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<p><a name="TOP"><b>Up:</b></a> <a
href="http://www.qhull.org">Home page</a> for Qhull (<a href="../index.htm">local</a>)<br>
<b>Up:</b> <a
href="http://www.qhull.org/news">News</a> about Qhull<br>
<b>Up:</b> <a href="http://github.com/qhull/qhull/wiki">Qhull Wiki</a> and
<a href="http://www.qhull.org/html/qh-faq.htm">FAQ</a> (<a href="qh-faq.htm">local</a>) <br>
<b>To:</b> <a href="#TOC">Qhull manual</a>: contents<br>
<b>To:</b> <a href="qh-quick.htm#programs">Programs</a>
• <a href="qh-quick.htm#options">Options</a>
• <a href="qh-opto.htm#output">Output</a>
• <a href="qh-optf.htm#format">Formats</a>
• <a href="qh-optg.htm#geomview">Geomview</a>
• <a href="qh-optp.htm#print">Print</a>
• <a href="qh-optq.htm#qhull">Qhull</a>
• <a href="qh-optc.htm#prec">Precision</a>
• <a href="qh-optt.htm#trace">Trace</a>
• <a href="http://www.qhull.org/src/libqhull_r/index.htm">Functions</a> (<a href="../src/libqhull_r/index.htm">local</a>)<br>
<hr>
<!-- Main text of document -->
<h1><a
href="http://www.geom.uiuc.edu/graphics/pix/Special_Topics/Computational_Geometry/fixed.html"><img
src="qh--rand.gif" alt="[random-fixed]" align="middle"
width="100" height="100"></a> Qhull manual </h1>
<p>Qhull is a general dimension code for computing convex hulls,
Delaunay triangulations, halfspace intersections about a point, Voronoi
diagrams, furthest-site Delaunay triangulations, and
furthest-site Voronoi diagrams. These structures have
applications in science, engineering, statistics, and
mathematics. See <a
href="http://www.cs.mcgill.ca/~fukuda/soft/polyfaq/polyfaq.html">Fukuda's
introduction</a> to convex hulls, Delaunay triangulations,
Voronoi diagrams, and linear programming. For a detailed
introduction, see O'Rourke [<a href="#orou94">'94</a>], <i>Computational
Geometry in C</i>.
</p>
<p>There are six programs. Except for rbox, they use
the same code. Each program includes instructions and examples.
<blockquote>
<ul>
<li><a href="qconvex.htm">qconvex</a> -- convex hulls
<li><a href="qdelaun.htm">qdelaunay</a> -- Delaunay triangulations and
furthest-site Delaunay triangulations
<li><a href="qhalf.htm">qhalf</a> -- halfspace intersections about a point
<li><a href="qhull.htm">qhull</a> -- all structures with additional options
<li><a href="qvoronoi.htm">qvoronoi</a> -- Voronoi diagrams and
furthest-site Voronoi diagrams
<li><a href="rbox.htm">rbox</a> -- generate point distributions for qhull
</ul>
</blockquote>
<p>Qhull implements the Quickhull algorithm for computing the
convex hull. Qhull includes options
for hull volume, facet area, multiple output formats, and
graphical output. It can approximate a convex hull. </p>
<p>Qhull handles roundoff errors from floating point
arithmetic. It generates a convex hull with "thick" facets.
A facet's outer plane is clearly above all of the points;
its inner plane is clearly below the facet's vertices. Any
exact convex hull must lie between the inner and outer plane.
<p>Qhull uses merged facets, triangulated output, or joggled
input. Triangulated output triangulates non-simplicial, merged
facets. Joggled input also
guarantees simplicial output, but it
is less accurate than merged facets. For merged facets, Qhull
reports the maximum outer and inner plane.
<p><i>Brad Barber, Arlington, MA</i></p>
<p><b>Copyright © 1995-2020 C.B. Barber</b></p>
<hr>
<h2><a href="#TOP">»</a><a name="TOC">Qhull manual</a>: contents</h2>
<ul>
<li><a href="#when">When</a> to use Qhull
<ul>
<li><a href="http://www.qhull.org/news">News</a> for Qhull
with new features and reported bugs.
