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<title>Qhull for Convex Hull, Delaunay Triangulation, Voronoi Diagram, and Halfspace Intersection about a Point</title>
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<h1>Qhull</h1>
<a
href="http://www.geom.uiuc.edu/graphics/pix/Special_Topics/Computational_Geometry/cone.html"><img
src="html/qh--cone.gif" alt="[CONE]" align="middle" width="100"
height="100"></a>
</td><td>
Qhull computes convex hulls, Delaunay triangulations,
halfspace intersections about a point, Voronoi diagrams, furthest-site Delaunay
triangulations, and furthest-site Voronoi diagrams. It runs in
2-d, 3-d, 4-d, and higher dimensions. It implements the Quickhull
algorithm for computing the convex hull. Qhull handles roundoff
errors from floating point arithmetic. It computes volumes,
surface areas, and approximations to the convex hull.</p>
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<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>
</td></tr></table>
<hr>
<ul>
<li><a href="http://www.qhull.org/news">News</a> about Qhull </li>
<li><a href="http://www.qhull.org/download">Download</a>
Qhull</li>
<li><a
href="http://www.geom.uiuc.edu/graphics/pix/Special_Topics/Computational_Geometry/welcome.html">Examples
</a>of Qhull output </li>
<li><a href=http://savannah.gnu.org/projects/qhull/>Qhull development</a> at Savannah
<p>
<li><a href="http://www.qhull.org/news/qhull-news.html#use">How</a> is Qhull used?</li>
<li><a href="http://citeseer.nj.nec.com/context/86585/83502">CiteSeer</a> references to Qhull
</p>
<li><a href=http://www.google.com/search?as_q=qhull&num=100>Google</a> Qhull,
Qhull <a href=http://images.google.com/images?q=qhull&num=100>Images</a>,
Qhull in <a href=http://groups.google.com/groups?as_q=qhull&num=100&as_scoring=d>Newsgroups</a>,
and <a href=http://www.googlism.com/who_is/q/qhull/>Who is</a> Qhull?
<p>
<li><a href=http://www.mathworks.com/>MATLAB</a> uses Qhull for their computational geometry functions: <a href=http://www.mathworks.com/access/helpdesk/help/techdoc/ref/convhulln.shtml>convhulln</a>
<a href=http://www.mathworks.com/access/helpdesk/help/techdoc/ref/delaunayn.shtml>delaunayn</a>
<a href=http://www.mathworks.com/access/helpdesk/help/techdoc/ref/griddata3.shtml>griddata3</a>
<a href=http://www.mathworks.com/access/helpdesk/help/techdoc/ref/griddatan.shtml>griddatan</a>
<a href=http://www.mathworks.com/access/helpdesk/help/techdoc/ref/tsearch.shtml>tsearch</a>
<a href=http://www.mathworks.com/access/helpdesk/help/techdoc/ref/tsearchn.shtml>tsearchn</a>
<a href=http://www.mathworks.com/access/helpdesk/help/techdoc/ref/voronoin.shtml>voronoin</a>. MATLAB R14 will upgrade to
Qhull 2003.1 and triangulated output ('Qt').
</li>
<li><a href=http://www.octave.org/>GNU Octave</a> uses Qhull for their
<a href=http://octave.sourceforge.net/index/analysis.html#Geometry>computational geometry functions</a>. Octave has partially upgraded
to Qhull 2002.1 and triangulated output ('Qt').
