qhull -- convex hull and related structures

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

> - qconvex == qhull
> - qdelaunay == qhull d Qbb
> - qhalf == qhull H
> - qvoronoi == qhull v Qbb

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

> - qconvex == qhull Qx
> - qdelaunay == qhull d Qbb Qx
> - qhalf == qhull H Qx
> - qvoronoi == qhull v Qbb Qx

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?

Copyright © 1995-2015 C.B. Barber

### » qhull synopsis

    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 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 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 hyperplane that defines the halfspace. See qhalf.

> For a full list of output options see

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

### » qhull controls

> For a full list of control options see

> > - Qhull control options
> > - Precision options
> > - Trace options

### » 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
        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
        Q12  - do not error on wide merge due to duplicate ridge and nearly coincident points

    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)
        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

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