<li><a href="http://www.qhull.org">Home</a> for Qhull with additional URLs
(<a href=../index.htm>local copy</a>)
<li><a href="http://github.com/qhull/qhull/wiki">Wiki</a> and <a href="http://www.qhull.org/html/qh-faq.htm">FAQ</a> (<a href="qh-faq.htm">local</a>)
<li><a href="http://www.qhull.org/download">Download</a> Qhull (<a href=qh-get.htm>local</a>)
<li><a href="https://github.com/qhull/qhull/wiki/Qhull-build-systems">Qhull build systems</a>
<li><a href="qh-quick.htm#programs">Quick</a> reference for Qhull and its <a href="qh-quick.htm#options">options</a>
<p>
<li><a href="../COPYING.txt">COPYING.txt</a> - copyright notice<br>
<li><a href="../REGISTER.txt">REGISTER.txt</a> - registration<br>
<li><a href="../README.txt">README.txt</a> - installation
instructions<br>
<li><a href="../src/Changes.txt">Changes.txt</a> - change history <br>
<li><a href="qhull.txt">qhull.txt</a> - Unix manual page
<li><a href="rbox.txt">rbox.txt</a> - Unix manual page
</ul>
<p>
<li><a href="#description">Description</a> of Qhull
<ul>
<li><a href="#definition">de</a>finition • <a
href="#input">in</a>put • <a href="#output">ou</a>tput
• <a href="#algorithm">al</a>gorithm • <a
href="#structure">da</a>ta structure </li>
<li><a href="qh-impre.htm">Imprecision</a> in Qhull</li>
<li><a href="qh-impre.htm#joggle">Merged facets</a> or joggled input
<li><a href="qh-eg.htm">Examples</a> of Qhull</li>
</ul>
<p>
<li><a href=qh-quick.htm#programs>Qhull programs</a>, with instructions and examples
<ul>
<li><a href="qconvex.htm">qconvex</a> -- convex hulls
<li><a href="qdelaun.htm">qdelaunay</a> -- Delaunay triangulation
<li><a href="qdelau_f.htm">qdelaunay Qu</a> -- furthest-site Delaunay triangulation
<li><a href="qhalf.htm">qhalf</a> -- halfspace intersections about a point
<li><a href="qhull.htm">qhull</a> -- all structures with additional options
<li><a href="qvoronoi.htm">qvoronoi</a> -- Voronoi diagram
<li><a href="qvoron_f.htm">qvoronoi Qu</a> -- furthest-site Voronoi diagram
<li><a href="rbox.htm">rbox</a> -- generate point distributions for qhull
</ul>
<p>
<li><a href="qh-quick.htm#options">Qhull options</a><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>
<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>
</li>
<p>
<li><a href="#geomview">Geomview</a>, Qhull's graphical viewer</li>
<ul>
<li><a href="#geomview-install">Installing Geomview</a></li>
<li><a href="#geomview-use">Using Geomview</a></li>
<li><a href="#geomview-win">Building Geomview for Windows</a></li>
</ul>
<p>
<li><a href="qh-code.htm">Qhull code</a><ul>
<li><a href="qh-code.htm#reentrant">Reentrant</a> Qhull</li>
<li><a href="qh-code.htm#convert">How to convert</a> code to reentrant Qhull</li>
<li><a href="qh-code.htm#64bit">Qhull</a> on 64-bit computers</li>
<li><a href="qh-code.htm#cpp">Calling</a> Qhull
from C++ programs</li>
<li><a href="qh-code.htm#library">Calling</a> Qhull
from C programs</li>
<li><a href="qh-code.htm#debug">How to debug</a> Qhull</li>
<li><a href="qh-code.htm#performance">Performance</a>
of Qhull</li>
<li><a href="qh-code.htm#profile">Profiling</a> and optimizing Qhull performance</li>
<li><a href="qh-code.htm#enhance">Enhancements</a> to
Qhull</li>
<li><a href="http://www.qhull.org/src/libqhull_r/index.htm">Function</a> index to Qhull (<a href="../src/libqhull_r/index.htm">local</a>) </li>
</ul>
</li>
<p>
<li>Related URLs
<ul>
<li><a href="news:comp.graphics.algorithms">Newsgroup</a>:
comp.graphics.algorithms
<li><a
href="http://www.faqs.org/faqs/graphics/algorithms-faq/">FAQ</a> for computer graphics algorithms and
Exaflop's <a href="http://web.archive.org/web/19990826064500/http://www.exaflop.org/docs/cgafaq/cga6.html">geometric</a> structures (archived).
<li>Amenta's <a href="http://www.geom.uiuc.edu/software/cglist">Directory
of Computational Geometry Software</a></li>
<li>Erickson's <a
href="http://jeffe.cs.illinois.edu/compgeom/">Computational Geometry Software</a> </li>
<li>Fukuda's <a
href="http://www.cs.mcgill.ca/~fukuda/soft/polyfaq/polyfaq.html">
introduction</a> to convex hulls, Delaunay triangulations,
Voronoi diagrams, and linear programming.
<li>Stony Brook's <a
href="http://www.cs.sunysb.edu/~algorith/major_section/1.6.shtml">Algorithm Repository</a> on computational geometry.
</li>
</ul>
<p>
<li><a href="#bugs">What to do</a> if something goes wrong</li>
<li><a href="#email">Email</a></li>
<li><a href="#authors">Authors</a></li>
<li><a href="#ref">References</a></li>
<li><a href="#acknowledge">Acknowledgments</a></li>
</ul>
<h2><a href="#TOC">»</a><a name="when">When to use Qhull</a></h2>
<blockquote>
<p>Qhull constructs convex hulls, Delaunay triangulations,
halfspace intersections about a point, Voronoi diagrams, furthest-site Delaunay
triangulations, and furthest-site Voronoi diagrams.</p>
<p>For convex hulls and halfspace intersections, Qhull may be used
for 2-d up to 8-d. For Voronoi diagrams and Delaunay triangulations, Qhull may be
used for 2-d up to 7-d. In higher dimensions, the size of the output
grows rapidly and Qhull does not work well with virtual memory.
If <i>n</i> is the size of
the input and <i>d</i> is the dimension (d>=3), the size of the output
and execution time
grows by <i>n^(floor(d/2)</i>
[see <a href=qh-code.htm#performance>Performance</a>]. For example, do
not try to build a 16-d convex hull of 1000 points. It will
have on the order of 1,000,000,000,000,000,000,000,000 facets.
<p>On a 1.7 GHz i7, Qhull computes the 2-d convex hull of
3,000,000 evenly spaced, cocircular points in 4.7 seconds.
It computes the
2-d Delaunay triangulation and 3-d convex hull of 600,000 points
in 5.9 seconds. It computes the
3-d Delaunay triangulation and 4-d convex hull of 120,000 points
in 4.6 seconds. It computes the
4-d Delaunay triangulation and 5-d convex hull of 30,000 points
in 5.1 seconds. It computes the
5-d Delaunay triangulation and 6-d convex hull of 6,000 points
in 7.8 seconds. It computes the
6-d Delaunay triangulation and 7-d convex hull of 1,000 points
in 7.6 seconds. It computes the
7-d Delaunay triangulation and 8-d convex hull of 300 points
in 8.6 seconds. It computes the
8-d Delaunay triangulation and 9-d convex hull of 120 points
in 5.7 seconds. It computes the
9-d Delaunay triangulation and 10-d convex hull of 70 points
in 4.8 seconds.
