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# qhull -- convex hull and related structures

The convex hull of a set of points is the smallest convex set containing the points. The Delaunay triangulation and furthest-site Delaunay triangulation are equivalent to a convex hull in one higher dimension. Halfspace intersection about a point is equivalent to a convex hull by polar duality.

The qhull program provides options to build these structures and to experiment with the process. Use the qconvex, qdelaunay, qhalf, and qvoronoi programs to build specific structures. You may use qhull instead. It takes the same options and uses the same code.

Example: rbox 1000 D3 | qhull C-1e-4 FO Ts
Compute the 3-d convex hull of 1000 random points. Centrums must be 10^-4 below neighboring hyperplanes. Print the options and precision constants. When done, print statistics. These options may be used with any of the Qhull programs.

Example: rbox 1000 D3 | qhull d Qbb R1e-4 Q0
Compute the 3-d Delaunay triangulation of 1000 random points. Randomly perturb all calculations by [0.9999,1.0001]. Do not correct precision problems. This leads to serious precision errors.

Use the following equivalences when calling qhull:

By default, Qhull merges coplanar facets. For example, the convex hull of a cube's vertices has six facets.

If you use 'Qt' (triangulated output), all facets will be simplicial (e.g., triangles in 2-d). For the cube example, it will have 12 facets. Some facets may be degenerate and have zero area.

If you use 'QJ' (joggled input), all facets will be simplicial. The corresponding vertices will be slightly perturbed. Joggled input is less accurate that triangulated output.See Merged facets or joggled input.

The output for 4-d convex hulls may be confusing if the convex hull contains non-simplicial facets (e.g., a hypercube). See Why are there extra points in a 4-d or higher convex hull?

### »qhull synopsis

```qhull -- compute convex hulls and related structures.
input (stdin): dimension, number of points, point coordinates
halfspace: use dim+1 and put offsets after coefficients

options:
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
-?   - this message
-V   - version

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
FA   - report total area and volume
Fp   - halfspace intersections
FS   - total area and volume
Fx   - extreme points (convex hull vertices)
G    - Geomview output (2-d, 3-d and 4-d)
m    - Mathematica output (2-d and 3-d)
o    - OFF format (if 'v', outputs Voronoi regions)
QVn  - print facets that include point n, -n if not
TI file - input file, may be enclosed in single quotes
TO file - output file, may be enclosed in single quotes

examples:
rbox D4 | qhull Tv                        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 d D2 | qhull Qc s f Fx | more      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 Qc                 rbox c | qhull n
```

### »qhull input

The input data on stdin consists of:

• dimension
• number of points
• point coordinates

Use I/O redirection (e.g., qhull < data.txt), a pipe (e.g., rbox 10 | qhull), or the 'TI' option (e.g., qhull TI data.txt).

Comments start with a non-numeric character. Error reporting is simpler if there is one point per line. Dimension and number of points may be reversed. For halfspace intersection, an interior point may be prepended (see qhalf input).

Here is the input for computing the convex hull of the unit cube. The output is the normals, one per facet.

rbox c > data

```3 RBOX c
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
```

qhull s n < data

```
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 | QHULL s n

Number of points processed: 8
Number of hyperplanes created: 11
Number of distance tests for qhull: 35
Number of merged facets: 6
Number of distance tests for merging: 84
CPU seconds to compute hull (after input): 0.081

4
6
0      0     -1   -0.5
0     -1      0   -0.5
1      0      0   -0.5
-1      0      0   -0.5
0      1      0   -0.5
0      0      1   -0.5
```

### »qhull outputs

These options control the output of qhull. They may be used individually or together.

General
qhull
compute the convex hull of the input points. See qconvex.
qhull d Qbb
compute the Delaunay triangulation by lifting the points to a paraboloid. Use option 'Qbb' to scale the paraboloid and improve numeric precision. See qdelaunay.
qhull v Qbb
compute the Voronoi diagram by computing the Delaunay triangulation. Use option 'Qbb' to scale the paraboloid and improve numeric precision. See qvoronoi.
qhull H
compute the halfspace intersection about a point via polar duality. The point is below the hyperplanes that defines the halfspace. See qhalf.

For a full list of output options see

### »qhull controls

For a full list of control options see

### »qhull options

```qhull -- compute convex hulls and related structures.
http://www.qhull.org

input (stdin):
first lines: dimension and number of points (or vice-versa).
other lines: point coordinates, best if one point per line
halfspaces:  use dim plus one and put offset after coefficients.
May be preceded 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)
Hn,n,... - halfspace intersection about point [n,n,0,...]
Qc   - keep coplanar points with nearest facet
Qi   - keep interior points with nearest facet
QJ   - joggled input instead of merged facets
Qt   - triangulated output
v    - Voronoi diagram (dual of the Delaunay triangulation)
v Qu - furthest-site Voronoi diagram

Qhull control options:
Qa   - allow input with fewer or more points than coordinates
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)
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
QVn  - good facet if it includes point n, -n if not
Qx   - exact pre-merges (skips coplanar and angle-coplanar facets)
Qz   - add point-at-infinity to Delaunay triangulation

Qhull extra options:
Qf   - partition point to furthest outside facet
Qg   - only build good facets (needs 'QGn', 'QVn', or 'PdD')
QGn  - good facet if visible from point n, -n for not visible
Qm   - only process points that would increase max_outside
Qr   - process random outside points instead of furthest ones
Qv   - test vertex neighbors for convexity
Qw   - allow option warnings
Q0   - turn off default premerge with 'C-0'/'Qx'
Q1   - merge by mergetype/angle instead of mergetype/distance
Q2   - merge all coplanar facets instead of merging 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
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
Q12  - allow wide facets and wide dupridge
Q14  - merge pinched vertices that create a dupridge
Q15  - check for duplicate ridges with the same vertices

T options:
TFn  - report summary when n or more facets created
TI file - input file, may be enclosed in single quotes
TO file - output file, may be enclosed in single quotes
Ts   - statistics
Tv   - verify result: structure, convexity, and point inclusion
Tz   - send all output to stdout

Trace options:
T4   - trace at level n, 4=all, 5=mem/gauss, -1= events
Ta   - annotate output with message codes
TAn  - stop qhull after adding n vertices
TCn - stop qhull after building cone for point n
TVn - stop qhull after adding point n, -n for before
Tc   - check frequently during execution
Tf   - flush each qh_fprintf for debugging segfaults
TPn  - turn on tracing when point n added to hull
TP-1  turn on tracing after qh_buildhull and qh_postmerge
TMn - turn on tracing at merge n
TWn - trace merge facets when width > n
TRn  - rerun qhull n times for statistics to adjust 'QJn'

Precision options:
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)
n    - normals with offsets
o    - OFF format (dim, points and facets; Voronoi regions)
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 output (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
Gc   - centrums
GDn  - drop dimension n in 3-d and 4-d output
Gh   - hyperplane intersections
Gi   - inner planes only
Gn  -  no planes
Go  -  outer planes only
Gr   - ridges
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
PFn  - keep facets whose area is at least n
Pg   - print good facets (needs 'QGn' or 'QVn')
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
-?   - help with examples
-V   - version
``` The Geometry Center Home Page