors: f17 f18 f25 f27 f33 f34
- p2(v2):  -0.5   0.5  -0.5
  neighbors: f17 f29 f33
- p0(v0):  -0.5  -0.5  -0.5
  neighbors: f17 f18 f21 f22 f29 f30
- p4(v1):   0.5  -0.5  -0.5
  neighbors: f18 f21 f25
- p5(v7):   0.5  -0.5   0.5
  neighbors: f21 f22 f25 f27 f37 f38
- p1(v3):  -0.5  -0.5   0.5
  neighbors: f22 f30 f37
- p7(v5):   0.5   0.5   0.5
  neighbors: f27 f34 f38
- p3(v8):  -0.5   0.5   0.5
  neighbors: f29 f30 f33 f34 f37 f38
- f17
    - flags: top simplicial tricoplanar seen keepcentrum
    - owner of normal & centrum is facet f17
    - normal:        -0       -0       -1
    - offset:       -0.5
    - center:      0      0   -0.5 
    - vertices: p6(v6) p2(v2) p0(v0)
    - neighboring facets: f29 f18 f33
- f18
    - flags: bottom simplicial tricoplanar seen
    - owner of normal & centrum is facet f17
    - normal:        -0       -0       -1
    - offset:       -0.5
    - center:      0      0   -0.5 
    - vertices: p6(v6) p4(v1) p0(v0)
    - neighboring facets: f21 f17 f25
- f21
    - flags: top simplicial tricoplanar keepcentrum
    - owner of normal & centrum is facet f21
    - normal:         0       -1        0
    - offset:       -0.5
    - center:      0   -0.5      0 
    - vertices: p5(v7) p4(v1) p0(v0)
    - neighboring facets: f18 f22 f25
- f22
    - flags: bottom simplicial tricoplanar seen
    - owner of normal & centrum is facet f21
    - normal:         0       -1        0
    - offset:       -0.5
    - center:      0   -0.5      0 
    - vertices: p5(v7) p1(v3) p0(v0)
    - neighboring facets: f30 f21 f37
- f25
    - flags: top simplicial tricoplanar keepcentrum
    - owner of normal & centrum is facet f25
    - normal:         1       -0       -0
    - offset:       -0.5
    - center:    0.5      0      0 
    - vertices: p5(v7) p6(v6) p4(v1)
    - neighboring facets: f18 f21 f27
- f27
    - flags: bottom simplicial tricoplanar seen
    - owner of normal & centrum is facet f25
    - normal:         1       -0       -0
    - offset:       -0.5
    - center:    0.5      0      0 
    - vertices: p5(v7) p6(v6) p7(v5)
    - neighboring facets: f34 f38 f25
- f29
    - flags: bottom simplicial tricoplanar keepcentrum
    - owner of normal & centrum is facet f29
    - normal:        -1       -0       -0
    - offset:       -0.5
    - center:   -0.5      0      0 
    - vertices: p3(v8) p2(v2) p0(v0)
    - neighboring facets: f17 f30 f33
- f30
    - flags: top simplicial tricoplanar
    - owner of normal & centrum is facet f29
    - normal:        -1       -0       -0
    - offset:       -0.5
    - center:   -0.5      0      0 
    - vertices: p3(v8) p1(v3) p0(v0)
    - neighboring facets: f22 f29 f37
- f33
    - flags: bottom simplicial tricoplanar keepcentrum
    - owner of normal & centrum is facet f33
    - normal:         0        1       -0
    - offset:       -0.5
    - center:      0    0.5      0 
    - vertices: p3(v8) p6(v6) p2(v2)
    - neighboring facets: f17 f29 f34
- f34
    - flags: top simplicial tricoplanar
    - owner of normal & centrum is facet f33
    - normal:         0        1       -0
    - offset:       -0.5
    - center:      0    0.5      0 
    - vertices: p3(v8) p6(v6) p7(v5)
    - neighboring facets: f27 f38 f33
- f37
    - flags: top simplicial tricoplanar keepcentrum
    - owner of normal & centrum is facet f37
    - normal:        -0       -0        1
    - offset:       -0.5
    - center:      0      0    0.5 
    - vertices: p3(v8) p5(v7) p1(v3)
    - neighboring facets: f22 f30 f38
- f38
    - flags: bottom simplicial tricoplanar
    - owner of normal & centrum is facet f37
    - normal:        -0       -0        1
    - offset:       -0.5
    - center:      0      0    0.5 
    - vertices: p3(v8) p5(v7) p7(v5)
    - neighboring facets: f27 f34 f37
12
6 2 0 
4 6 0 
5 4 0 
1 5 0 
5 6 4 
6 5 7 
2 3 0 
3 1 0 
6 3 2 
3 6 7 
3 5 1 
5 3 7 
rbox c | qhull Qt m FM n
{
Polygon[{{      0.50000000,       0.50000000,      -0.50000000},
{     -0.50000000,       0.50000000,      -0.50000000},
{     -0.50000000,      -0.50000000,      -0.50000000}}],
Polygon[{{      0.50000000,      -0.50000000,      -0.50000000},
{      0.50000000,       0.50000000,      -0.50000000},
{     -0.50000000,      -0.50000000,      -0.50000000}}],
Polygon[{{      0.50000000,      -0.50000000,       0.50000000},
{      0.50000000,      -0.50000000,      -0.50000000},
{     -0.50000000,      -0.50000000,      -0.50000000}}],
Polygon[{{     -0.50000000,      -0.50000000,       0.50000000},
{      0.50000000,      -0.50000000,       0.50000000},
{     -0.50000000,      -0.50000000,      -0.50000000}}],
Polygon[{{      0.50000000,      -0.50000000,       0.50000000},
{      0.50000000,       0.50000000,      -0.50000000},
{      0.50000000,      -0.50000000,      -0.50000000}}],
Polygon[{{      0.50000000,       0.50000000,      -0.50000000},
{      0.50000000,      -0.50000000,       0.50000000},
{      0.50000000,       0.50000000,       0.50000000}}],
Polygon[{{     -0.50000000,       0.50000000,      -0.50000000},
{     -0.50000000,       0.50000000,       0.50000000},
{     -0.50000000,      -0.50000000,      -0.50000000}}],
Polygon[{{     -0.50000000,       0.50000000,       0.50000000},
{     -0.50000000,      -0.50000000,       0.50000000},
{     -0.50000000,      -0.50000000,      -0.50000000}}],
Polygon[{{      0.50000000,       0.50000000,      -0.50000000},
{     -0.50000000,       0.50000000,       0.50000000},
{     -0.50000000,       0.50000000,      -0.50000000}}],
Polygon[{{     -0.50000000,       0.50000000,       0.50000000},
{      0.50000000,       0.50000000,      -0.50000000},
{      0.50000000,       0.50000000,       0.50000000}}],
Polygon[{{     -0.50000000,       0.50000000,       0.50000000},
{      0.50000000,      -0.50000000,       0.50000000},
{     -0.50000000,      -0.50000000,       0.50000000}}],
Polygon[{{      0.50000000,      -0.50000000,       0.50000000},
{     -0.50000000,       0.50000000,       0.50000000},
{      0.50000000,       0.50000000,       0.50000000}}]}
PLOT3D(POLYGONS(
[[      0.50000000,       0.50000000,      -0.50000000],
[     -0.50000000,       0.50000000,      -0.50000000],
[     -0.50000000,      -0.50000000,      -0.50000000]],
[[      0.50000000,      -0.50000000,      -0.50000000],
[      0.50000000,       0.50000000,      -0.50000000],
[     -0.50000000,      -0.50000000,      -0.50000000]],
[[      0.50000000,      -0.50000000,       0.50000000],
[      0.50000000,      -0.50000000,      -0.50000000],
[     -0.50000000,      -0.50000000,      -0.50000000]],
[[     -0.50000000,      -0.50000000,       0.50000000],
[      0.50000000,      -0.50000000,       0.50000000],
[     -0.50000000,      -0.50000000,      -0.50000000]],
[[      0.50000000,      -0.50000000,       0.50000000],
[      0.50000000,       0.50000000,      -0.50000000],
[      0.50000000,      -0.50000000,      -0.50000000]],
[[      0.50000000,       0.50000000,      -0.50000000],
[      0.50000000,      -0.50000000,       0.50000000],
[      0.50000000,       0.50000000,       0.50000000]],
[[     -0.50000000,       0.50000000,      -0.50000000],
[     -0.50000000,       0.50000000,       0.50000000],
[     -0.50000000,      -0.50000000,      -0.50000000]],
[[     -0.50000000,       0.50000000,       0.50000000],
[     -0.50000000,      -0.50000000,       0.50000000],
[     -0.50000000,      -0.50000000,      -0.50000000]],
[[      0.50000000,       0.50000000,      -0.50000000],
[     -0.50000000,       0.50000000,       0.50000000],
[     -0.50000000,       0.50000000,      -0.50000000]],
[[     -0.50000000,       0.50000000,       0.50000000],
[      0.50000000,       0.50000000,      -0.50000000],
[      0.50000000,       0.50000000,       0.50000000]],
[[     -0.50000000,       0.50000000,       0.50000000],
[      0.50000000,      -0.50000000,       0.50000000],
[     -0.50000000,      -0.50000000,       0.50000000]],
[[      0.50000000,      -0.50000000,       0.50000000],
[     -0.50000000,       0.50000000,       0.50000000],
[      0.50000000,       0.50000000,       0.50000000]]));
4
12
    -0     -0     -1   -0.5 
    -0     -0     -1   -0.5 
     0     -1      0   -0.5 
     0     -1      0   -0.5 
     1     -0     -0   -0.5 
     1     -0     -0   -0.5 
    -1     -0     -0   -0.5 
    -1     -0     -0   -0.5 
     0      1     -0   -0.5 
     0      1     -0   -0.5 
    -0     -0      1   -0.5 
    -0     -0      1   -0.5 
rbox c | qhull Qt p o
3
8
  -0.5   -0.5   -0.5 
  -0.5   -0.5    0.5 
  -0.5    0.5   -0.5 
  -0.5    0.5    0.5 
   0.5   -0.5   -0.5 
   0.5   -0.5    0.5 
   0.5    0.5   -0.5 
   0.5    0.5    0.5 
3
8 12 18
  -0.5   -0.5   -0.5 
  -0.5   -0.5    0.5 
  -0.5    0.5   -0.5 
  -0.5    0.5    0.5 
   0.5   -0.5   -0.5 
   0.5   -0.5    0.5 
   0.5    0.5   -0.5 
   0.5    0.5    0.5 
3 6 2 0 
3 4 6 0 
3 5 4 0 
3 1 5 0 
3 5 6 4 
3 6 5 7 
3 2 3 0 
3 3 1 0 
3 6 3 2 
3 3 6 7 
3 3 5 1 
3 5 3 7 
rbox c | qhull Qt Fx
8
0
1
2
3
4
5
6
7
rbox c | qhull Qt FA s Fa
12
   0.5 
   0.5 
   0.5 
   0.5 
   0.5 
   0.5 
   0.5 
   0.5 
   0.5 
   0.5 
   0.5 
   0.5 

Convex hull of 8 points in 3-d:

  Number of vertices: 8
  Number of facets: 12
  Number of triangulated facets: 6

Statistics for: rbox c | qhull Qt FA s Fa

  Number of points processed: 8
  Number of hyperplanes created: 11
  Number of distance tests for qhull: 35
  Number of distance tests for merging: 108
  Number of distance tests for checking: 56
  Number of merged facets: 6
  CPU seconds to compute hull (after input):  0
  Approximate facet area:    6
  Approximate volume:        1

rbox 6 r s c G0.1 D2 | qhull Qt d FA Tcv

Delaunay triangulation by the convex hull of 10 points in 3-d:

  Number of input sites: 10
  Number of Delaunay regions: 12
  Number of triangulated facets: 3

Statistics for: rbox 6 r s c G0.1 D2 | qhull Qt d FA Tcv

  Number of points processed: 10
  Number of hyperplanes created: 20
  Number of facets in hull: 16
  Number of distance tests for qhull: 51
  Number of distance tests for merging: 257
  Number of distance tests for checking: 100
  Number of merged facets: 7
  CPU seconds to compute hull (after input):  0
  Approximate facet area:   0.64951905


Output completed.  Verifying that all points are below outer planes of
all facets.  Will make 120 distance computations.
rbox 6 r s c G0.1 D2 | qhull d FA Tcv

Delaunay triangulation by the convex hull of 10 points in 3-d:

  Number of input sites: 10
  Number of Delaunay regions: 9
  Number of non-simplicial Delaunay regions: 3

Statistics for: rbox 6 r s c G0.1 D2 | qhull d FA Tcv

  Number of points processed: 10
  Number of hyperplanes created: 20
  Number of facets in hull: 10
  Number of distance tests for qhull: 51
  Number of distance tests for merging: 239
  Number of distance tests for checking: 100
  Number of merged facets: 7
  CPU seconds to compute hull (after input):  0
  Approximate facet area:   0.64951905


Output completed.  Verifying that all points are below outer planes of
all facets.  Will make 90 distance computations.
rbox 6 r s c G0.1 D2 | qhull Qt v p Tcv
2
12
-0.3359969878447703 -0.1939879513790814 
-0.3359969878447704 0.1939879513790817 
-0.2875      0 
0.3359969878447702 -0.1939879513790819 
0.3359969878447707 0.1939879513790815 
0.2875      0 
-2.220446049250313e-016 -0.3453321730569308 
-2.220446049250313e-016 -0.3453321730569308 
1.110223024625157e-016 0.3453321730569308 
1.110223024625157e-016 0.3453321730569308 
     0      0 
     0      0 

Output completed.  Verifying that all points are below outer planes of
all facets.  Will make 120 distance computations.
rbox c | qhull Qt C-0.1 Qc FF s FQ Fi n Fo FQ FI Fm Fn FN FO FO FQ Fs FS FV Fv Tcv
Vertices and facets:
- p6(v6):   0.5   0.5  -0.5
  neighbors: f17 f18 f25 f27 f33 f34
- p2(v2):  -0.5   0.5  -0.5
  neighbors: f17 f29 f33
- p0(v0):  -0.5  -0.5  -0.5
  neighbors: f17 f18 f21 f22 f29 f30
- p4(v1):   0.5  -0.5  -0.5
  neighbors: f18 f21 f25
- p5(v7):   0.5  -0.5   0.5
  neighbors: f21 f22 f25 f27 f37 f38
- p1(v3):  -0.5  -0.5   0.5
  neighbors: f22 f30 f37
- p7(v5):   0.5   0.5   0.5
  neighbors: f27 f34 f38
- p3(v8):  -0.5   0.5   0.5
  neighbors: f29 f30 f33 f34 f37 f38
- f17
    - flags: top simplicial tricoplanar keepcentrum
    - area: 0.5
    - normal:        -0       -0       -1
    - offset:       -0.5
    - center:      0      0   -0.5 
    - vertices: p6(v6) p2(v2) p0(v0)
    - neighboring facets: f29 f18 f33
- f18
    - flags: bottom simplicial tricoplanar
    - area: 0.5
    - normal:        -0       -0       -1
    - offset:       -0.5
    - center:      0      0   -0.5 
    - vertices: p6(v6) p4(v1) p0(v0)
    - neighboring facets: f21 f17 f25
- f21
    - flags: top simplicial tricoplanar keepcentrum
    - area: 0.5
    - normal:         0       -1        0
    - offset:       -0.5
    - center:      0   -0.5      0 
    - vertices: p5(v7) p4(v1) p0(v0)
    - neighboring facets: f18 f22 f25
- f22
    - flags: bottom simplicial tricoplanar
    - area: 0.5
    - normal:         0       -1        0
    - offset:       -0.5
    - center:      0   -0.5      0 
    - vertices: p5(v7) p1(v3) p0(v0)
    - neighboring facets: f30 f21 f37
- f25
    - flags: top simplicial tricoplanar keepcentrum
    - area: 0.5
    - normal:         1       -0       -0
    - offset:       -0.5
    - center:    0.5      0      0 
    - vertices: p5(v7) p6(v6) p4(v1)
    - neighboring facets: f18 f21 f27
- f27
    - flags: bottom simplicial tricoplanar
    - area: 0.5
    - normal:         1       -0       -0
    - offset:       -0.5
    - center:    0.5      0      0 
    - vertices: p5(v7) p6(v6) p7(v5)
    - neighboring facets: f34 f38 f25
- f29
    - flags: bottom simplicial tricoplanar keepcentrum
    - area: 0.5
    - normal:        -1       -0       -0
    - offset:       -0.5
    - center:   -0.5      0      0 
    - vertices: p3(v8) p2(v2) p0(v0)
    - neighboring facets: f17 f30 f33
- f30
    - flags: top simplicial tricoplanar
    - area: 0.5
    - normal:        -1       -0       -0
    - offset:       -0.5
    - center:   -0.5      0      0 
    - vertices: p3(v8) p1(v3) p0(v0)
    - neighboring facets: f22 f29 f37
- f33
    - flags: bottom simplicial tricoplanar keepcentrum
    - area: 0.5
    - normal:         0        1       -0
    - offset:       -0.5
    - center:      0    0.5      0 
    - vertices: p3(v8) p6(v6) p2(v2)
    - neighboring facets: f17 f29 f34
- f34
    - flags: top simplicial tricoplanar
    - area: 0.5
    - normal:         0        1       -0
    - offset:       -0.5
    - center:      0    0.5      0 
    - vertices: p3(v8) p6(v6) p7(v5)
    - neighboring facets: f27 f38 f33
- f37
    - flags: top simplicial tricoplanar keepcentrum
    - area: 0.5
    - normal:        -0       -0        1
    - offset:       -0.5
    - center:      0      0    0.5 
    - vertices: p3(v8) p5(v7) p1(v3)
    - neighboring facets: f22 f30 f38
- f38
    - flags: bottom simplicial tricoplanar
    - area: 0.5
    - normal:        -0       -0        1
    - offset:       -0.5
    - center:      0      0    0.5 
    - vertices: p3(v8) p5(v7) p7(v5)
    - neighboring facets: f27 f34 f37
rbox c | qhull Qt C-0.1 Qc FF s FQ Fi n Fo FQ FI Fm Fn FN FO FO FQ Fs FS FV Fv Tcv
4
12
    -0     -0     -1 -0.4999999999999993 
    -0     -0     -1 -0.4999999999999993 
     0     -1      0 -0.4999999999999993 
     0     -1      0 -0.4999999999999993 
     1     -0     -0 -0.4999999999999993 
     1     -0     -0 -0.4999999999999993 
    -1     -0     -0 -0.4999999999999993 
    -1     -0     -0 -0.4999999999999993 
     0      1     -0 -0.4999999999999993 
     0      1     -0 -0.4999999999999993 
    -0     -0      1 -0.4999999999999993 
    -0     -0      1 -0.4999999999999993 
4
12
    -0     -0     -1   -0.5 
    -0     -0     -1   -0.5 
     0     -1      0   -0.5 
     0     -1      0   -0.5 
     1     -0     -0   -0.5 
     1     -0     -0   -0.5 
    -1     -0     -0   -0.5 
    -1     -0     -0   -0.5 
     0      1     -0   -0.5 
     0      1     -0   -0.5 
    -0     -0      1   -0.5 
    -0     -0      1   -0.5 
4
12
    -0     -0     -1 -0.5000000000000013 
    -0     -0     -1 -0.5000000000000013 
     0     -1      0 -0.5000000000000013 
     0     -1      0 -0.5000000000000013 
     1     -0     -0 -0.5000000000000013 
     1     -0     -0 -0.5000000000000013 
    -1     -0     -0 -0.5000000000000013 
    -1     -0     -0 -0.5000000000000013 
     0      1     -0 -0.5000000000000013 
     0      1     -0 -0.5000000000000013 
    -0     -0      1 -0.5000000000000013 
    -0     -0      1 -0.5000000000000013 
rbox c | qhull Qt C-0.1 Qc FF s FQ Fi n Fo FQ FI Fm Fn FN FO FO FQ Fs FS FV Fv Tcv
12
17
18
21
22
25
27
29
30
33
34
37
38
12
0
0
0
0
0
0
0
0
0
0
0
0
12
3 6 1 8
3 2 0 4
3 1 3 4
3 7 2 10
3 1 2 5
3 9 11 4
3 0 7 8
3 3 6 10
3 0 6 9
3 5 11 8
3 3 7 11
3 5 9 10
8
6 7 3 2 1 0 6
3 10 3 7
3 8 0 6
6 11 9 8 6 7 10
3 4 1 2
6 11 5 4 2 3 10
6 9 5 4 1 0 8
3 11 5 9
Options selected for Qhull 2011.2 2011/11/21:
  run-id 586733301  Qtriangulate  Centrum-premerge- 0.1  Qcoplanar-keep
  FFacets-xridge  summary  FQhull  Finner  normals  Fouter  FQhull  FIDs
  Fmerges  Fneighbors  FNeighbors-vertex  FOptions  FQhull  Fsummary  FSize
  FVertex-average  Tcheck-frequently  Tverify  Fvertices  _max-width  1
  Error-roundoff 6.9e-016  _one-merge 0.3  _near-inside 1.5
  Visible-distance 0.1  U-coplanar-distance 0.1  Width-outside 0.2
  _wide-facet 0.6
rbox c | qhull Qt C-0.1 Qc FF s FQ Fi n Fo FQ FI Fm Fn FN FO FO FQ Fs FS FV Fv Tcv
10 3 8 8 12 8 12 0 0 0 6
2 1.387359992873471e-015 -6.936799964367356e-016
0
2      6      1 
3 1
     0      0      0 
12
3 6 2 0
3 6 4 0
3 5 4 0
3 5 1 0
3 5 6 4
3 5 6 7
3 3 2 0
3 3 1 0
3 3 6 2
3 3 6 7
3 3 5 1
3 3 5 7
Options selected for Qhull 2011.2 2011/11/21:
  run-id 586733301  Qtriangulate  Centrum-premerge- 0.1  Qcoplanar-keep
  FFacets-xridge  summary  FQhull  Finner  normals  Fouter  FQhull  FIDs
  Fmerges  Fneighbors  FNeighbors-vertex  FOptions  FQhull  Fsummary  FSize
  FVertex-average  Tcheck-frequently  Tverify  Fvertices  _max-width  1
  Error-roundoff 6.9e-016  _one-merge 0.3  _near-inside 1.5
  Visible-distance 0.1  U-coplanar-distance 0.1  Width-outside 0.2
  _wide-facet 0.6

Convex hull of 8 points in 3-d:

  Number of vertices: 8
  Number of facets: 12
  Number of triangulated facets: 6

Statistics for: rbox c | qhull Qt C-0.1 Qc FF s FQ Fi n Fo FQ FI Fm Fn FN FO FO FQ Fs FS FV Fv Tcv

  Number of points processed: 8
  Number of hyperplanes created: 11
  Number of distance tests for qhull: 35
  Number of distance tests for merging: 198
  Number of distance tests for checking: 56
  Number of merged facets: 6
  CPU seconds to compute hull (after input):  0
  Approximate facet area:    6
  Approximate volume:        1


Output completed.  Verifying that all points are below outer planes of
all facets.  Will make 96 distance computations.
rbox 6 r s c G0.1 D2 P0.1,0.1 | qhull s FP d FO Qt
1
0 10 36      0 
Options selected for Qhull 2011.2 2011/11/21:
  run-id 586733301  summary  FPoint-nearest  delaunay  Qtriangulate  _pre-merge
  _zero-centrum  Pgood  Qcoplanar  _max-width  1  Error-roundoff 6.9e-016
  _one-merge 4.9e-015  _near-inside 2.4e-014  Visible-distance 1.4e-015
  U-coplanar-distance 1.4e-015  Width-outside 2.8e-015  _wide-facet 8.3e-015

Delaunay triangulation by the convex hull of 11 points in 3-d:

  Number of input sites: 10
  Number of nearly incident points: 1
  Number of Delaunay regions: 12
  Number of triangulated facets: 3

Statistics for: rbox 6 r s c G0.1 D2 P0.1,0.1 | qhull s FP d FO Qt

  Number of points processed: 10
  Number of hyperplanes created: 20
  Number of facets in hull: 16
  Number of distance tests for qhull: 83
  Number of distance tests for merging: 166
  Number of distance tests for checking: 111
  Number of merged facets: 6
  CPU seconds to compute hull (after input):  0

rbox 100 W0 | qhull Tv Q11 

Convex hull of 100 points in 3-d:

  Number of vertices: 45
  Number of facets: 86
  Number of triangulated facets: 33

Statistics for: rbox 100 W0 | qhull Tv Q11

  Number of points processed: 52
  Number of hyperplanes created: 172
  Number of distance tests for qhull: 1576
  Number of distance tests for merging: 1498
  Number of distance tests for checking: 2172
  Number of merged facets: 41
  CPU seconds to compute hull (after input):  0


Output completed.  Verifying that all points are below outer planes of
all facets.  Will make 8600 distance computations.

echo === test unbounded intersection ${d:-`date`}
date
=== test unbounded intersection Wed Nov 23 22:04:47 EST 2011
rbox c | qhull PD0:0.5 n | qhull H0 Fp Tcv
3
5
  -0.5    0.5   -0.5 
  -0.5   -0.5   -0.5 
  -0.5    0.5    0.5 
  -0.5   -0.5    0.5 
-10.101 -10.101 -10.101 

Output completed.  Verifying that all points are below outer planes of
all facets.  Will make 25 distance computations.
rbox 1000 W1e-3 D3 | qhull PA8 Fa FS s n Tcv 
8
0.1446087232094601 
0.1622198334884075 
0.2057158590759609 
0.1567143851824352 
0.188985866819172 
0.1580674614313253 
0.1616733826473112 
0.1767351337533958 
0
2 5.881743214947514 0.9968516315221345 
4
8
1.009144997325421e-005 0.9999999999463923 2.319049754624652e-006 -0.4999875785488853 
-1.229609861584452e-005 8.587946880918045e-005 0.9999999962367613 -0.500004859549355 
-1.383340239672453e-005 -0.9999999998842694 6.332306547224587e-006 -0.4999917156571327 
1.802476239613105e-005 -1.901476191096464e-005 0.9999999996567733 -0.4999909156385879 
0.9999999996788914 1.780140020603311e-005 -1.803683546259361e-005 -0.4999971349013697 
3.470111681088057e-006 -3.668365732095583e-005 -0.9999999993211337 -0.4999893476858696 
-1.633273154317779e-005 -0.9999999998072314 1.089857916466569e-005 -0.4999921708899001 
-0.9999999997878146 -2.055073083002867e-005 -1.427680287414111e-006 -0.4999990322171606 

Convex hull of 1000 points in 3-d:

  Number of vertices: 156
  Number of facets: 308
  Number of 'good' facets: 8

Statistics for: rbox 1000 W1e-3 D3 | qhull PA8 Fa FS s n Tcv

  Number of points processed: 203
  Number of hyperplanes created: 1043
  Number of distance tests for qhull: 21470
  CPU seconds to compute hull (after input):  0
  Total facet area:   5.8817432
  Total volume:       0.99685163


Output completed.  Verifying that all points are below outer planes of
all facets.  Will make 8000 distance computations.
rbox 1000 W1e-3 D3 | qhull C-0.01 PM10 Fm n Tcv Qc
10
1
1
8
0
6
7
10
0
11
6
4
10
0.6345454980886096 0.2737427089707436 0.7227841587506171 -0.7458921924784656 
-0.00253733560834815 0.666653414016696 0.7453635270845322 -0.6767722083805112 
-0.0004022623770974538 0.9999997831336711 0.0005214571799133833 -0.5000167453415887 
-0.5357402012449896 -0.5359428531021374 -0.6524934443952053 -0.811263040246498 
-0.005480509661076106 -0.9999701072857251 0.005454223027194167 -0.4992904789191579 
0.001196850675493193 0.0003334814784797861 0.9999992281689842 -0.4989807987213126 
0.9999992961453315 0.00118637926097064 1.459763172590229e-005 -0.4998611663332402 
-0.9129791746952519 0.1462545158779729 -0.3808919048208773 -0.6797481567213483 
-0.999601787347817 0.02140818392627282 0.01838359029211947 -0.4985486000821886 
-0.003237401631533091 0.002761552371517554 -0.9999909464886048 -0.4969398742962436 

