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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 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` in 2-d to 4-d (a 3-d
Delaunay triangulation is a 4-d convex hull):

Use the following equivalences when calling `qhull` in 5-d and higher (a 4-d
Delaunay triangulation is a 5-d convex hull):

If you use 'Qt' (triangulated output), all furthest-site Delaunay regions will be simplicial (e.g., triangles in 2-d). Some regions may be degenerate and have zero area. Qhull can identify coincident points. With facet merging, cocircular and cospherical extreme points will lead to non-simplicial output (e.g., a square).

If you use 'QJ' (joggle), all furthest-site Delaunay regions will be simplicial (e.g., triangles in 2-d). Duplicate points will create small regions since the points are joggled apart. See Joggled input or merged facets.

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?

**Copyright © 1995-2002 The Geometry Center, Minneapolis MN**

qhull- compute convex hulls and related structures. 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 - joggle input instead of merging 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 (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 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

The input data on

stdinconsists 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 0.5qhull 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

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

General- qhull d
- compute the Delaunay triangulation by lifting the points to a paraboloid. See qdelaunay.
- qhull v
- compute the Voronoi diagram by computing the Delaunay triangulation. See qvoronoi.
- qhull H
- compute the halfspace intersection about a point via polar duality. See qhalf.
For a full list of output options see

- Output formats
- Additional I/O formats
- Geomview output options
- Print options

For a full list of control options see

qhull- compute convex hulls and related structures. http://www.geom.umn.edu/software/qhull 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 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) 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 - joggle input instead of merging 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 anglomaniacs 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 p remerge 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 Towpaths 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 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) 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

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Comments to: qhull@geom.umn.edu

Created: Sept. 25, 1995 --- Last modified: see top