The 10-d convex hull of 70 cospherical points has about 375,000 facets.
<!-- duplicated in index.htm and html/index.htm -->
<p>Qhull does <i>not</i> support constrained Delaunay
triangulations, triangulation of non-convex surfaces, mesh
generation of non-convex objects, or medium-sized inputs in 9-D
and higher. </p>
<p>This is a big package with many options. For 32-bit code, it is one of the
fastest available. It is the only 3-d and higher code that handles precision
problems due to floating point arithmetic. For example, it
implements the identity function for extreme points (see <a
href="qh-impre.htm">Imprecision in Qhull</a>). </p>
<p>Qhull does not handle all precision problems (<a href="../html/qh-impre.htm#limit">Limitations of merged facets</a>).
For example, errors may occur for the 4-D Delaunay triangulation of nearly adjacent points in the unit cube. If every
input site has a nearly adjacent point within a 10^-13 ball, precision errors and topological errors are likely to
occur. A bounding box may alleviate these errors. Nearly adjacent points within substantially smaller or larger balls
are OK.</p>
<p>Joggled input, option '<a href="qh-optq.htm#QJn">QJn</a>', is an alternative to merged facets (<a href="../html/qh-impre.htm#limit">Merged facets or joggled input</a>).
It randomly perturbs the input. If precision problems occur, Qhull restarts with a larger, random perturbation. Joggled input
is a simple solution without the limitations of merged facets. For example, Qhull constructs a convex hull even if all of the input points
are identical.</p>
<p>If you need a short code for convex hull, Delaunay
triangulation, or Voronoi volumes consider Clarkson's <a
href="http://www.netlib.org/voronoi/hull.html">hull
program</a>. If you need 2-d Delaunay triangulations consider
Shewchuk's <a href="http://www.cs.cmu.edu/~quake/triangle.html">triangle
program</a>. It is much faster than Qhull and it allows
constraints. Both programs use exact arithmetic. They are in <a
href="http://www.netlib.org/voronoi/">http://www.netlib.org/voronoi/</a>.
<p>If your input is in general position (i.e., no coplanar or colinear points),
<a href="https://github.com/tomilov/quickhull/blob/master/include/quickhull.hpp">Tomilov's quickhull.hpp</a> (<a href="http://habrahabr.ru/post/245221/">documentation-ru</a>),
or Qhull <a
href="http://www.qhull.org/download">version 1.0</a> may meet your needs. Both programs detect precision problems,
but do not handle them.</p>
<p><a href=http://www.cgal.org>CGAL</a> is a library of efficient and reliable
geometric algorithms. It uses C++ templates and the Boost library to produce dimension-specific
code. This allows more efficient use of memory than Qhull's general-dimension
code. For 64-bit code, CGAL uses significantly less memory than Qhull and runs faster.
CGAL simulates arbitrary precision while Qhull handles round-off error
with thick facets. Compare the two approaches with <a href="http://doc.cgal.org/latest/Manual/devman_robustness.html">Robustness Issues in CGAL</a>,
and <a href="qh-impre.htm">Imprecision in Qhull</a>.
<p><a href=http://www.algorithmic-solutions.com/index.php/products/leda-for-c>Leda</a> is a
library for writing computational
geometry programs and other combinatorial algorithms. It
includes routines for computing 3-d convex
hulls, 2-d Delaunay triangulations, and 3-d Delaunay triangulations.
It provides rational arithmetic and graphical output. It runs on most
platforms.
<p>If your problem is in high dimensions with a few,
non-simplicial facets, try Fukuda's <a
href="http://www.cs.mcgill.ca/~fukuda/soft/cdd_home/cdd.html">cdd</a>.
It is much faster than Qhull for these distributions. </p>
<p>Custom software for 2-d and 3-d convex hulls may be faster
than Qhull. Custom software should use less memory. Qhull uses
general-dimension data structures and code. The data structures
support non-simplicial facets.</p>
<p>Qhull is not suitable for mesh generation or triangulation of
arbitrary surfaces. You may use Qhull if the surface is convex or
completely visible from an interior point (e.g., a star-shaped
polyhedron). First, project each site to a sphere that is
centered at the interior point. Then, compute the convex hull of
the projected sites. The facets of the convex hull correspond to
a triangulation of the surface. For mesh generation of arbitrary
surfaces, see <a
href="http://www.robertschneiders.de/meshgeneration/meshgeneration.html">Schneiders'
Finite Element Mesh Generation</a>.</p>
<p>Qhull is not suitable for constrained Delaunay triangulations.
With a lot of work, you can write a program that uses Qhull to
add constraints by adding additional points to the triangulation.</p>
<p>Qhull is not suitable for the subdivision of arbitrary
objects. Use <tt>qdelaunay</tt> to subdivide a convex object.</p>
</blockquote>
<h2><a href="#TOC">»</a><a name="description">Description of Qhull</a></h2>
<blockquote>
<h3><a href="#TOC">»</a><a name="definition">definition</a></h3>
<blockquote>
<p>The <i>convex hull</i> of a point set <i>P</i> is the smallest
convex set that contains <i>P</i>. If <i>P</i> is finite, the
convex hull defines a matrix <i>A</i> and a vector <i>b</i> such
that for all <i>x</i> in <i>P</i>, <i>Ax+b <= [0,...]</i>. </p>
<p>Qhull computes the convex hull in 2-d, 3-d, 4-d, and higher
dimensions. Qhull represents a convex hull as a list of facets.
