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			<p><a name="TOP"><b>Up:</b></a> <a
			href="http://www.qhull.org">Home page</a> for Qhull
			(http://www.qhull.org)<br>
			<b>Up:</b> <A href="index.htm#TOC">Qhull manual</A>: Table of Contents<br>
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			<b>To:</b> <A href="#TOC">FAQ: Table of Contents</A> (please
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			<hr>
			<!-- Main text of document -->
			<h1><a
			href="http://www.geom.uiuc.edu/graphics/pix/Special_Topics/Computational_Geometry/4dcube.html"><IMG
			align=middle alt ="[4-d cube]"
			height=100 src="qh--4d.gif" width=100 ></a> Frequently Asked Questions about Qhull</h1><!--
			<p><i>This is the most recent FAQ. It was updated Sept. 13, 1999.</i>
			-->
			<p>If your question does not appear here, see: </p>

			<ul>
			<li><a href="http://www.qhull.org/news">News</a> about Qhull
			<li><A href="index.htm#TOC">Qhull manual:</A> table of contents
			<li><A href="../README.txt">Installation</A> instructions for Qhull and rbox

			<li><A href="mailto:qhull@qhull.org">Send e-mail</A> to
			qhull@qhull.org
			<li><A href="mailto:qhull_bug@qhull.org">Report bugs</A>
			to qhull_bug@qhull.org </li>
			</ul>

			<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. For a detailed introduction, see O'Rourke [<A
			href="index.htm#orou94" >'94</A>], <i>Computational Geometry in C</i>.
			</p>

			<p>There are separate programs for each application of
			Qhull. These programs disable experimental and inappropriate
			options. If you prefer, you may use Qhull directly. All programs
			run the same code.

			<p>Version 3.1 added triangulated output ('<A href="qh-optq.htm#Qt">Qt</A>').
			It should be used for Delaunay triangulations instead of
			using joggled input ('<A href="qh-optq.htm#QJn">QJ</A>').

			<p><i>Brad Barber, Arlington MA,
			2010/01/09 <!--
			--> </i></p>

			<p><b>Copyright © 1998-2015 C.B. Barber</b></p>

			<hr>

			<h2><A href="#TOP">»</A><a name="TOC">FAQ: Table of Contents </a></h2>

			<p>Within each category, the most recently asked questions are
			first.
			<ul>
			<li>Startup questions <ul>
			<li><A href="#console">How</A> do I run Qhull from Windows?
			<li><A href="#input">How</A> do I enter points for Qhull?
			<li><A href="#learn">How</A> do I learn to use Qhull?</li>
			</ul>
			<li>Convex hull questions<ul>
			<li><A href="#area">How</A> do I report just the area and volume of a
			convex hull?
			<li><A href="#extra">Why</A> are there extra points in a 4-d or higher
			convex hull?
			<li><A href="#dup">How</A> do I report duplicate
			vertices? </li>
			</ul>
			<li>Delaunay triangulation questions<ul>
			<li><A href="#flat">How</A> do I get rid of nearly flat Delaunay
			triangles?
			<li><A href="#vclosest">How</A> do I find the Delaunay triangle or Voronoi
			region that is closest to a point?

			<li><A href="#mesh">How</A> do I compute the Delaunay triangulation of a
			non-convex object?

			<li><A href="#mesh">How</A> do I mesh a volume from a set of triangulated
			surface points?

			<li><A href="#constrained">Can</A> Qhull produce a triangular mesh for an
			object?

			<li><A href="#dridges">For</A> 3-d Delaunay triangulations, how do I
			report the triangles of each tetrahedron?

			<li><A href="#3dd">How</A> do I construct a 3-d Delaunay triangulation?
			<li><A href="#2d">How</A> do I get the triangles for a 2-d Delaunay
			triangulation and the vertices of its Voronoi diagram?
			<li><A href="#big">Can </A>Qhull triangulate a
			hundred 16-d points?</li>
			</ul>

			<li>Voronoi diagram questions<ul>
			<li>See also "Delaunay diagram questions". Qhull computes the Voronoi diagram from the Delaunay triagulation.
			<li><a href="#volume">How</a> do I compute the volume of a Voronoi region?
			<li><a href="#maxsphere">How</a> do I get the radii of the empty
			spheres for each Voronoi vertex?

