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<h1><a
href="http://www.geom.uiuc.edu/graphics/pix/Special_Topics/Computational_Geometry/cone.html"><img
src="qh--cone.gif" alt="[cone]" align="middle" width="100"
height="100"></a>qhull -- convex hull and related structures</h1>

<p>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.

<p>The <tt>qhull</tt> program provides options to build these
structures and to experiment with the process.  Use the
<a href=qconvex.htm>qconvex</a>,
<a href=qdelaun.htm>qdelaunay</a>, <a href=qhalf.htm>qhalf</a>,
and <a href=qvoronoi.htm>qvoronoi</a> programs
to build specific structures.  You may use <tt>qhull</tt> instead.
It takes the same options and uses the same code.
<blockquote>
<dl>
    <dt><b>Example:</b> rbox 1000 D3 | qhull
         <a href="qh-optc.htm#Cn">C-1e-4</a>
         <a href="qh-optf.htm#FO">FO</a>
         <a href="qh-optt.htm#Ts">Ts</a>
         </dt>
    <dd>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.</dd>
    <dt>&nbsp;</dt>
    <dt><b>Example:</b> rbox 1000 D3 | qhull <a href=qhull.htm#outputs>d</a>
         <a href="qh-optq.htm#Qbb">Qbb</a>
         <a href="qh-optc.htm#Rn">R1e-4</a>
         <a href="qh-optq.htm#Q0">Q0</a></dt>
    <dd>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.</dd>
</dl>
</blockquote>
<p>Use the following equivalences when calling <tt>qhull</tt> in 2-d to 4-d (a 3-d
Delaunay triangulation is a 4-d convex hull):
<blockquote>
<ul>
<li>
<a href="qconvex.htm">qconvex</a> == qhull
<li>
<a href=qdelaun.htm>qdelaunay</a> == qhull d <a href="qh-optq.htm#Qbb">Qbb</a>
<li>
<a href=qhalf.htm>qhalf</a> == qhull H
<li>
<a href=qvoronoi.htm>qvoronoi</a> == qhull v <a href="qh-optq.htm#Qbb">Qbb</a>
</ul>
</blockquote>

<p>Use the following equivalences when calling <tt>qhull</tt> in 5-d and higher (a 4-d
Delaunay triangulation is a 5-d convex hull):
<blockquote>
<ul>
<li>
<a href="qconvex.htm">qconvex</a> == qhull <a href="qh-optq.htm#Qx">Qx</a>
<li>
<a href=qdelaun.htm>qdelaunay</a> == qhull d <a href="qh-optq.htm#Qbb">Qbb</a> <a href="qh-optq.htm#Qx">Qx</a>
<li>
<a href=qhalf.htm>qhalf</a> == qhull H <a href="qh-optq.htm#Qx">Qx</a>
<li>
<a href=qvoronoi.htm>qvoronoi</a> == qhull v <a href="qh-optq.htm#Qbb">Qbb</a> <a href="qh-optq.htm#Qx">Qx</a>
</ul>
</blockquote>


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

<p>If you use '<a href="qh-optq.htm#Qt">Qt</a>' (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.

<p>If you use '<a href="qh-optq.htm#QJn">QJ</a>' (joggled input),
all facets will be simplicial.  The corresponding vertices will be
slightly perturbed.  Joggled input is less accurate that triangulated
output.See <a
href="qh-impre.htm#joggle">Merged facets or joggled input</a>. </p>

<p>The output for 4-d convex hulls may be confusing if the convex
hull contains non-simplicial facets (e.g., a hypercube). See
<a href=qh-faq.htm#extra>Why
are there extra points in a 4-d or higher convex hull?</a><br>
</p>

<p><b>Copyright &copy; 1995-2015 C.B. Barber</b></p>

<hr>

<h3><a href="#TOP">&#187;</a><a name="synopsis">qhull synopsis</a></h3>
<pre>
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
</pre>

<h3><a href="#TOP">&#187;</a><a name="input">qhull input</a></h3>
<blockquote>

<p>The input data on <tt>stdin</tt> consists of:</p>
<ul>
    <li>dimension
    <li>number of points</li>
    <li>point coordinates</li>
</ul>

<p>Use I/O redirection (e.g., qhull &lt; data.txt), a pipe (e.g., rbox 10 | qhull),
or the '<a href=qh-optt.htm#TI>TI</a>' option (e.g., qhull TI data.txt).

<p>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 <a href=qhalf.htm#input>qhalf input</a>).

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

<blockquote>
    <p>rbox c &gt; data </p>
    <pre>
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
</pre>
    <p>qhull s n &lt; data</p>
    <pre>

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
</pre>
</blockquote>

</blockquote>
<h3><a href="#TOP">&#187;</a><a name="outputs">qhull outputs</a></h3>
<blockquote>

<p>These options control the output of qhull. They may be used
individually or together.</p>
<blockquote>
<dl compact>
    <dt>&nbsp;</dt>
    <dd><b>General</b></dd>
    <dt><a>qhull</a></dt>
    <dd>compute the convex hull of the input points.
        See <a href=qconvex.htm>qconvex</a>.</dd>
    <dt><a name="d">qhull d Qbb</a></dt>
    <dd>compute the Delaunay triangulation by lifting the points
        to a paraboloid.  Use option '<a href="qh-optq.htm#Qbb">Qbb</a>'
        to scale the paraboloid and improve numeric precision.
        See <a href=qdelaun.htm>qdelaunay</a>.</dd>
    <dt><a name="v">qhull v Qbb</a></dt>
    <dd>compute the Voronoi diagram by computing the Delaunay
        triangulation.    Use option '<a href="qh-optq.htm#Qbb">Qbb</a>'
        to scale the paraboloid and improve numeric precision.
        See <a href=qvoronoi.htm>qvoronoi</a>.</dd>
    <dt><a name="H">qhull H</a></dt>
    <dd>compute the halfspace intersection about a point via polar
        duality.  The point is below the hyperplane that defines the halfspace.
        See <a href=qhalf.htm>qhalf</a>.</dd>
</dl>
</blockquote>

<p>For a full list of output options see
<blockquote>
<ul>
            <li><a href="qh-opto.htm#output">Output</a> formats</li>
            <li><a href="qh-optf.htm#format">Additional</a> I/O
                formats</li>
            <li><a href="qh-optg.htm#geomview">Geomview</a>
                output options</li>
            <li><a href="qh-optp.htm#print">Print</a> options</li>
</ul>
</blockquote>

</blockquote>
<h3><a href="#TOP">&#187;</a><a name="controls">qhull controls</a></h3>
<blockquote>

<p>For a full list of control options see
<blockquote>
<ul>
            <li><a href="qh-optq.htm#qhull">Qhull</a> control
                options</li>
            <li><a href="qh-optc.htm#prec">Precision</a> options</li>
            <li><a href="qh-optt.htm#trace">Trace</a> options</li>
</ul>
</blockquote>

</blockquote>
<h3><a href="#TOP">&#187;</a><a name="options">qhull options</a></h3>

<pre>
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] &lt;= 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
</pre>

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<p><a href="http://www.geom.uiuc.edu/"><img src="qh--geom.gif"
align="middle" width="40" height="40"></a><i>The Geometry Center
Home Page </i></p>

<p>Comments to: <a href=mailto:qhull@qhull.org>qhull@qhull.org</a>
</a><br>
Created: Sept. 25, 1995 --- <!-- hhmts start --> Last modified: see top <!-- hhmts end --> </p>
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