Each facet has a set of vertices, a set of neighboring facets,
and a halfspace. A halfspace is defined by a unit normal and an
offset (i.e., a row of <i>A</i> and an element of <i>b</i>). </p>
<p>Qhull accounts for round-off error. It returns
"thick" facets defined by two parallel hyperplanes. The
outer planes contain all input points. The inner planes exclude
all output vertices. See <a href="qh-impre.htm#imprecise">Imprecise
convex hulls</a>.</p>
<p>Qhull may be used for the Delaunay triangulation or the
Voronoi diagram of a set of points. It may be used for the
intersection of halfspaces. </p>
</blockquote>
<h3><a href="#TOC">»</a><a name="input">input format</a></h3>
<blockquote>
<p>The input data on <tt>stdin</tt> consists of:</p>
<ul>
<li>first line contains the dimension</li>
<li>second line contains the number of input points</li>
<li>remaining lines contain point coordinates</li>
</ul>
<p>For example: </p>
<pre>
3 #sample 3-d input
5
0.4 -0.5 1.0
1000 -1e-5 -100
0.3 0.2 0.1
1.0 1.0 1.0
0 0 0
</pre>
<p>Input may be entered by hand. End the input with a control-D
(^D) character. </p>
<p>To input data from a file, use I/O redirection or '<a
href="qh-optt.htm#TI">TI file</a>'. The filename may not
include spaces or quotes.</p>
<p>A comment starts with a non-numeric character and continues to
the end of line. The first comment is reported in summaries and
statistics. With multiple <tt>qhull</tt> commands, use option '<a
href="qh-optf.htm#FQ">FQ</a>' to place a comment in the output.</p>
<p>The dimension and number of points can be reversed. Comments
and line breaks are ignored. Error reporting is better if there
is one point per line.</p>
</blockquote>
<h3><a href="#TOC">»</a><a name="option">option format</a></h3>
<blockquote>
<p>Use options to specify the output formats and control
Qhull. The <tt>qhull</tt> program takes all options. The
other programs use a subset of the options. They disallow
experimental and inappropriate options.
<blockquote>
<ul>
<li>
qconvex == qhull
<li>
qdelaunay == qhull d Qbb
<li>
qhalf == qhull H
<li>
qvoronoi == qhull v Qbb
</ul>
</blockquote>
<p>Single letters are used for output formats and precision
constants. The other options are grouped into menus for formats
('<a href="qh-optf.htm#format">F</a>'), Geomview ('<a
href="qh-optg.htm#geomview">G</a>'), printing ('<a
href="qh-optp.htm#print">P</a>'), Qhull control ('<a
href="qh-optq.htm#qhull">Q</a>'), and tracing ('<a
href="qh-optt.htm#trace">T</a>'). The menu options may be listed
together (e.g., 'GrD3' for 'Gr' and 'GD3'). Options may be in any
order. Capitalized options take a numeric argument (except for '<a
href="qh-optp.htm#PG">PG</a>' and '<a href="qh-optf.htm#format">F</a>'
options). Use option '<a href="qh-optf.htm#FO">FO</a>' to print
the selected options.</p>
<p>Qhull uses zero-relative indexing. If there are <i>n</i>
points, the index of the first point is <i>0</i> and the index of
the last point is <i>n-1</i>.</p>
<p>The default options are:</p>
<ul>
<li>summary output ('<a href="qh-opto.htm#s">s</a>') </li>
<li>merged facets ('<a href="qh-optc.htm#C0">C-0</a>' in 2-d,
3-d, 4-d; '<a href="qh-optq.htm#Qx">Qx</a>' in 5-d and
up)</li>
</ul>
<p>Except for bounding box
('<a href="qh-optq.htm#Qbk">Qbk:n</a>', etc.), drop facets
('<a href="qh-optp.htm#Pdk">Pdk:n</a>', etc.), and
Qhull command ('<a href="qh-optf.htm#FQ">FQ</a>'), only the last
occurence of an option counts.
Bounding box and drop facets may be repeated for each dimension.
Option 'FQ' may be repeated any number of times.
<p>The Unix <tt>tcsh</tt> and <tt>ksh </tt>shells make it easy to
try out different options. In Windows 95, use a command window with <tt>doskey</tt>
and a window scroller (e.g., <tt>peruse</tt>). </p>
</blockquote>
<h3><a href="#TOC">»</a><a name="output">output format</a></h3>
<blockquote>
<p>To write the results to a file, use I/O redirection or '<a
href="qh-optt.htm#TO">TO file</a>'. Windows 95 users should use
'TO file' or the console. If a filename is surrounded by single quotes,
it may include spaces.
</p>
<p>The default output option is a short summary ('<a
href="qh-opto.htm#s">s</a>') to <tt>stdout</tt>. There are many
others (see <a href="qh-opto.htm">output</a> and <a
href="qh-optf.htm">formats</a>). You can list vertex incidences,
vertices and facets, vertex coordinates, or facet normals. You
can view Qhull objects with Geomview, Mathematica, or Maple. You can
print the internal data structures. You can call Qhull from your
application (see <a href="qh-code.htm#library">Qhull library</a>).</p>
<p>For example, 'qhull <a href="qh-opto.htm#o">o</a>' lists the
vertices and facets of the convex hull. </p>
<p>Error messages and additional summaries ('<a
href="qh-opto.htm#s">s</a>') go to <tt>stderr</tt>. Unless
redirected, <tt>stderr</tt> is the console.</p>
</blockquote>
<h3><a href="#TOC">»</a><a name="algorithm">algorithm</a></h3>
<blockquote>
<p>Qhull implements the Quickhull algorithm for convex hull
[Barber et al. <a href="#bar-dob96">'96</a>]. This algorithm
combines the 2-d Quickhull algorithm with the <em>n</em>-d
beneath-beyond algorithm [c.f., Preparata & Shamos <a
href="#pre-sha85">'85</a>]. It is similar to the randomized
algorithms of Clarkson and others [Clarkson & Shor <a
href="#cla-sho89">'89</a>; Clarkson et al. <a href="#cla-meh93">'93</a>;
Mulmuley <a href="#mulm94">'94</a>]. For a demonstration, see <a
href="qh-eg.htm#how">How Qhull adds a point</a>. The main
advantages of Quickhull are output sensitive performance (in
terms of the number of extreme points), reduced space
requirements, and floating-point error handling. </p>
</blockquote>
<h3><a href="#TOC">»</a><a name="structure">data structures</a></h3>
<blockquote>
<p>Qhull produces the following data structures for dimension <i>d</i>:
</p>
<ul>
<li>A <em>coordinate</em> is a real number in floating point
format. </li>
<li>A <em>point</em> is an array of <i>d</i> coordinates.