<li><a href=http://www.wolfram.com/products/mathematica/>Mathematica</a>'s Delaunay interface: <a href=http://www.mathsource.com/Content/Enhancements/Geometry/0211-251>qh-math</a>
<li><a href=http://www.geomview.org>Geomview</a> for 3-D and 4-D visualization of Qhull output
</ul>
<p><b>Introduction</b>
<ul>
<li><a
href="http://www.ifor.math.ethz.ch/staff/fukuda/polyfaq/polyfaq.html">Fukuda's
introduction</a> to convex hulls, Delaunay
triangulations, Voronoi diagrams, and linear programming</li>
<li><a
href="http://www.cse.unsw.edu.au/~lambert/java/3d/hull.html">Lambert's
Java</a> visualization of convex hull algorithms </li>
<li><a
href="http://www.cs.sunysb.edu/~algorith/major_section/1.6.shtml">Stony
Brook</a> Algorithm Repository, computational geometry</li>
</ul>
<p><b>Qhull Documentation and Support</b>
<ul>
<li><a href="html/index.htm">Manual</a> for Qhull and rbox
<ul>
<li><a href="html/qconvex.htm">qconvex</a> -- convex hull
<li><a href="html/qdelaun.htm">qdelaunay</a> -- Delaunay triangulation
<li><a href="html/qvoronoi.htm">qvoronoi</a> -- Voronoi diagram
<li><a href="html/qhalf.htm">qhalf</a> -- halfspace intersection about a point
<li><a href="html/rbox.htm">rbox</a> -- generate point distributions
<li><a href=src/index.htm>Qhull functions</a>, macros, and data structures with source
</ul>
<li><a href="http://www.qhull.org/html/qh-faq.htm">Frequently</a> asked
questions about Qhull</li>
<li>Send e-mail to <a href=mailto:qhull@qhull.org>qhull@qhull.org</a> </li>
<li>Report bugs to <a
href="mailto:qhull_bug@qhull.org">qhull_bug@qhull.org</a>
</ul>
<p><b>Related URLs</b>
<ul>
<li><a href="http://www.geom.uiuc.edu/software/cglist">Amenta's directory</a> of
computational geometry software </li>
<li><a href=http://www.boost.org/libs/graph/doc/table_of_contents.html>BGL</a>
Boost Graph Library provides C++ classes for graph data structures
and algorithms,
<li><a
href="http://netlib.bell-labs.com/netlib/voronoi/hull.html">Clarkson's
hull </a>program with exact arithmetic for convex hulls, Delaunay triangulations,
Voronoi volumes, and alpha shapes. </li>
<li><a href="http://compgeom.cs.uiuc.edu/~jeffe/compgeom/compgeom.html">Erickson's
Computational</a> Geometry Pages and
<a href="http://compgeom.cs.uiuc.edu/~jeffe/compgeom/code.html">Software</a>
<li><a
href="http://www.cs.mcgill.ca/~fukuda/soft/cdd_home/cdd.html">Fukuda's
cdd</a> program for halfspace intersection and convex hulls</li>
<li><a href="http://www.inf.ethz.ch/personal/gaertner/miniball.html">Gartner's
Miniball</a> for fast and robust smallest enclosing balls (up to 20-d)
<li><a href=http://directory.google.com/Top/Science/Math/Geometry/Computational_Geometry/Software/>Google's directory</a> for
Science > Math > Geometry > Computational Geometry > Software
<li><a href=http://www.algorithmic-solutions.com/enleda.htm>Leda</a>
and <a href=http://www.cgal.org/>CGAL</a> libraries for writing computational
geometry programs and other combinatorial algorithms
<li><a href=http://www.magic-software.com>Magic Software</a> source code for computer
graphics, image analysis, and numerical methods
<li><a href=http://www.mathtools.net/>Mathtools.net</a> of scientific and engineering
software
<li><a href="http://www.andrew.cmu.edu/user/sowen/mesh.html">Owen's Meshing</a> Research Corner
<li><a
href="http://www-users.informatik.rwth-aachen.de/~roberts/meshgeneration.html">Schneiders'
Finite Element</a> Mesh Generation page</li>
<li><a href="http://www.cs.cmu.edu/~quake/triangle.html">Shewchuk's
triangle </a>program for 2-d Delaunay</li>
<li><a href=ftp://ftp.zib.de/pub/Packages/mathprog/index.html>Skorobohatyj's Mathprog@ZIB</a> for