Output completed.  Verifying that all points are below outer planes of
all facets.  Will make 10000 distance computations.
rbox 1000 W1e-3 D3 | qhull C-0.01 PA8 PG n Tcv Qc
4
11
0.6345454980886096 0.2737427089707436 0.7227841587506171 -0.7458921924784656 
0.8278306348810541 -0.3414688798813042 -0.4450791435520636 -0.7665474225756105 
0.7383776593319754 -0.4655216164679658 0.4879426777814023 -0.7788991125303444 
0.2008161614085187 -0.8823413491644904 0.425613219803773 -0.6982307124493008 
-0.4745617604911457 -0.2935495843214587 0.8298311738084099 -0.7523676152838523 
-0.8259135249806853 -0.559220008247317 0.07169261907511661 -0.6812000445406896 
0.41932306725662 0.6721295192910486 -0.6102541065524131 -0.7755692997856908 
-0.5357402012449896 -0.5359428531021374 -0.6524934443952053 -0.811263040246498 
-0.3304054030140146 0.4364026747636864 -0.8368900615482577 -0.7495022949733237 
0.6755431502353132 -0.2163920762108173 0.7048517017950092 -0.7475007704997935 
-0.9129791746952519 0.1462545158779729 -0.3808919048208773 -0.6797481567213483 

Output completed.  Verifying that all points are below outer planes of
all facets.  Will make 8000 distance computations.
rbox 10 | qhull FO Tz TO q_test.log.1
cat q_test.log.1
Options selected for Qhull 2011.2 2011/11/21:
  run-id 586733301  Tz-stdout  TOutput-file  q_test.log.1  _pre-merge
  _zero-centrum  _max-width 0.92  Error-roundoff 6.8e-016  _one-merge 4.8e-015
  _near-inside 2.4e-014  Visible-distance 1.4e-015
  U-coplanar-distance 1.4e-015  Width-outside 2.7e-015  _wide-facet 8.2e-015

Convex hull of 10 points in 3-d:

  Number of vertices: 10
  Number of facets: 16

Statistics for: rbox 10 | qhull FO Tz TO q_test.log.1

  Number of points processed: 10
  Number of hyperplanes created: 27
  Number of distance tests for qhull: 44
  CPU seconds to compute hull (after input):  0


echo === check Delaunay/Voronoi ${d:-`date`}
date
=== check Delaunay/Voronoi Wed Nov 23 22:04:47 EST 2011
rbox 10 D2 | qhull d Tcv

Delaunay triangulation by the convex hull of 10 points in 3-d:

  Number of input sites: 10
  Number of Delaunay regions: 14

Statistics for: rbox 10 D2 | qhull d Tcv

  Number of points processed: 10
  Number of hyperplanes created: 28
  Number of facets in hull: 16
  Number of distance tests for qhull: 48
  CPU seconds to compute hull (after input):  0


Output completed.  Verifying that all points are below 2.1e-015 of
all facets.  Will make 140 distance computations.
rbox 10 D2 | qhull d Tcv Qz

Delaunay triangulation by the convex hull of 11 points in 3-d:

  Number of input sites and at-infinity: 11
  Number of Delaunay regions: 14

Statistics for: rbox 10 D2 | qhull d Tcv Qz

  Number of points processed: 11
  Number of hyperplanes created: 31
  Number of facets in hull: 18
  Number of distance tests for qhull: 56
  CPU seconds to compute hull (after input):  0


Output completed.  Verifying that all points are below 2.1e-015 of
all facets.  Will make 154 distance computations.
rbox 10 D3 | qhull d Tcv

Delaunay triangulation by the convex hull of 10 points in 4-d:

  Number of input sites: 10
  Number of Delaunay regions: 18

Statistics for: rbox 10 D3 | qhull d Tcv

  Number of points processed: 10
  Number of hyperplanes created: 44
  Number of facets in hull: 27
  Number of distance tests for qhull: 67
  CPU seconds to compute hull (after input):  0


Output completed.  Verifying that all points are below 2.9e-015 of
all facets.  Will make 180 distance computations.
rbox c | qhull d Qz Ft Tcv
3
10 12 3
  -0.5   -0.5   -0.5 
  -0.5   -0.5    0.5 
  -0.5    0.5   -0.5 
  -0.5    0.5    0.5 
   0.5   -0.5   -0.5 
   0.5   -0.5    0.5 
   0.5    0.5   -0.5 
   0.5    0.5    0.5 
     0      0      0 
     0      0      0 
4 9 1 2 0 
4 9 2 4 0 
4 9 4 1 0 
4 9 5 7 1 
4 9 7 6 2 
4 9 2 6 4 
4 9 6 7 4 
4 9 5 1 4 
4 9 7 5 4 
4 9 1 3 2 
4 9 3 7 2 
4 9 7 3 1 

Output completed.  Verifying that all points are below outer planes of
all facets.  Will make  9 distance computations.
rbox 10 s D2 c | qhull d Tcv 

Delaunay triangulation by the convex hull of 14 points in 3-d:

  Number of input sites: 14
  Number of Delaunay regions: 15
  Number of non-simplicial Delaunay regions: 1

Statistics for: rbox 10 s D2 c | qhull d Tcv

  Number of points processed: 14
  Number of hyperplanes created: 33
  Number of facets in hull: 16
  Number of distance tests for qhull: 116
  Number of distance tests for merging: 395
  Number of distance tests for checking: 182
  Number of merged facets: 8
  CPU seconds to compute hull (after input):  0


Output completed.  Verifying that all points are below outer planes of
all facets.  Will make 210 distance computations.
rbox 10 s D2 | qhull d Tcv Q8 Qz

Delaunay triangulation by the convex hull of 11 points in 3-d:

  Number of input sites and at-infinity: 11
  Number of Delaunay regions: 1
  Number of non-simplicial Delaunay regions: 1

Statistics for: rbox 10 s D2 | qhull d Tcv Q8 Qz

  Number of points processed: 11
  Number of hyperplanes created: 19
  Number of facets in hull: 11
  Number of distance tests for qhull: 55
  Number of distance tests for merging: 278
  Number of distance tests for checking: 132
  Number of merged facets: 7
  CPU seconds to compute hull (after input):  0

QH7077 qhull input warning: no outer plane check ('Q5') or no processing of
near-inside points ('Q8').  Verify may report that a point is outside
of a facet.

Output completed.  Verifying that all points are below outer planes of
all facets.  Will make 11 distance computations.
rbox 10 D2 | qhull d Tcv p
3
10
-0.02222276248244826 -0.4979727817680433 0.2484707425541546 
-0.4285431913366012 0.4745826469497594 0.4088779556267987 
0.3105396575392593 0.2400179190933871 0.1540434803905201 
-0.01883958887200765 0.3630260628303755 0.13214285240299 
0.3790312361708201 0.3779794437605696 0.2865331378987296 
-0.2994955874043476 0.3776609263174803 0.2323253821416526 
0.3471817493878135 0.08365533089605659 0.1275333814953112 
-0.00485819764887746 0.3482682405489201 0.121314369459436 
0.3443122672329771 -0.1437312230875075 0.1392096018573439 
0.309330780347186 -0.07758103877080702 0.1017043492469565 

Output completed.  Verifying that all points are below 2.1e-015 of
all facets.  Will make 140 distance computations.
rbox 10 D2 | qhull d Tcv i
14
9 7 0 
8 9 0 
3 4 1 
3 7 4 
7 2 4 
0 5 1 
7 5 0 
5 3 1 
3 5 7 
6 7 9 
6 2 7 
8 6 9 
6 8 4 
2 6 4 

Output completed.  Verifying that all points are below 2.1e-015 of
all facets.  Will make 140 distance computations.
rbox 10 D2 | qhull d Tcv o
3
10 14 21
-0.02222276248244826 -0.4979727817680433 0.2484707425541546 
-0.4285431913366012 0.4745826469497594 0.4088779556267987 
0.3105396575392593 0.2400179190933871 0.1540434803905201 
-0.01883958887200765 0.3630260628303755 0.13214285240299 
0.3790312361708201 0.3779794437605696 0.2865331378987296 
-0.2994955874043476 0.3776609263174803 0.2323253821416526 
0.3471817493878135 0.08365533089605659 0.1275333814953112 
-0.00485819764887746 0.3482682405489201 0.121314369459436 
0.3443122672329771 -0.1437312230875075 0.1392096018573439 
0.309330780347186 -0.07758103877080702 0.1017043492469565 
3 9 7 0 
3 8 9 0 
3 3 4 1 
3 3 7 4 
3 7 2 4 
3 0 5 1 
3 7 5 0 
3 5 3 1 
3 3 5 7 
3 6 7 9 
3 6 2 7 
3 8 6 9 
3 6 8 4 
3 2 6 4 

Output completed.  Verifying that all points are below 2.1e-015 of
all facets.  Will make 140 distance computations.
rbox 10 D2 | qhull v Tcv o
2
15 10 1
-10.101 -10.101 
-0.1294381801544404 -0.07247409101984714 
0.08267689532419748 -0.2397644955865706 
0.1295260566906465 1.716033573116838 
0.1740355150742391 0.5317519038435655 
0.1851415205797575 0.3882545794457364 
-0.9065939866848107 -0.2962957610652136 
-0.1954805620516266 -0.07111892482963184 
-0.1407581310832468 0.7233857048236082 
-0.1676297826663962 0.2080621273999375 
0.05868313821742954 0.06632066014880154 
0.08806341399736994 0.1054080604689985 
0.4761588899009253 -0.03168366595227294 
3.094213357897477 -0.064721945677682 
0.5410515627308725 0.211561543495592 
5 7 1 2 0 6
4 8 3 0 6
3 14 5 11
4 9 4 3 8
6 14 5 4 3 0 13
4 9 7 6 8
5 14 11 10 12 13
7 11 5 4 9 7 1 10
4 13 0 2 12
4 12 2 1 10

Output completed.  Verifying that all points are below 2.1e-015 of
all facets.  Will make 140 distance computations.
rbox 10 D2 | qhull v Tcv p
2
14
-0.1294381801544404 -0.07247409101984714 
0.08267689532419748 -0.2397644955865706 
0.1295260566906465 1.716033573116838 
0.1740355150742391 0.5317519038435655 
0.1851415205797575 0.3882545794457364 
-0.9065939866848107 -0.2962957610652136 
-0.1954805620516266 -0.07111892482963184 
-0.1407581310832468 0.7233857048236082 
-0.1676297826663962 0.2080621273999375 
0.05868313821742954 0.06632066014880154 
0.08806341399736994 0.1054080604689985 
0.4761588899009253 -0.03168366595227294 
3.094213357897477 -0.064721945677682 
0.5410515627308725 0.211561543495592 

Output completed.  Verifying that all points are below 2.1e-015 of
all facets.  Will make 140 distance computations.
rbox 10 D2 | qhull v Tcv G
{appearance {linewidth 3} LIST # rbox 10 D2 | qhull v Tcv G
{appearance {+edge -face} OFF 15 10 1 # Voronoi centers and cells
     0      0  0 # infinity not used
# 1 f8
-0.1294381801544404 -0.07247409101984714  0
# 2 f11
0.08267689532419748 -0.2397644955865706  0
# 3 f12
0.1295260566906465 1.716033573116838  0
# 4 f14
0.1740355150742391 0.5317519038435655  0
# 5 f15
0.1851415205797575 0.3882545794457364  0
# 6 f19
-0.9065939866848107 -0.2962957610652136  0
# 7 f20
-0.1954805620516266 -0.07111892482963184  0
# 8 f21
-0.1407581310832468 0.7233857048236082  0
# 9 f22
-0.1676297826663962 0.2080621273999375  0
# 10 f23
0.05868313821742954 0.06632066014880154  0
# 11 f24
0.08806341399736994 0.1054080604689985  0
# 12 f25
0.4761588899009253 -0.03168366595227294  0
# 13 f26
3.094213357897477 -0.064721945677682  0
# 14 f27
0.5410515627308725 0.211561543495592  0
4 7 1 2 6 # p0(v2)
3 8 3 6 # p1(v0)
3 14 5 11 # p2(v8)
4 9 4 3 8 # p3(v7)
5 14 5 4 3 13 # p4(v1)
4 9 7 6 8 # p5(v9)
5 14 11 10 12 13 # p6(v10)
7 11 5 4 9 7 1 10 # p7(v5)
3 13 2 12 # p8(v6)
4 12 2 1 10 # p9(v3)
}
}

Output completed.  Verifying that all points are below 2.1e-015 of
all facets.  Will make 140 distance computations.
rbox 10 D2 | qhull v Tcv Fv
23
4 0 1 0 6
4 0 7 1 7
4 0 9 1 2
4 0 8 0 2
4 0 5 6 7
4 1 4 0 3
4 1 3 3 8
4 1 5 6 8
4 2 7 5 11
4 2 4 5 14
4 2 6 11 14
4 3 4 3 4
4 3 7 4 9
4 3 5 8 9
4 4 8 0 13
4 4 7 4 5
4 4 6 13 14
4 5 7 7 9
4 6 7 10 11
4 6 9 10 12
4 6 8 12 13
4 7 9 1 10
4 8 9 2 12

Output completed.  Verifying that all points are below 2.1e-015 of
all facets.  Will make 140 distance computations.
rbox 10 D2 | qhull v Tcv Fi
19
5 0 7 0.02051532578114336 0.9997895385570373 0.07511430445246037 
5 0 9 0.619259133056969 0.7851866823409139 0.1370614663104577 
5 0 5 -0.301880639203215 0.9533457293522945 0.008789126238507583 
5 1 3 0.9648718030091262 -0.2627211520946079 0.3258622775065156 
5 1 5 0.7995952824356682 -0.6005392446015695 0.5469710423089692 
5 2 7 -0.9458410914974847 0.3246299888101017 0.04907537812572171 
5 2 4 0.444671031232542 0.8956939622340812 -0.4300843534994401 
5 2 6 0.2281595097543261 -0.9736237661995858 0.08253528745668784 
5 3 4 0.9992944873266901 0.03755699133966539 -0.1938837324602516 
5 3 7 0.6877523324625972 -0.7259454037269313 0.2663295190946441 
5 3 5 -0.9986432053419565 0.05207445078292259 -0.1782370644858237 
5 4 7 -0.9970183790688239 -0.07716444646969993 0.2145489484590235 
5 4 6 -0.1075842248297359 -0.9941959739245502 0.2685422477498993 
5 5 7 0.9950609206245129 -0.09926612839179765 0.187455367716064 
5 6 7 -0.7993641842155074 0.6008468199079334 0.007060641163779212 
5 6 9 -0.2285415587894343 -0.973534157544611 0.07797696388863314 
5 6 8 -0.01261839651625044 -0.9999203848653946 -0.02567278177543585 
5 7 9 0.5936952885926361 -0.8046899429612046 0.01852766555277016 
5 8 9 -0.4674785097727932 0.8840044360186256 0.2506025495170936 

Voronoi ridge statistics
     19 bounded ridges
4.2e-017   ave. distance of midpoint to ridge
1.7e-016   max. distance of midpoint to ridge
     19 bounded ridges with ok normal
1.2e-017   ave. angle to ridge
1.1e-016   max. angle to ridge

Output completed.  Verifying that all points are below 2.1e-015 of
all facets.  Will make 140 distance computations.
rbox 10 D2 | qhull v Tcv Fo
4
5 0 1 -0.3854955578888161 0.9227096915324941 -0.07609298438083822 
5 0 8 0.719062520784207 0.6949453872092842 0.1071733734620186 
5 1 4 0.9929212365203526 -0.1187746524595787 0.07521211888503937 
5 4 8 -0.06640144810487483 -0.9977929883946747 0.1408811441173877 

Output completed.  Verifying that all points are below 2.1e-015 of
all facets.  Will make 140 distance computations.
rbox 10 D2 | qhull v Qu o Fv Fi Fo Tcv
2
3 10 1
-10.101 -10.101 
-0.06934933486244874 0.05349366577784781 
-0.3051278694340224 0.161498505581022 
3 0 2 1
2 0 1
0
0
3 0 1 2
0
0
0
2 0 2
0
5
4 0 8 0 2
4 0 4 1 2
4 0 1 0 1
4 1 4 0 1
4 4 8 0 2
1
5 0 4 0.4164624817835529 0.9091529031283372 -0.01975252543405325 
4
5 0 8 0.7190625207842071 0.6949453872092841 0.1071733734620185 
5 0 1 -0.3854955578888162 0.9227096915324941 -0.07609298438083828 
5 1 4 0.9929212365203528 -0.1187746524595786 0.07521211888503934 
5 4 8 -0.06640144810487485 -0.9977929883946747 0.1408811441173877 

Voronoi ridge statistics
      1 bounded ridges
1.2e-016   ave. distance of midpoint to ridge
1.2e-016   max. distance of midpoint to ridge
      1 bounded ridges with ok normal
      0   max. angle to ridge

Output completed.  Verifying that all points are below 2.1e-015 of
all facets.  Will make 20 distance computations.
rbox 10 D3 | qhull v Fv Tcv
32
6 0 1 0 17 15 3
7 0 9 0 7 12 11 2
6 0 6 0 7 18 17
5 0 7 0 2 3
6 0 2 2 3 15 11
5 0 8 7 18 12
7 0 5 11 12 18 17 15
5 1 7 0 5 3
6 1 2 3 5 0 15
5 1 5 15 0 17
6 2 3 1 4 13 9
6 2 9 1 2 11 9
7 2 7 1 2 3 5 4
6 2 4 4 5 0 13
7 2 5 9 11 15 0 13
7 3 9 0 6 10 9 1
5 3 7 0 4 1
5 3 6 0 8 6
7 3 4 0 8 14 13 4
6 3 8 6 8 14 10
6 3 5 9 10 14 13
5 4 7 0 5 4
5 4 6 0 16 8
5 4 8 8 16 14
6 4 5 13 0 16 14
6 5 9 9 11 12 10
7 5 8 10 12 18 16 14
6 5 6 0 17 18 16
5 6 9 0 6 7
7 6 8 6 7 18 16 8
5 7 9 0 1 2
6 8 9 6 7 12 10

Output completed.  Verifying that all points are below 2.9e-015 of
all facets.  Will make 180 distance computations.
rbox 10 D3 | qhull v Fi Tcv
13
6 0 2 0.1026012494065407 0.8309156055795224 -0.5468566905731633 0.2399036185798247 
6 0 8 -0.1644981128676531 0.5492243080463513 -0.8193247404503221 0.122023739903044 
6 0 5 -0.1640386761614814 0.6276090456203117 -0.7610507201092119 0.1709974344854013 
6 2 3 0.8628264983391891 0.1277424492346306 -0.4890933453107607 -0.1660057988150949 
6 2 9 0.514615149546644 -0.7556910962035636 -0.4050953159143434 -0.3000434746332833 
6 2 7 0.9862472983740704 0.008356748990702902 0.1650649302430821 -0.2195655184060844 
6 3 8 -0.9471370887617876 -0.1545163381156413 -0.281169052968456 0.0437290219976465 
6 3 5 -0.9850113492204077 -0.1553182373520525 -0.07502590921037534 0.09265233510028302 
6 4 8 -0.7177135399298968 -0.2457215366779669 -0.6515429387338296 -0.02974683605231299 
6 5 9 0.7110332662678565 -0.6935177833709648 -0.1160378317993181 -0.270606286195582 
6 5 8 -0.1310037283001487 -0.04490613421135449 -0.9903643078593113 -0.2004293237314498 
6 6 8 0.5636530776159621 0.07101685912338829 0.8229531054770393 0.4221671959133706 
6 8 9 0.7278413449934531 -0.6852671087526033 0.02561183671604039 -0.2412830130907243 

Voronoi ridge statistics
     15 non-simplicial Voronoi vertices for all ridges
4.5e-017   ave. distance to ridge
1.1e-016   max. distance to ridge
     13 bounded ridges
5.1e-017   ave. distance of midpoint to ridge
1.1e-016   max. distance of midpoint to ridge
     13 bounded ridges with ok normal
 6e-017   ave. angle to ridge
2.2e-016   max. angle to ridge

Output completed.  Verifying that all points are below 2.9e-015 of
all facets.  Will make 180 distance computations.
rbox 10 D3 | qhull v Fo Tcv
19
6 0 1 -0.1769702595802887 0.8331351847724079 0.523991689932513 0.004105600879712693 
6 0 9 0.5856843296472445 -0.1558732679166011 -0.7954102025713887 -0.1264594278193165 
6 0 6 -0.305501768024369 0.3812318858994052 -0.8725427891556765 -0.04822326899809229 
6 0 7 0.6633910639781613 0.6672883530766242 -0.3385831509130508 0.06175731395259272 
6 1 7 0.7533877382120365 0.334939238848763 -0.5658821628140998 0.06152842458597928 
6 1 2 0.2077539373736715 0.4344067294949107 -0.8764297432625574 0.2556318087147513 
6 1 5 -0.08886729274769867 0.2768317827550707 -0.9568002760954578 0.1656561460661368 
6 2 4 0.8896724708236493 0.4529391082738924 -0.05769799697211265 -0.124969949041603 
6 2 5 -0.6080629223625024 -0.1069684332549472 -0.7866493734408968 -0.05602222006342443 
6 3 9 0.1157648604151236 -0.9728190902550374 -0.2005530222370839 -0.2615067759720444 
6 3 7 0.3131056731324619 -0.1651787990807294 0.9352437125084113 -0.1060546443225209 
6 3 6 -0.8429664438017094 -0.1288333013498811 -0.5223117412882753 -0.1470675860428305 
6 3 4 -0.6546392846048743 0.1665291828386517 0.7373706247980789 0.1638426312062793 
6 4 7 0.8997625993863546 -0.3161631359012113 0.3007792151107228 -0.2534488584590283 
6 4 6 -0.6610636619910347 -0.1756381951302929 -0.7294834194184101 -0.1931488689248214 
6 4 5 -0.8177140658972856 -0.2799433900977096 -0.5029666040348318 0.01966883326066575 
6 5 6 -0.4656867325018191 -0.06587339986788444 -0.8824945112359766 -0.374728122564582 
6 6 9 0.7885740013077054 -0.5314391513195501 0.309392102204163 -0.04784058563315544 
6 7 9 -0.0125726543874506 -0.8388634914110149 -0.5441966291143082 -0.2018836580171401 

Voronoi ridge statistics
     14 non-simplicial Voronoi vertices for all ridges
5.4e-017   ave. distance to ridge
1.2e-016   max. distance to ridge

Output completed.  Verifying that all points are below 2.9e-015 of
all facets.  Will make 180 distance computations.
rbox 10 D3 | qhull v Qu o Fv Fi Fo Tcv
3
10 10 1
-10.101 -10.101 -10.101 
-0.04510412048603351 -0.1962229636993575 -0.06746145990413344 
-0.2032838761823815 -0.3188041900902701 0.1251484339163768 
0.1359785067833994 0.06954805978697726 -0.07249043373758879 
-0.9550934192592715 0.3815223390696589 -0.9370156532984113 
-0.01637948561987637 -0.8836250780525146 -0.03718065630611937 
0.09499885792019704 -0.7038538868165176 -0.181396220925921 
2.128743389186444 -4.06034590041635 -1.216246894786401 
2.789531685277142 -5.39366622961672 -1.494034092231972 
2.640156785602008 -4.796303906482421 -1.459799746992416 
3 0 3 4
7 0 1 3 4 6 7 9
4 0 7 8 9
3 0 2 5
5 0 5 6 7 8
3 0 8 9
9 0 1 2 3 5 6 7 8 9
6 0 1 2 4 5 6
0
5 0 1 2 3 4
18
5 0 1 0 3 4
5 0 9 0 3 4
8 1 6 0 3 1 6 7 9
5 1 2 0 7 9
5 1 4 0 6 7
6 1 7 0 4 1 6
5 1 9 1 3 4
5 2 5 0 9 8
5 2 4 0 7 8
5 2 6 7 9 8
5 3 6 0 5 2
5 3 7 0 2 5
7 4 6 0 5 6 7 8
5 4 7 0 6 5
5 5 6 0 9 8
6 6 9 0 3 1 2
6 6 7 1 2 5 6
6 7 9 0 2 1 4
3
6 1 9 0.5123553720802024 -0.363812082853534 -0.7779027838170592 -0.100757404152973 
6 2 6 -0.5221667006142706 -0.08192192666458514 -0.8488997200501182 -0.2535440955968324 
6 6 7 0.752213645322669 0.06033708575303686 0.6561509489997043 0.09003235761431797 
15
6 0 1 -0.1769702595802886 0.8331351847724078 0.5239916899325134 0.004105600879712498 
6 0 9 0.5856843296472447 -0.1558732679166012 -0.7954102025713884 -0.1264594278193166 
6 1 6 -0.2436653478720987 0.147883680687048 -0.9585184480399009 -0.04663519095132841 
6 1 2 0.207753937373674 0.4344067294949119 -0.8764297432625561 0.2556318087147511 
6 1 4 0.444220078847362 0.4986447037065236 -0.744326528490203 0.1737545932855555 
6 1 7 0.7533877382120362 0.3349392388487629 -0.5658821628141001 0.06152842458597942 
6 2 5 -0.6080629223625025 -0.1069684332549475 -0.7866493734408965 -0.05602222006342417 
6 2 4 0.8896724708236504 0.4529391082738911 -0.05769799697210674 -0.1249699490416038 
6 3 6 -0.8429664438017096 -0.1288333013498811 -0.5223117412882751 -0.1470675860428304 
6 3 7 0.3131056731324614 -0.1651787990807291 0.9352437125084115 -0.1060546443225207 
6 4 6 -0.661063661991035 -0.1756381951302929 -0.7294834194184101 -0.1931488689248215 
6 4 7 0.8997625993863546 -0.3161631359012115 0.3007792151107228 -0.2534488584590282 
6 5 6 -0.4656867325018209 -0.06587339986788572 -0.8824945112359753 -0.3747281225645818 
6 6 9 0.7885740013077054 -0.5314391513195503 0.3093921022041629 -0.04784058563315553 
6 7 9 -0.01257265438745033 -0.8388634914110151 -0.5441966291143079 -0.2018836580171402 

Voronoi ridge statistics
      9 non-simplicial Voronoi vertices for all ridges
7.9e-017   ave. distance to ridge
2.2e-016   max. distance to ridge
      3 bounded ridges
1.1e-016   ave. distance of midpoint to ridge
1.4e-016   max. distance of midpoint to ridge
      3 bounded ridges with ok normal
7.4e-017   ave. angle to ridge
2.2e-016   max. angle to ridge

Output completed.  Verifying that all points are below 2.9e-015 of
all facets.  Will make 90 distance computations.
rbox 5 D2 | qhull v f FnN o
Vertices and facets:
- p3(v5):  0.43 -0.11  0.19
  neighbors: f5 f6 f7 f8
- p2(v1):  0.47 -0.11  0.24
  neighbors: f1 f5 f7
- p4(v0): -0.39  0.45  0.36
  neighbors: f1 f3 f5 f6
- p0(v3): -0.34  0.23  0.17
  neighbors: f3 f6 f8
- p1(v2):  0.32 -0.23  0.16
  neighbors: f1 f3 f7 f8
- f5
    - flags: top simplicial
    - normal:     0.437    0.773  -0.4599
    - offset: -0.01170704
    - center: 0.4751017705052502 0.8404705693422864 
    - vertices: p3(v5) p2(v1) p4(v0)
    - neighboring facets: f1 f6 f7
- f6
    - flags: bottom simplicial
    - normal:    0.3235   0.6691   -0.669
    - offset:     0.0657
    - center: 0.24177356134713 0.5000766029034823 
    - vertices: p3(v5) p0(v3) p4(v0)
    - neighboring facets: f3 f5 f8
- f7
    - flags: top simplicial
    - normal:    0.6288  -0.3317  -0.7033
    - offset: -0.1682885
    - center: 0.447029359505762 -0.2357835939937328 
    - vertices: p3(v5) p1(v2) p2(v1)
    - neighboring facets: f1 f5 f8
- f8
    - flags: top simplicial
    - normal:    0.1177   0.1841  -0.9758
    - offset:    0.15929
    - center: 0.06029550723695781 0.09433023738340306 
    - vertices: p3(v5) p0(v3) p1(v2)
    - neighboring facets: f3 f7 f6
4
3 -1 1 2
3 -3 0 3
3 -1 0 3
3 -3 2 1
5
3 3 -3 1
4 3 -3 -1 2
3 2 -1 0
4 3 1 0 2
4 1 -3 -1 0
2
5 5 1
-10.101 -10.101 
0.4751017705052502 0.8404705693422864 
0.24177356134713 0.5000766029034823 
0.447029359505762 -0.2357835939937328 
0.06029550723695781 0.09433023738340306 
3 2 4 0
3 3 4 0
3 1 3 0
4 3 4 2 1
3 1 2 0

echo === check Halfspace ${d:-`date`}
date
=== check Halfspace Wed Nov 23 22:04:48 EST 2011
rbox 100 s D4 | qhull FA FV n s Tcv | qhull H Fp Tcv | qhull FA  Tcv

Convex hull of 100 points in 4-d:

  Number of vertices: 100
  Number of facets: 584

Statistics for: rbox 100 s D4 | qhull FA FV n s Tcv

  Number of points processed: 100
  Number of hyperplanes created: 1718
  Number of distance tests for qhull: 4091
  CPU seconds to compute hull (after input):  0
  Total facet area:   1.7919449
  Total volume:       0.18640279


Output completed.  Verifying that all points are below 2.8e-015 of
all facets.  Will make 58400 distance computations.