			<li><A href="#square">What</A> is the Voronoi diagram of a square?

			<li><A href="#vsphere">How</A> do I construct the Voronoi diagram of
			cospherical points?
			<li><A href="#rays">Can</A> Qhull compute the unbounded rays of the
			Voronoi diagram?
			</ul>
			<li>Approximation questions<ul>
			<li><A href="#simplex">How</A> do I approximate data
			with a simplex?</li>
			</ul>
			<li>Halfspace questions<ul>
			<li><A href="#halfspace">How</A> do I compute the
			intersection of halfspaces with Qhull?</li>
			</ul>
			<li><a name="library">Qhull library</a> questions<ul>
			<li><A href="#math">Is</A> Qhull available for Mathematica, Matlab, or
			Maple?

			<li><A href="#ridges">Why</A> are there too few ridges?
			<li><A href="#call">Can</A> Qhull use coordinates without placing them in
			a data file?
			<li><A href="#size">How</A> large are Qhull's data structures?
			<li><A href="#inc">Can</A> Qhull construct convex hulls and Delaunay
			triangulations one point at a time?
			<li><A href="#ridges2">How</A> do I visit the ridges of a Delaunay
			triangulation?
			<li><A href="#listd">How</A> do I visit the Delaunay facets?
			<LI><A
			href="#outside">When</A> is a point outside or inside a facet?
			<li><A href="#closest">How</A> do I find the facet that is closest to a
			point?
			<li><A href="#vclosest">How</A> do I find the Delaunay triangle or Voronoi
			region that is closest to a point?
			<li><A href="#vertices">How</A> do I list the vertices?
			<li><A href="#test">How</A> do I test code that uses the Qhull library?
			<li><A href="#orient">When</A> I compute a plane
			equation from a facet, I sometimes get an
			outward-pointing normal and sometimes an
			inward-pointing normal</li>
			</ul>
			</li>
			</ul>

			<hr>

			<h2><A href="#TOC">»</A><a name="startup">Startup</a> questions</h2>

			<h4><A href="#TOC">»</A><a name="console">How</a> do I run Qhull
			from Windows?</h4><blockquote>

			<p>Qhull is a console program. You will first need a command window
			(i.e., a "command prompt"). You can double click on
			'eg\Qhull-go.bat'. </p>

			<blockquote><ul>
			<li>Type 'qconvex', 'qdelaunay', 'qhalf', 'qvoronoi,
			'qhull', and 'rbox' for a synopsis of each program.

			<li>Type 'rbox c D2 \| qconvex s i' to compute the
			convex hull of a square.

			<li>Type 'rbox c D2 \| qconvex s i TO results.txt' to
			write the results to the file 'results.txt'. A summary is still printed on
			the the console.

			<li>Type 'rbox c D2' to see the input format for
			qconvex.

			<li>Type 'qconvex < data.txt s i TO results.txt' to
			read input data from 'data.txt'.

			<li>If you want to enter data by hand, type 'qconvex s i TO
			results.txt' to read input data from the console. Type in
			the numbers and end with a ctrl-D. </li>
			</ul></blockquote>

			</blockquote><h4><A href="#TOC">»</A><a name="input">How</a> do I enter
			points for Qhull?</h4><blockquote>

			<p>Qhull takes its data from standard input. For example, create
			a file named 'data.txt' with the following contents: </p>

			<blockquote>
			<pre>
			2 #sample 2-d input
			5 #number of points
			1 2 #coordinates of points
			-1.1 3
			3 2.2
			4 5
			-10 -10
			</pre>
			</blockquote>

			<p>Then call qconvex with 'qconvex < data.txt'. It will print a
			summary of the convex hull. Use 'qconvex < data.txt o' to print
			the vertices and edges. See also <A href="index.htm#input">input
			format</A>. </p>