With option '<a href="qh-optq.htm#QJ">QJ</a>', the
coordinates are joggled by a small amount. </li>
<li>A <em>vertex</em> is an input point. </li>
<li>A <em>hyperplane</em> is <i>d</i> normal coefficients and
an offset. The length of the normal is one. The
hyperplane defines a halfspace. If <i>V</i> is a normal, <i>b</i>
is an offset, and <i>x</i> is a point inside the convex
hull, then <i>Vx+b <0</i>.</li>
<li>An <em>outer plane</em> is a positive
offset from a hyperplane. When Qhull is done, all points
will be below all outer planes.</li>
<li>An <em>inner plane</em> is a negative
offset from a hyperplane. When Qhull is done, all
vertices will be above the corresponding inner planes.</li>
<li>An <em>orientation</em> is either 'top' or 'bottom'. It is the
topological equivalent of a hyperplane's geometric
orientation. </li>
<li>A <em>simplicial facet</em> is a set of
<i>d</i> neighboring facets, a set of <i>d</i> vertices, a
hyperplane equation, an inner plane, an outer plane, and
an orientation. For example in 3-d, a simplicial facet is
a triangle. </li>
<li>A <em>centrum</em> is a point on a facet's hyperplane. A
centrum is the average of a facet's vertices. Neighboring
facets are <em>convex</em> if each centrum is below the
neighbor facet's hyperplane. </li>
<li>A <em>ridge</em> is a set of <i>d-1</i> vertices, two
neighboring facets, and an orientation. For example in
3-d, a ridge is a line segment. </li>
<li>A <em>non-simplicial facet</em> is a set of ridges, a
hyperplane equation, a centrum, an outer plane, and an
inner plane. The ridges determine a set of neighboring
facets, a set of vertices, and an orientation. Qhull
produces a non-simplicial facet when it merges two facets
together. For example, a cube has six non-simplicial
facets. </li>
</ul>
<p>For examples, use option '<a href="qh-opto.htm#f">f</a>'. See <a
href="http://www.qhull.org/src/libqhull_r/qh-poly_r.htm">polyhedron operations</a> (<a href="../src/libqhull_r/qh-poly_r.htm">local</a>) for further
design documentation. </p>
</blockquote>
<h3><a href="#TOC">»</a>Imprecision in Qhull</h3>
<blockquote>
<p>See <a href="qh-impre.htm">Imprecision in Qhull</a> and <a href="qh-impre.htm#joggle">Merged facets or joggled input</a></p>
</blockquote>
<h3><a href="#TOC">»</a>Examples of Qhull</h3>
<blockquote>
<p>See <a href="qh-eg.htm">Examples of Qhull</a>. Most of these examples require <a href="#geomview">Geomview</a>.
Some of the examples have <a
href="http://www.geom.uiuc.edu/graphics/pix/Special_Topics/Computational_Geometry/welcome.html">pictures
</a>.</p>
</blockquote>
</blockquote>
<h2><a href="#TOC">»</a>Options for using Qhull </h2>
<blockquote>
<p>See <a href="qh-quick.htm#options">Qhull options</a>.</p>
</blockquote>
<h2><a href="#TOC">»</a>Qhull code and internals</h2>
<blockquote>
<p>See <a href="qh-code.htm">Qhull code</a>.</p>
</blockquote>
<h2><a href="#TOC">»</a><a name="geomview">Geomview, Qhull's
graphical viewer</a></h2>
<blockquote>
<p><a href="http://www.geomview.org">Geomview</a>
is an interactive geometry viewing program.
Geomview provides a good visualization of Qhull's 2-d and 3-d results.
<p>Qhull includes <a href="qh-eg.htm">Examples of Qhull</a> that may be viewed with Geomview.
<p>Geomview can help visulalize a 3-d Delaunay triangulation or the surface of a 4-d convex hull,
Use option '<a href="qh-optq.htm#QVn">QVn</a>' to select the 3-D facets adjacent to a vertex.
<p>You may use Geomview to create movies that animate your objects (c.f., <a href="http://www.geomview.org/FAQ/answers.shtml#mpeg">How can I create a video animation?</a>).
Geomview helped create the <a href="http://www.geom.uiuc.edu/video/">mathematical videos</a> "Not Knot", "Outside In", and "The Shape of Space" from the Geometry Center.
<h3><a href="#TOC">»</a><a name="geomview-install">Installing Geomview</a></h3>
<blockquote>
<p>Geomview is an <a href=http://sourceforge.net/projects/geomview>open source project</a>
under SourceForge.
<p>
For build instructions see
<a href="http://www.geomview.org/download/">Downloading Geomview</a>.
Geomview builds under Linux, Unix, Macintosh OS X, and Windows.
<p>Geomview has <a href="https://packages.debian.org/search?keywords=geomview">installable packages</a> for Debian and Ubuntu.
The OS X build needs Xcode, an X11 SDK, and Lesstif or Motif.
The Windows build uses Cygwin (see <a href="#geomview-win">Building Geomview</a> below for instructions).