Output completed.  Verifying that all points are below outer planes of
all facets.  Will make 58400 distance computations.

Convex hull of 100 points in 4-d:

  Number of vertices: 100
  Number of facets: 584

Statistics for:  | qhull FA Tcv

  Number of points processed: 100
  Number of hyperplanes created: 1718
  Number of distance tests for qhull: 4091
  CPU seconds to compute hull (after input):  0
  Total facet area:   1.7919449
  Total volume:       0.18640279


Output completed.  Verifying that all points are below 2.8e-015 of
all facets.  Will make 58400 distance computations.
rbox d D3 | qhull FQ n s FD Tcv | qhull Fd H0.1,0.1 Fp FQ Tcv

Convex hull of 6 points in 3-d:

  Number of vertices: 6
  Number of facets: 8

Statistics for: rbox d D3 | qhull FQ n s FD Tcv

  Number of points processed: 6
  Number of hyperplanes created: 11
  Number of distance tests for qhull: 11
  Number of distance tests for merging: 59
  Number of distance tests for checking: 54
  Number of merged facets: 1
  CPU seconds to compute hull (after input):  0


Output completed.  Verifying that all points are below outer planes of
all facets.  Will make 48 distance computations.
3
6
-1.387778780781446e-017 -1.387778780781446e-017    0.5 
-2.775557561562891e-017    0.5      0 
  -0.5      0      0 
   0.5 -2.775557561562891e-017      0 
     0   -0.5      0 
-1.387778780781446e-017 1.387778780781446e-017 -0.4999999999999999 
rbox d D3 | qhull FQ n s FD Tcv | qhull Fd H0.1,0.1 Fp FQ Tcv

Output completed.  Verifying that all points are below outer planes of
all facets.  Will make 48 distance computations.
rbox 5 r D2 | qhull s n Tcv | qhull H0 Fp Tcv

Convex hull of 5 points in 2-d:

  Number of vertices: 5
  Number of facets: 5

Statistics for: rbox 5 r D2 | qhull s n Tcv

  Number of points processed: 5
  Number of hyperplanes created: 8
  Number of distance tests for qhull: 8
  CPU seconds to compute hull (after input):  0


Output completed.  Verifying that all points are below 1.3e-015 of
all facets.  Will make 25 distance computations.
2
5
-0.4045084971874738 -0.2938926261462365 
   0.5 -1.528090898784888e-016 
0.1545084971874736 -0.4755282581475768 
-0.4045084971874737 0.2938926261462366 
0.1545084971874737 0.4755282581475768 

Output completed.  Verifying that all points are below 6.4e-015 of
all facets.  Will make 25 distance computations.

echo === check qhull ${d:-`date`}
date
=== check qhull Wed Nov 23 22:04:48 EST 2011
rbox 10 s D3 | qhull Tcv

Convex hull of 10 points in 3-d:

  Number of vertices: 10
  Number of facets: 16

Statistics for: rbox 10 s D3 | qhull Tcv

  Number of points processed: 10
  Number of hyperplanes created: 29
  Number of distance tests for qhull: 49
  CPU seconds to compute hull (after input):  0


Output completed.  Verifying that all points are below 2e-015 of
all facets.  Will make 160 distance computations.
rbox 10 s D3 | qhull f Pd0:0.5 Pd2 Tcv
Vertices and facets:
- p7(v8):   0.2 0.093  0.45
- p9(v5):  0.29 -0.16  0.38
- p3(v1):  0.48 -0.13 0.022
- p8(v10):  0.32 0.025 -0.38
- f16
    - flags: bottom simplicial
    - normal:     0.858   0.1759   0.4826
    - offset: -0.4016457
    - vertices: p7(v8) p9(v5) p3(v1)
    - neighboring facets: f7 f27 f23
- f27
    - flags: bottom simplicial
    - normal:     0.646   0.7625  0.03668
    - offset: -0.2138755
    - vertices: p8(v10) p7(v8) p3(v1)
    - neighboring facets: f16 f26 f28

Output completed.  Verifying that all points are below 2e-015 of
all facets.  Will make 20 distance computations.
rbox 10 s D3 | qhull f Tcv PD2:-0.5
Vertices and facets:
- p4(v7):  0.23 -0.32 -0.31
- p1(v3): -0.072 0.057 -0.49
- p6(v2): -0.13 -0.47 -0.09
- p0(v9): -0.37 -0.27  0.19
- p8(v10):  0.32 0.025 -0.38
- p5(v6): 0.028  0.16 -0.47
- f12
    - flags: top simplicial
    - normal:   -0.2354  -0.5691  -0.7879
    - offset: -0.3715833
    - vertices: p4(v7) p1(v3) p6(v2)
    - neighboring facets: f21 f11 f25
- f21
    - flags: bottom simplicial
    - normal:   -0.8148  -0.2922  -0.5006
    - offset: -0.2881945
    - vertices: p0(v9) p1(v3) p6(v2)
    - neighboring facets: f12 f19 f20
- f24
    - flags: top simplicial
    - normal:    0.2636 -0.06941  -0.9621
    - offset: -0.4497787
    - vertices: p8(v10) p5(v6) p1(v3)
    - neighboring facets: f9 f25 f28
- f25
    - flags: bottom simplicial
    - normal:    0.2402  -0.2624  -0.9346
    - offset: -0.4268812
    - vertices: p8(v10) p4(v7) p1(v3)
    - neighboring facets: f12 f24 f26

Output completed.  Verifying that all points are below 2e-015 of
all facets.  Will make 40 distance computations.
rbox 10 s D3 | qhull QR-1

Convex hull of 10 points in 3-d:

  Number of vertices: 10
  Number of facets: 16

Statistics for: rbox 10 s D3 | qhull QR-1 QR-1322103889

  Number of points processed: 10
  Number of hyperplanes created: 29
  Number of distance tests for qhull: 49
  CPU seconds to compute hull (after input):  0

rbox 10 s D3 | qhull QR-40

Convex hull of 10 points in 3-d:

  Number of vertices: 10
  Number of facets: 16

Statistics for: rbox 10 s D3 | qhull QR-40 QR-40

  Number of points processed: 10
  Number of hyperplanes created: 29
  Number of distance tests for qhull: 49
  CPU seconds to compute hull (after input):  0

rbox 1000 D3 | qhull Tcvs


 qhull invoked by: rbox 1000 D3 | qhull Tcvs
2011.2 2011/11/21 with options:
  run-id 586766915  Tcheck-frequently  Tverify  Tstatistics  _pre-merge
  _zero-centrum  _max-width  1  Error-roundoff 6.9e-016  _one-merge 4.9e-015
  _near-inside 2.4e-014  Visible-distance 1.4e-015
  U-coplanar-distance 1.4e-015  Width-outside 2.8e-015  _wide-facet 8.3e-015

precision constants:
    0.5 max. abs. coordinate in the (transformed) input('Qbd:n')
 6.9e-016 max. roundoff error for distance computation('En')
 6.7e-016 max. roundoff error for angle computations
 2.8e-015 min. distance for outside points ('Wn')
 1.4e-015 min. distance for visible facets ('Vn')
 1.4e-015 max. distance for coplanar facets ('Un')
 8.3e-015 max. facet width for recomputing centrum and area
 2.4e-014 max. distance for near-inside points
 1.4e-015 radius of pre-merge centrum
 4.9e-015 max. distance for merging two simplicial facets
 2.2e-016 max. roundoff error for arithmetic operations
 1.1e-308 min. denominator for divisions
  zero diagonal for Gauss: 8.88e-015 1.78e-014 2.66e-014 


precision statistics
3.7e-017 ave. distance of a new vertex to a facet(!0s)
1.4e-016 max. distance of a new vertex to a facet
1.1e-016 max. distance of an output vertex to a facet
-1.4e-016 min. distance of an output vertex to a facet
 0.0026 min. denominator in hyperplane computation

summary information
     73 number of vertices in output
    142 number of facets in output
      3 average number of neighbors per facet
      3 maximum number of neighbors
      3 average number of vertices per facet
      3 maximum number of vertices
     82 vertices created altogether
    368 facets created altogether
      0   maximum merges for a facet(at most 511)
   0.93 average angle(cosine) of facet normals for all ridges
      1   maximum angle(cosine) of facet normals across a ridge
   0.21   minimum angle(cosine) of facet normals across a ridge

build hull statistics
     81 points processed
     73 max. vertices at any one time
   2.78 ave. visible facets per iteration
   0.21   ave. visible facets without an horizon neighbor
   2.78   ave. facets deleted per iteration
      8     maximum
   5.43 ave. visible vertices per iteration
     11     maximum
   4.38 ave. horizon facets per iteration
   4.48 ave. new or merged facets per iteration
     10     maximum(includes initial simplex)
  -0.77 average new facet balance
    1.5   standard deviation
     77   number of trials
      7 determinants computed(area & initial hull)
    504 distance tests for facet visibility
     73 points checked for facets' outer planes
   12.4   ave. distance tests per check
   1629   max visit_id/2
    619   max vertex_visit/2

partitioning statistics(see previous for outer planes)
      8 total vertices deleted
      1     maximum vertices deleted per iteration
    135 calls to findbest
   2.59  ave. facets tested
      7  max. facets tested
   2045 calls to findbestnew
   4.47  ave. facets tested
     13  max. facets tested
     36  calls due to qh_sharpnewfacets
    927 calls to findhorizon
   3.02  ave. facets tested
      6  max. facets tested
      5  horizon facets better than bestfacet
    919 inside points
      8   inside points that were coplanar with a facet
-3.1e-017 difference in max_outside at final check
   3747 distance tests for initial partition
   3132 partitions of a point
   8558 distance tests for partitioning
    513 distance tests for checking flipped facets
   1162 distance tests for statistics
  16990 total number of distance tests
      8 partitions of coplanar points or deleted vertices
     75    distance tests for these partitions

statistics for matching ridges
    726 total lookups for matching ridges of new facets
  0.563 average number of tests to match a ridge

statistics for determining merges
   1089 distance tests for checking simplicial convexity

statistics for merging
1.4e-016 max distance of vertex or coplanar point above facet(w/roundoff)
-6.9e-016 max distance of merged vertex below facet(or roundoff)

memory usage statistics(in bytes)
  21584 for facets 
Convex hull of 1000 points in 3-d:

  Number of vertices: 73
  Number of facets: 142

Statistics for: rbox 1000 D3 | qhull Tcvs

  Number of points processed: 81
  Number of hyperplanes created: 368
  Number of distance tests for qhull: 12884
  CPU seconds to compute hull (after input):  0

and their normals, neighbor and vertex sets
   1752 for vertices and their neighbor sets
  30368 for input points and outside and coplanar sets

memory statistics:
   1166 quick allocations
    673 short allocations
     69 long allocations
   1184 short frees
     68 long frees
  25312 bytes of short memory in use
    680 bytes of short memory in freelists
 105072 bytes of dropped short memory
   3100 bytes of unused short memory (estimated)
  10916 bytes of long memory allocated (max, except for input)
     96 bytes of long memory in use (in 1 pieces)
 131064 bytes of short memory buffers (minus links)
  65536 bytes per short memory buffer (initially 131072 bytes)
     73 calls to qh_setlarger
     15     average copy size
  freelists(bytes->count): 16->1 24->13 88->4

    size in bytes: merge 24 ridge 16 vertex 24 facet 88
         normal 24 ridge vertices 16 facet vertices or neighbors 20

Output completed.  Verifying that all points are below outer planes of
all facets.  Will make 142000 distance computations.
rbox 100 D3 | qhull T8 Tz TO q_test.log.1
tail -n -10 q_test.log.1
qh_mem 00270170 n     1022 free short: 88 bytes (tot 8984 cnt 228)
qh_mem 002701C8 n     1023 free short: 88 bytes (tot 8896 cnt 227)
qh_mem 00270220 n     1024 free short: 88 bytes (tot 8808 cnt 226)
qh_mem 00270278 n     1025 free short: 88 bytes (tot 8720 cnt 225)
qh_mem 00263260 n       30 free long: 96 bytes (tot 0 cnt 0)
qh_mem 002702D0 n     1026 free short: 16 bytes (tot 8704 cnt 224)
qh_mem 00270050 n     1027 free short: 88 bytes (tot 8616 cnt 223)
qh_mem 002700A8 n     1028 free short: 88 bytes (tot 8528 cnt 222)
qh_mem 00270100 n     1029 free short: 88 bytes (tot 8440 cnt 221)
qh_freebuffers: finished
rm q_test.log.1
rbox 100 s D3 | qhull TcvV-2

At a premature exit due to 'TVn', 'TCn', 'TRn', or precision error with 'QJn'.
Convex hull of 100 points in 3-d:

  Number of vertices: 78
  Number of facets: 152

Statistics for: rbox 100 s D3 | qhull TcvV-2

  Number of points processed: 78
  Number of hyperplanes created: 390
  Number of distance tests for qhull: 1489
  CPU seconds to compute hull (after input):  0

rbox 100 s D3 | qhull TcvC2

At a premature exit due to 'TVn', 'TCn', 'TRn', or precision error with 'QJn'.
Convex hull of 100 points in 3-d:

  Number of vertices: 79
  Number of facets: 154

Statistics for: rbox 100 s D3 | qhull TcvC2

  Number of points processed: 79
  Number of hyperplanes created: 395
  Number of distance tests for qhull: 1496
  CPU seconds to compute hull (after input):  0

rbox 100 s D3 | qhull TcvV2

At a premature exit due to 'TVn', 'TCn', 'TRn', or precision error with 'QJn'.
Convex hull of 100 points in 3-d:

  Number of vertices: 79
  Number of facets: 154

Statistics for: rbox 100 s D3 | qhull TcvV2

  Number of points processed: 79
  Number of hyperplanes created: 395
  Number of distance tests for qhull: 1496
  CPU seconds to compute hull (after input):  0

rbox 100 s D3 | qhull T1cvV2P2

At a premature exit due to 'TVn', 'TCn', 'TRn', or precision error with 'QJn'.
Convex hull of 100 points in 3-d:

  Number of vertices: 79
  Number of facets: 154

Statistics for: rbox 100 s D3 | qhull T1cvV2P2

  Number of points processed: 79
  Number of hyperplanes created: 395
  Number of distance tests for qhull: 1496
  CPU seconds to compute hull (after input):  0

qh_readpoints: read in 100 3-dimensional points
qh_initqhull_globals: for rbox 100 s D3 | qhull T1cvV2P2

Trace level 1 for rbox 100 s D3 | qhull T1cvV2P2
Options selected for Qhull 2011.2 2011/11/21:
  run-id 586766915  Tcheck-frequently  Tverify  TV-stop-after-point 2
  Trace-point 2  _pre-merge  _zero-centrum  _max-width 0.94
  Error-roundoff 6.7e-016  _one-merge 4.7e-015  _near-inside 2.3e-014
  Visible-distance 1.3e-015  U-coplanar-distance 1.3e-015
  Width-outside 2.7e-015  _wide-facet 8.1e-015
qh_findbest: point p2 starting at f7 isnewfacets? 1, unless 0 exit if > 2.7e-015
  testhorizon? 1 noupper? 0  Last point added was p41.  Last merge was #0.  max_outside 5.6e-017
qh_findbest: point p2 starting at f15 isnewfacets? 1, unless 0 exit if > 2.7e-015
  testhorizon? 1 noupper? 0  Last point added was p52.  Last merge was #0.  max_outside 1.1e-016
qh_findbest: point p2 starting at f37 isnewfacets? 1, unless 0 exit if > 2.7e-015
  testhorizon? 1 noupper? 0  Last point added was p73.  Last merge was #0.  max_outside 1.1e-016
qh_findbest: point p2 starting at f52 isnewfacets? 1, unless 0 exit if > 2.7e-015
  testhorizon? 1 noupper? 0  Last point added was p87.  Last merge was #0.  max_outside 1.1e-016
qh_findbest: point p2 starting at f72 isnewfacets? 1, unless 0 exit if > 2.7e-015
  testhorizon? 1 noupper? 0  Last point added was p76.  Last merge was #0.  max_outside 1.1e-016
qh_findbest: point p2 starting at f120 isnewfacets? 1, unless 0 exit if > 2.7e-015
  testhorizon? 1 noupper? 0  Last point added was p84.  Last merge was #0.  max_outside 1.7e-016
qh_findbest: point p2 starting at f211 isnewfacets? 1, unless 0 exit if > 2.7e-015
  testhorizon? 1 noupper? 0  Last point added was p10.  Last merge was #0.  max_outside 1.7e-016
qh_findbest: point p2 starting at f289 isnewfacets? 1, unless 0 exit if > 2.7e-015
  testhorizon? 1 noupper? 0  Last point added was p6.  Last merge was #0.  max_outside 1.7e-016
qh_addpoint: add p2(v79) to hull of 152 facets(0.014 above f288) and 22 outside at    0 CPU secs.  Previous was p56.
qh_findhorizon: find horizon for point p2 facet f288
qh_findhorizon: 5 horizon facets(good 5), 3 visible(good 3), 0 coplanar
qh_makenewfacets: created 5 new facets from point p2 to horizon
qh_matchnewfacets: match neighbors for new facets.
qh_partitionvisible: partitioned 0 points from outsidesets and 0 points from coplanarsets
qh_deletevisible: delete 3 visible facets and 0 vertices
qh_checkpolygon: check all facets from f390
qh_buildhull: completed the hull construction
Qhull: algorithm completed
qh_checkpolygon: check all facets from f62
qh_checkconvex: check all ridges are convex
qh_freeqhull2: free global memory
qh_freebuild: free memory from qh_inithull and qh_buildhull
rbox 100 s D3 | qhull TcvF100

Convex hull of 100 points in 3-d:

  Number of vertices: 100
  Number of facets: 196

Statistics for: rbox 100 s D3 | qhull TcvF100

  Number of points processed: 100
  Number of hyperplanes created: 514
  Number of distance tests for qhull: 1691
  CPU seconds to compute hull (after input):  0


At 22:04:49 &  0 CPU secs, qhull has created 102 facets and merged 0.
 The current hull contains 46 facets and 25 vertices.  There are 75
 outside points.  Next is point p55(v26), 0.035 above f44.

At 22:04:49 &  0 CPU secs, qhull has created 203 facets and merged 0.
 The current hull contains 86 facets and 45 vertices.  There are 55
 outside points.  Next is point p82(v46), 0.067 above f117.

At 22:04:49 &  0 CPU secs, qhull has created 306 facets and merged 0.
 The current hull contains 122 facets and 63 vertices.  There are 37
 outside points.  Next is point p50(v64), 0.042 above f218.

At 22:04:49 &  0 CPU secs, qhull has created 412 facets and merged 0.
 The current hull contains 160 facets and 82 vertices.  There are 18
 outside points.  Next is point p60(v83), 0.031 above f307.

Output completed.  Verifying that all points are below 2e-015 of
all facets.  Will make 19600 distance computations.
rbox 100 s D3 | qhull Qf Tcv

Convex hull of 100 points in 3-d:

  Number of vertices: 100
  Number of facets: 196

Statistics for: rbox 100 s D3 | qhull Qf Tcv

  Number of points processed: 100
  Number of hyperplanes created: 504
  Number of distance tests for qhull: 4423
  CPU seconds to compute hull (after input):  0


Output completed.  Verifying that all points are below 2e-015 of
all facets.  Will make 19600 distance computations.
rbox 100 D3 | qhull Tcv

Convex hull of 100 points in 3-d:

  Number of vertices: 26
  Number of facets: 48

Statistics for: rbox 100 D3 | qhull Tcv

  Number of points processed: 28
  Number of hyperplanes created: 110
  Number of distance tests for qhull: 1164
  CPU seconds to compute hull (after input):  0


Output completed.  Verifying that all points are below outer planes of
all facets.  Will make 4800 distance computations.
rbox 100 D3 | qhull Qs Tcv

Convex hull of 100 points in 3-d:

  Number of vertices: 26
  Number of facets: 48

Statistics for: rbox 100 D3 | qhull Qs Tcv

  Number of points processed: 29
  Number of hyperplanes created: 105
  Number of distance tests for qhull: 1176
  CPU seconds to compute hull (after input):  0


Output completed.  Verifying that all points are below outer planes of
all facets.  Will make 4800 distance computations.
rbox 100 D5 | qhull Qs Tcv

Convex hull of 100 points in 5-d:

  Number of vertices: 78
  Number of facets: 1174

Statistics for: rbox 100 D5 | qhull Qs Tcv

  Number of points processed: 85
  Number of hyperplanes created: 4015
  Number of distance tests for qhull: 8584
  CPU seconds to compute hull (after input): 0.015

QH7076 qhull input warning: exact merge ('Qx').  Verify may report that a point
is outside of a facet.  See qh-optq.htm#Qx

Output completed.  Verifying that all points are below outer planes of
all facets.  Will make 117400 distance computations.
rbox 100 D3 | qhull Qr Tcv

Convex hull of 100 points in 3-d:

  Number of vertices: 26
  Number of facets: 48

Statistics for: rbox 100 D3 | qhull Qr Tcv

  Number of points processed: 56
  Number of hyperplanes created: 294
  Number of distance tests for qhull: 2129


Output completed.  Verifying that all points are below outer planes of
all facets.  Will make 4800 distance computations.
rbox 100 D3 | qhull Qxv Tcv

Convex hull of 100 points in 3-d:

  Number of vertices: 26
  Number of facets: 48

Statistics for: rbox 100 D3 | qhull Qxv Tcv

  Number of points processed: 28
  Number of hyperplanes created: 110
  Number of distance tests for qhull: 1164
  CPU seconds to compute hull (after input):  0

QH7076 qhull input warning: exact merge ('Qx').  Verify may report that a point
is outside of a facet.  See qh-optq.htm#Qx

Output completed.  Verifying that all points are below outer planes of
all facets.  Will make 4800 distance computations.
rbox 100 D3 | qhull Qi f Pd0 Pd1 Pd2 Tcv
Vertices and facets:
- p11(v14): -0.31  0.47  0.07
- p59(v8):  0.44  0.42 -0.11
- p33(v2): -0.067  0.49 -0.11
- p48(v25):  0.25  0.24  0.44
- p37(v15):  0.45  0.26  0.04
- p38(v5):  0.42 -0.25  0.42
- f42
    - flags: bottom simplicial
    - normal:    0.1292   0.9495   0.2858
    - offset: -0.4221543
    - coplanar set(furthest p70):
      p70:   0.04762   0.3507 -0.06712
      furthest distance= -0.1
    - vertices: p11(v14) p59(v8) p33(v2)
    - neighboring facets: f52 f41 f93
- f93
    - flags: top simplicial
    - normal:     0.142   0.9273   0.3463
    - offset: -0.4120445
    - coplanar set(furthest p28):
      p27:    0.1182   0.3419    0.068
      p25:    0.1663   0.2352   0.3516
      p15:    0.2743   0.2665  0.07375
      p44:   -0.1321   0.3861    0.167
      p28:   -0.1003   0.4263  0.07335
      furthest distance= -0.0056
    - vertices: p48(v25) p11(v14) p59(v8)
    - neighboring facets: f42 f94 f95
- f94
    - flags: bottom simplicial
    - normal:    0.7783   0.4631   0.4241
    - offset: -0.4905394
    - vertices: p48(v25) p37(v15) p59(v8)
    - neighboring facets: f47 f93 f97
- f97
    - flags: top simplicial
    - normal:    0.8488   0.2736   0.4523
    - offset: -0.4744876
    - vertices: p48(v25) p37(v15) p38(v5)
    - neighboring facets: f46 f99 f94

Output completed.  Verifying that all points are below outer planes of
all facets.  Will make 400 distance computations.
rbox c d | qhull Qc f Tcv
Vertices and facets:
- p6(v6):   0.5   0.5  -0.5
  neighbors: f1 f9 f10
- p2(v2):  -0.5   0.5  -0.5
  neighbors: f1 f3 f10
- p4(v1):   0.5  -0.5  -0.5
  neighbors: f1 f2 f9
- p0(v0):  -0.5  -0.5  -0.5
  neighbors: f1 f2 f3
- p5(v7):   0.5  -0.5   0.5
  neighbors: f2 f9 f13
- p1(v3):  -0.5  -0.5   0.5
  neighbors: f2 f3 f13
- p7(v5):   0.5   0.5   0.5
  neighbors: f9 f10 f13
- p3(v8):  -0.5   0.5   0.5
  neighbors: f3 f10 f13
- f1
    - flags: bottom tested seen coplanar
    - merges: 1
    - normal:        -0       -0       -1
    - offset:       -0.5
    - center:      0      0   -0.5 
    - coplanar set(furthest p8):
      p8:         0        0     -0.5
      furthest distance=  0
    - vertices: p6(v6) p2(v2) p4(v1) p0(v0)
    - neighboring facets: f2 f3 f9 f10
    - ridges:
     - r4 tested
           vertices: p2(v2) p0(v0)
           between f1 and f3
     - r3 tested
           vertices: p4(v1) p0(v0)
           between f2 and f1
     - r1 tested
           vertices: p6(v6) p4(v1)
           between f9 and f1
     - r2 tested
           vertices: p6(v6) p2(v2)
           between f1 and f10
- f2
    - flags: top tested seen coplanar
    - merges: 1
    - normal:         0       -1        0
    - offset:       -0.5
    - center:      0   -0.5      0 
    - coplanar set(furthest p10):
      p10:         0     -0.5        0
      furthest distance=  0
    - vertices: p5(v7) p1(v3) p4(v1) p0(v0)
    - neighboring facets: f1 f3 f9 f13
    - ridges:
     - r3 tested
           vertices: p4(v1) p0(v0)
           between f2 and f1
     - r8 tested
           vertices: p1(v3) p0(v0)
           between f3 and f2
     - r7 tested
           vertices: p5(v7) p1(v3)
           between f13 and f2
     - r6 tested
           vertices: p5(v7) p4(v1)
           between f2 and f9
- f9
    - flags: bottom tested seen coplanar
    - merges: 1
    - normal:         1       -0       -0
    - offset:       -0.5
    - center:    0.5      0      0 
    - coplanar set(furthest p13):
      p13:       0.5        0        0
      furthest distance=  0
    - vertices: p5(v7) p6(v6) p7(v5) p4(v1)
    - neighboring facets: f2 f1 f10 f13
    - ridges:
     - r1 tested
           vertices: p6(v6) p4(v1)
           between f9 and f1
     - r6 tested
           vertices: p5(v7) p4(v1)
           between f2 and f9
     - r10 tested
           vertices: p5(v7) p7(v5)
           between f9 and f13
     - r11 tested
           vertices: p6(v6) p7(v5)
           between f10 and f9
- f3
    - flags: bottom tested seen coplanar
    - merges: 1
    - normal:        -1       -0       -0
    - offset:       -0.5
    - center:   -0.5      0      0 
    - coplanar set(furthest p12):
      p12:      -0.5        0        0
      furthest distance=  0
    - vertices: p3(v8) p1(v3) p2(v2) p0(v0)
    - neighboring facets: f1 f2 f10 f13
    - ridges:
     - r4 tested
           vertices: p2(v2) p0(v0)
           between f1 and f3
     - r13 tested
           vertices: p3(v8) p2(v2)
           between f10 and f3
     - r14 tested
           vertices: p3(v8) p1(v3)
           between f3 and f13
     - r8 tested
           vertices: p1(v3) p0(v0)
           between f3 and f2
- f10
    - flags: top tested seen coplanar
    - merges: 1
    - normal:         0        1       -0
    - offset:       -0.5
    - center:      0    0.5      0 
    - coplanar set(furthest p11):
      p11:         0      0.5        0
      furthest distance=  0
    - vertices: p3(v8) p6(v6) p7(v5) p2(v2)
    - neighboring facets: f3 f1 f9 f13
    - ridges:
     - r2 tested
           vertices: p6(v6) p2(v2)
           between f1 and f10
     - r11 tested
           vertices: p6(v6) p7(v5)
           between f10 and f9
     - r16 tested
           vertices: p3(v8) p7(v5)
           between f13 and f10
     - r13 tested
           vertices: p3(v8) p2(v2)
           between f10 and f3
- f13
    - flags: bottom tested seen coplanar
    - merges: 1
    - normal:        -0       -0        1
    - offset:       -0.5
    - center:      0      0    0.5 
    - coplanar set(furthest p9):
      p9:         0        0      0.5
      furthest distance=  0
    - vertices: p3(v8) p5(v7) p7(v5) p1(v3)
    - neighboring facets: f10 f2 f9 f3
    - ridges:
     - r7 tested
           vertices: p5(v7) p1(v3)
           between f13 and f2
     - r14 tested
           vertices: p3(v8) p1(v3)
           between f3 and f13
     - r16 tested
           vertices: p3(v8) p7(v5)
           between f13 and f10
     - r10 tested
           vertices: p5(v7) p7(v5)
           between f9 and f13