			<p>You can generate sample data with rbox, e.g., 'rbox 10'
			generates 10 random points in 3-d. Use a pipe ('\|') to run rbox
			and qhull together, e.g., </p>

			<blockquote>
			<p>rbox c \| qconvex o </p>
			</blockquote>

			<p>computes the convex hull of a cube. </p>

			</blockquote><h4><A href="#TOC">»</A><a name="learn">How</a> do I learn to
			use Qhull?</h4><blockquote>

			<p>First read: </p>

			<ul>
			<li><A href="index.htm">Introduction</A> to Qhull
			<li><A href="index.htm#when">When</A> to use Qhull
			<li><A href="qconvex.htm">qconvex</A> -- convex hull
			<li><A href="qdelaun.htm">qdelaunay</A> -- Delaunay triangulation
			<li><A href="qhalf.htm">qhalf</A> -- half-space intersection about a point

			<li><A href="qvoronoi.htm">qvoronoi</A> -- Voronoi diagram
			<li><A href="rbox.htm">Rbox</A>, for sample inputs
			<li><A href="qh-eg.htm">Examples</A> of Qhull</li>
			</ul>

			<p>Look at Qhull's on-line documentation: </p>

			<ul>
			<li>'qconvex' gives a synopsis of qconvex and its options

			<li>'rbox' lists all of the options for generating point
			sets
			<li>'qconvex - \| more' lists the options for qconvex
			<li>'qconvex .' gives a concise list of options
			<li>'qdelaunay', 'qhalf', 'qvoronoi', and 'qhull' also have a synopsis and option list</li>
			</ul>

			<p>Then try out the Qhull programs on small examples. </p>

			<ul>
			<li>'rbox c' lists the vertices of a cube
			<li>'rbox c \| qconvex' is the convex hull of a cube
			<li>'rbox c \| qconvex o' lists the vertices and facets of
			a cube
			<li>'rbox c \| qconvex Qt o' triangulates the cube
			<li>'rbox c \| qconvex QJ o' joggles the input and
			triangulates the cube
			<li>'rbox c D2 \| qconvex' generates the convex hull of a
			square
			<li>'rbox c D4 \| qconvex' generates the convex hull of a
			hypercube
			<li>'rbox 6 s D2 \| qconvex p Fx' lists 6 random points in
			a circle and lists the vertices of their convex hull in order
			<li>'rbox c D2 c G2 \| qdelaunay' computes the Delaunay
			triangulation of two embedded squares. It merges the cospherical facets.
			<li>'rbox c D2 c G2 \| qdelaunay Qt' computes the Delaunay
			triangulation of two embedded squares. It triangulates the cospherical facets.
			<li>'rbox c D2 c G2 \| qvoronoi o' computes the
			corresponding Voronoi vertices and regions.
			<li>'rbox c D2 c G2 \| qvoronio Fv' shows the Voronoi diagram
			for the previous example. Each line is one edge of the
			diagram. The first number is 4, the next two numbers list
			a pair of input sites, and the last two numbers list the
			corresponding pair of Voronoi vertices. </li>
			</ul>

			<p>Install <a href="http://www.geomview.org">Geomview</a>
			if you are running SGI Irix, Solaris, SunOS, Linux, HP, IBM
			RS/6000, DEC Alpha, or Next. You can then visualize the output of
			Qhull. Qhull comes with Geomview <A href="qh-eg.htm">examples</A>.
			</p>

			<p>Then try Qhull with a small example of your application. Work
			out the results by hand. Then experiment with Qhull's options to
			find the ones that you need. </p>

			<p>You will need to decide how Qhull should handle precision
			problems. It can triangulate the output ('<A
			href="qh-optq.htm#Qt" >Qt</A>'), joggle the input ('<A
			href="qh-optq.htm#QJn" >QJ</A>'), or merge facets (the default). </p>

			<ul>
			<li>With joggle, Qhull produces simplicial (i.e.,
			triangular) output by joggling the input. After joggle,
			no points are cocircular or cospherical.
			<li>With facet merging, Qhull produces a better
			approximation and does not modify the input.
			<li>With triangulated output, Qhull merges facets and triangulates
			the result.</li>
			<li>See <A href="qh-impre.htm#joggle">Merged facets or joggled input</A>. </li>
			</ul>