<p>If using Xforms (e.g., for Geomview's <a href="http://www.geomview.org/docs/html/Modules.html">External Modules</a>), install the 'libXpm-devel' package from cygwin and move the xforms directory into your geomview directory, e.g.,<br><tt>mv xforms-1.2.4 geomview-1.9.5/xforms</tt>
<p>Geomview's <a href="http://www.geom.uiuc.edu/software/geomview/docs/NDview/manpagehelp.html">ndview</a> provides multiple views into 4-d and higher objects.
This module is out-of-date (<a href="http://sourceforge.net/p/geomview/mailman/message/2004152/">geomview-users: 4dview</a>).
Download NDview-sgi.tar.Z at <a href="ftp://www.geom.uiuc.edu/pub/software/geomview/newpieces/sgi">newpieces</a> and 4dview at <a href="https://stuff.mit.edu/afs/sipb/project/3d/arch/sgi_62/lib/Geomview/modules/">Geomview/modules</a>.
</blockquote>
<h3><a href="#TOC">»</a><a name="geomview-use">Using Geomview</a></h3>
<blockquote>
<p>Use Geomview to view <a href="qh-eg.htm">Examples of Qhull</a>. You can spin the convex hull, fly a camera through its facets,
and see how Qhull produces thick facets in response to round-off error.
<p>Follow these instructions to view 'eg,01.cube' from Examples of Qhull
<ol>
<li>Launch an XTerm command shell
<ul>
<li>If needed, start the X terminal server, Use 'xinit' or 'startx' in /usr/X11R6/bin<br><tt>xinit -- -multiwindow -clipboard</tt><br><tt>startx</tt>
<li>Start an XTerm command shell. In Windows, click the Cygwin/bash icon on your desktop.
<li>Set the DISPLAY variable, e.g.,<br><tt>export DISPLAY=:0</tt><br><tt>export DISPLAY=:0 >>~/.bashenv</tt>
</ul>
<li>Use Qhull's <a href="qh-optg.htm">Geomview options</a> to create a geomview object
<ul>
<li><tt>rbox c D3 | qconvex G >eg.01.cube</tt>
<li>On windows, convert the output to Unix text format with 'd2u'<br><tt>rbox c D3 | qconvex G | d2u >eg.01.cube</tt><br><tt>d2u eg.*</tt>
</ul>
<li>Run Geomview
<ul>
<li>Start Geomview with your example<br><tt>./geomview eg.01.cube</tt>
<li>Follow the instructions in <a href="http://www.geomview.org/docs/html/Tutorial.html">Gemoview Tutorial</a>
<li>Geomview creates the <i>Geomview control panel</i> with Targets and External Module, the <i>Geomview toolbar</i> with buttons for controlling Geomview, and the <i>Geomview camera window</i> showing a cube.
<li>Clear the camera window by selecting your object in the Targets list and 'Edit > Delete' or 'dd'
<li>Load the <i>Geomview files panel</i>. Select 'Open' in the 'File' menu.
<li>Set 'Filter' in the files panel to your example directory followed by '/*' (e.g., '/usr/local/qhull-2015.2/eg/*')
<li>Click 'Filter' in the files panel to view your examples in the 'Files' list.
<li>Load another example into the camera window by selecting it and clicking 'OK'.
<li>Review the instructions for <a href="http://www.geomview.org/docs/html/Interaction.html">Interacting with Geomview</a>
<li>When viewing multiple objects at once, you may want to turn off normalization. In the 'Inspect > Apperance' control panel, set 'Normalize' to 'None'.
</ul>
</ol>
<p>Geomview defines GCL (a textual API for controlling Geomview) and OOGL (a textual file format for defining objects).
<ul>
<li>To control Geomview, you may use any program that reads and writes from stdin and stdout. For example, it could report Qhull's information about a vertex identified by a double-click 'pick' event.
<li><a href="http://www.geomview.org/docs/html/GCL.html">GCL</a> command language for controlling Geomview
<li><a href="http://www.geomview.org/docs/html/OOGL-File-Formats.html">OOGL</a> file format for defining objects (<a href="http://www.geomview.org/docs/oogltour.html">tutorial</a>).
<li><a href="http://www.geomview.org/docs/html/Modules.html">External Modules</a> for interacting with Geomview via GCL
<li>Interact with your objects via <a href="http://www.geomview.org/docs/html/pick.html">pick</a> commands in response to right-mouse double clicks. Enable pick events with the <a href="http://www.geomview.org/docs/html/interest.html">interest</a> command.
</ul>
</blockquote>
<h3><a href="#TOC">»</a><a name="geomview-win">Building Geomview for Windows</a></h3>
<blockquote>
<p>Compile Geomview under Cygwin. For detailed instructions, see
<a href="http://www.ee.surrey.ac.uk/Personal/L.Wood/software/SaVi/building-under-Windows/"
>Building Savi and Geomview under Windows</a>. These instructions are somewhat out-of-date. Updated
instructions follow.
<p>How to compile Geomview under 32-bit Cygwin (October 2015)</p>
<ol>
<li><b>Note:</b> L. Wood has run into multiple issues with Geomview on Cygwin. He recommends Virtualbox/Ubuntu
and a one-click install of geomview via the Ubuntu package. See his Savi/Geomview link above.
<li>Install 32-bit <a href="http://cygwin.com/">Cygwin</a> as follows.
For additional guidance, see Cygwin's <a href="https://cygwin.com/install.html">Installing and Updating Cygwin Packages</a>
and <a href="http://www.qhull.org/road/road-faq/xml/cmdline.xml#setup-cygwin">Setup cygwin</a>.
<ul>
<li>Launch the cygwin installer.
<li>Select a mirror from <a href="http://cygwin.com/mirrors.html">Cygwin mirrors</a> (e.g., http://mirrors.kernel.org/sourceware/cygwin/ in California).