Output completed.  Verifying that all points are below outer planes of
all facets.  Will make 84 distance computations.
rbox c d | qhull Qc p Tcv
3
14
  -0.5   -0.5   -0.5 
  -0.5   -0.5    0.5 
  -0.5    0.5   -0.5 
  -0.5    0.5    0.5 
   0.5   -0.5   -0.5 
   0.5   -0.5    0.5 
   0.5    0.5   -0.5 
   0.5    0.5    0.5 
     0      0   -0.5 
     0      0    0.5 
     0   -0.5      0 
     0    0.5      0 
  -0.5      0      0 
   0.5      0      0 

Output completed.  Verifying that all points are below outer planes of
all facets.  Will make 84 distance computations.
rbox 100 D3 | qhull QbB FO Tcv

Convex hull of 100 points in 3-d:

  Number of vertices: 26
  Number of facets: 48

Statistics for: rbox 100 D3 | qhull QbB FO Tcv

  Number of points processed: 28
  Number of hyperplanes created: 110
  Number of distance tests for qhull: 1164
  CPU seconds to compute hull (after input):  0

Options selected for Qhull 2011.2 2011/11/21:
  run-id 586766915  QbBound-unit-box 0.5  Tcheck-frequently  Tverify
  _pre-merge  _zero-centrum  _max-width  1  Error-roundoff 6.9e-016
  _one-merge 4.9e-015  _near-inside 2.4e-014  Visible-distance 1.4e-015
  U-coplanar-distance 1.4e-015  Width-outside 2.8e-015  _wide-facet 8.3e-015

Output completed.  Verifying that all points are below outer planes of
all facets.  Will make 4800 distance computations.
rbox 1000 D2 B1e6 | qhull d Qbb FO Tcv

Delaunay triangulation by the convex hull of 1000 points in 3-d:

  Number of input sites: 1000
  Number of Delaunay regions: 1982

Statistics for: rbox 1000 D2 B1e6 | qhull d Qbb FO Tcv

  Number of points processed: 1000
  Number of hyperplanes created: 5422
  Number of facets in hull: 1996
  Number of distance tests for qhull: 27767
  CPU seconds to compute hull (after input): 0.015

Options selected for Qhull 2011.2 2011/11/21:
  run-id 586766915  delaunay  Qbbound-last  Tcheck-frequently  Tverify
  _pre-merge  _zero-centrum  Pgood  _max-width 2e+006  Error-roundoff 2.8e-009
  _one-merge 1.9e-008  _near-inside 9.7e-008  Visible-distance 5.5e-009
  U-coplanar-distance 5.5e-009  Width-outside 1.1e-008  _wide-facet 3.3e-008

qhull output completed.  Verifying that 1000 points are
below 8.3e-009 of the nearest facet.
rbox 10 D3 | qhull QbB p Tcv
3
9
-0.08928142660407056 -0.3194671263098747 0.3605569202297553 
-0.1534778158747162 0.03516440701453921    0.5 
-0.01663016790622804 0.3709299907861695 0.0764863283135003 
0.3932651765570926 0.4421392241401818 -0.09396622763110282 
0.199424728304684    0.5 0.06620718432588452 
-0.2331094527352667 0.3262436269787957 -0.1289660986727641 
  -0.5 0.2819439085196089   -0.5 
0.4999999999999999 0.3760666748352666 0.1399187075693636 
0.4888102268053206   -0.5 -0.2153953634886358 

Output completed.  Verifying that all points are below 2.1e-015 of
all facets.  Will make 140 distance computations.
rbox 10 D3 | qhull Qbb p Tcv
3
9
-0.0222149361131852 -0.366434993563625 0.5972263834933854 
-0.06676722137887703 -0.1566931052661437 0.6939998615477234 
0.02820502736438535 0.04189077954915421 0.4000814320337247 
0.3126723396709863 0.08400649026409401 0.2817873818077145 
0.1781470954214661 0.1182274414396169 0.3929477075294902 
-0.1220315663349177 0.01546165115708642 0.2574974761506134 
-0.3072535691850387 -0.01073880122111998      0 
0.3867462923626847 0.04492879989084675 0.4441034944549365 
0.3789805913148268 -0.4732086509216658 0.1975155783347269 
QH7040 qhull input warning: option 'Qbb' (scale-last-coordinate) is normally used with 'd' or 'v'

Output completed.  Verifying that all points are below 2.9e-015 of
all facets.  Will make 140 distance computations.
rbox 10 D3 | qhull Qb0:-10B2:20 p Tcv
3
9
-5.733970380575339 -0.366434993563625 17.14322461142018 
-6.400761988815289 -0.1566931052661437     20 
-4.97936018866004 0.04189077954915421 11.32346018178494 
-0.7218812392989191 0.08400649026409401 7.831392539952951 
-2.735252796494545 0.1182274414396169 11.11287098934976 
-7.227875597731384 0.01546165115708642 7.114348923589699 
   -10 -0.01073880122111998 -0.4870359524963758 
0.3867462923626847 0.04492879989084675 12.62300161615219 
0.2705209571204694 -0.4732086509216658 5.343669467959106 

Output completed.  Verifying that all points are below 7.5e-014 of
all facets.  Will make 140 distance computations.
rbox 10 D3 | qhull Qb0:-10B2:20 p Tcv | qhull QbB p Tcv

Output completed.  Verifying that all points are below 7.5e-014 of
all facets.  Will make 140 distance computations.
3
9
-0.08928142660407051 -0.3194671263098747 0.3605569202297555 
-0.1534778158747163 0.03516440701453921    0.5 
-0.01663016790622796 0.3709299907861695 0.07648632831350066 
0.3932651765570927 0.4421392241401818 -0.09396622763110274 
0.1994247283046841    0.5 0.06620718432588446 
-0.2331094527352668 0.3262436269787957 -0.128966098672764 
  -0.5 0.2819439085196089 -0.4999999999999999 
   0.5 0.3760666748352666 0.1399187075693635 
0.4888102268053207   -0.5 -0.2153953634886357 

Output completed.  Verifying that all points are below 2.1e-015 of
all facets.  Will make 126 distance computations.
rbox 10 D3 | qhull Qb1:0B1:0 d Tcv Q8

Delaunay triangulation by the convex hull of 10 points in 3-d:

  Number of input sites: 10
  Number of Delaunay regions: 14

Statistics for: rbox 10 D3 | qhull Qb1:0B1:0 d Tcv Q8

  Number of points processed: 10
  Number of hyperplanes created: 32
  Number of facets in hull: 16
  Number of distance tests for qhull: 62
  CPU seconds to compute hull (after input):  0

QH7077 qhull input warning: no outer plane check ('Q5') or no processing of
near-inside points ('Q8').  Verify may report that a point is outside
of a facet.

Output completed.  Verifying that all points are below 2e-015 of
all facets.  Will make 140 distance computations.
rbox 10 D3 | qhull Qb1:0B1:0B2:0 d Tcv Q8

Delaunay triangulation by the convex hull of 10 points in 3-d:

  Number of input sites: 10
  Number of Delaunay regions: 14

Statistics for: rbox 10 D3 | qhull Qb1:0B1:0B2:0 d Tcv Q8

  Number of points processed: 10
  Number of hyperplanes created: 32
  Number of facets in hull: 16
  Number of distance tests for qhull: 62
  CPU seconds to compute hull (after input):  0

QH7077 qhull input warning: no outer plane check ('Q5') or no processing of
near-inside points ('Q8').  Verify may report that a point is outside
of a facet.

Output completed.  Verifying that all points are below 2e-015 of
all facets.  Will make 140 distance computations.
rbox 10 D3 | qhull Qb1:0 d Tcv 

Delaunay triangulation by the convex hull of 10 points in 4-d:

  Number of input sites: 10
  Number of Delaunay regions: 18

Statistics for: rbox 10 D3 | qhull Qb1:0 d Tcv

  Number of points processed: 10
  Number of hyperplanes created: 44
  Number of facets in hull: 27
  Number of distance tests for qhull: 67
  CPU seconds to compute hull (after input):  0


Output completed.  Verifying that all points are below 2.9e-015 of
all facets.  Will make 180 distance computations.
rbox 10 D3 | qhull Qb1:0B1:0  Tcv

Convex hull of 10 points in 2-d:

  Number of vertices: 4
  Number of facets: 4

Statistics for: rbox 10 D3 | qhull Qb1:0B1:0 Tcv

  Number of points processed: 4
  Number of hyperplanes created: 6
  Number of distance tests for qhull: 49
  CPU seconds to compute hull (after input):  0


Output completed.  Verifying that all points are below 1.3e-015 of
all facets.  Will make 40 distance computations.
echo "== next command will error ${d:-`date`} =="
date
== next command will error Wed Nov 23 22:04:50 EST 2011 ==
rbox 10 D2 | qhull Qb1:1B1:1 Tcv
QH6154 qhull precision error: initial facet 1 is coplanar with the interior point
ERRONEOUS FACET:
- f1
    - flags: bottom simplicial flipped
    - normal:         0        1
    - offset:         -1
    - vertices: p4(v1) p1(v0)
    - neighboring facets: f2 f3

While executing: rbox 10 D2 | qhull Qb1:1B1:1 Tcv
Options selected for Qhull 2011.2 2011/11/21:
  run-id 586783722  Qbound-dim-low 1  1  QBound-dim-high 1  1
  Tcheck-frequently  Tverify  _pre-merge  _zero-centrum  _max-width 0.81
  Error-roundoff 8.6e-016  _one-merge 4.3e-015  _near-inside 2.1e-014
  Visible-distance 1.7e-015  U-coplanar-distance 1.7e-015
  Width-outside 3.4e-015  _wide-facet 1e-014

precision problems (corrected unless 'Q0' or an error)
      2 flipped facets

The input to qhull appears to be less than 2 dimensional, or a
computation has overflowed.

Qhull could not construct a clearly convex simplex from points:
- p0(v2): -0.022     1
- p4(v1):  0.38     1
- p1(v0): -0.43     1

The center point is coplanar with a facet, or a vertex is coplanar
with a neighboring facet.  The maximum round off error for
computing distances is 8.6e-016.  The center point, facets and distances
to the center point are as follows:

center point -0.02391        1

facet p4 p1 distance=    0
facet p0 p1 distance=    0
facet p0 p4 distance=    0

These points either have a maximum or minimum x-coordinate, or
they maximize the determinant for k coordinates.  Trial points
are first selected from points that maximize a coordinate.

The min and max coordinates for each dimension are:
  0:   -0.4285     0.379  difference= 0.8076
  1:         1         1  difference=    0

If the input should be full dimensional, you have several options that
may determine an initial simplex:
  - use 'QJ'  to joggle the input and make it full dimensional
  - use 'QbB' to scale the points to the unit cube
  - use 'QR0' to randomly rotate the input for different maximum points
  - use 'Qs'  to search all points for the initial simplex
  - use 'En'  to specify a maximum roundoff error less than 8.6e-016.
  - trace execution with 'T3' to see the determinant for each point.

If the input is lower dimensional:
  - use 'QJ' to joggle the input and make it full dimensional
  - use 'Qbk:0Bk:0' to delete coordinate k from the input.  You should
    pick the coordinate with the least range.  The hull will have the
    correct topology.
  - determine the flat containing the points, rotate the points
    into a coordinate plane, and delete the other coordinates.
  - add one or more points to make the input full dimensional.
rbox 200 L20 D2 t | qhull FO Tcv C-0

Convex hull of 200 points in 2-d:

  Number of vertices: 200
  Number of facets: 200

Statistics for: rbox 200 L20 D2 t1322103890 | qhull FO Tcv C-0

  Number of points processed: 200
  Number of hyperplanes created: 398
  Number of distance tests for qhull: 2094
  CPU seconds to compute hull (after input):  0

Options selected for Qhull 2011.2 2011/11/21:
  run-id 586783722  Tcheck-frequently  Tverify  Centrum-premerge-  0
  _zero-centrum  _max-width 0.99  Error-roundoff 3.4e-016  _one-merge 1.7e-015
  _near-inside 8.5e-015  Visible-distance 6.8e-016
  U-coplanar-distance 6.8e-016  Width-outside 1.4e-015  _wide-facet 4.1e-015

Output completed.  Verifying that all points are below 1e-015 of
all facets.  Will make 40000 distance computations.
rbox 1000 L20 t | qhull FO Tcv C-0

Convex hull of 1000 points in 3-d:

  Number of vertices: 61
  Number of facets: 118

Statistics for: rbox 1000 L20 t1322103890 | qhull FO Tcv C-0

  Number of points processed: 63
  Number of hyperplanes created: 273
  Number of distance tests for qhull: 12767
  CPU seconds to compute hull (after input):  0

Options selected for Qhull 2011.2 2011/11/21:
  run-id 586783722  Tcheck-frequently  Tverify  Centrum-premerge-  0
  _zero-centrum  _max-width  1  Error-roundoff 6.9e-016  _one-merge 4.9e-015
  _near-inside 2.4e-014  Visible-distance 1.4e-015
  U-coplanar-distance 1.4e-015  Width-outside 2.8e-015  _wide-facet 8.3e-015

Output completed.  Verifying that all points are below outer planes of
all facets.  Will make 118000 distance computations.
rbox 200 L20 D4 t | qhull FO Tcv C-0

Convex hull of 200 points in 4-d:

  Number of vertices: 53
  Number of facets: 243

Statistics for: rbox 200 L20 D4 t1322103890 | qhull FO Tcv C-0

  Number of points processed: 64
  Number of hyperplanes created: 716
  Number of distance tests for qhull: 5329
  CPU seconds to compute hull (after input):  0

Options selected for Qhull 2011.2 2011/11/21:
  run-id 586783722  Tcheck-frequently  Tverify  Centrum-premerge-  0
  _zero-centrum  _max-width  1  Error-roundoff 1e-015  _one-merge 9.1e-015
  _near-inside 4.5e-014  Visible-distance 6e-015  U-coplanar-distance 6e-015
  Width-outside 1.2e-014  _wide-facet 3.6e-014

Output completed.  Verifying that all points are below outer planes of
all facets.  Will make 48600 distance computations.
rbox 200 L20 D5 t | qhull FO Tcv Qx

Convex hull of 200 points in 5-d:

  Number of vertices: 101
  Number of facets: 1478

Statistics for: rbox 200 L20 D5 t1322103890 | qhull FO Tcv Qx

  Number of points processed: 116
  Number of hyperplanes created: 5040
  Number of distance tests for qhull: 17467
  CPU seconds to compute hull (after input): 0.015

Options selected for Qhull 2011.2 2011/11/21:
  run-id 586783722  Tcheck-frequently  Tverify  Qxact-merge  _zero-centrum
  _max-width 0.99  Error-roundoff 1.4e-015  _one-merge 1.5e-014
  _near-inside 7.5e-014  Visible-distance 8.2e-015
  U-coplanar-distance 8.2e-015  Width-outside 1.6e-014  _wide-facet 4.9e-014
QH7076 qhull input warning: exact merge ('Qx').  Verify may report that a point
is outside of a facet.  See qh-optq.htm#Qx

Output completed.  Verifying that all points are below outer planes of
all facets.  Will make 295600 distance computations.
rbox 1000 W1e-3 s D2 t | qhull d FO Tcv Qu Q0

Furthest-site Delaunay triangulation by the convex hull of 1000 points in 3-d:

  Number of input sites: 1000
  Number of Delaunay regions: 271

Statistics for: rbox 1000 W1e-3 s D2 t1322103890 | qhull d FO Tcv Qu Q0

  Number of points processed: 1000
  Number of hyperplanes created: 4791
  Number of facets in hull: 1996
  Number of distance tests for qhull: 27988
  CPU seconds to compute hull (after input):  0

Options selected for Qhull 2011.2 2011/11/21:
  run-id 586783722  delaunay  Tcheck-frequently  Tverify  QupperDelaunay
  Q0-no-premerge  Pgood  _max-width  1  Error-roundoff 6.9e-016
  Visible-distance 6.9e-016  U-coplanar-distance 6.9e-016
  Width-outside 1.4e-015  _wide-facet 4.2e-015

Output completed.  Verifying that all points are below 2.1e-015 of
all facets.  Will make 271000 distance computations.
rbox 1000 W1e-3 s D2 t | qhull d FO Tcv Qu C-0

Furthest-site Delaunay triangulation by the convex hull of 1000 points in 3-d:

  Number of input sites: 1000
  Number of Delaunay regions: 271

Statistics for: rbox 1000 W1e-3 s D2 t1322103890 | qhull d FO Tcv Qu C-0

  Number of points processed: 1000
  Number of hyperplanes created: 4791
  Number of facets in hull: 1996
  Number of distance tests for qhull: 27988
  CPU seconds to compute hull (after input): 0.015

Options selected for Qhull 2011.2 2011/11/21:
  run-id 586783722  delaunay  Tcheck-frequently  Tverify  QupperDelaunay
  Centrum-premerge-  0  _zero-centrum  Pgood  _max-width  1
  Error-roundoff 6.9e-016  _one-merge 4.9e-015  _near-inside 2.4e-014
  Visible-distance 1.4e-015  U-coplanar-distance 1.4e-015
  Width-outside 2.8e-015  _wide-facet 8.3e-015

Output completed.  Verifying that all points are below 2.1e-015 of
all facets.  Will make 271000 distance computations.

echo === check joggle and TRn ${d:-`date`}
date
=== check joggle and TRn Wed Nov 23 22:04:50 EST 2011
rbox 100 W0 | qhull QJ1e-14 Qc TR100 Tv

At a premature exit due to 'TVn', 'TCn', 'TRn', or precision error with 'QJn'.
Convex hull of 100 points in 3-d:

  Number of vertices: 15
  Number of coplanar points: 4
  Number of facets: 25

Statistics for: rbox 100 W0 | qhull QJ1e-14 Qc TR100 Tv

  Number of points processed: 15
  Number of hyperplanes created: 11270
  Number of distance tests for qhull: 120689
  CPU seconds to compute hull (after input): 0.015
  Percentage of runs with precision errors: 100.0


precision problems (corrected unless 'Q0' or an error)
    100 coplanar horizon facets for new vertices
    400 coplanar points during partitioning
rbox 100 W0 | qhull QJ1e-13 Qc TR100 Tv

At a premature exit due to 'TVn', 'TCn', 'TRn', or precision error with 'QJn'.
Convex hull of 100 points in 3-d:

  Number of vertices: 28
  Number of coplanar points: 12
  Number of facets: 50

Statistics for: rbox 100 W0 | qhull QJ1e-13 Qc TR100 Tv

  Number of points processed: 28
  Number of hyperplanes created: 24800
  Number of distance tests for qhull: 197857
  CPU seconds to compute hull (after input): 0.046
  Percentage of runs with precision errors: 63.0


precision problems (corrected unless 'Q0' or an error)
     10 coplanar half ridges in output
     53 coplanar horizon facets for new vertices
    119 coplanar points during partitioning
rbox 100 W0 | qhull QJ1e-12 Qc TR100 Tv

At a premature exit due to 'TVn', 'TCn', 'TRn', or precision error with 'QJn'.
Convex hull of 100 points in 3-d:

  Number of vertices: 40
  Number of coplanar points: 16
  Number of facets: 75

Statistics for: rbox 100 W0 | qhull QJ1e-12 Qc TR100 Tv

  Number of points processed: 40
  Number of hyperplanes created: 29205
  Number of distance tests for qhull: 217398
  CPU seconds to compute hull (after input): 0.046
  Percentage of runs with precision errors:  8.0


precision problems (corrected unless 'Q0' or an error)
      2 coplanar half ridges in output
      6 coplanar horizon facets for new vertices
     17 coplanar points during partitioning
rbox 100 W0 | qhull QJ1e-11 Qc TR100 Tv

Convex hull of 100 points in 3-d:

  Number of vertices: 66
  Number of coplanar points: 34
  Number of facets: 128

Statistics for: rbox 100 W0 | qhull QJ1e-11 Qc TR100 Tv

  Number of points processed: 75
  Number of hyperplanes created: 29630
  Number of distance tests for qhull: 219104
  CPU seconds to compute hull (after input): 0.031
  Percentage of runs with precision errors:  0.0


precision problems (corrected unless 'Q0' or an error)
      3 coplanar points during partitioning

Output completed.  Verifying that all points are below 2.1e-015 of
all facets.  Will make 12800 distance computations.
rbox 100 W0 | qhull QJ1e-10 Qc TR100 Tv

Convex hull of 100 points in 3-d:

  Number of vertices: 66
  Number of coplanar points: 34
  Number of facets: 128

Statistics for: rbox 100 W0 | qhull QJ1e-10 Qc TR100 Tv

  Number of points processed: 75
  Number of hyperplanes created: 29636
  Number of distance tests for qhull: 218960
  CPU seconds to compute hull (after input): 0.046
  Percentage of runs with precision errors:  0.0


Output completed.  Verifying that all points are below 2.1e-015 of
all facets.  Will make 12800 distance computations.
rbox 100 | qhull d QJ Qb0:1e4 QB0:1e5 Qb1:1e4 QB1:1e6 Qb2:1e5 QB2:1e7 FO Tv

Delaunay triangulation by the convex hull of 100 points in 4-d:

  Number of input sites: 100
  Number of Delaunay regions: 488

Statistics for: rbox 100 | qhull d QJ Qb0:1e4 QB0:1e5 Qb1:1e4 QB1:1e6 Qb2:1e5 QB2:1e7 FO Tv

  Number of points processed: 100
  Number of hyperplanes created: 1578
  Number of facets in hull: 560
  Number of distance tests for qhull: 3924
  CPU seconds to compute hull (after input):  0
  Input joggled by: 6.7e+003

Options selected for Qhull 2011.2 2011/11/21:
  run-id 586783722  delaunay  Qbound-dim-low 0 1e+004  QBound-dim-high 0 1e+005
  Qbound-dim-low 1 1e+004  QBound-dim-high 1 1e+006  Qbound-dim-low 2 1e+005
  QBound-dim-high 2 1e+007  Tverify  Pgood  _run 1  QJoggle 6.7e+003
  _joggle-seed 16807  _max-width 1e+014  Error-roundoff 0.11
  Visible-distance 0.11  U-coplanar-distance 0.11  Width-outside 0.23
  _wide-facet 0.68

Output completed.  Verifying that all points are below 0.34 of
all facets.  Will make 48800 distance computations.

echo === check precision options ${d:-`date`}
date
=== check precision options Wed Nov 23 22:04:50 EST 2011
rbox 100 D3 s | qhull E0.01 Qx Tcv FO

Convex hull of 100 points in 3-d:

  Number of vertices: 45
  Number of facets: 42
  Number of non-simplicial facets: 28

Statistics for: rbox 100 D3 s | qhull E0.01 Qx Tcv FO

  Number of points processed: 45
  Number of hyperplanes created: 160
  Number of distance tests for qhull: 3992
  Number of distance tests for merging: 2006
  Number of distance tests for checking: 1937
  Number of merged facets: 57
  CPU seconds to compute hull (after input):  0
  Maximum distance of point above facet: 0.048 (0.6x)
  Maximum distance of vertex below facet: -0.072 (0.9x)

Options selected for Qhull 2011.2 2011/11/21:
  run-id 586783722  Distance-roundoff 0.01  Qxact-merge  Tcheck-frequently
  Tverify  _zero-centrum  _max-width 0.98  _one-merge 0.07  _near-inside 0.35
  Visible-distance 0.02  U-coplanar-distance 0.02  Width-outside 0.04
  _wide-facet 0.12
QH7076 qhull input warning: exact merge ('Qx').  Verify may report that a point
is outside of a facet.  See qh-optq.htm#Qx

Output completed.  Verifying that all points are below outer planes of
all facets.  Will make 4200 distance computations.
rbox 100 D3 W1e-1 | qhull W1e-3 Tcv

Convex hull of 100 points in 3-d:

  Number of vertices: 33
  Number of facets: 62

Statistics for: rbox 100 D3 W1e-1 | qhull W1e-3 Tcv

  Number of points processed: 37
  Number of hyperplanes created: 170
  Number of distance tests for qhull: 1580
  CPU seconds to compute hull (after input):  0
  Maximum distance of point above facet: 0.00084


Output completed.  Verifying that all points are below outer planes of
all facets.  Will make 6200 distance computations.
rbox 100 D3 W1e-1 | qhull W1e-2 Tcv Q0

Convex hull of 100 points in 3-d:

  Number of vertices: 28
  Number of facets: 52

Statistics for: rbox 100 D3 W1e-1 | qhull W1e-2 Tcv Q0

  Number of points processed: 29
  Number of hyperplanes created: 128
  Number of distance tests for qhull: 1442
  CPU seconds to compute hull (after input):  0


precision problems (corrected unless 'Q0' or an error)
     22 coplanar points during partitioning

Output completed.  Verifying that all points are below 0.0097 of
all facets.  Will make 5200 distance computations.
rbox 100 D3 W1e-1 | qhull W1e-2 Tcv 

Convex hull of 100 points in 3-d:

  Number of vertices: 29
  Number of facets: 54

Statistics for: rbox 100 D3 W1e-1 | qhull W1e-2 Tcv

  Number of points processed: 30
  Number of hyperplanes created: 132
  Number of distance tests for qhull: 1768
  CPU seconds to compute hull (after input):  0
  Maximum distance of point above facet: 0.0057


Output completed.  Verifying that all points are below outer planes of
all facets.  Will make 5400 distance computations.
rbox 100 D3 W1e-1 | qhull W1e-1 Tcv

Convex hull of 100 points in 3-d:

  Number of vertices: 13
  Number of facets: 22

Statistics for: rbox 100 D3 W1e-1 | qhull W1e-1 Tcv

  Number of points processed: 13
  Number of hyperplanes created: 46
  Number of distance tests for qhull: 2521
  CPU seconds to compute hull (after input):  0
  Maximum distance of point above facet: 0.098


Output completed.  Verifying that all points are below outer planes of
all facets.  Will make 2200 distance computations.
rbox 15 D2 P0 P1e-14,1e-14 | qhull d Quc Tcv

Furthest-site Delaunay triangulation by the convex hull of 17 points in 3-d:

  Number of input sites: 17
  Number of Delaunay regions: 5

Statistics for: rbox 15 D2 P0 P1e-14,1e-14 | qhull d Quc Tcv

  Number of points processed: 17
  Number of hyperplanes created: 64
  Number of facets in hull: 28
  Number of distance tests for qhull: 142
  Number of distance tests for merging: 527
  Number of distance tests for checking: 194
  Number of merged facets: 7
  CPU seconds to compute hull (after input):  0
  Maximum distance of point above facet: 1.6e-015 (0.3x)


Output completed.  Verifying that all points are below outer planes of
all facets.  Will make 85 distance computations.
rbox 15 D3 P0 P1e-12,1e-14,1e-14 | qhull d Qcu Tcv

Furthest-site Delaunay triangulation by the convex hull of 17 points in 4-d:

  Number of input sites: 17
  Number of Delaunay regions: 14

Statistics for: rbox 15 D3 P0 P1e-12,1e-14,1e-14 | qhull d Qcu Tcv

  Number of points processed: 17
  Number of hyperplanes created: 124
  Number of facets in hull: 60
  Number of distance tests for qhull: 225
  CPU seconds to compute hull (after input):  0


Output completed.  Verifying that all points are below 2.8e-015 of
all facets.  Will make 238 distance computations.
rbox 1000 s D3 | qhull C-0.01 Tcv Qc

