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<title>Qhull code 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>
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Qhull computes the convex hull, Delaunay triangulation, Voronoi diagram,
halfspace intersection about a point, furthest-site Delaunay
triangulation, and furthest-site Voronoi diagram. The source code runs in
2-d, 3-d, 4-d, and higher dimensions. Qhull implements the Quickhull
algorithm for computing the convex hull. It 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 triangulation of non-convex surfaces, mesh
generation of non-convex objects, medium-sized inputs in 9-D
and higher, alpha shapes, weighted Voronoi diagrams, Voronoi volumes, or
constrained Delaunay triangulations,  </p>

<p>Qhull 2015.2 introduces reentrant Qhull.  It allows concurrent Qhull runs and simplifies the C++ interface to Qhull.
If you call Qhull from your program, you should use reentrant Qhull (libqhull_r) instead of qh_QHpointer (libqhull).
If you use Qhull 2003.1. please upgrade or apply <a href="http://www.qhull.org/download/poly.c-qh_gethash.patch">poly.c-qh_gethash.patch</a>.
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    <li><a href="http://www.qhull.org/news">News</a> and
        <a href="http://www.qhull.org/news/qhull-news.html#bugs">Bugs</a>
        about Qhull 2015.2  2016/01/18</li>
         <li><a href="http://www.qhull.org/download">Download</a> Qhull (<a href="http://www.qhull.org/src/Changes.txt">changes</a>)</li>
        <li><a href=http://github.com/qhull/qhull/wiki>GitHub</a> C++ interface to Qhull
        <li><a href="html/index.htm">Manual</a> for Qhull and rbox
        <li><a href="html/index.htm#geomview">Geomview</a> for 3-D and 4-D visualization of Qhull output
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                www.qhull.org
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    <li><a href="http://www.qhull.org/news/qhull-news.html#users">How</a> is Qhull used?</li>
        <li><a href="http://scholar.google.com/scholar?cites=13151392091060773178&as_sdt=40000005">Google Scholar</a>,
        <a href="http://libra.msra.cn/Publication/232063/the-quickhull-algorithm-for-convex-hulls">Microsoft Academic</a>,
        and <a href="http://citeseerx.ist.psu.edu/showciting?doi=10.1.1.117.405&sort=cite">CiteSeer</a>
        references to Qhull
    <li>
    <a href=http://www.google.com/search?as_q=qhull+-debian+-cvs+-gentoo+-pool+-mirrors&num=100>Google</a> Qhull,
         <a href="http://images.google.com/images?q=qhull&num=100">Images</a>,
         <a href="http://www.google.com/#q=qhull&tbm=bks">Books</a>,
         <a href="http://www.google.com/search?q=qhull&tbm=pts">Patents</a>,
         <a href="http://groups.google.com/groups?as_q=qhull&num=100&as_scoring=d">Newsgroups</a>,
         <a href="http://www.google.com/search?q=qhull&tbm=blg">Blogs</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 n-d computational geometry functions:
        <a href=http://www.mathworks.com/help/techdoc/ref/convhulln.html>convhulln</a>
        <a href=http://www.mathworks.com/help/techdoc/ref/delaunayn.html>delaunayn</a>
        <a href=http://www.mathworks.com/help/techdoc/ref/griddatan.html>griddatan</a>
        <a href=http://www.mathworks.com/help/techdoc/ref/voronoin.html>voronoin</a>.
    </li>
        <li>The <a href="http://cran.r-project.org/web/packages/geometry/geometry.pdf">geometry</a> package of <a href="http://www.r-project.org/">R</a> provides <a href="http://geometry.r-forge.r-project.org/">Qhull in R</a>.
        <li>The <a href="http://packages.debian.org/sid/octave3.2">Debian build</a> of
        <a href=http://www.octave.org/>GNU Octave</a> includes Qhull for <a href="http://www.gnu.org/software/octave/doc/interpreter/Geometry.html">computational geometry<a>.
        <li><a href=http://www.wolfram.com/products/mathematica/>Mathematica</a>'s Delaunay interface <a href=http://library.wolfram.com/infocenter/MathSource/1160/>qh-math</a>
        and <a href="http://portal.uni-freiburg.de/imteksimulation/downloads/ims">QHullInterface</a>
     <li>The <a href="http://docs.scipy.org/doc/scipy/reference/tutorial/spatial.html">scipy.spatial</a> of <a href="http://scipy.org/">SciPy</a> is implemented with Qhull.  It includes
     support for Delaunay triangulations,
</ul>
</form>