			</blockquote>
			<h2><A href="#TOC">»</A><a name="convex">Convex hull questions</a></h2>

			<h4><A href="#TOC">»</A><a name="area">How</a> do I report just the area
			and volume of a convex hull?</h4><blockquote>

			Use option 'FS'. For example,

			<blockquote><pre>
			C:\qhull>rbox 10 \| qconvex FS
			0
			2 2.192915621644613 0.2027867899638665

			C:\qhull>rbox 10 \| qconvex FA

			Convex hull of 10 points in 3-d:

			Number of vertices: 10
			Number of facets: 16

			Statistics for: RBOX 10 \| QCONVEX FA

			Number of points processed: 10
			Number of hyperplanes created: 28
			Number of distance tests for qhull: 44
			CPU seconds to compute hull (after input): 0
			Total facet area: 2.1929156
			Total volume: 0.20278679
			</pre></blockquote>

			</blockquote><h4><A href="#TOC">»</A><a name="extra">Why</a> are there extra
			points in a 4-d or higher convex hull?</h4><blockquote>

			<p>You may see extra points if you use options '<A
			href="qh-opto.htm#i" >i</A>' or '<A href="qh-optf.htm#Ft">Ft</A>'
			without using triangulated output ('<A href="qh-optq.htm#Qt">Qt</A>').
			The extra points occur when a facet is non-simplicial (i.e., a
			facet with more than <i>d</i> vertices). For example, Qhull
			reports the following for one facet of the convex hull of a hypercube.
			Option 'Pd0:0.5' returns the facet along the positive-x axis: </p>

			<blockquote>
			<pre>
			rbox c D4 \| qconvex i Pd0:0.5
			12
			17 13 14 15
			17 13 12 14
			17 11 13 15
			17 14 11 15
			17 10 11 14
			17 14 12 8
			17 12 13 8
			17 10 14 8
			17 11 10 8
			17 13 9 8
			17 9 11 8
			17 11 9 13
			</pre>
			</blockquote>

			<p>The 4-d hypercube has 16 vertices; so point "17" was
			added by qconvex. Qhull adds the point in order to report a
			simplicial decomposition of the facet. The point corresponds to
			the "centrum" which Qhull computes to test for
			convexity. </p>

			<p>Triangulate the output ('<A href="qh-optq.htm#Qt">Qt</A>') to avoid the extra points.
			Since the hypercube is 4-d, each simplicial facet is a tetrahedron.
			<blockquote>
			<pre>
			C:\qhull3.1>rbox c D4 \| qconvex i Pd0:0.5 Qt
			9
			9 13 14 15
			12 9 13 14
			9 11 13 15
			11 9 14 15
			9 10 11 14
			12 9 14 8
			9 12 13 8
			9 10 14 8
			10 9 11 8
			</pre>
			</blockquote>

			<p>Use the '<A href="qh-optf.htm#Fv">Fv</A>' option to print the
			vertices of simplicial and non-simplicial facets. For example,
			here is the same hypercube facet with option 'Fv' instead of 'i':
			</p>

			<blockquote>
			<pre>
			C:\qhull>rbox c D4 \| qconvex Pd0:0.5 Fv
			1
			8 9 10 12 11 13 14 15 8
			</pre>
			</blockquote>

			<p>The coordinates of the extra point are printed with the '<A
			href="qh-optf.htm#Ft" >Ft</A>' option. </p>

			<blockquote>
			<pre>
			rbox c D4 \| qconvex Pd0:0.5 Ft
			4
			17 12 3
			-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
			-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
			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 0 0
			4 16 13 14 15
			4 16 13 12 14
			4 16 11 13 15
			4 16 14 11 15
			4 16 10 11 14
			4 16 14 12 8
			4 16 12 13 8
			4 16 10 14 8
			4 16 11 10 8
			4 16 13 9 8
			4 16 9 11 8
			4 16 11 9 13
			</pre>
			</blockquote>