Convex hull of 1000 points in 3-d:

  Number of vertices: 107
  Number of coplanar points: 893
  Number of facets: 96
  Number of non-simplicial facets: 77

Statistics for: rbox 1000 s D3 | qhull C-0.01 Tcv Qc

  Number of points processed: 112
  Number of hyperplanes created: 449
  Number of distance tests for qhull: 64526
  Number of distance tests for merging: 8425
  Number of distance tests for checking: 23411
  Number of merged facets: 256
  CPU seconds to compute hull (after input):  0
  Maximum distance of point above facet: 0.02 (0.7x)
  Maximum distance of vertex below facet: -0.035 (1.2x)


Output completed.  Verifying that all points are below outer planes of
all facets.  Will make 96000 distance computations.
rbox 1000 s D3 | qhull C-0.01 V0 Qc Tcv

Convex hull of 1000 points in 3-d:

  Number of vertices: 120
  Number of coplanar points: 880
  Number of facets: 82
  Number of non-simplicial facets: 71

Statistics for: rbox 1000 s D3 | qhull C-0.01 V0 Qc Tcv

  Number of points processed: 445
  Number of hyperplanes created: 2451
  Number of distance tests for qhull: 172077
  Number of distance tests for merging: 52430
  Number of distance tests for checking: 19885
  Number of merged facets: 1741
  CPU seconds to compute hull (after input): 0.016
  Maximum distance of point above facet: 0.032 (1.1x)
  Maximum distance of vertex below facet: -0.064 (2.1x)


Output completed.  Verifying that all points are below outer planes of
all facets.  Will make 82000 distance computations.
rbox 1000 s D3 | qhull C-0.01 U0 Qc Tcv

Convex hull of 1000 points in 3-d:

  Number of vertices: 107
  Number of coplanar points: 893
  Number of facets: 98
  Number of non-simplicial facets: 77

Statistics for: rbox 1000 s D3 | qhull C-0.01 U0 Qc Tcv

  Number of points processed: 112
  Number of hyperplanes created: 497
  Number of distance tests for qhull: 64717
  Number of distance tests for merging: 9106
  Number of distance tests for checking: 23621
  Number of merged facets: 254
  CPU seconds to compute hull (after input): 0.015
  Maximum distance of point above facet: 0.02 (0.7x)
  Maximum distance of vertex below facet: -0.035 (1.2x)


Output completed.  Verifying that all points are below outer planes of
all facets.  Will make 98000 distance computations.
rbox 1000 s D3 | qhull C-0.01 V0 Qcm Tcv

Convex hull of 1000 points in 3-d:

  Number of vertices: 106
  Number of coplanar points: 894
  Number of facets: 94
  Number of non-simplicial facets: 71

Statistics for: rbox 1000 s D3 | qhull C-0.01 V0 Qcm Tcv

  Number of points processed: 114
  Number of hyperplanes created: 597
  Number of distance tests for qhull: 54389
  Number of distance tests for merging: 10179
  Number of distance tests for checking: 18597
  Number of merged facets: 288
  CPU seconds to compute hull (after input): 0.015
  Maximum distance of point above facet: 0.025 (0.8x)
  Maximum distance of vertex below facet: -0.07 (2.3x)


Output completed.  Verifying that all points are below outer planes of
all facets.  Will make 94000 distance computations.
rbox 1000 s D3 | qhull C-0.01 Qcm Tcv

Convex hull of 1000 points in 3-d:

  Number of vertices: 103
  Number of coplanar points: 897
  Number of facets: 96
  Number of non-simplicial facets: 69

Statistics for: rbox 1000 s D3 | qhull C-0.01 Qcm Tcv

  Number of points processed: 106
  Number of hyperplanes created: 434
  Number of distance tests for qhull: 64355
  Number of distance tests for merging: 7947
  Number of distance tests for checking: 23460
  Number of merged facets: 234
  CPU seconds to compute hull (after input): 0.015
  Maximum distance of point above facet: 0.022 (0.7x)
  Maximum distance of vertex below facet: -0.035 (1.2x)


Output completed.  Verifying that all points are below outer planes of
all facets.  Will make 96000 distance computations.
rbox 1000 s D3 | qhull C-0.01 Q1 FO Tcv Qc

Convex hull of 1000 points in 3-d:

  Number of vertices: 105
  Number of coplanar points: 895
  Number of facets: 94
  Number of non-simplicial facets: 70

Statistics for: rbox 1000 s D3 | qhull C-0.01 Q1 FO Tcv Qc

  Number of points processed: 110
  Number of hyperplanes created: 457
  Number of distance tests for qhull: 66717
  Number of distance tests for merging: 8730
  Number of distance tests for checking: 23168
  Number of merged facets: 258
  CPU seconds to compute hull (after input): 0.015
  Maximum distance of point above facet: 0.02 (0.7x)
  Maximum distance of vertex below facet: -0.035 (1.2x)

Options selected for Qhull 2011.2 2011/11/21:
  run-id 586800529  Centrum-premerge- 0.01  Q1-no-angle-sort  Tcheck-frequently
  Tverify  Qcoplanar-keep  _max-width  1  Error-roundoff 6.9e-016
  _one-merge 0.03  _near-inside 0.15  Visible-distance 0.01
  U-coplanar-distance 0.01  Width-outside 0.02  _wide-facet 0.06

Output completed.  Verifying that all points are below outer planes of
all facets.  Will make 94000 distance computations.
rbox 1000 s D3 | qhull C-0.01 Q2 FO Tcv Qc

Convex hull of 1000 points in 3-d:

  Number of vertices: 106
  Number of coplanar points: 894
  Number of facets: 91
  Number of non-simplicial facets: 74

Statistics for: rbox 1000 s D3 | qhull C-0.01 Q2 FO Tcv Qc

  Number of points processed: 112
  Number of hyperplanes created: 454
  Number of distance tests for qhull: 64909
  Number of distance tests for merging: 8602
  Number of distance tests for checking: 22777
  Number of merged facets: 267
  CPU seconds to compute hull (after input):  0
  Maximum distance of point above facet: 0.02 (0.7x)
  Maximum distance of vertex below facet: -0.049 (1.6x)

Options selected for Qhull 2011.2 2011/11/21:
  run-id 586800529  Centrum-premerge- 0.01  Q2-no-merge-independent
  Tcheck-frequently  Tverify  Qcoplanar-keep  _max-width  1
  Error-roundoff 6.9e-016  _one-merge 0.03  _near-inside 0.15
  Visible-distance 0.01  U-coplanar-distance 0.01  Width-outside 0.02
  _wide-facet 0.06

Output completed.  Verifying that all points are below outer planes of
all facets.  Will make 91000 distance computations.
rbox 1000 s D3 | qhull C-0.01 Q3 FO Tcv Qc

Convex hull of 1000 points in 3-d:

  Number of vertices: 112
  Number of coplanar points: 888
  Number of facets: 95
  Number of non-simplicial facets: 78

Statistics for: rbox 1000 s D3 | qhull C-0.01 Q3 FO Tcv Qc

  Number of points processed: 112
  Number of hyperplanes created: 453
  Number of distance tests for qhull: 64248
  Number of distance tests for merging: 8529
  Number of distance tests for checking: 23283
  Number of merged facets: 260
  CPU seconds to compute hull (after input):  0
  Maximum distance of point above facet: 0.02 (0.7x)
  Maximum distance of vertex below facet: -0.035 (1.2x)

Options selected for Qhull 2011.2 2011/11/21:
  run-id 586800529  Centrum-premerge- 0.01  Q3-no-merge-vertices
  Tcheck-frequently  Tverify  Qcoplanar-keep  _max-width  1
  Error-roundoff 6.9e-016  _one-merge 0.03  _near-inside 0.15
  Visible-distance 0.01  U-coplanar-distance 0.01  Width-outside 0.02
  _wide-facet 0.06

Output completed.  Verifying that all points are below outer planes of
all facets.  Will make 95000 distance computations.
rbox 1000 s D3 | qhull C-0.01 Q4 FO Tcv Qc

Convex hull of 1000 points in 3-d:

  Number of vertices: 107
  Number of coplanar points: 893
  Number of facets: 96
  Number of non-simplicial facets: 77

Statistics for: rbox 1000 s D3 | qhull C-0.01 Q4 FO Tcv Qc

  Number of points processed: 112
  Number of hyperplanes created: 449
  Number of distance tests for qhull: 64526
  Number of distance tests for merging: 8425
  Number of distance tests for checking: 23411
  Number of merged facets: 256
  CPU seconds to compute hull (after input):  0
  Maximum distance of point above facet: 0.02 (0.7x)
  Maximum distance of vertex below facet: -0.035 (1.2x)

Options selected for Qhull 2011.2 2011/11/21:
  run-id 586800529  Centrum-premerge- 0.01  Q4-avoid-old-into-new
  Tcheck-frequently  Tverify  Qcoplanar-keep  _max-width  1
  Error-roundoff 6.9e-016  _one-merge 0.03  _near-inside 0.15
  Visible-distance 0.01  U-coplanar-distance 0.01  Width-outside 0.02
  _wide-facet 0.06

Output completed.  Verifying that all points are below outer planes of
all facets.  Will make 96000 distance computations.
echo === this may generate an error ${d:-`date`}
date
=== this may generate an error Wed Nov 23 22:04:51 EST 2011
rbox 1000 s D3 | qhull C-0.01 Q5 FO Tcv 

Convex hull of 1000 points in 3-d:

  Number of vertices: 107
  Number of facets: 96
  Number of non-simplicial facets: 77

Statistics for: rbox 1000 s D3 | qhull C-0.01 Q5 FO Tcv

  Number of points processed: 112
  Number of hyperplanes created: 449
  Number of distance tests for qhull: 38064
  Number of distance tests for merging: 8425
  Number of distance tests for checking: 0
  Number of merged facets: 256
  CPU seconds to compute hull (after input):  0
  Maximum distance of point above facet: 0.022 (0.7x)
  Maximum distance of vertex below facet: -0.035 (1.2x)

Options selected for Qhull 2011.2 2011/11/21:
  run-id 586800529  Centrum-premerge- 0.01  Q5-no-check-outer
  Tcheck-frequently  Tverify  _max-width  1  Error-roundoff 6.9e-016
  _one-merge 0.03  Visible-distance 0.01  U-coplanar-distance 0.01
  Width-outside 0.02  _wide-facet 0.06
QH7077 qhull input warning: no outer plane check ('Q5') or no processing of
near-inside points ('Q8').  Verify may report that a point is outside
of a facet.

Output completed.  Verifying that all points are below 0.022 of
all facets.  Will make 96000 distance computations.
echo === this should generate an error ${d:-`date`}
date
=== this should generate an error Wed Nov 23 22:04:51 EST 2011
rbox 1000 s D3 | qhull C-0.01 Q6 FO Po Tcv Qc

Convex hull of 1000 points in 3-d:

  Number of vertices: 114
  Number of coplanar points: 886
  Number of facets: 182
  Number of non-simplicial facets: 40

Statistics for: rbox 1000 s D3 | qhull C-0.01 Q6 FO Po Tcv Qc

  Number of points processed: 114
  Number of hyperplanes created: 447
  Number of distance tests for qhull: 53157
  Number of distance tests for merging: 3458
  Number of distance tests for checking: 27470
  Number of merged facets: 88
  CPU seconds to compute hull (after input):  0
  Maximum distance of point above facet: 0.02 (0.7x)
  Maximum distance of vertex below facet: -0.009 (0.3x)

Options selected for Qhull 2011.2 2011/11/21:
  run-id 586800529  Centrum-premerge- 0.01  Q6-no-concave-merge  Poutput-forced
  Tcheck-frequently  Tverify  Qcoplanar-keep  _max-width  1
  Error-roundoff 6.9e-016  _one-merge 0.03  _near-inside 0.15
  Visible-distance 0.01  U-coplanar-distance 0.01  Width-outside 0.02
  _wide-facet 0.06
QH6117 qhull precision error: f324 is concave to f325.  Centrum of f324 is 0.004134 above f325
QH6117 qhull precision error: f325 is concave to f324.  Centrum of f325 is 0.001027 above f324
rbox 1000 s D3 | qhull C-0.01 Q7 FO Tcv Qc

Convex hull of 1000 points in 3-d:

  Number of vertices: 100
  Number of coplanar points: 900
  Number of facets: 94
  Number of non-simplicial facets: 73

Statistics for: rbox 1000 s D3 | qhull C-0.01 Q7 FO Tcv Qc

  Number of points processed: 103
  Number of hyperplanes created: 448
  Number of distance tests for qhull: 70481
  Number of distance tests for merging: 8005
  Number of distance tests for checking: 21158
  Number of merged facets: 233
  CPU seconds to compute hull (after input): 0.015
  Maximum distance of point above facet: 0.02 (0.7x)
  Maximum distance of vertex below facet: -0.042 (1.4x)

Options selected for Qhull 2011.2 2011/11/21:
  run-id 586800529  Centrum-premerge- 0.01  Q7-no-breadth-first
  Tcheck-frequently  Tverify  Qcoplanar-keep  _max-width  1
  Error-roundoff 6.9e-016  _one-merge 0.03  _near-inside 0.15
  Visible-distance 0.01  U-coplanar-distance 0.01  Width-outside 0.02
  _wide-facet 0.06

Output completed.  Verifying that all points are below outer planes of
all facets.  Will make 94000 distance computations.
rbox 1000 s D3 | qhull C-0.01 Qx Tcv Qc

Convex hull of 1000 points in 3-d:

  Number of vertices: 114
  Number of coplanar points: 886
  Number of facets: 103
  Number of non-simplicial facets: 77

Statistics for: rbox 1000 s D3 | qhull C-0.01 Qx Tcv Qc

  Number of points processed: 114
  Number of hyperplanes created: 447
  Number of distance tests for qhull: 90887
  Number of distance tests for merging: 5483
  Number of distance tests for checking: 20583
  Number of merged facets: 167
  CPU seconds to compute hull (after input):  0
  Maximum distance of point above facet: 0.02 (0.7x)
  Maximum distance of vertex below facet: -0.034 (1.1x)

QH7076 qhull input warning: exact merge ('Qx').  Verify may report that a point
is outside of a facet.  See qh-optq.htm#Qx

Output completed.  Verifying that all points are below outer planes of
all facets.  Will make 103000 distance computations.
echo === this may generate an error e.g., t1263080158 ${d:-`date`}
date
=== this may generate an error e.g., t1263080158 Wed Nov 23 22:04:51 EST 2011
rbox 100 s D3 t | qhull R1e-3 Tcv Qc

Convex hull of 100 points in 3-d:

  Number of vertices: 99
  Number of coplanar points: 1
  Number of facets: 169
  Number of non-simplicial facets: 22

Statistics for: rbox 100 s D3 t1322103891 | qhull R1e-3 Tcv Qc

  Number of points processed: 100
  Number of hyperplanes created: 460
  Number of distance tests for qhull: 1616
  Number of distance tests for merging: 3342
  Number of distance tests for checking: 1306
  Number of merged facets: 42
  CPU seconds to compute hull (after input):  0
  Maximum distance of point above facet: 0.0015 (0.4x)
  Maximum distance of vertex below facet: -0.0042 (1.1x)


Output completed.  Verifying that all points are below outer planes of
all facets.  Will make 16900 distance computations.
rbox 100 s D3 t | qhull R1e-2 Tcv Qc

Convex hull of 100 points in 3-d:

  Number of vertices: 66
  Number of coplanar points: 34
  Number of facets: 67
  Number of non-simplicial facets: 36

Statistics for: rbox 100 s D3 t1322103891 | qhull R1e-2 Tcv Qc

  Number of points processed: 69
  Number of hyperplanes created: 287
  Number of distance tests for qhull: 3414
  Number of distance tests for merging: 4589
  Number of distance tests for checking: 2142
  Number of merged facets: 128
  CPU seconds to compute hull (after input):  0
  Maximum distance of point above facet: 0.039 (1.0x)
  Maximum distance of vertex below facet: -0.063 (1.6x)


Output completed.  Verifying that all points are below outer planes of
all facets.  Will make 6700 distance computations.
rbox 500 s D3 t | qhull R0.05 A-1 Tcv Qc

Convex hull of 500 points in 3-d:

  Number of vertices: 21
  Number of coplanar points: 479
  Number of facets: 19
  Number of non-simplicial facets: 12

Statistics for: rbox 500 s D3 t1322103891 | qhull R0.05 A-1 Tcv Qc

  Number of points processed: 25
  Number of hyperplanes created: 82
  Number of distance tests for qhull: 23372
  Number of distance tests for merging: 1576
  Number of distance tests for checking: 10032
  Number of merged facets: 51
  CPU seconds to compute hull (after input):  0
  Maximum distance of point above facet: 0.15 (0.3x)
  Maximum distance of vertex below facet: -0.2 (0.3x)


Output completed.  Verifying that all points are below outer planes of
all facets.  Will make 9500 distance computations.
rbox 100 W0 D3 t | qhull R1e-3 Tcv Qc

Convex hull of 100 points in 3-d:

  Number of vertices: 40
  Number of coplanar points: 60
  Number of facets: 52
  Number of non-simplicial facets: 7

Statistics for: rbox 100 W0 D3 t1322103891 | qhull R1e-3 Tcv Qc

  Number of points processed: 48
  Number of hyperplanes created: 166
  Number of distance tests for qhull: 1981
  Number of distance tests for merging: 1672
  Number of distance tests for checking: 1094
  Number of merged facets: 42
  CPU seconds to compute hull (after input):  0
  Maximum distance of point above facet: 0.0024 (0.6x)
  Maximum distance of vertex below facet: -0.0018 (0.5x)


Output completed.  Verifying that all points are below outer planes of
all facets.  Will make 5200 distance computations.
rbox 100 W0 D3 t | qhull R1e-3 Qx Tcv Qc

Convex hull of 100 points in 3-d:

  Number of vertices: 40
  Number of coplanar points: 60
  Number of facets: 50
  Number of non-simplicial facets: 9

Statistics for: rbox 100 W0 D3 t1322103891 | qhull R1e-3 Qx Tcv Qc

  Number of points processed: 48
  Number of hyperplanes created: 164
  Number of distance tests for qhull: 2163
  Number of distance tests for merging: 1593
  Number of distance tests for checking: 1089
  Number of merged facets: 47
  CPU seconds to compute hull (after input):  0
  Maximum distance of point above facet: 0.0024 (0.6x)
  Maximum distance of vertex below facet: -0.0049 (1.2x)

QH7076 qhull input warning: exact merge ('Qx').  Verify may report that a point
is outside of a facet.  See qh-optq.htm#Qx

Output completed.  Verifying that all points are below outer planes of
all facets.  Will make 5000 distance computations.
rbox 100 W0 D3 t | qhull R1e-2 Tcv Qc

Convex hull of 100 points in 3-d:

  Number of vertices: 24
  Number of coplanar points: 76
  Number of facets: 24
  Number of non-simplicial facets: 11

Statistics for: rbox 100 W0 D3 t1322103891 | qhull R1e-2 Tcv Qc

  Number of points processed: 35
  Number of hyperplanes created: 113
  Number of distance tests for qhull: 2677
  Number of distance tests for merging: 1708
  Number of distance tests for checking: 1182
  Number of merged facets: 56
  CPU seconds to compute hull (after input): 0.015
  Maximum distance of point above facet: 0.031 (0.8x)
  Maximum distance of vertex below facet: -0.036 (0.9x)


Output completed.  Verifying that all points are below outer planes of
all facets.  Will make 2400 distance computations.
rbox 100 W0 D3 t | qhull R1e-2 Qx Tcv Qc

Convex hull of 100 points in 3-d:

  Number of vertices: 26
  Number of coplanar points: 74
  Number of facets: 25
  Number of non-simplicial facets: 12

Statistics for: rbox 100 W0 D3 t1322103891 | qhull R1e-2 Qx Tcv Qc

  Number of points processed: 38
  Number of hyperplanes created: 121
  Number of distance tests for qhull: 3183
  Number of distance tests for merging: 1304
  Number of distance tests for checking: 1158
  Number of merged facets: 53
  CPU seconds to compute hull (after input):  0
  Maximum distance of point above facet: 0.025 (0.6x)
  Maximum distance of vertex below facet: -0.049 (1.2x)

QH7076 qhull input warning: exact merge ('Qx').  Verify may report that a point
is outside of a facet.  See qh-optq.htm#Qx

Output completed.  Verifying that all points are below outer planes of
all facets.  Will make 2500 distance computations.
rbox 500 W0 D3 t | qhull R0.05 A-1 Tcv Qc

Convex hull of 500 points in 3-d:

  Number of vertices: 8
  Number of coplanar points: 492
  Number of facets: 6
  Number of non-simplicial facets: 6

Statistics for: rbox 500 W0 D3 t1322103892 | qhull R0.05 A-1 Tcv Qc

  Number of points processed: 13
  Number of hyperplanes created: 33
  Number of distance tests for qhull: 13798
  Number of distance tests for merging: 558
  Number of distance tests for checking: 3864
  Number of merged facets: 21
  CPU seconds to compute hull (after input):  0
  Maximum distance of point above facet: 0.11 (0.2x)
  Maximum distance of vertex below facet: -0.14 (0.2x)


Output completed.  Verifying that all points are below outer planes of
all facets.  Will make 3000 distance computations.
rbox 500 W0 D3 t | qhull R0.05 Qx Tcv Qc

Convex hull of 500 points in 3-d:

  Number of vertices: 11
  Number of coplanar points: 489
  Number of facets: 8
  Number of non-simplicial facets: 6

Statistics for: rbox 500 W0 D3 t1322103892 | qhull R0.05 Qx Tcv Qc

  Number of points processed: 15
  Number of hyperplanes created: 36
  Number of distance tests for qhull: 19163
  Number of distance tests for merging: 514
  Number of distance tests for checking: 4512
  Number of merged facets: 23
  CPU seconds to compute hull (after input):  0
  Maximum distance of point above facet: 0.098 (0.5x)
  Maximum distance of vertex below facet: -0.14 (0.7x)

QH7076 qhull input warning: exact merge ('Qx').  Verify may report that a point
is outside of a facet.  See qh-optq.htm#Qx

Output completed.  Verifying that all points are below outer planes of
all facets.  Will make 4000 distance computations.
rbox 1000 W1e-20 t | qhull Tcv Qc

Convex hull of 1000 points in 3-d:

  Number of vertices: 75
  Number of coplanar points: 925
  Number of facets: 89
  Number of non-simplicial facets: 6

Statistics for: rbox 1000 W1e-20 t1322103892 | qhull Tcv Qc

  Number of points processed: 90
  Number of hyperplanes created: 289
  Number of distance tests for qhull: 18808
  Number of distance tests for merging: 3738
  Number of distance tests for checking: 28507
  Number of merged facets: 87
  CPU seconds to compute hull (after input):  0


Output completed.  Verifying that all points are below outer planes of
all facets.  Will make 89000 distance computations.
rbox 1000 W1e-20 D4 t | qhull Tcv Qc

Convex hull of 1000 points in 4-d:

  Number of vertices: 279
  Number of coplanar points: 721
  Number of facets: 1054
  Number of non-simplicial facets: 8

Statistics for: rbox 1000 W1e-20 D4 t1322103892 | qhull Tcv Qc

  Number of points processed: 367
  Number of hyperplanes created: 5167
  Number of distance tests for qhull: 79310
  Number of distance tests for merging: 76496
  Number of distance tests for checking: 146719
  Number of merged facets: 1181
  CPU seconds to compute hull (after input): 0.094


Output completed.  Verifying that all points are below outer planes of
all facets.  Will make 1054000 distance computations.
rbox 500 W1e-20 D5 t | qhull Tv Qc

Convex hull of 500 points in 5-d:

  Number of vertices: 344
  Number of coplanar points: 156
  Number of facets: 5286
  Number of non-simplicial facets: 10

Statistics for: rbox 500 W1e-20 D5 t1322103892 | qhull Tv Qc

  Number of points processed: 403
  Number of hyperplanes created: 25485
  Number of distance tests for qhull: 129457
  Number of distance tests for merging: 287684
  Number of distance tests for checking: 228920
  Number of merged facets: 3222
  CPU seconds to compute hull (after input): 0.109

QH7076 qhull input warning: exact merge ('Qx').  Verify may report that a point
is outside of a facet.  See qh-optq.htm#Qx

Output completed.  Verifying that all points are below outer planes of
all facets.  Will make 2643000 distance computations.
rbox 100 W1e-20 D6 t | qhull Tv Qc

Convex hull of 100 points in 6-d:

  Number of vertices: 100
  Number of facets: 5773
  Number of non-simplicial facets: 10

Statistics for: rbox 100 W1e-20 D6 t1322103892 | qhull Tv Qc

  Number of points processed: 100
  Number of hyperplanes created: 21227
  Number of distance tests for qhull: 34101
  Number of distance tests for merging: 195353
  Number of distance tests for checking: 119302
  Number of merged facets: 151
  CPU seconds to compute hull (after input): 0.078

QH7076 qhull input warning: exact merge ('Qx').  Verify may report that a point
is outside of a facet.  See qh-optq.htm#Qx

Output completed.  Verifying that all points are below outer planes of
all facets.  Will make 577300 distance computations.
rbox 50 W1e-20 D6 t | qhull Qv Tv Qc

Convex hull of 50 points in 6-d:

  Number of vertices: 50
  Number of facets: 2106

Statistics for: rbox 50 W1e-20 D6 t1322103892 | qhull Qv Tv Qc

  Number of points processed: 50
  Number of hyperplanes created: 6276
  Number of distance tests for qhull: 9174
  CPU seconds to compute hull (after input): 0.125

QH7076 qhull input warning: exact merge ('Qx').  Verify may report that a point
is outside of a facet.  See qh-optq.htm#Qx

Output completed.  Verifying that all points are below outer planes of
all facets.  Will make 105300 distance computations.
rbox 10000 D4 t | qhull QR0 Qc C-0.01 A0.3 Tv

Convex hull of 10000 points in 4-d:

  Number of vertices: 16
  Number of coplanar points: 8737
  Number of facets: 8
  Number of non-simplicial facets: 8

Statistics for: rbox 10000 D4 t1322103892 | qhull QR0 Qc C-0.01 A0.3 Tv QR1322103892

  Number of points processed: 57
  Number of hyperplanes created: 436
  Number of distance tests for qhull: 1765751
  Number of distance tests for merging: 15578
  Number of distance tests for checking: 99750
  Number of merged facets: 492
  CPU seconds to compute hull (after input): 0.078
  Maximum distance of point above facet: 0.11
  Maximum distance of vertex below facet: -0.18 (0.1x)


Output completed.  Verifying that all points are below outer planes of
all facets.  Will make 80000 distance computations.
rbox 1000 D2 t | qhull d QR0 Qc C-1e-8 Qu Tv

Furthest-site Delaunay triangulation by the convex hull of 1000 points in 3-d:

  Number of input sites: 1000
  Number of Delaunay regions: 19

Statistics for: rbox 1000 D2 t1322103893 | qhull d QR0 Qc C-1e-8 Qu Tv QR1322103893

  Number of points processed: 1000
  Number of hyperplanes created: 5656
  Number of facets in hull: 1996
  Number of distance tests for qhull: 32009
  CPU seconds to compute hull (after input): 0.015


Output completed.  Verifying that all points are below outer planes of
all facets.  Will make 19000 distance computations.
rbox 300 D5 t |qhull A-0.999 Qx Qc Tcv

Convex hull of 300 points in 5-d:

  Number of vertices: 157
  Number of coplanar points: 37
  Number of facets: 1720
  Number of non-simplicial facets: 436

Statistics for: rbox 300 D5 t1322103893 | qhull A-0.999 Qx Qc Tcv

  Number of points processed: 197
  Number of hyperplanes created: 12414
  Number of distance tests for qhull: 193411
  Number of distance tests for merging: 57505
  Number of distance tests for checking: 20584
  Number of merged facets: 1170
  CPU seconds to compute hull (after input): 0.046
  Maximum distance of vertex below facet: -0.028 (0.3x)

QH7076 qhull input warning: exact merge ('Qx').  Verify may report that a point
is outside of a facet.  See qh-optq.htm#Qx

Output completed.  Verifying that all points are below outer planes of
all facets.  Will make 516000 distance computations.
rbox 100 D6 t |qhull A-0.9999 Qx Qc Tcv

Convex hull of 100 points in 6-d:

  Number of vertices: 89
  Number of coplanar points: 2
  Number of facets: 3869
  Number of non-simplicial facets: 247

Statistics for: rbox 100 D6 t1322103893 | qhull A-0.9999 Qx Qc Tcv

  Number of points processed: 95
  Number of hyperplanes created: 15448
  Number of distance tests for qhull: 31318
  Number of distance tests for merging: 82887
  Number of distance tests for checking: 83510
  Number of merged facets: 586
  CPU seconds to compute hull (after input): 0.062
  Maximum distance of vertex below facet: -0.0072 (0.2x)

QH7076 qhull input warning: exact merge ('Qx').  Verify may report that a point
is outside of a facet.  See qh-optq.htm#Qx

Output completed.  Verifying that all points are below outer planes of
all facets.  Will make 386900 distance computations.
rbox 50 D7 t |qhull A-0.99999 Qx Qc Tcv W0.1

Convex hull of 50 points in 7-d:

  Number of vertices: 45
  Number of coplanar points: 5
  Number of facets: 4321
  Number of non-simplicial facets: 71

Statistics for: rbox 50 D7 t1322103893 | qhull A-0.99999 Qx Qc Tcv W0.1

  Number of points processed: 45
  Number of hyperplanes created: 12524
  Number of distance tests for qhull: 24913
  Number of distance tests for merging: 86623
  Number of distance tests for checking: 127750
  Number of merged facets: 205
  CPU seconds to compute hull (after input): 0.062
  Maximum distance of point above facet: 0.13
  Maximum distance of vertex below facet: -0.0018 (0.2x)

QH7076 qhull input warning: exact merge ('Qx').  Verify may report that a point
is outside of a facet.  See qh-optq.htm#Qx

Output completed.  Verifying that all points are below outer planes of
all facets.  Will make 216050 distance computations.

echo === check bad cases for Qhull.  May cause errors ${d:-`date`}
date
=== check bad cases for Qhull. May cause errors Wed Nov 23 22:04:53 EST 2011
rbox 1000 L100000 s G1e-6 t | qhull Tv

Convex hull of 1000 points in 3-d:

  Number of vertices: 994
  Number of facets: 1506
  Number of non-simplicial facets: 6

Statistics for: rbox 1000 L100000 s G1e-6 t1322103893 | qhull Tv

  Number of points processed: 995
  Number of hyperplanes created: 3480
  Number of distance tests for qhull: 28411
  Number of distance tests for merging: 136762
  Number of distance tests for checking: 124433
  Number of merged facets: 479
  CPU seconds to compute hull (after input): 0.016
  Maximum distance of point above facet: 1.8e-015 (0.3x)
  Maximum distance of vertex below facet: -3.5e-015 (0.6x)

qhull precision warning: 
The initial hull is narrow (cosine of min. angle is 1.0000000000000000).
Is the input lower dimensional (e.g., on a plane in 3-d)?  Qhull may
produce a wide facet.  Options 'QbB' (scale to unit box) or 'Qbb' (scale
last coordinate) may remove this warning.  Use 'Pp' to skip this warning.
See 'Limitations' in qh-impre.htm.