<p><b>Introduction</b>
<ul>
    <li><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</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.algorithmic-solutions.info/leda_guide/geometryalgorithms.html"
                >LEDA Guide</a> to geometry algorithms
        <li><a
            href="http://mathworld.wolfram.com/ComputationalGeometry.html"
                >MathWorld's</a> Computational Geometry from Wolfram Research
        <li><a
                href="http://www.cs.sunysb.edu/~algorith/major_section/1.6.shtml"
                >Skiena's</a> Computational Geometry from his <i>Algorithm Design Manual</i>.
    <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
   <table><tr><td>
        <ul>
          <li><a href="html/index.htm#description">Description</a> of Qhull
                <li><a href="html/qh-impre.htm">Imprecision</a> in Qhull
                <li><a href="html/qh-quick.htm#programs">Programs</a> and <a href="html/qh-quick.htm#options">Options</a>
                    quick reference
                <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
        </ul></td><td><ul>
            <li><a href="http://www.qhull.org/html/qh-faq.htm">Frequently</a> asked
                questions about Qhull</li>
            <li><a href="html/index.htm#geomview">Geomview</a> for visualizing Qhull
            <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="html/qh-code.htm">Calling Qhull</a> from your program
            <li><a href="src/libqhull_r/index.htm">Reentrant</a> Qhull functions, macros, and data structures with source
            </ul>
        </td></tr></table>
    <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://www.netlib.org/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 (<a
        href="http://www.csb.ethz.ch/tools/polco">Polco/Java</a>)</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://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.mathtools.net/>Mathtools.net</a> of scientific and engineering
        software
    <li><a href="http://www.imr.sandia.gov/papers/topics.html">Owen's International Meshing</a> Roundtable
    <li><a
        href="http://www.robertschneiders.de/meshgeneration/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=http://www.voronoi.com>Voronoi Web Site</a> for all things Voronoi
        <li>Young's <a href="http://homepage.usask.ca/~ijm451/finite/fe_resources/">Internet Finite Element Resources</a>
        <li><a href="http://www.uic.nnov.ru/~zny/skeleton/">Zolotykh's Skeleton</a> generates all extreme rays of a polyhedral cone using the Double Description Method</li>
        <li><a href="https://github.com/tomilov/quickhull/blob/master/include/quickhull.hpp">Tomilov's quickhull.hpp</a> (<a href="http://habrahabr.ru/post/245221/">doc-ru</a>) implements the Quickhull algorithm for points in general position.
</ul>

<p><b>FAQs and Newsgroups</b>
<ul>
    <li><a
        href="http://www.faqs.org/faqs/graphics/algorithms-faq/">FAQ</a>
        for computer graphics algorithms
                (Exaflop.org: <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.vtk.org/>VTK</a>.</p>

<p>For an article about Qhull, download from
  <a href="http://dl.acm.org/authorize?89250">ACM</a> or <a
    href="http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.117.405">CiteSeer</a>:
</p>

<blockquote>
    <p>Barber, C.B., Dobkin, D.P., and Huhdanpaa, H.T., &quot;The
    Quickhull algorithm for convex hulls,&quot; <i>ACM Trans. on
    Mathematical Software</i>, 22(4):469-483, Dec 1996, http://www.qhull.org</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>Comments to: <a href="mailto:qhull@qhull.org">qhull@qhull.org</a>
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