<li>Select the packages to install. Besides the cygwin packages listed in the Savi/Windows instructions consider adding
<ul>
<li><b>Default</b> -- libXm-devel (required for /usr/include/Xm/Xm.h)
<li><b>Devel</b> -- bashdb, gcc-core (in place of gcc), gdb
<li><b>Lib</b> -- libGL-devel, libGLU1 (required, obsolete), libGLU-devel (required, obsolete), libjpeg-devel(XForms), libXext-devel (required), libXpm-devel (Xforms)
libGL and lib
<li><b>Math</b> -- bc
<li><b>Net</b> -- autossh, inetutils, openssh
<li><b>System</b> -- chere
<li><b>Utils</b> -- dos2unix (required for qhull), keychain
<li>If installing perl, ActiveState Perl may be a better choice than cygwin's perl. Perl is not used by Geomview or Qhull.
<li><a href="https://cygwin.com/cgi-bin2/package-grep.cgi">Cygwin Package Search</a> -- Search for cygwin programs and packages
</ul>
<li>Click 'Next' to download and install the packages.
<li>If the download is incomplete, try again.
<li>If you try again after a successful install, cygwin will uninstall and reinstall all modules..
<li>Click on the 'Cywin Terminal' icon on the Desktop. It sets up a user directory in /home from /etc/skel/...
<li>Mount your disk drives<br>mount c: /c # Ignore the warning /c does not exist
</ul>
<li>Consider installing the <a href="http://www.qhull.org/bash/doc/road-bash.html">Road Bash</a> scripts (/etc/road-*) from <a href="http://www.qhull.org/road/">Road</a>.
They define aliases and functions for Unix command shells (Unix, Linux, Mac OS X, Windows),
<ul>
<li>Download Road Bash and unzip the downloaded file
<li>Copy .../bash/etc/road-* to the Cywin /etc directory (by default, C:\cygwin\etc).
<li>Using the cygwin terminal, convert the road scripts to Unix format<br>d2u /etc/road-*
<li>Try it<br>source /etc/road-home.bashrc
<li>Install it<br>cp /etc/road-home.bashrc ~/.bashrc
</ul>
<li>Launch the X terminal server from '<tt>Start > All programs > Cygwin-X > Xwin Server</tt>'. Alternatively, run 'startx'
<li>Launch an XTerm shell
<ul>
<li>Right click the Cywin icon on the system tray in the Windows taskbar.
<li>Select '<tt>System Tools > XTerm</tt>'
</ul>
<li>Download and extract Geomview -- <a href="http://www.geomview.org/download/">Downloading Geomview</a>
<li>Compile Geomview
<ul>
<li>./configure
<li>make
</ul>
<li>If './configure' fails, check 'config.log' at the failing step. Look carefully for missing libraries, etc. The <a href="http://www.geomview.org/FAQ/answers.shtml">Geomview FAQ</a> contains suggestions (e.g., "configure claims it can't find OpenGl").
<li>If 'make' fails, read the output carefully for error messages. Usually it is a missing include file or package. Locate and install the missing cygwin packages
(<a href="https://cygwin.com/cgi-bin2/package-grep.cgi">Cygwin Package Search</a>).
</ol>
</blockquote>
<h2><a href="#TOC">»</a><a name="bugs">What to do if something
goes wrong</a></h2>
<blockquote>
<p>Please report bugs to <a href=mailto:qhull_bug@qhull.org>qhull_bug@qhull.org</a>.
Please report if Qhull crashes. Please report if Qhull
generates an "internal error". Please report if Qhull
produces a poor approximate hull in 2-d, 3-d or 4-d. Please
report documentation errors. Please report missing or incorrect
links.</p>
<p>If you do not understand something, try a small example. The <a
href="rbox.htm">rbox</a> program is an easy way to generate
test cases. The <a href="#geomview">Geomview</a> program helps to
visualize the output from Qhull.</p>
<p>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. Qhull produces a compiler error
if __STDC__ is not defined. You may need to set a flag (e.g.,
'-A' or '-ansi').</p>
<p>If Qhull compiles but crashes on the test case (rbox D4),
there's still incompatibility between your system and ours.
Sometimes it is due to memory management. This can be turned off
with qh_NOmem in mem.h. Please let us know if you figure out how
to fix these problems. </p>
<p>If you doubt the output from Qhull, add option '<a
href="qh-optt.htm#Tv">Tv</a>'. It checks that every point is
inside the outer planes of the convex hull. It checks that every
facet is convex with its neighbors. It checks the topology of the
convex hull.</p>
<p>Qhull resolves most precision issues. It reports
a precision error for lower dimensional inputs. It may report a precision
error for non-simplicial facets with multiple merges or nearly
adjacent vertices. See <a href=qh-impre.htm#limit>Limitations of merged facets</a> for
more information.