Output completed.  Verifying that all points are below outer planes of
all facets.  Will make 1506000 distance computations.
rbox 1000 L100000 s G1e-6 t | qhull Tv Q10

Convex hull of 1000 points in 3-d:

  Number of vertices: 994
  Number of facets: 1506
  Number of non-simplicial facets: 6

Statistics for: rbox 1000 L100000 s G1e-6 t1322103893 | qhull Tv Q10

  Number of points processed: 995
  Number of hyperplanes created: 3480
  Number of distance tests for qhull: 28410
  Number of distance tests for merging: 136762
  Number of distance tests for checking: 124433
  Number of merged facets: 479
  CPU seconds to compute hull (after input): 0.016
  Maximum distance of point above facet: 1.8e-015 (0.3x)
  Maximum distance of vertex below facet: -3.5e-015 (0.6x)


Output completed.  Verifying that all points are below outer planes of
all facets.  Will make 1506000 distance computations.
rbox 1000 s Z1 G1e-13 t | qhull Tv

Convex hull of 1001 points in 3-d:

  Number of vertices: 1001
  Number of facets: 1211
  Number of non-simplicial facets: 67

Statistics for: rbox 1000 s Z1 G1e-13 t1322103893 | qhull Tv

  Number of points processed: 1001
  Number of hyperplanes created: 9428
  Number of distance tests for qhull: 34359
  Number of distance tests for merging: 428992
  Number of distance tests for checking: 562916
  Number of merged facets: 6824
  CPU seconds to compute hull (after input): 0.062
  Maximum distance of point above facet: 1.5e-014 (2.7x)
  Maximum distance of vertex below facet: -1.8e-014 (3.2x)


Output completed.  Verifying that all points are below outer planes of
all facets.  Will make 1212211 distance computations.
rbox 1000 s W1e-13 P0 t | qhull d Qbb Qc Tv

Delaunay triangulation by the convex hull of 1001 points in 4-d:

  Number of input sites: 1001
  Number of Delaunay regions: 1996

Statistics for: rbox 1000 s W1e-13 P0 t1322103893 | qhull d Qbb Qc Tv

  Number of points processed: 1001
  Number of hyperplanes created: 207266
  Number of facets in hull: 2007
  Number of distance tests for qhull: 296838
  Number of distance tests for merging: 15668744
  Number of distance tests for checking: 1996340
  Number of merged facets: 201257
  CPU seconds to compute hull (after input): 3.594
  Maximum distance of point above facet: 2.1e-012 (104.8x)
  Maximum distance of vertex below facet: -1.8e-012 (89.6x)


Output completed.  Verifying that all points are below outer planes of
all facets.  Will make 1997996 distance computations.
rbox 1000 s W1e-13 t | qhull d Tv

Delaunay triangulation by the convex hull of 1000 points in 4-d:

  Number of input sites: 67
  Total number of deleted points due to merging: 794
  Total number of nearly incident points: 139
  Number of Delaunay regions: 51
  Number of non-simplicial Delaunay regions: 20

Statistics for: rbox 1000 s W1e-13 t1322103897 | qhull d Tv

  Number of points processed: 861
  Number of hyperplanes created: 11744
  Number of facets in hull: 93
  Number of distance tests for qhull: 3016025
  Number of distance tests for merging: 480695
  Number of distance tests for checking: 88799
  Number of merged facets: 10769
  CPU seconds to compute hull (after input): 0.25
  Maximum distance of point above facet: 4e-014 (3.9x)
  Maximum distance of vertex below facet: -4.1e-014 (4.1x)

qhull precision warning: 
The initial hull is narrow (cosine of min. angle is 1.0000000000000000).
Is the input lower dimensional (e.g., on a plane in 3-d)?  Qhull may
produce a wide facet.  Options 'QbB' (scale to unit box) or 'Qbb' (scale
last coordinate) may remove this warning.  Use 'Pp' to skip this warning.
See 'Limitations' in qh-impre.htm.

Output completed.  Verifying that all points are below outer planes of
all facets.  Will make 51000 distance computations.
rbox 1000 s W1e-13 t D2 | qhull d Tv

Delaunay triangulation by the convex hull of 1000 points in 3-d:

  Number of input sites: 984
  Total number of deleted points due to merging: 7
  Total number of nearly incident points: 9
  Number of Delaunay regions: 756
  Number of non-simplicial Delaunay regions: 119

Statistics for: rbox 1000 s W1e-13 D2 t1322103897 | qhull d Tv

  Number of points processed: 991
  Number of hyperplanes created: 3871
  Number of facets in hull: 1491
  Number of distance tests for qhull: 138800
  Number of distance tests for merging: 44507
  Number of distance tests for checking: 205672
  Number of merged facets: 787
  CPU seconds to compute hull (after input): 0.015
  Maximum distance of point above facet: 5.6e-014 (10.0x)
  Maximum distance of vertex below facet: -7.2e-015 (1.3x)

qhull precision warning: 
The initial hull is narrow (cosine of min. angle is 1.0000000000000000).
Is the input lower dimensional (e.g., on a plane in 3-d)?  Qhull may
produce a wide facet.  Options 'QbB' (scale to unit box) or 'Qbb' (scale
last coordinate) may remove this warning.  Use 'Pp' to skip this warning.
See 'Limitations' in qh-impre.htm.

Output completed.  Verifying that all points are below outer planes of
all facets.  Will make 756000 distance computations.

echo =======================================================
=======================================================
echo =======================================================
=======================================================
echo === The following commands may cause errors ${d:-`date`}
date
=== The following commands may cause errors Wed Nov 23 22:04:57 EST 2011
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rbox c D7 | qhull Q0 Tcv

Convex hull of 128 points in 7-d:

  Number of vertices: 128
  Number of facets: 8666

Statistics for: rbox c D7 | qhull Q0 Tcv

  Number of points processed: 128
  Number of hyperplanes created: 12287
  Number of distance tests for qhull: 32936
  CPU seconds to compute hull (after input): 0.031


precision problems (corrected unless 'Q0' or an error)
  41888 coplanar half ridges in output
  10185 coplanar horizon facets for new vertices
   8389 nearly singular or axis-parallel hyperplanes
   8389 zero divisors during back substitute
  11903 zero divisors during gaussian elimination

qhull output completed.  Verifying that 128 points are
below 6.6e-015 of the nearest facet.
rbox 100 s D3 | qhull Q0 E1e-3 Tc Po

Convex hull of 100 points in 3-d:

  Number of vertices: 100
  Number of facets: 196

Statistics for: rbox 100 s D3 | qhull Q0 E1e-3 Tc Po

  Number of points processed: 100
  Number of hyperplanes created: 508
  Number of distance tests for qhull: 1728
  CPU seconds to compute hull (after input):  0

QH6115 qhull precision error: f100 is concave to f247, since p44(v13) is 0.002232 above
QH6115 qhull precision error: f139 is concave to f274, since p90(v32) is 0.007128 above
QH6115 qhull precision error: f163 is concave to f232, since p5(v6) is 0.001226 above
QH6115 qhull precision error: f293 is concave to f389, since p11(v61) is 0.002161 above
QH6115 qhull precision error: f307 is concave to f459, since p75(v21) is 0.01537 above

precision problems (corrected unless 'Q0' or an error)
     16 coplanar half ridges in output
      5 concave half ridges in output
     18 coplanar horizon facets for new vertices
rbox 100 s D3 | qhull Q0 E1e-2 Tc Po

Convex hull of 100 points in 3-d:

  Number of vertices: 72
  Number of facets: 140

Statistics for: rbox 100 s D3 | qhull Q0 E1e-2 Tc Po

  Number of points processed: 72
  Number of hyperplanes created: 321
  Number of distance tests for qhull: 1693
  CPU seconds to compute hull (after input):  0

QH6115 qhull precision error: f123 is concave to f235, since p85(v8) is 0.06092 above
QH6115 qhull precision error: f177 is concave to f272, since p98(v20) is 0.03571 above
QH6115 qhull precision error: f246 is concave to f252, since p7(v57) is 0.02104 above
QH6115 qhull precision error: f261 is concave to f305, since p82(v60) is 0.01847 above
QH6115 qhull precision error: f266 is concave to f317, since p82(v60) is 0.02592 above

precision problems (corrected unless 'Q0' or an error)
     52 coplanar half ridges in output
      5 concave half ridges in output
     54 coplanar horizon facets for new vertices
     27 coplanar points during partitioning
rbox 100 s D3 | qhull Q0 E1e-1 Tc Po

Convex hull of 100 points in 3-d:

  Number of vertices: 10
  Number of facets: 16

Statistics for: rbox 100 s D3 | qhull Q0 E1e-1 Tc Po

  Number of points processed: 10
  Number of hyperplanes created: 27
  Number of distance tests for qhull: 977
  CPU seconds to compute hull (after input):  0


precision problems (corrected unless 'Q0' or an error)
      3 coplanar half ridges in output
      3 coplanar horizon facets for new vertices
     59 coplanar points during partitioning
rbox 100 s D3 | qhull Q0 R1e-3 Tc Po

Convex hull of 100 points in 3-d:

  Number of vertices: 100
  Number of facets: 196

Statistics for: rbox 100 s D3 | qhull Q0 R1e-3 Tc Po

  Number of points processed: 100
  Number of hyperplanes created: 513
  Number of distance tests for qhull: 1693
  CPU seconds to compute hull (after input):  0

QH6115 qhull precision error: f229 is concave to f300, since p90(v34) is 0.007895 above
QH6115 qhull precision error: f323 is concave to f421, since p11(v67) is 0.001894 above
QH6115 qhull precision error: f343 is concave to f493, since p75(v21) is  0.011 above
QH6115 qhull precision error: f493 is concave to f343, since p25(v97) is 0.000937 above

precision problems (corrected unless 'Q0' or an error)
     14 coplanar half ridges in output
      4 concave half ridges in output
      9 coplanar horizon facets for new vertices
rbox 100 s D3 | qhull Q0 R1e-2 Tc Po

Convex hull of 100 points in 3-d:

  Number of vertices: 87
  Number of facets: 170

Statistics for: rbox 100 s D3 | qhull Q0 R1e-2 Tc Po

  Number of points processed: 87
  Number of hyperplanes created: 405
  Number of distance tests for qhull: 1734
  CPU seconds to compute hull (after input):  0

QH6136 qhull precision error: facet f326 is flipped, distance= 0.51437088552
QH6136 qhull precision error: facet f326 is flipped, distance= 0.511098236347
QH6115 qhull precision error: f76 is concave to f201, since p76(v21) is 0.0101 above
QH6115 qhull precision error: f96 is concave to f279, since p38(v22) is 0.008383 above
QH6115 qhull precision error: f132 is concave to f140, since p57(v0) is 0.02117 above
QH6115 qhull precision error: f140 is concave to f132, since p96(v34) is 0.00703 above
QH6115 qhull precision error: f150 is concave to f230, since p11(v37) is 0.04121 above
QH6115 qhull precision error: f161 is concave to f326, since p26(v32) is 0.01279 above
QH6115 qhull precision error: f194 is concave to f281, since p73(v12) is 0.03149 above
QH6115 qhull precision error: f230 is concave to f150, since p47(v52) is 0.008464 above
QH6115 qhull precision error: f263 is concave to f156, since p15(v60) is 0.01055 above
QH6115 qhull precision error: f277 is concave to f353, since p89(v36) is 0.005387 above
QH6115 qhull precision error: f281 is concave to f194, since p66(v63) is 0.005996 above
QH6115 qhull precision error: f319 is concave to f341, since p82(v70) is 0.07818 above
QH6115 qhull precision error: f324 is concave to f368, since p41(v5) is 0.01436 above
QH6113 qhull precision error: f326 is flipped(interior point is outside)
QH6115 qhull precision error: f344 is concave to f352, since p29(v75) is 0.009226 above
QH6115 qhull precision error: f348 is concave to f351, since p27(v76) is 0.008815 above
QH6115 qhull precision error: f349 is concave to f350, since p15(v60) is 0.01089 above
QH6115 qhull precision error: f352 is concave to f344, since p69(v77) is 0.008279 above
QH6115 qhull precision error: f353 is concave to f277, since p69(v77) is 0.006627 above
QH6115 qhull precision error: f385 is concave to f404, since p18(v73) is 0.007494 above
QH6115 qhull precision error: f404 is concave to f385, since p13(v87) is 0.005049 above

precision problems (corrected unless 'Q0' or an error)
     45 coplanar half ridges in output
     20 concave half ridges in output
      1 flipped facets
     47 coplanar horizon facets for new vertices
     13 coplanar points during partitioning
rbox 100 s D3 | qhull Q0 R0.05 Tc
QH6136 qhull precision error: facet f160 is flipped, distance= 0.41168102504
ERRONEOUS FACET:
- f160
    - flags: bottom simplicial flipped
    - normal:   -0.3615  -0.1924   0.9123
    - offset:  0.4644952
    - vertices: p94(v40) p19(v28) p56(v27)
    - neighboring facets: f107 f158 f202
QH6136 qhull precision error: facet f203 is flipped, distance= 0.448767103644
ERRONEOUS FACET:
- f203
    - flags: top simplicial flipped
    - normal:   -0.2281  0.05702   -0.972
    - offset:  0.4136177
    - vertices: p40(v51) p83(v19) p48(v6)
    - neighboring facets: f123 f204 f205

A flipped facet occurs when its distance to the interior point is
greater than -0.024, the maximum roundoff error.

While executing: rbox 100 s D3 | qhull Q0 R0.05 Tc
Options selected for Qhull 2011.2 2011/11/21:
  run-id 586918178  Q0-no-premerge  Random_perturb 0.05  Tcheck-frequently
  _max-width 0.94  Error-roundoff 0.024  Visible-distance 0.024
  U-coplanar-distance 0.024  Width-outside 0.048  _wide-facet 0.15
Last point added to hull was p40.
Qhull has finished constructing the hull.

At error exit:

Convex hull of 100 points in 3-d:

  Number of vertices: 51
  Number of facets: 98

Statistics for: rbox 100 s D3 | qhull Q0 R0.05 Tc

  Number of points processed: 51
  Number of hyperplanes created: 206
  Number of distance tests for qhull: 2041
  CPU seconds to compute hull (after input):  0


precision problems (corrected unless 'Q0' or an error)
      2 flipped facets
     50 coplanar horizon facets for new vertices
     46 coplanar points during partitioning

Precision problems were detected during construction of the convex hull.
This occurs because convex hull algorithms assume that calculations are
exact, but floating-point arithmetic has roundoff errors.

To correct for precision problems, do not use 'Q0'.  By default, Qhull
selects 'C-0' or 'Qx' and merges non-convex facets.  With option 'QJ',
Qhull joggles the input to prevent precision problems.  See "Imprecision
in Qhull" (qh-impre.htm).

If you use 'Q0', the output may include
coplanar ridges, concave ridges, and flipped facets.  In 4-d and higher,
Qhull may produce a ridge with four neighbors or two facets with the same 
vertices.  Qhull reports these events when they occur.  It stops when a
concave ridge, flipped facet, or duplicate facet occurs.

If you need triangular output:
  - use option 'Qt' to triangulate the output
  - use option 'QJ' to joggle the input points and remove precision errors
  - use option 'Ft'.  It triangulates non-simplicial facets with added points.

If you must use 'Q0',
try one or more of the following options.  They can not guarantee an output.
  - use 'QbB' to scale the input to a cube.
  - use 'Po' to produce output and prevent partitioning for flipped facets
  - use 'V0' to set min. distance to visible facet as 0 instead of roundoff
  - use 'En' to specify a maximum roundoff error less than 0.024.
  - options 'Qf', 'Qbb', and 'QR0' may also help

To guarantee simplicial output:
  - use option 'Qt' to triangulate the output
  - use option 'QJ' to joggle the input points and remove precision errors
  - use option 'Ft' to triangulate the output by adding points
  - use exact arithmetic (see "Imprecision in Qhull", qh-impre.htm)
rbox 100 s D3 | qhull Q0 R0.05 Tc Po

Convex hull of 100 points in 3-d:

  Number of vertices: 51
  Number of facets: 98

Statistics for: rbox 100 s D3 | qhull Q0 R0.05 Tc Po

  Number of points processed: 51
  Number of hyperplanes created: 209
  Number of distance tests for qhull: 2072
  CPU seconds to compute hull (after input):  0

QH6136 qhull precision error: facet f105 is flipped, distance= 0.531034433025
QH6136 qhull precision error: facet f172 is flipped, distance= 0.351989788321
QH6136 qhull precision error: facet f105 is flipped, distance= 0.524438872687
QH6136 qhull precision error: facet f172 is flipped, distance= 0.311891927779
QH6115 qhull precision error: f30 is concave to f172, since p90(v11) is 0.05852 above
QH6115 qhull precision error: f41 is concave to f70, since p28(v14) is 0.03835 above
QH6115 qhull precision error: f65 is concave to f128, since p18(v19) is 0.03961 above
QH6115 qhull precision error: f77 is concave to f201, since p89(v21) is 0.03681 above
QH6115 qhull precision error: f99 is concave to f192, since p85(v8) is 0.1891 above
QH6113 qhull precision error: f105 is flipped(interior point is outside)
QH6115 qhull precision error: f143 is concave to f179, since p3(v2) is 0.0956 above
QH6115 qhull precision error: f150 is concave to f185, since p70(v13) is 0.03157 above
QH6115 qhull precision error: f161 is concave to f171, since p41(v5) is 0.07964 above
QH6115 qhull precision error: f166 is concave to f108, since p96(v41) is 0.02537 above
QH6115 qhull precision error: f169 is concave to f129, since p60(v42) is 0.02492 above
QH6113 qhull precision error: f172 is flipped(interior point is outside)
QH6115 qhull precision error: f183 is concave to f205, since p44(v37) is 0.1309 above
QH6115 qhull precision error: f186 is concave to f207, since p5(v25) is 0.1066 above
QH6115 qhull precision error: f187 is concave to f151, since p75(v47) is 0.04347 above
QH6115 qhull precision error: f199 is concave to f200, since p12(v3) is 0.08495 above
QH6115 qhull precision error: f201 is concave to f77, since p35(v50) is 0.02603 above

precision problems (corrected unless 'Q0' or an error)
     61 coplanar half ridges in output
     15 concave half ridges in output
      2 flipped facets
     44 coplanar horizon facets for new vertices
     46 coplanar points during partitioning
rbox 1000 W1e-7 | qhull Q0 Tc Po

Convex hull of 1000 points in 3-d:

  Number of vertices: 164
  Number of facets: 324

Statistics for: rbox 1000 W1e-7 | qhull Q0 Tc Po

  Number of points processed: 221
  Number of hyperplanes created: 1160
  Number of distance tests for qhull: 22647
  CPU seconds to compute hull (after input):  0

rbox 50 s | qhull Q0 V0.05 W0.01 Tc Po
QH6136 qhull precision error: facet f105 is flipped, distance= 0.328228630954
QH6136 qhull precision error: facet f125 is flipped, distance= 0.389476173148
QH6136 qhull precision error: facet f152 is flipped, distance= 0.420187947128
QH6136 qhull precision error: facet f155 is flipped, distance= 0.396217639489
QH6136 qhull precision error: facet f161 is flipped, distance= 0.0543980568868
QH6136 qhull precision error: facet f164 is flipped, distance= 0.448464152152
QH6136 qhull precision error: facet f170 is flipped, distance= 0.430073300444
QH6136 qhull precision error: facet f105 is flipped, distance= 0.328228630954
QH6136 qhull precision error: facet f125 is flipped, distance= 0.389476173148
QH6136 qhull precision error: facet f152 is flipped, distance= 0.420187947128
QH6136 qhull precision error: facet f155 is flipped, distance= 0.396217639489
QH6136 qhull precision error: facet f161 is flipped, distance= 0.0543980568868
QH6136 qhull precision error: facet f164 is flipped, distance= 0.448464152152
QH6136 qhull precision error: facet f170 is flipped, distance= 0.430073300444
QH6115 qhull precision error: f25 is concave to f60, since p41(v8) is 0.1672 above
QH6115 qhull precision error: f25 is concave to f155, since p19(v2) is 0.01771 above
QH6115 qhull precision error: f38 is concave to f70, since p0(v13) is 0.06094 above
QH6115 qhull precision error: f39 is concave to f145, since p8(v11) is 0.2589 above
QH6115 qhull precision error: f45 is concave to f72, since p5(v15) is 0.02091 above
QH6115 qhull precision error: f47 is concave to f98, since p5(v15) is 0.05573 above
QH6115 qhull precision error: f47 is concave to f96, since p42(v7) is 0.05284 above
QH6115 qhull precision error: f50 is concave to f59, since p49(v16) is 0.01059 above
QH6115 qhull precision error: f54 is concave to f84, since p28(v17) is 0.03013 above
QH6115 qhull precision error: f54 is concave to f152, since p29(v1) is 0.2143 above
QH6115 qhull precision error: f59 is concave to f50, since p1(v19) is 0.01736 above
QH6115 qhull precision error: f60 is concave to f25, since p1(v19) is 0.02263 above
QH6115 qhull precision error: f62 is concave to f119, since p14(v5) is 0.02783 above
QH6115 qhull precision error: f65 is concave to f120, since p16(v10) is 0.003387 above
QH6115 qhull precision error: f70 is concave to f38, since p9(v22) is 0.04722 above
QH6115 qhull precision error: f71 is concave to f73, since p9(v22) is 0.04568 above
QH6115 qhull precision error: f72 is concave to f45, since p31(v23) is 0.02334 above
QH6115 qhull precision error: f73 is concave to f71, since p31(v23) is 0.04141 above
QH6115 qhull precision error: f78 is concave to f125, since p38(v24) is 0.01386 above
QH6115 qhull precision error: f84 is concave to f54, since p25(v26) is 0.01696 above
QH6115 qhull precision error: f87 is concave to f93, since p14(v5) is 0.01836 above
QH6115 qhull precision error: f88 is concave to f112, since p4(v27) is 0.04742 above
QH6115 qhull precision error: f91 is concave to f161, since p26(v28) is 0.3121 above
QH6115 qhull precision error: f93 is concave to f87, since p26(v28) is 0.01572 above
QH6115 qhull precision error: f96 is concave to f47, since p21(v29) is 0.02763 above
QH6115 qhull precision error: f98 is concave to f47, since p20(v30) is 0.03086 above
QH6113 qhull precision error: f105 is flipped(interior point is outside)
QH6115 qhull precision error: f106 is concave to f149, since p29(v1) is 0.0618 above
QH6115 qhull precision error: f112 is concave to f88, since p6(v34) is 0.02917 above
QH6115 qhull precision error: f119 is concave to f62, since p39(v36) is 0.006072 above
QH6115 qhull precision error: f120 is concave to f65, since p39(v36) is 0.001213 above
QH6113 qhull precision error: f125 is flipped(interior point is outside)
QH6115 qhull precision error: f134 is concave to f139, since p46(v40) is 0.07699 above
QH6115 qhull precision error: f139 is concave to f134, since p47(v41) is 0.01871 above
QH6115 qhull precision error: f141 is concave to f144, since p47(v41) is 0.1017 above
QH6115 qhull precision error: f144 is concave to f14
Convex hull of 50 points in 3-d:

  Number of vertices: 50
  Number of facets: 96

Statistics for: rbox 50 s | qhull Q0 V0.05 W0.01 Tc Po

  Number of points processed: 50
  Number of hyperplanes created: 171
  Number of distance tests for qhull: 552
  CPU seconds to compute hull (after input):  0

1, since p35(v42) is 0.03263 above
QH6115 qhull precision error: f145 is concave to f39, since p2(v43) is 0.04048 above
QH6115 qhull precision error: f149 is concave to f106, since p18(v44) is 0.009672 above
QH6113 qhull precision error: f152 is flipped(interior point is outside)
QH6113 qhull precision error: f155 is flipped(interior point is outside)
QH6113 qhull precision error: f161 is flipped(interior point is outside)
QH6113 qhull precision error: f164 is flipped(interior point is outside)
QH6115 qhull precision error: f166 is concave to f170, since p16(v10) is 0.04552 above
QH6113 qhull precision error: f170 is flipped(interior point is outside)

precision problems (corrected unless 'Q0' or an error)
     37 concave half ridges in output
      7 flipped facets
     82 coplanar horizon facets for new vertices
rbox 100 s D5 | qhull Q0 R1e-2 Tc Po

Convex hull of 100 points in 5-d:

  Number of vertices: 37
  Number of facets: 500

Statistics for: rbox 100 s D5 | qhull Q0 R1e-2 Tc Po

  Number of points processed: 37
  Number of hyperplanes created: 1198
  Number of distance tests for qhull: 2764

QH6107 qhull precision error: facets f1150, f1191 and f1152 meet at a ridge with more than 2 neighbors.  Can not continue.
ERRONEOUS FACET:
- f1150
    - flags: top simplicial new
    - normal:   -0.7563   0.3207  -0.3738   0.2956   0.3133
    - offset: -0.2362615
    - vertices: p70(v37) p59(v25) p47(v17) p33(v15) p6(v9)
    - neighboring facets: f519 f1148 f1151 f1152 f1153
ERRONEOUS OTHER FACET:
- f1191
    - flags: top simplicial new
    - normal:    0.7615  -0.2592   0.3014  -0.3662  -0.3577
    - offset:    0.27682
    - vertices: p70(v37) p59(v25) p47(v17) p6(v9) p54(v0)
    - neighboring facets: f519 f1148
ERRONEOUS and NEIGHBORING FACETS to output

While executing: rbox 100 s D5 | qhull Q0 R1e-2 Tc Po
Options selected for Qhull 2011.2 2011/11/21:
  run-id 586918178  Q0-no-premerge  Random_perturb 0.01  Tcheck-frequently
  Poutput-forced  _max-width 0.88  Error-roundoff 0.0045
  Visible-distance 0.0045  U-coplanar-distance 0.0045  Width-outside 0.009
  _wide-facet 0.027
Last point added to hull was p70.

precision problems (corrected unless 'Q0' or an error)
     50 coplanar horizon facets for new vertices

Precision problems were detected during construction of the convex hull.
This occurs because convex hull algorithms assume that calculations are
exact, but floating-point arithmetic has roundoff errors.