			</blockquote><h4><A href="#TOC">»</A><a name="dup">How</a> do I report
			duplicate vertices?</h4><blockquote>

			<p>There's no direct way. You can use option
			'<A href="qh-optf.htm#FP">FP</A>' to
			report the distance to the nearest vertex for coplanar input
			points. Select the minimum distance for a duplicated vertex, and
			locate all input sites less than this distance. </p>

			<p>For Delaunay triangulations, all coplanar points are nearly
			incident to a vertex. If you want a report of coincident input
			sites, do not use option '<A href="qh-optq.htm#QJn">QJ</A>'. By
			adding a small random quantity to each input coordinate, it
			prevents coincident input sites. </p>

			</blockquote>
			<h2><A href="#TOC">»</A><a name="delaunay">Delaunay triangulation questions</a></h2>

			<h4><A href="#TOC">»</A><a name="flat">How</a> do I get rid of
			nearly flat Delaunay triangles?</h4><blockquote>

			<p>Nearly flat triangles occur when boundary points are nearly
			collinear or coplanar. They also occur for nearly coincident
			points. Both events can easily occur when using joggle. For example
			(rbox 10 W0 D2 \| qdelaunay QJ Fa) lists the areas of the Delaunay
			triangles of 10 points on the boundary of a square. Some of
			these triangles are nearly flat. This occurs when one point
			is joggled inside of two other points. In this case, nearly flat
			triangles do not occur with triangulated output (rbox 10 W0 D2 \| qdelaunay Qt Fa).


			<p>Another example, (rbox c P0 P0 D2 \| qdelaunay QJ Fa), computes the
			areas of the Delaunay triangles for the unit square and two
			instances of the origin. Four of the triangles have an area
			of 0.25 while two have an area of 2.0e-11. The later are due to
			the duplicated origin. With triangulated output (rbox c P0 P0 D2 \| qdelaunay Qt Fa)
			there are four triangles of equal area.

			<p>Nearly flat triangles also occur without using joggle. For
			example, (rbox c P0 P0,0.4999999999 \| qdelaunay Fa), computes
			the areas of the Delaunay triangles for the unit square,
			a nearly collinear point, and the origin. One triangle has an
			area of 3.3e-11.

			<p>Unfortunately, none of Qhull's merging options remove nearly
			flat Delaunay triangles due to nearly collinear or coplanar boundary
			points.
			The merging options concern the empty circumsphere
			property of Delaunay triangles. This is independent of the area of
			the Delaunay triangles. Qhull does handle nearly coincident points.

			<p>You can handle collinear or coplanar boundary points by
			enclosing the points in a box. For example,
			(rbox c P0 P0,0.4999999999 c G1 \| qdelaunay Fa), surrounds the
			previous points with [(1,1), (1,-1), (-1,-1), (-1, 1)].
			Its Delaunay triangulation does not include a
			nearly flat triangle. The box also simplifies the graphical
			output from Qhull.

			<p>Without joggle, Qhull lists coincident points as "coplanar"
			points. For example, (rbox c P0 P0 D2 \| qdelaunay Fa), ignores
			the duplicated origin and lists four triangles of size 0.25.
			Use 'Fc' to list the coincident points (e.g.,
			rbox c P0 P0 D2 \| qdelaunay Fc).

			<p>There is no easy way to determine coincident points with joggle.
			Joggle removes all coincident, cocircular, and cospherical points
			before running Qhull. Instead use facet merging (the default)
			or triangulated output ('<A href="qh-optq.htm#Qt">Qt</A>').

			</blockquote><h4><A href="#TOC">»</A><a name="mesh">How</a> do I compute
			the Delaunay triangulation of a non-convex object?</h4><blockquote>

			<p>A similar question is
			"How do I mesh a volume from a set of triangulated surface points?"

			<p>This is an instance of the constrained Delaunay Triangulation
			problem. Qhull does not handle constraints. The boundary of the
			Delaunay triangulation is always convex. But if the input set
			contains enough points, the triangulation will include the
			boundary. The number of points needed depends on the input.