mathematical software
<li><a href=http://www.voronoi.com>Voronoi Web Site</a> for all things Voronoi
</ul>
<p><b>FAQs and Newsgroups</b>
<ul>
<li><a
href="http://exaflop.org/docs/cgafaq/">FAQ</a>
for computer graphics algorithms
(<a href="http://exaflop.org/docs/cgafaq/cga6.html">geometric</a> structures)
</li>
<li><a
href="http://www-unix.mcs.anl.gov/otc/Guide/faq/linear-programming-faq.html">FAQ
</a>for linear programming </li>
<li><a href="news:comp.graphics.algorithms">Newsgroup</a>:
comp.graphics.algorithms </li>
<li><a href="news:comp.soft-sys.matlab">Newsgroup</a>:
comp.soft-sys.matlab</li>
<li><a href="news:sci.math.num-analysis">Newsgroup</a>:
sci.math.num-analysis </li>
<li><a href="news:sci.op-research">Newsgroup</a>:
sci.op-research </li>
</ul>
</blockquote>
<hr>
<p>The program includes options for input transformations,
randomization, tracing, multiple output formats, and execution
statistics. The program can be called from within your
application. </p>
<p>You can view the results in 2-d, 3-d and 4-d with <a
href="http://www.geomview.org">Geomview</a>. An alternative
is <a href=http://www.kitware.com/vtk.html>VTK</a>.</p>
<p>For an article about Qhull, download from <a
href="http://citeseer.nj.nec.com/83502.html">CiteSeer</a>:
</p>
<blockquote>
<p>Barber, C.B., Dobkin, D.P., and Huhdanpaa, H.T., "The
Quickhull algorithm for convex hulls," <i>ACM Trans. on
Mathematical Software</i>, Dec 1996. <a
href="http://www.acm.org/pubs/citations/journals/toms/1996-22-4/p469-barber/">http://www.acm.org</a>;
</p>
</blockquote>
<p>Abstract: </p>
<blockquote>
<p>The convex hull of a set of points is the smallest convex
set that contains the points. This article presents a
practical convex hull algorithm that combines the
two-dimensional Quickhull Algorithm with the general
dimension Beneath-Beyond Algorithm. It is similar to the
randomized, incremental algorithms for convex hull and
Delaunay triangulation. We provide empirical evidence that
the algorithm runs faster when the input contains non-extreme
points, and that it uses less memory. </p>
<p>Computational geometry algorithms have traditionally
assumed that input sets are well behaved. When an algorithm
is implemented with floating point arithmetic, this
assumption can lead to serious errors. We briefly describe a
solution to this problem when computing the convex hull in
two, three, or four dimensions. The output is a set of
"thick" facets that contain all possible exact convex hulls
of the input. A variation is effective in five or more
dimensions. </p>
</blockquote>
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<p><b>Up:</b> <a href="http://www.geom.uiuc.edu/software/past-projects.html"><i>Past Software
Projects of the Geometry Center</i></a> <br>
<b>URL:</b> <a href="http://www.qhull.org">http://www.qhull.org</a>
<br><b>To:</b>
<a href="http://www.qhull.org/news">News</a>
• <a href="http://www.qhull.org/download">Download</a>
• <a href="http://citeseer.nj.nec.com/83502.html">CiteSeer</a>
• <a href=http://images.google.com/images?q=qhull&num=100>Images</a>
• <a href="html/index.htm#TOC">Manual</a>
• <a href="http://www.qhull.org/html/qh-faq.htm">FAQ</a>
• <a href="html/qh-quick.htm#programs">Programs</a>
• <a href="html/qh-quick.htm#options">Options</a>
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<p><a href="http://www.geom.uiuc.edu/"><img src="html/qh--geom.gif" alt="[HOME]"
align="middle"></a> <i>The Geometry Center Home Page</i> </p>
<p>Comments to: <a href="mailto:qhull@qhull.org">qhull@qhull.org</a>
<br>
Created: May 17 1995 --- <!-- hhmts start -->
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