To correct for precision problems, do not use 'Q0'.  By default, Qhull
selects 'C-0' or 'Qx' and merges non-convex facets.  With option 'QJ',
Qhull joggles the input to prevent precision problems.  See "Imprecision
in Qhull" (qh-impre.htm).

If you use 'Q0', the output may include
coplanar ridges, concave ridges, and flipped facets.  In 4-d and higher,
Qhull may produce a ridge with four neighbors or two facets with the same 
vertices.  Qhull reports these events when they occur.  It stops when a
concave ridge, flipped facet, or duplicate facet occurs.

If you need triangular output:
  - use option 'Qt' to triangulate the output
  - use option 'QJ' to joggle the input points and remove precision errors
  - use option 'Ft'.  It triangulates non-simplicial facets with added points.

If you must use 'Q0',
try one or more of the following options.  They can not guarantee an output.
  - use 'QbB' to scale the input to a cube.
  - use 'Po' to produce output and prevent partitioning for flipped facets
  - use 'V0' to set min. distance to visible facet as 0 instead of roundoff
  - use 'En' to specify a maximum roundoff error less than 0.0045.
  - options 'Qf', 'Qbb', and 'QR0' may also help

To guarantee simplicial output:
  - use option 'Qt' to triangulate the output
  - use option 'QJ' to joggle the input points and remove precision errors
  - use option 'Ft' to triangulate the output by adding points
  - use exact arithmetic (see "Imprecision in Qhull", qh-impre.htm)
qhull

qhull- compute convex hulls and related structures.  Qhull 2011.2 2011/11/21
    input (stdin): dimension, n, point coordinates
    comments start with a non-numeric character
    halfspace: use dim+1 and put offsets after coefficients

options (qh-quick.htm):
    d    - Delaunay triangulation by lifting points to a paraboloid
    d Qu - furthest-site Delaunay triangulation (upper convex hull)
    v    - Voronoi diagram as the dual of the Delaunay triangulation
    v Qu - furthest-site Voronoi diagram
    H1,1 - Halfspace intersection about [1,1,0,...] via polar duality
    Qt   - triangulated output
    QJ   - joggled input instead of merged facets
    Tv   - verify result: structure, convexity, and point inclusion
    .    - concise list of all options
    -    - one-line description of each option

Output options (subset):
    s    - summary of results (default)
    i    - vertices incident to each facet
    n    - normals with offsets
    p    - vertex coordinates (if 'Qc', includes coplanar points)
           if 'v', Voronoi vertices
    Fp   - halfspace intersections
    Fx   - extreme points (convex hull vertices)
    FA   - compute total area and volume
    o    - OFF format (if 'v', outputs Voronoi regions)
    G    - Geomview output (2-d, 3-d and 4-d)
    m    - Mathematica output (2-d and 3-d)
    QVn  - print facets that include point n, -n if not
    TO file- output results to file, may be enclosed in single quotes

examples:
    rbox c d D2 | qhull Qc s f Fx | more      rbox 1000 s | qhull Tv s FA
    rbox 10 D2 | qhull d QJ s i TO result     rbox 10 D2 | qhull v Qbb Qt p
    rbox 10 D2 | qhull d Qu QJ m              rbox 10 D2 | qhull v Qu QJ o
    rbox c | qhull n                          rbox c | qhull FV n | qhull H Fp
    rbox d D12 | qhull QR0 FA                 rbox c D7 | qhull FA TF1000
    rbox y 1000 W0 | qhull                    rbox 10 | qhull v QJ o Fv

qhull .

Qhull 2011.2 2011/11/21.
Except for 'F.' and 'PG', upper-case options take an argument.

 delaunay       voronoi        Geomview       Halfspace      facet_dump
 incidences     mathematica    normals        OFF_format     points
 summary

 Farea          FArea-total    Fcoplanars     FCentrums      Fd-cdd-in
 FD-cdd-out     FF-dump-xridge Finner         FIDs           Fmerges
 Fneighbors     FNeigh-vertex  Fouter         FOptions       Fpoint-intersect
 FPoint_near    FQhull         Fsummary       FSize          Ftriangles
 Fvertices      Fvoronoi       FVertex-ave    Fxtremes       FMaple

 Gvertices      Gpoints        Gall_points    Gno_planes     Ginner
 Gcentrums      Ghyperplanes   Gridges        Gouter         GDrop_dim
 Gtransparent

 PArea-keep     Pdrop d0:0D0   Pgood          PFacet_area_keep
 PGood_neighbors PMerge-keep   Poutput_forced Pprecision_not

 QbBound 0:0.5  Qbk:0Bk:0_drop QbB-scale-box  Qbb-scale-last Qcoplanar
 Qfurthest      Qgood_only     QGood_point    Qinterior      Qmax_out
 QJoggle        Qrandom        QRotate        Qsearch_1st    Qtriangulate
 QupperDelaunay QVertex_good   Qvneighbors    Qxact_merge    Qzinfinite

 Q0_no_premerge Q1_no_angle    Q2_no_independ Q3_no_redundant Q4_no_old
 Q5_no_check_out Q6_no_concave Q7_depth_first Q8_no_near_in  Q9_pick_furthest
 Q10_no_narrow  Q11_trinormals

 T4_trace       Tannotate      Tcheck_often   Tstatistics    Tverify
 Tz_stdout      TFacet_log     TInput_file    TPoint_trace   TMerge_trace
 TOutput_file   TRerun         TWide_trace    TVertex_stop   TCone_stop

 Angle_max      Centrum_size   Error_round    Random_dist    Visible_min
 Ucoplanar_max  Wide_outside
qhull -

qhull- compute convex hulls and related structures.
    http://www.qhull.org  2011.2 2011/11/21

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
    halfspaces:  use dim plus one and put offset after coefficients.
                 May be preceeded by a single interior point ('H').

options:
    d    - Delaunay triangulation by lifting points to a paraboloid
    d Qu - furthest-site Delaunay triangulation (upper convex hull)
    v    - Voronoi diagram (dual of the Delaunay triangulation)
    v Qu - furthest-site Voronoi diagram
    Hn,n,... - halfspace intersection about point [n,n,0,...]
    Qt   - triangulated output
    QJ   - joggled input instead of merged facets
    Qc   - keep coplanar points with nearest facet
    Qi   - keep interior points with nearest facet

Qhull control options:
    Qbk:n   - scale coord k so that low bound is n
      QBk:n - scale coord k so that upper bound is n (QBk is 0.5)
    QbB  - scale input to unit cube centered at the origin
    Qbb  - scale last coordinate to [0,m] for Delaunay triangulations
    Qbk:0Bk:0 - remove k-th coordinate from input
    QJn  - randomly joggle input in range [-n,n]
    QRn  - random rotation (n=seed, n=0 time, n=-1 time/no rotate)
    Qf   - partition point to furthest outside facet
    Qg   - only build good facets (needs 'QGn', 'QVn', or 'PdD')
    Qm   - only process points that would increase max_outside
    Qr   - process random outside points instead of furthest ones
    Qs   - search all points for the initial simplex
    Qu   - for 'd' or 'v', compute upper hull without point at-infinity
              returns furthest-site Delaunay triangulation
    Qv   - test vertex neighbors for convexity
    Qx   - exact pre-merges (skips coplanar and angle-coplanar facets)
    Qz   - add point-at-infinity to Delaunay triangulation
    QGn  - good facet if visible from point n, -n for not visible
    QVn  - good facet if it includes point n, -n if not
    Q0   - turn off default premerge with 'C-0'/'Qx'
    Q1     - sort merges by type instead of angle
    Q2   - merge all non-convex at once instead of independent sets
    Q3   - do not merge redundant vertices
    Q4   - avoid old->new merges
    Q5   - do not correct outer planes at end of qhull
    Q6   - do not pre-merge concave or coplanar facets
    Q7   - depth-first processing instead of breadth-first
    Q8   - do not process near-inside points
    Q9   - process furthest of furthest points
    Q10  - no special processing for narrow distributions
    Q11  - copy normals and recompute centrums for tricoplanar facets

Topts- Trace options:
    T4   - trace at level n, 4=all, 5=mem/gauss, -1= events
    Ta   - annotate output with message codes
    Tc   - check frequently during execution
    Ts   - print statistics
    Tv   - verify result: structure, convexity, and point inclusion
    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
    TRn  - rerun qhull n times.  Use with 'QJn'
    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
    En   - max roundoff error for distance computation
    Rn   - randomly perturb computations by a factor of [1-n,1+n]
    Vn   - min distance above plane for a visible facet (default 3C-n or En)
    Un   - max distance below plane for a new, coplanar point (default Vn)
    Wn   - min facet width for outside point (before roundoff, default 2Vn)

Output formats (may be combined; if none, produces a summary to stdout):
    f    - facet dump
    G    - Geomview output (see below)
    i    - vertices incident to each facet
    m    - Mathematica output (2-d and 3-d)
    o    - OFF format (dim, points and facets; Voronoi regions)
    n    - normals with offsets
    p    - vertex coordinates or Voronoi vertices (coplanar points if 'Qc')
    s    - summary (stderr)

More formats:
    Fa   - area for each facet
    FA   - compute total area and volume for option 's'
    Fc   - count plus coplanar points for each facet
           use 'Qc' (default) for coplanar and 'Qi' for interior
    FC   - centrum or Voronoi center for each facet
    Fd   - use cdd format for input (homogeneous with offset first)
    FD   - use cdd format for numeric output (offset first)
    FF   - facet dump without ridges
    Fi   - inner plane for each facet
           for 'v', separating hyperplanes for bounded Voronoi regions
    FI   - ID of each facet
    Fm   - merge count for each facet (511 max)
    FM   - Maple output (2-d and 3-d)
    Fn   - count plus neighboring facets for each facet
    FN   - count plus neighboring facets for each point
    Fo   - outer plane (or max_outside) for each facet
           for 'v', separating hyperplanes for unbounded Voronoi regions
    FO   - options and precision constants
    Fp   - dim, count, and intersection coordinates (halfspace only)
    FP   - nearest vertex and distance for each coplanar point
    FQ   - command used for qhull
    Fs   - summary: #int (8), dimension, #points, tot vertices, tot facets,
                      output: #vertices, #facets, #coplanars, #nonsimplicial
                    #real (2), max outer plane, min vertex
    FS   - sizes:   #int (0)
                    #real (2) tot area, tot volume
    Ft   - triangulation with centrums for non-simplicial facets (OFF format)
    Fv   - count plus vertices for each facet
           for 'v', Voronoi diagram as Voronoi vertices for pairs of sites
    FV   - average of vertices (a feasible point for 'H')
    Fx   - extreme points (in order for 2-d)

Geomview options (2-d, 3-d, and 4-d; 2-d Voronoi)
    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
    Gt   - for 3-d 'd', transparent outer ridges

Print options:
    PAn  - keep n largest facets by area
    Pdk:n - drop facet if normal[k] <= n (default 0.0)
    PDk:n - drop facet if normal[k] >= n
    Pg   - print good facets (needs 'QGn' or 'QVn')
    PFn  - keep facets whose area is at least n
    PG   - print neighbors of good facets
    PMn  - keep n facets 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
rbox

-rbox- generate various point distributions.  Default is random in cube.

args (any order, space separated):                    Version: 2001/06/24
  3000    number of random points in cube, lens, spiral, sphere or grid
  D3      dimension 3-d
  c       add a unit cube to the output ('c G2.0' sets size)
  d       add a unit diamond to the output ('d G2.0' sets size)
  l       generate a regular 3-d spiral
  r       generate a regular polygon, ('r s Z1 G0.1' makes a cone)
  s       generate cospherical points
  x       generate random points in simplex, may use 'r' or 'Wn'
  y       same as 'x', plus simplex
  Pn,m,r  add point [n,m,r] first, pads with 0

  Ln      lens distribution of radius n.  Also 's', 'r', 'G', 'W'.
  Mn,m,r  lattice(Mesh) rotated by [n,-m,0], [m,n,0], [0,0,r], ...
          '27 M1,0,1' is {0,1,2} x {0,1,2} x {0,1,2}.  Try 'M3,4 z'.
  W0.1    random distribution within 0.1 of the cube's or sphere's surface
  Z0.5 s  random points in a 0.5 disk projected to a sphere
  Z0.5 s G0.6 same as Z0.5 within a 0.6 gap

  Bn      bounding box coordinates, default 0.5
  h       output as homogeneous coordinates for cdd
  n       remove command line from the first line of output
  On      offset coordinates by n
  t       use time as the random number seed(default is command line)
  tn      use n as the random number seed
  z       print integer coordinates, default 'Bn' is 1e+006
cat html/qhull.txt html/rbox.txt



qhull(1)                                                 qhull(1)


NAME
       qhull  - convex hull, Delaunay triangulation, Voronoi dia-
       gram, halfspace intersection about a point, hull volume, facet area

SYNOPSIS
       qhull- compute convex hulls and related structures
           input (stdin): dimension, #points, point coordinates
           first comment (non-numeric) is listed in the summary
           halfspace: use dim plus one with offsets after coefficients

       options (qh-quick.htm):
           d      - Delaunay triangulation by lifting points to a paraboloid
           v      - Voronoi diagram via the Delaunay triangulation
           H1,1   - Halfspace intersection about [1,1,0,...]
           d Qu   - Furthest-site Delaunay triangulation (upper convex hull)
           v Qu   - Furthest-site Voronoi diagram
           QJ     - Joggle the input to avoid precision problems
           .      - concise list of all options
           -      - one-line description of all options

       Output options (subset):
           FA     - compute total area and volume
           Fx     - extreme points (convex hull vertices)
           G      - Geomview output (2-d, 3-d and 4-d)
           Fp     - halfspace intersection coordinates
           m      - Mathematica output (2-d and 3-d)
           n      - normals with offsets
           o      - OFF file format (if Voronoi, outputs regions)
           TO file- output results to file, may be enclosed in single quotes
           f      - print all fields of all facets
           s      - summary of results (default)
           Tv     - verify result: structure, convexity, and point inclusion
           p      - vertex coordinates
           i      - vertices incident to each facet

       example:
           rbox 1000 s | qhull Tv s FA

        - html manual:    index.htm
        - installation:   README.txt
        - see also:       COPYING.txt, REGISTER.txt, Changes.txt
        - WWW:            <http://www.qhull.org>
        - GIT:            <git@gitorious.org:qhull/qhull.git>
        - mirror:         <http://www6.uniovi.es/ftp/pub/mirrors/geom.umn.edu/software/ghindex.html>
        - news:           <http://www.qhull.org/news>
        - Geomview:       <http://www.geomview.org>
        - news group:     <news:comp.graphics.algorithms>
        - FAQ:            <http://exaflop.org/docs/cgafaq/cga6.html>
        - email:          qhull@qhull.org
        - bug reports:    qhull_bug@qhull.org




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qhull(1)                                                 qhull(1)


       The sections are:
        - INTRODUCTION
        - DESCRIPTION, a description of Qhull
        - IMPRECISION, how Qhull handles imprecision
        - OPTIONS
        -    Input and output options
        -    Additional input/output formats
        -    Precision options
        -    Geomview options
        -    Print options
        -    Qhull options
        -    Trace options
        - BUGS
        - E-MAIL
        - SEE ALSO
        - AUTHORS
        - ACKNOWLEGEMENTS

       This man page briefly describes all Qhull options.  Please
       report  any  mismatches  with  Qhull's  html  manual  (qh-
       man.htm).



INTRODUCTION
       Qhull  is  a  general  dimension code for computing convex
       hulls, Delaunay triangulations, Voronoi diagram, furthest-
       site  Voronoi  diagram,  furthest-site Delaunay triangula-
       tions, and halfspace  intersections  about  a  point.   It
       implements  the Quickhull algorithm for computing the con-
       vex hull.  Qhull handles round-off  errors  from  floating
       point arithmetic.  It can approximate a convex hull.

       The  program includes options for hull volume, facet area,
       partial hulls, input transformations, randomization, trac-
       ing,  multiple  output  formats, and execution statistics.
       The program can be called from  within  your  application.
       You  can  view  the  results  in  2-d,  3-d  and  4-d with
       Geomview.


DESCRIPTION
       The format of input is the following: first line  contains
       the  dimension,  second  line contains the number of input
       points, and point coordinates follow.  The  dimension  and
       number  of  points  can  be  reversed.   Comments and line
       breaks are ignored.  A comment starts with  a  non-numeric
       character  and  continues  to  the end of line.  The first
       comment is reported in summaries  and  statistics.   Error
       reporting is better if there is one point per line.

       The  default printout option is a short summary. There are
       many other output formats.




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qhull(1)                                                 qhull(1)


       Qhull implements the Quickhull algorithm for convex  hull.
       This  algorithm  combines the 2-d Quickhull algorithm with
       the n-d beneath-beyond algorithm [c.f., Preparata & Shamos
       '85].   It  is  similar  to  the  randomized algorithms of
       Clarkson and others  [Clarkson  et  al.  '93].   The  main
       advantages  of Quickhull are output sensitive performance,
       reduced space requirements, and automatic handling of pre-
       cision problems.

       The data structure produced by Qhull consists of vertices,
       ridges, and facets.  A vertex is a point of the input set.
       A ridge is a set of d vertices and two neighboring facets.
       For example in 3-d, a ridge is an edge of the  polyhedron.
       A facet is a set of ridges, a set of neighboring facets, a
       set of incident vertices, and a hyperplane equation.   For
       simplicial  facets, the ridges are defined by the vertices
       and neighboring facets.  When Qhull merges two facets,  it
       produces  a  non-simplicial facet.  A non-simplicial facet
       has more than d neighbors and  may  share  more  than  one
       ridge with a neighbor.


IMPRECISION
       Since Qhull uses floating point arithmetic, roundoff error
       may occur for each calculation.  This causes  problems for
       most geometric algorithms.

       Qhull  automatically  sets  option  'C-0' in 2-d, 3-d, and
       4-d, or option 'Qx' in 5-d and higher.  These options han-
       dle  precision problems by merging facets.  Alternatively,
       use option 'QJ' to joggle the input.

       With 'C-0', Qhull  merges  non-convex  facets  while  con-
       structing  the hull. The remaining facets are clearly con-
       vex. With 'Qx',  Qhull  merges  coplanar  horizon  facets,
       flipped  facets, concave facets and duplicated ridges.  It
       merges coplanar facets after constructing the hull.   With
       'Qx',  coplanar points may be missed, but it appears to be
       unlikely.

       To guarantee triangular  output,  joggle  the  input  with
       option 'QJ'.  Facet merging will not occur.

OPTIONS
       To  get  a  list  of  the  most important options, execute
       'qhull' by itself.  To get a  complete  list  of  options,
       execute  'qhull  -'.   To  get a complete, concise list of
       options, execute 'qhull .'.

       Options can be in any order.  Capitalized options take  an
       argument  (except  'PG'  and 'F' options).  Single letters
       are used for output formats and precision constants.   The
       other options are grouped into menus for other output for-
       mats ('F'), Geomview output ('G'), printing  ('P'),  Qhull



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qhull(1)                                                 qhull(1)


       control ('Q'), and tracing ('T').

       Main options:

       default
              Compute  the  convex  hull  of  the  input  points.
              Report a summary of the result.

       d      Compute the Delaunay triangulation by  lifting  the
              input  points  to  a  paraboloid.   The  'o' option
              prints the  input  points  and  facets.   The  'QJ'
              option  guarantees  triangular  output.   The  'Ft'
              option prints a triangulation.  It adds points (the
              centrums) to non-simplicial facets.

       v      Compute  the Voronoi diagram from the Delaunay tri-
              angulation.  The 'p' option prints the Voronoi ver-
              tices.   The 'o' option prints the Voronoi vertices
              and the vertices in each Voronoi region.  It  lists
              regions  in  site id order.  The 'Fv' option prints
              each ridge of the Voronoi diagram.   The  first  or
              zero'th  vertex indicates the infinity vertex.  Its
              coordinates are qh_INFINITE  (-10.101).   It  indi-
              cates   unbounded  Voronoi  regions  or  degenerate
              Delaunay triangles.

       Hn,n,...
              Compute halfspace intersection  about  [n,n,0,...].
              The  input  is  a  set of halfspaces defined in the
              same format as 'n', 'Fo', and 'Fi'.   Use  'Fp'  to
              print  the  intersection  points.  Use 'Fv' to list
              the intersection points for  each  halfspace.   The
              other  output formats display the dual convex hull.

              The point [n,n,n,...] is a feasible point  for  the
              halfspaces, i.e., a point that is inside all of the
              halfspaces (Hx+b <=  0).   The  default  coordinate
              value is 0.

              The  input may start with a feasible point.  If so,
              use 'H' by itself.  The input starts with a  feasi-
              ble  point  when the first number is the dimension,
              the second number is "1", and the coordinates  com-
              plete  a line.  The 'FV' option produces a feasible
              point for a convex hull.

       d Qu   Compute the  furthest-site  Delaunay  triangulation
              from  the upper convex hull.  The 'o' option prints
              the input points and facets.  The 'QJ' option guar-
              antees triangular otuput.  You can also use facets.

       v Qu   Compute the furthest-site Voronoi diagram.  The 'p'
              option prints the Voronoi vertices.  The 'o' option
              prints the Voronoi vertices  and  the  vertices  in



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qhull(1)                                                 qhull(1)


              each  Voronoi  region.  The 'Fv' option prints each
              ridge of the Voronoi diagram.  The first or zero'th
              vertex  indicates  the infinity vertex at infinity.
              Its  coordinates  are  qh_INFINITE  (-10.101).   It
              indicates  unbounded Voronoi regions and degenerate
              Delaunay triangles.

       Qt     Triangulated output.


       Input/Output options:

       f      Print out all facets and all fields of each  facet.

       G      Output  the hull in Geomview format.  For imprecise
              hulls, Geomview displays the inner and outer  hull.
              Geomview can also display points, ridges, vertices,
              coplanar  points,  and  facet  intersections.   See
              below for a list of options.

              For  Delaunay triangulations, 'G' displays the cor-
              responding paraboloid.  For halfspace intersection,
              'G' displays the dual polytope.

       i      Output the incident vertices for each facet.  Qhull
              prints the number of facets followed  by  the  ver-
              tices  of  each  facet.   One  facet is printed per
              line.  The numbers are the  0-relative  indices  of
              the  corresponding  input  points.   The facets are
              oriented.

              In 4-d and higher, Qhull  triangulates  non-simpli-
              cial  facets.   Each  apex  (the first vertex) is a
              created point that corresponds to the facet's  cen-
              trum.  Its index is greater than the indices of the
              input points.  Each base corresponds to  a  simpli-
              cial  ridge  between two facets.  To print the ver-
              tices without triangulation, use option 'Fv'.

       m      Output  the  hull  in  Mathematica  format.   Qhull
              writes  a  Mathematica  file for 2-d and 3-d convex
              hulls and for 2-d Delaunay triangulations.    Qhull
              produces a list of objects that you can assign to a
              variable in Mathematica,  for  example:  "list=  <<
              <outputfilename> ". If the object is 2-d, it can be
              visualized  by  "Show[Graphics[list]]  ".  For  3-d
              objects the command is "Show[Graphics3D[list]]".

       n      Output  the  normal equation for each facet.  Qhull
              prints the dimension  (plus  one),  the  number  of
              facets,  and  the  normals  for  each  facet.   The
              facet's offset follows its normal coefficients.

       o      Output the facets in OFF file format.  Qhull prints
              the  dimension, number of points, number of facets,
              and  number  of  ridges.   Then   it   prints   the



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qhull(1)                                                 qhull(1)


              coordinates  of  the  input points and the vertices
              for each facet.  Each facet is on a separate  line.
              The  first  number  is the number of vertices.  The
              remainder are  the  indices  of  the  corresponding
              points.  The vertices are oriented in 2-d, 3-d, and
              in simplicial facets.

              For 2-d Voronoi diagrams, the vertices  are  sorted
              by adjacency, but not oriented.  In 3-d and higher,
              the Voronoi vertices are sorted by index.  See  the
              'v' option for more information.

       p      Output the coordinates of each vertex point.  Qhull
              prints the dimension, the number of points, and the
              coordinates  for  each  vertex.   With the 'Gc' and
              'Gi' options, it also prints coplanar and  interior
              points.   For Voronoi diagrams, it prints the coor-
              dinates of each Voronoi vertex.

       s      Print a summary to stderr.  If  no  output  options
              are  specified  at  all,  a summary goes to stdout.
              The summary lists the number of input  points,  the
              dimension,  the  number  of  vertices in the convex
              hull, the number of facets in the convex hull,  the
              number of good facets (if 'Pg'), and statistics.

              The  last  two  statistics  (if needed) measure the
              maximum distance from a point or vertex to a facet.
              The number in parenthesis (e.g., 2.1x) is the ratio
              between the maximum  distance  and  the  worst-case
              distance due to merging two simplicial facets.


       Precision options

       An     Maximum  angle  given  as  a  cosine.  If the angle
              between a pair of facet normals is greater than n, Qhull
              merges  one  of the facets into a neighbor.  If 'n'
              is negative, Qhull tests angles after  adding  each
              point  to  the hull (pre-merging).  If 'n' is posi-
              tive, Qhull tests  angles  after  constructing  the
              hull  (post-merging).   Both  pre- and post-merging
              can be defined.

              Option 'C0' or 'C-0' is set  if  the  corresponding
              'Cn' or 'C-n' is not set.  If 'Qx' is set, then 'A-
              n' and 'C-n' are checked after  the  hull  is  con-
              structed and before 'An' and 'Cn' are checked.

       Cn     Centrum  radius.  If a centrum is less than n below
              a  neighboring  facet,  Qhull  merges  one  of  the
              facets.   If  'n'  is negative or '-0', Qhull tests
              and merges facets after adding each  point  to  the
              hull.   This  is  called  "pre-merging".  If 'n' is



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qhull(1)                                                 qhull(1)


              positive, Qhull  tests  for  convexity  after  con-
              structing the hull ("post-merging").  Both pre- and
              post-merging can be defined.

              For 5-d and higher, 'Qx' should be used instead  of
              'C-n'.  Otherwise, most or all facets may be merged
              together.

       En     Maximum roundoff error for distance computations.

       Rn     Randomly perturb distance computations up to +/-  n
              *  max_coord.  This option perturbs every distance,
              hyperplane, and angle computation.  To use time  as
              the random number seed, use option 'QR-1'.

       Vn     Minimum  distance  for  a  facet  to be visible.  A
              facet is visible if the distance from the point  to
              the facet is greater than 'Vn'.

              Without  merging, the default value for 'Vn' is the
              round-off error ('En').  With merging, the  default
              value  is  the  pre-merge centrum ('C-n') in 2-d or
              3--d, or three times that in other dimensions.   If
              the outside width is specified ('Wn'), the maximum,
              default value for 'Vn' is 'Wn'.

       Un     Maximum distance below a facet for a  point  to  be
              coplanar  to the facet.  The default value is 'Vn'.

       Wn     Minimum outside width  of  the  hull.   Points  are
              added  to  the convex hull only if they are clearly
              outside of a facet.  A point is outside of a  facet
              if  its distance to the facet is greater than 'Wn'.
              The normal value for 'Wn' is  'En'.   If  the  user
              specifies  pre-merging  and does not set 'Wn', than
              'Wn'  is  set  to  the  premerge  'Cn'  and  maxco-
              ord*(1-An).


       Additional input/output formats

       Fa     Print area for each facet.  For Delaunay triangula-
              tions, the area is the area of the  triangle.   For
              Voronoi  diagrams, the area is the area of the dual
              facet.   Use  'PAn'  for  printing  the  n  largest
              facets, and option 'PFn' for printing facets larger
              than 'n'.

              The area for non-simplicial facets is  the  sum  of
              the areas for each ridge to the centrum.   Vertices
              far below the facet's hyperplane are ignored.   The
              reported  area  may  be significantly less than the
              actual area.




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qhull(1)                                                 qhull(1)


       FA     Compute the total area and volume for  option  's'.
              It  is  an  approximation for non-simplicial facets
              (see 'Fa').