<p>Qhull reports precision
errors if you turn off merged facets with option '<a
href="qh-optq.htm#Q0">Q0</a>'. This can get as bad as facets with
flipped orientation or two facets with the same vertices. You'll
get a long help message if you run into such a case. They are
easy to generate with <tt>rbox</tt>.</p>
<p>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 ('<a href="qh-optt.htm#Tn">T4</a>') as
a guide. This is especially true if you're incorporating Qhull
into your own program. </p>
<p>When you report an error, please attach a data set to the end
of your message. Include the options that you used with Qhull,
the results of option '<a href="qh-optf.htm#FO">FO</a>', and any
messages generated by Qhull. This allows me to see the error for
myself. Qhull is maintained part-time. </p>
</blockquote>
<h2><a href="#TOC">»</a><a name="email">Email</a></h2>
<blockquote>
<p>Please send correspondence to Brad Barber at <a href=mailto:qhull@qhull.org>qhull@qhull.org</a>
and report bugs to <a href=mailto:qhull_bug@qhull.org>qhull_bug@qhull.org</a>
</a>. Let me know how you use Qhull. If you mention it in a
paper, please send a reference and abstract.</p>
<p>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. For Internet news about geometric algorithms
and convex hulls, look at comp.graphics.algorithms and
sci.math.num-analysis. For Qhull news look at <a
href="http://www.qhull.org/news">qhull-news.html</a>.</p>
</blockquote>
<h2><a href="#TOC">»</a><a name="authors">Authors</a></h2>
<blockquote>
<pre>
C. Bradford Barber Hannu Huhdanpaa
bradb@shore.net hannu@qhull.org
</pre>
</blockquote>
<h2><a href="#TOC">»</a><a name="acknowledge">Acknowledgments</a></h2>
<blockquote>
<p>A special thanks to David Dobkin for his guidance. A special
thanks to Albert Marden, Victor Milenkovic, the Geometry Center,
and Harvard University for supporting this work.</p>
<p>A special thanks to Mark Phillips, Robert Miner, and Stuart Levy for running the Geometry
Center web site long after the Geometry Center closed.
Stuart moved the web site to the University of Illinois at Champaign-Urbana.
Mark and Robert are founders of <a href=http://www.geomtech.com>Geometry Technologies</a>.
Mark, Stuart, and Tamara Munzner are the original authors of <a href=http://www.geomview.org>Geomview</a>.
<p>A special thanks to <a href="http://www.endocardial.com/">Endocardial
Solutions, Inc.</a> of St. Paul, Minnesota for their support of the
code documentation (<a href=http://www.qhull.org/src/libqhull_r/index.htm>src/libqhull_r/index.htm</a>, <a href="../src/libqhull_r/index.htm">local</a>). They use Qhull to build 3-d models of
heart chambers.</p>
<p>Qhull 1.0 and 2.0 were developed under National Science Foundation
grants NSF/DMS-8920161 and NSF-CCR-91-15793 750-7504. If you find
it useful, please let us know.</p>
<p>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.</p>
</blockquote>
<h2><a href="#TOC">»</a><a name="ref">References</a></h2>
<blockquote>
<p><a name="aure91">Aurenhammer</a>, F., "Voronoi diagrams
-- A survey of a fundamental geometric data structure," <i>ACM
Computing Surveys</i>, 1991, 23:345-405. </p>
<p>
<a name="bar-dob96">Barber</a>, C. B., D.P. Dobkin, and H.T.
Huhdanpaa, "The Quickhull Algorithm for Convex Hulls," <i>ACM
Transactions on Mathematical Software</i>, 22(4):469-483, Dec 1996, www.qhull.org
[<a
href="http://portal.acm.org/citation.cfm?doid=235815.235821">http://portal.acm.org</a>;
<a href="http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.117.405">http://citeseerx.ist.psu.edu</a>].
</p>
<p><a name="cla-sho89">Clarkson</a>, K.L. and P.W. Shor,
"Applications of random sampling in computational geometry,
II", <i>Discrete Computational Geometry</i>, 4:387-421, 1989</p>
<p><a name="cla-meh93">Clarkson</a>, K.L., K. Mehlhorn, and R.
Seidel, "Four results on randomized incremental
construction," <em>Computational Geometry: Theory and
Applications</em>, vol. 3, p. 185-211, 1993.</p>
<p><a name="devi01">Devillers</a>, et. al.,
"Walking in a triangulation," <i>ACM Symposium on
Computational Geometry</i>, June 3-5,2001, Medford MA.
<p><a name="dob-kir90">Dobkin</a>, D.P. and D.G. Kirkpatrick,
"Determining the separation of preprocessed polyhedra--a
unified approach," in <i>Proc. 17th Inter. Colloq. Automata
Lang. Program.</i>, in <i>Lecture Notes in Computer Science</i>,
Springer-Verlag, 443:400-413, 1990. </p>
<p><a name="edel01">Edelsbrunner</a>, H, <i>Geometry and Topology for Mesh Generation</i>,
Cambridge University Press, 2001.
<p><a name="gart99">Gartner, B.</a>, "Fast and robust smallest enclosing balls", <i>Algorithms - ESA '99</i>, LNCS 1643.
<p><a name="golub83">Golub, G.H. and van Loan, C.F.</a>, <i>Matric Computations</i>, Baltimore, Maryland, USA: John Hopkins Press, 1983
<p><a name="fort93">Fortune, S.</a>, "Computational
geometry," in R. Martin, editor, <i>Directions in Geometric
Computation</i>, Information Geometers, 47 Stockers Avenue,
Winchester, SO22 5LB, UK, ISBN 1-874728-02-X, 1993.</p>
<p><a name="mile93">Milenkovic, V.</a>, "Robust polygon
modeling," Computer-Aided Design, vol. 25, p. 546-566,
September 1993. </p>
<p><a name="muck96">Mucke</a>, E.P., I. Saias, B. Zhu, <i>Fast
randomized point location without preprocessing in Two- and
Three-dimensional Delaunay Triangulations</i>, ACM Symposium on
Computational Geometry, p. 274-283, 1996 [<a
href="http://www.geom.uiuc.edu/software/cglist/GeomDir/">GeomDir</a>].
</p>
<p><a name="mulm94">Mulmuley</a>, K., <i>Computational Geometry,
An Introduction Through Randomized Algorithms</i>, Prentice-Hall,
NJ, 1994.</p>
<p><a name="orou94">O'Rourke</a>, J., <i>Computational Geometry
in C</i>, Cambridge University Press, 1994.</p>
<p><a name="pre-sha85">Preparata</a>, F. and M. Shamos, <i>Computational
Geometry</i>, Springer-Verlag, New York, 1985.</p>
</blockquote>
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