       Fc     Print coplanar points for each facet.   The  output
              starts  with the number of facets.  Then each facet
              is printed one per line.  Each line is  the  number
              of  coplanar  points  followed  by  the  point ids.
              Option 'Qi' includes  the  interior  points.   Each
              coplanar  point (interior point) is assigned to the
              facet it is furthest above (resp., least below).

       FC     Print centrums for each facet.  The  output  starts
              with  the  dimension  followed  by  the  number  of
              facets.  Then each facet centrum  is  printed,  one
              per line.

       Fd     Read  input  in cdd format with homogeneous points.
              The input starts with comments.  The first  comment
              is  reported  in  the summary.  Data starts after a
              "begin" line.  The  next  line  is  the  number  of
              points  followed  by  the dimension+1 and "real" or
              "integer".  Then the  points  are  listed   with  a
              leading  "1" or "1.0".  The data ends with an "end"
              line.

              For halfspaces ('Fd Hn,n,...'), the input format is
              the  same.   Each halfspace starts with its offset.
              The sign of the offset is the opposite  of  Qhull's
              convention.

       FD     Print  normals ('n', 'Fo', 'Fi') or points ('p') in
              cdd format.  The first line  is  the  command  line
              that  invoked  Qhull.   Data  starts with a "begin"
              line.  The next line is the number  of  normals  or
              points  followed  by  the  dimension+1  and "real".
              Then the normals or points  are  listed   with  the
              offset  before  the  coefficients.   The offset for
              points is 1.0.  The  offset  for  normals  has  the
              opposite sign.  The data ends with an "end" line.

       FF     Print  facets  (as  in  'f')  without  printing the
              ridges.

       Fi     Print inner planes for each facet.  The inner plane
              is below all vertices.

       Fi     Print  separating  hyperplanes  for  bounded, inner
              regions of the Voronoi diagram.  The first line  is
              the  number  of  ridges.   Then  each hyperplane is
              printed, one per line.  A line starts with the num-
              ber  of  indices  and floats.  The first pair lists
              adjacent input sites, the next  d  floats  are  the
              normalized coefficients for the hyperplane, and the



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qhull(1)                                                 qhull(1)


              last float is the offset.  The hyperplane  is  ori-
              ented  toward  verify that the hyperplanes are per-
              pendicular  bisectors.   Use  'Fo'  for   unbounded
              regions,  and  'Fv'  for  the corresponding Voronoi
              vertices.

       FI     Print facet identifiers.

       Fm     Print number of merges for each facet.  At most 511
              merges  are  reported  for  a facet.  See 'PMn' for
              printing the facets with the most merges.

       FM     Output  the  hull  in  Maple format.  See 'm'

       Fn     Print neighbors for each facet.  The output  starts
              with  the  number  of  facets.   Then each facet is
              printed one per line.  Each line is the  number  of
              neighbors  followed  by an index for each neighbor.
              The indices match the other facet output formats.

              A negative index indicates an unprinted  facet  due
              to  printing  only  good  facets ('Pg').  It is the
              negation of the  facet's  id  (option  'FI').   For
              example,  negative  indices are used for facets "at
              infinity" in the Delaunay triangulation.

       FN     Print vertex neighbors or coplanar facet  for  each
              point.   The  first  line  is the number of points.
              Then each point is printed, one per line.   If  the
              point  is coplanar, the line is "1" followed by the
              facet's id.  If the point is not a selected vertex,
              the  line is "0".  Otherwise, each line is the num-
              ber of  neighbors  followed  by  the  corresponding
              facet indices (see 'Fn').

       Fo     Print  outer planes for each facet in the same for-
              mat as 'n'.  The outer plane is above all points.

       Fo     Print separating hyperplanes for  unbounded,  outer
              regions  of the Voronoi diagram.  The first line is
              the number of  ridges.   Then  each  hyperplane  is
              printed, one per line.  A line starts with the num-
              ber of indices and floats.  The  first  pair  lists
              adjacent  input  sites,  the  next d floats are the
              normalized coefficients for the hyperplane, and the
              last  float  is the offset.  The hyperplane is ori-
              ented toward verify that the hyperplanes  are  per-
              pendicular   bisectors.    Use   'Fi'  for  bounded
              regions, and 'Fv'  for  the  corresponding  Voronoi
              vertices.

       FO     List  all  options to stderr, including the default
              values.  Additional 'FO's are printed to stdout.

       Fp     Print points for  halfspace  intersections  (option
              'Hn,n,...').   Each  intersection  corresponds to a



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qhull(1)                                                 qhull(1)


              facet of the dual polytope.  The  "infinity"  point
              [-10.101,-10.101,...]    indicates   an   unbounded
              intersection.

       FP     For each coplanar point ('Qc') print the  point  id
              of  the nearest vertex, the point id, the facet id,
              and the distance.

       FQ     Print command used for qhull and input.

       Fs     Print a summary.  The first line  consists  of  the
              number  of  integers  ("7"), followed by the dimen-
              sion, the number of points, the number of vertices,
              the  number  of  facets,  the  number  of  vertices
              selected for output, the number of facets  selected
              for  output, the number of coplanar points selected
              for output.

              The second line consists of  the  number  of  reals
              ("2"),  followed by the maxmimum offset to an outer
              plane and and minimum offset  to  an  inner  plane.
              Roundoff  is included.  Later versions of Qhull may
              produce additional integers or reals.

       FS     Print the size of the hull.  The  first  line  con-
              sists  of the number of integers ("0").  The second
              line consists of the number of  reals  ("2"),  fol-
              lowed  by  the total facet area, and the total vol-
              ume.  Later versions of  Qhull  may  produce  addi-
              tional integers or reals.

              The  total volume measures the volume of the inter-
              section of the halfspaces defined  by  each  facet.
              Both  area  and  volume are approximations for non-
              simplicial facets.  See option 'Fa'.

       Ft     Print a triangulation with added  points  for  non-
              simplicial facets.  The first line is the dimension
              and the second line is the number of points and the
              number of facets.  The points follow, one per line,
              then the facets follow as a list of point  indices.
              With option points include the point-at-infinity.

       Fv     Print  vertices  for each facet.  The first line is
              the number of facets.  Then each facet is  printed,
              one  per line.  Each line is the number of vertices
              followed by the corresponding point ids.   Vertices
              are listed in the order they were added to the hull
              (the last one is first).

       Fv     Print all ridges of a Voronoi diagram.   The  first
              line  is  the number of ridges.  Then each ridge is
              printed, one per line.  A line starts with the num-
              ber  of  indices.   The  first  pair lists adjacent



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qhull(1)                                                 qhull(1)


              input sites, the  remaining  indices  list  Voronoi
              vertices.   Vertex  '0'  indicates  the  vertex-at-
              infinity (i.e., an unbounded  ray).   In  3-d,  the
              vertices  are  listed  in order.  See 'Fi' and 'Fo'
              for separating hyperplanes.

       FV     Print average vertex.  The average vertex is a fea-
              sible point for halfspace intersection.

       Fx     List  extreme points (vertices) of the convex hull.
              The first line is the number of points.  The  other
              lines give the indices of the corresponding points.
              The first point is '0'.  In 2-d, the  points  occur
              in counter-clockwise order; otherwise they occur in
              input order.   For  Delaunay  triangulations,  'Fx'
              lists  the  extreme points of the input sites.  The
              points are unordered.


       Geomview options

       G      Produce a file for viewing with Geomview.   Without
              other  options,  Qhull displays edges in 2-d, outer
              planes in 3-d, and ridges in 4-d.  A ridge  can  be
              explicit or implicit.  An explicit ridge is a dim-1
              dimensional simplex between two  facets.   In  4-d,
              the explicit ridges are triangles.  When displaying
              a ridge in 4-d, Qhull projects the ridge's vertices
              to  one  of  its  facets' hyperplanes.  Use 'Gh' to
              project ridges to the intersection of  both  hyper-
              planes.

       Ga     Display all input points as dots.

       Gc     Display  the  centrum  for  each facet in 3-d.  The
              centrum is defined by a green radius sitting  on  a
              blue  plane.   The plane corresponds to the facet's
              hyperplane.  The radius  is  defined  by  'C-n'  or
              'Cn'.

       GDn    Drop  dimension  n  in 3-d or 4-d.  The result is a
              2-d or 3-d object.

       Gh     Display hyperplane intersections in  3-d  and  4-d.
              In  3-d, the intersection is a black line.  It lies
              on two  neighboring  hyperplanes  (c.f.,  the  blue
              squares  associated with centrums ('Gc')).  In 4-d,
              the ridges are projected  to  the  intersection  of
              both hyperplanes.

       Gi     Display  inner  planes  in  2-d and 3-d.  The inner
              plane of a facet is below all of its vertices.   It
              is  parallel  to the facet's hyperplane.  The inner
              plane's color is the opposite (1-r,1-g,1-b) of  the



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qhull(1)                                                 qhull(1)


              outer  plane.  Its edges are determined by the ver-
              tices.

       Gn     Do not display inner or outer planes.  By  default,
              Geomview displays the precise plane (no merging) or
              both inner  and  output  planes  (merging).   Under
              merging,  Geomview does not display the inner plane
              if the the difference between inner  and  outer  is
              too small.

       Go     Display  outer  planes  in  2-d and 3-d.  The outer
              plane of a facet is above all input points.  It  is
              parallel  to  the facet's hyperplane.  Its color is
              determined by the facet's normal, and its edges are
              determined by the vertices.

       Gp     Display  coplanar  points and vertices as radii.  A
              radius defines a  ball  which  corresponds  to  the
              imprecision  of  the point.  The imprecision is the
              maximum of the roundoff error, the centrum  radius,
              and  maxcoord  * (1-An).  It is at least 1/20'th of
              the maximum coordinate, and ignores post-merging if
              pre-merging is done.

       Gr     Display  ridges  in  3-d.  A ridge connects the two
              vertices that are  shared  by  neighboring  facets.
              Ridges are always displayed in 4-d.

       Gt     A  3-d  Delaunay  triangulation looks like a convex
              hull with interior facets.  Option 'Gt' removes the
              outside  ridges to reveal the outermost facets.  It
              automatically sets options 'Gr' and 'GDn'.

       Gv     Display vertices as spheres.   The  radius  of  the
              sphere  corresponds to the imprecision of the data.
              See 'Gp' for determining the radius.


       Print options

       PAn    Only the n  largest  facets  are  marked  good  for
              printing.   Unless  'PG'  is set, 'Pg' is automati-
              cally set.

       Pdk:n  Drop facet from output  if  normal[k]  <=  n.   The
              option 'Pdk' uses the default value of 0 for n.

       PDk:n  Drop  facet  from  output  if  normal[k] >= n.  The
              option 'PDk' uses the default value of 0 for n.

       PFn    Only facets with area at least 'n' are marked  good
              for printing.  Unless 'PG' is set, 'Pg' is automat-
              ically set.




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qhull(1)                                                 qhull(1)


       Pg     Print only good facets.  A  good  facet  is  either
              visible from a point (the 'QGn' option) or includes
              a point (the 'QVn'  option).   It  also  meets  the
              requirements  of  'Pdk'  and 'PDk' options.  Option
              'Pg' is automatically set  for  options  'PAn'  and
              'PFn'.

       PG     Print neighbors of good facets.

       PMn    Only  the  n facets with the most merges are marked
              good for printing.  Unless 'PG'  is  set,  'Pg'  is
              automatically set.

       Po     Force output despite precision problems.  Verify  ('Tv')  does    not check
              coplanar points.  Flipped facets are  reported  and
              concave  facets  are  counted.   If  'Po'  is used,
              points are not partitioned into flipped facets  and
              a  flipped  facet  is  always  visible  to a point.
              Also, if an error occurs before the  completion  of
              Qhull  and  tracing  is  not active, 'Po' outputs a
              neighborhood of the erroneous facets (if any).

       Pp     Do not report precision problems.


       Qhull control options

       Qbk:0Bk:0
              Drop dimension  k  from  the  input  points.   This
              allows  the user to take convex hulls of sub-dimen-
              sional objects.  It happens before the Delaunay and
              Voronoi transformation.

       QbB    Scale the input points to fit the unit cube.  After
              scaling, the lower bound will be -0.5 and the upper
              bound  +0.5  in  all  dimensions.  For Delaunay and
              Voronoi diagrams, scaling happens after  projection
              to the paraboloid.  Under precise arithmetic, scal-
              ing does not change  the  topology  of  the  convex
              hull.

       Qbb    Scale  the last coordinate to [0, m] where m is the
              maximum absolute value of  the  other  coordinates.
              For  Delaunay and Voronoi diagrams, scaling happens
              after projection to  the  paraboloid.   It  reduces
              roundoff error for inputs with integer coordinates.
              Under precise arithmetic, scaling does  not  change
              the topology of the convex hull.

       Qbk:n  Scale  the  k'th  coordinate  of  the input points.
              After scaling, the lower bound of the input  points
              will be n.  'Qbk' scales to -0.5.



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qhull(1)                                                 qhull(1)


       QBk:n  Scale  the  k'th  coordinate  of  the input points.
              After scaling, the upper bound will  be  n.   'QBk'
              scales to +0.5.

       Qc     Keep  coplanar points with the nearest facet.  Out-
              put formats 'p', 'f', 'Gp', 'Fc',  'FN',  and  'FP'
              will print the points.

       Qf     Partition points to the furthest outside facet.

       Qg     Only  build  good  facets.   With  the 'Qg' option,
              Qhull will only build those facets that it needs to
              determine  the  good  facets  in  the  output.  See
              'QGn', 'QVn', and 'PdD' for defining  good  facets,
              and  'Pg'  and  'PG'  for  printing good facets and
              their neighbors.

       QGn    A facet is good (see 'Qg' and 'Pg') if it is  visi-
              ble  from point n.  If n < 0, a facet is good if it
              is not visible from point n.  Point n is not  added
              to  the  hull  (unless 'TCn' or 'TPn').  With rbox,
              use the 'Pn,m,r' option to define  your  point;  it
              will be point 0 (QG0).

       Qi     Keep  interior points with the nearest facet.  Out-
              put formats 'p', 'f', 'Gp', 'FN',  'FP',  and  'Fc'
              will print the points.

       QJn    Joggle  each  input  coordinate  by adding a random
              number in [-n,n].  If  a  precision  error  occurs,
              then  qhull  increases  n and tries again.  It does
              not increase n beyond a certain value, and it stops
              after  a  certain  number of attempts [see user.h].
              Option 'QJ' selects a default  value  for  n.   The
              output will be simplicial.  For Delaunay triangula-
              tions, 'QJn' sets 'Qbb' to scale the  last  coordi-
              nate (not if 'Qbk:n' or 'QBk:n' is set).  'QJn' is
              deprecated for Voronoi diagrams.  See also 'Qt'.

       Qm     Only  process  points that would otherwise increase
              max_outside.  Other points are treated as  coplanar
              or interior points.

       Qr     Process  random  outside points instead of furthest
              ones.  This makes Qhull equivalent to  the  random-
              ized  incremental  algorithms.   CPU  time  is  not
              reported since the randomization is inefficient.

       QRn    Randomly rotate the input points.  If n=0, use time
              as  the  random  number seed.  If n>0, use n as the
              random number seed.  If n=-1, don't rotate but  use
              time  as the random number seed.  For Delaunay tri-
              angulations ('d' and 'v'), rotate  about  the  last
              axis.




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qhull(1)                                                 qhull(1)


       Qs     Search all points for the initial simplex.

       Qt     Triangulated output.  Triangulate non-simplicial
              facets. 'Qt' is deprecated for Voronoi diagrams.
              See also 'QJn'

       Qv     Test  vertex  neighbors  for  convexity after post-
              merging.  To use the 'Qv' option, you also need  to
              set a merge option (e.g., 'Qx' or 'C-0').

       QVn    A  good facet (see 'Qg' and 'Pg') includes point n.
              If n<0, then a good facet does not include point n.
              The point is either in the initial simplex or it is
              the first point added to the  hull.   Option  'QVn'
              may not be used with merging.

       Qx     Perform  exact merges while building the hull.  The
              "exact" merges are merging a point into a  coplanar
              facet  (defined  by 'Vn', 'Un', and 'C-n'), merging
              concave facets, merging duplicate ridges, and merg-
              ing  flipped  facets.   Coplanar  merges  and angle
              coplanar merges ('A-n') are not performed.  Concav-
              ity testing is delayed until a merge occurs.

              After  the  hull  is built, all coplanar merges are
              performed (defined by 'C-n' and 'A-n'), then  post-
              merges are performed (defined by 'Cn' and 'An').

       Qz     Add  a  point  "at  infinity"  that  is  above  the
              paraboloid for Delaunay triangulations and  Voronoi
              diagrams.   This  reduces  precision  problems  and
              allows the triangulation of cospherical points.


       Qhull experiments and speedups

       Q0     Turn off pre-merging as  a  default  option.   With
              'Q0'/'Qx'  and  without explicit pre-merge options,
              Qhull ignores precision issues  while  constructing
              the  convex  hull.   This  may  lead  to  precision
              errors.  If so, a descriptive warning is generated.

       Q1     With  'Q1',  Qhull  sorts merges by type (coplanar,
              angle coplanar, concave) instead of by angle.

       Q2     With 'Q2', Qhull merges all facets at once  instead
              of  using  independent  sets  of  merges  and  then
              retesting.

       Q3     With 'Q3', Qhull does  not  remove  redundant  ver-
              tices.

       Q4     With 'Q4', Qhull avoids merges of an old facet into
              a new facet.

       Q5     With 'Q5', Qhull does not correct outer  planes  at
              the  end.  The maximum outer plane is used instead.




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qhull(1)                                                 qhull(1)


       Q6     With 'Q6', Qhull  does  not  pre-merge  concave  or
              coplanar facets.

       Q7     With  'Q7',  Qhull  processes facets in depth-first
              order instead of breadth-first order.

       Q8     With 'Q8' and merging, Qhull does not retain  near-
              interior  points  for adjusting outer planes.  'Qc'
              will probably retain all points that  adjust  outer
              planes.

       Q9     With 'Q9', Qhull processes the furthest of all out-
              side sets at each iteration.

       Q10    With 'Q10', Qhull does not use special processing
              for narrow distributions.

       Q11    With 'Q11', Qhull copies normals and recomputes
              centrums for tricoplanar facets.

       Trace options

       Tn     Trace at level n.  Qhull  includes  full  execution
              tracing.   'T-1'  traces  events.   'T1' traces the
              overall execution of the program.   'T2'  and  'T3'
              trace overall execution and geometric and topologi-
              cal  events.   'T4'  traces  the  algorithm.   'T5'
              includes  information  about  memory allocation and
              Gaussian elimination.

       Ta     Annotate output with codes that identify the
              corresponding qh_fprintf() statement.

       Tc     Check frequently during execution.  This will catch
              most inconsistency errors.

       TCn    Stop  Qhull  after  building the cone of new facets
              for point n.  The output for 'f' includes the  cone
              and the old hull.  See also 'TVn'.

       TFn    Report  progress  whenever  more  than n facets are
              created During post-merging, 'TFn' reports progress
              after more than n/2 merges.

       TI file
              Input data from 'file'.  The filename may not include
                  spaces or quotes.

       TO file
              Output results to 'file'.  The name may be enclosed
              in single quotes.

       TPn    Turn on tracing when point n is added to the  hull.
          Trace partitions of point n. If used with TWn, turn off
          tracing after adding point n to the hull.

       TRn    Rerun  qhull  n  times.  Usually used with 'QJn' to
              determine the probability that a given joggle  will
              fail.

       Ts     Collect  statistics  and print to stderr at the end
              of execution.

       Tv     Verify the convex hull.  This checks the  topologi-
              cal  structure,  facet  convexity, and point inclu-
              sion.  If precision problems occurred,  facet  con-
              vexity  is  tested whether or not 'Tv' is selected.
              Option 'Tv'  does  not  check  point  inclusion  if



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qhull(1)                                                 qhull(1)


              forcing output with 'Po', or if 'Q5' is set.

              For  point  inclusion  testing, Qhull verifies that
              all points are below all outer planes  (facet->max-
              outside).  Point inclusion is exhaustive if merging
              or if the facet-point product is small enough; oth-
              erwise  Qhull  verifies  each point with a directed
              search (qh_findbest).

              Point inclusion testing occurs after producing out-
              put.   It  prints a message to stderr unless option
              'Pp' is used.  This allows the  user  to  interrupt
              Qhull without changing the output.

       TVn    Stop  Qhull  after  adding point n.  If n < 0, stop
              Qhull before adding point n.  Output shows the hull
              at this time.  See also 'TCn'

       TMn    Turn on tracing at n'th merge.

       TWn    Trace  merge  facets when the width is greater than
              n.

       Tz     Redirect stderr to stdout.


BUGS
       Please    report    bugs     to     Brad     Barber     at
       qhull_bug@qhull.org.

       If Qhull does not compile, it is due to an incompatibility
       between your system and ours.  The first thing to check is
       that  your compiler is ANSI standard.  If it is, check the
       man page for the best options, or  find  someone  to  help
       you.  If you locate the cause of your problem, please send
       email since it might help others.

       If Qhull compiles but crashes on the test case (rbox  D4),
       there's  still  incompatibility  between  your  system and
       ours.  Typically it's been due to mem.c and memory  align-
       ment.   You  can  use qh_NOmem in mem.h to turn off memory
       management.  Please let us know if you figure out  how  to
       fix these problems.

       If  you  do  find  a  problem,  try  to simplify it before
       reporting the error.  Try different size inputs to  locate
       the  smallest one that causes an error.  You're welcome to
       hunt through the code  using  the  execution  trace  as  a
       guide.   This  is  especially true if you're incorporating
       Qhull into your own program.

       When you do report an error, please attach a data  set  to
       the  end of your message.  This allows us to see the error
       for ourselves.  Qhull is maintained part-time.



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qhull(1)                                                 qhull(1)


E-MAIL
       Please  send  correspondence  to  qhull@qhull.org   and
       report  bugs  to  qhull_bug@qhull.org.  Let us know how
       you use Qhull.  If you mention it in a paper, please  send
       the reference and an abstract.

       If  you would like to get Qhull announcements (e.g., a new
       version) and news (any bugs that get fixed, etc.), let  us
       know  and  we  will  add  you to our mailing list.  If you
       would like to communicate with other Qhull users, we  will
       add you to the qhull_users alias.  For Internet news about
       geometric algorithms and convex hulls, look at comp.graph-
       ics.algorithms and sci.math.num-analysis


SEE ALSO
       rbox(1)

       Barber,  C.  B.,  D.P.  Dobkin,  and  H.T. Huhdanpaa, "The
       Quickhull Algorithm for Convex Hulls," ACM Trans. on Math-
       ematical Software, 22(4):469-483, Dec. 1996.
       http://portal.acm.org/citation.cfm?doid=235815.235821
       http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.117.405


       Clarkson, K.L., K. Mehlhorn, and R. Seidel, "Four  results
       on  randomized  incremental  construction,"  Computational
       Geometry: Theory and Applications,  vol.  3,  p.  185-211,
       1993.

       Preparata,  F.  and  M.  Shamos,  Computational  Geometry,
       Springer-Verlag, New York, 1985.



AUTHORS
         C. Bradford Barber                    Hannu Huhdanpaa
         bradb@shore.net                       hannu@qhull.org



ACKNOWLEDGEMENTS
       A special thanks to Albert Marden, Victor Milenkovic,  the
       Geometry Center, Harvard University, and Endocardial Solu-
       tions, Inc. for supporting this work.

       Qhull 1.0 and 2.0 were developed under National Science Foundation
       grants NSF/DMS-8920161 and NSF-CCR-91-15793 750-7504.  David Dobkin



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qhull(1)                                                 qhull(1)


       guided the original work at Princeton University.  If you find it
       useful, please let us know.

       The Geometry Center was supported by grant DMS-8920161 from the National
       Science Foundation, by grant DOE/DE-FG02-92ER25137 from the Department
       of Energy, by the University of Minnesota, and by Minnesota Technology, Inc.

       Qhull is available from http://www.qhull.org













































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rbox(1)                                                   rbox(1)


NAME
       rbox - generate point distributions for qhull

SYNOPSIS
       Command "rbox" (w/o arguments) lists the options.

DESCRIPTION
       rbox  generates random or regular points according to the
       options given, and  outputs  the  points  to  stdout.  The
       points  are  generated in a cube, unless 's' or given. The
       format of the output is the following: first line contains
       the dimension and a comment, second line contains the num-
       ber of points, and the following lines contain the points,
       one  point per line. Points are represented by their coor-
       dinate values.

EXAMPLES
       rbox 10
              10 random points in the unit cube centered  at  the
              origin.

       rbox 10 s D2
              10 random points on a 2-d circle.

       rbox 100 W0
              100 random points on the surface of a cube.

       rbox 1000 s D4
              1000 random points on a 4-d sphere.

       rbox c D5 O0.5
              a 5-d hypercube with one corner at the origin.

       rbox d D10
              a 10-d diamond.

       rbox x 1000 r W0
              100 random points on the surface of a fixed simplex

       rbox y D12
              a 12-d simplex.

       rbox l 10
              10 random points along a spiral

       rbox l 10 r
              10 regular points  along  a  spiral  plus  two  end
              points

       rbox 1000 L10000 D4 s
              1000 random points on the surface of a narrow lens.

       rbox c G2 d G3
              a cube with coordinates +2/-2 and  a  diamond  with



Geometry Center          August 10, 1998                        1





rbox(1)                                                   rbox(1)


              coordinates +3/-3.

       rbox 64 M3,4 z
              a  rotated,  {0,1,2,3} x {0,1,2,3} x {0,1,2,3} lat-
              tice (Mesh) of integer points.

       rbox P0 P0 P0 P0 P0
              5 copies of the origin in 3-d.  Try 'rbox P0 P0  P0
              P0 P0 | qhull QJ'.

       r 100 s Z1 G0.1
              two  cospherical  100-gons plus another cospherical
              point.

       100 s Z1
              a cone of points.

       100 s Z1e-7
              a narrow cone of points with many precision errors.

OPTIONS
       n      number of points

       Dn     dimension n-d (default 3-d)

       Bn     bounding box coordinates (default 0.5)

       l      spiral distribution, available only in 3-d

       Ln     lens  distribution  of  radius n.  May be used with
              's', 'r', 'G', and 'W'.

       Mn,m,r lattice  (Mesh)  rotated  by  {[n,-m,0],   [m,n,0],
              [0,0,r],  ...}.   Use  'Mm,n'  for a rigid rotation
              with r = sqrt(n^2+m^2).  'M1,0'  is  an  orthogonal
              lattice.   For  example,  '27  M1,0'  is  {0,1,2} x
              {0,1,2} x {0,1,2}.

       s      cospherical points randomly generated in a cube and
              projected to the unit sphere

       x      simplicial  distribution.   It  is fixed for option
              'r'.  May be used with 'W'.

       y      simplicial distribution plus a simplex.   Both  'x'
              and 'y' generate the same points.

       Wn     restrict  points  to distance n of the surface of a
              sphere or a cube

       c      add a unit cube to the output

       c Gm   add a cube with all combinations of +m  and  -m  to
              the output



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rbox(1)                                                   rbox(1)


       d      add a unit diamond to the output.

       d Gm   add a diamond made of 0, +m and -m to the output

       Pn,m,r add point [n,m,r] to the output first.  Pad coordi-
              nates with 0.0.

       n      Remove the command line from the first line of out-
              put.

       On     offset the data by adding n to each coordinate.

       t      use  time  in  seconds  as  the  random number seed
              (default is command line).

       tn     set the random number seed to n.

       z      generate integer coordinates.  Use 'Bn'  to  change
              the  range.   The  default  is 'B1e6' for six-digit
              coordinates.  In R^4, seven-digit coordinates  will
              overflow hyperplane normalization.

       Zn s   restrict points to a disk about the z+ axis and the
              sphere (default Z1.0).  Includes the opposite pole.
              'Z1e-6'  generates  degenerate  points under single
              precision.

       Zn Gm s
              same as Zn with an empty center (default G0.5).

       r s D2 generate a regular polygon

       r s Z1 G0.1
              generate a regular cone

BUGS
       Some combinations of arguments generate odd results.

       Report bugs to  qhull_bug@qhull.org,     other  correspon-
       dence to qhull@qhull.org

SEE ALSO
       qhull(1)

AUTHOR
       C. Bradford Barber
       bradb@shore.net
       






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# end of q_test