			<p>Shewchuk has developed a theory of constrained Delaunay triangulations.
			See his
			<a href="http://www.cs.cmu.edu/~jrs/jrspapers.html#cdt">paper</a> at the
			1998 Computational Geometry Conference. Using these ideas, constraints
			could be added to Qhull. They would have many applications.

			<p>There is a large literature on mesh generation and many commercial
			offerings. For pointers see
			<a href="http://morden.csee.usf.edu/dragon/kpalbrec/mesh.html">Owen's Meshing
			Research Corner</a> and
			<a href="http://www-users.informatik.rwth-aachen.de/~roberts/meshgeneration.html">Schneiders'
			Finite Element Mesh Generation page</a>.</p>

			</blockquote><h4><A href="#TOC">»</A><a name="constrained">Can</a> Qhull
			produce a triangular mesh for an object?</h4><blockquote>

			<p>Yes for convex objects, no for non-convex objects. For
			non-convex objects, it triangulates the concavities. Unless the
			object has many points on its surface, triangles may cross the
			surface. </p>

			</blockquote><h4><A href="#TOC">»</A><a name="dridges">For</a> 3-d Delaunay
			triangulations, how do I report the triangles of each
			tetrahedron?</h4><blockquote>

			<p>For points in general position, a 3-d Delaunay triangulation
			generates tetrahedron. Each face of a tetrahedron is a triangle.
			For example, the 3-d Delaunay triangulation of random points on
			the surface of a cube, is a cellular structure of tetrahedron. </p>

			<p>Use triangulated output ('qdelaunay Qt i') or joggled input ('qdelaunay QJ i')
			to generate the Delaunay triangulation.
			Option 'i' reports each tetrahedron. The triangles are
			every combination of 3 vertices. Each triangle is a
			"ridge" of the Delaunay triangulation. </p>

			<p>For example, </p>

			<pre>
			rbox 10 \| qdelaunay Qt i
			14
			9 5 8 7
			0 9 8 7
			5 3 8 7
			3 0 8 7
			5 4 8 1
			4 6 8 1
			2 9 5 8
			4 2 5 8
			4 2 9 5
			6 2 4 8
			9 2 0 8
			2 6 0 8
			2 4 9 1
			2 6 4 1
			</pre>

			<p>is the Delaunay triangulation of 10 random points. Ridge 9-5-8
			occurs twice. Once for tetrahedron 9 5 8 7 and the other for
			tetrahedron 2 9 5 8. </p>

			<p>You can also use the Qhull library to generate the triangles.
			See "<A href="#ridges2">How</A> do I visit the ridges of a
			Delaunay triangulation?"</p>

			</blockquote><h4><A href="#TOC">»</A><a name="3dd">How</a> do I construct a
			3-d Delaunay triangulation?</h4><blockquote>

			<p>For 3-d Delaunay triangulations with cospherical input sites,
			use triangulated output ('<A href="qh-optq.htm#Qt">Qt</A>') or
			joggled input ('<A href="qh-optq.htm#QJn">QJ</A>'). Otherwise
			option 'i' will
			triangulate non-simplicial facets by adding a point to the facet.

			<p>If you want non-simplicial output for cospherical sites, use
			option
			'<A href="qh-optf.htm#Fv">Fv</A>' or '<A href="qh-opto.htm#o">o</A>'.
			For option 'o', ignore the last coordinate. It is the lifted
			coordinate for the corresponding convex hull in 4-d.

			<p>The following example is a cube
			inside a tetrahedron. The 8-vertex facet is the cube. Ignore the
			last coordinates. </p>

			<blockquote>
			<pre>
			C:\qhull>rbox r y c G0.1 \| qdelaunay Fv
			4
			12 20 44
			0.5 0 0 0.3055555555555555
			0 0.5 0 0.3055555555555555
			0 0 0.5 0.3055555555555555
			-0.5 -0.5 -0.5 0.9999999999999999
			-0.1 -0.1 -0.1 -6.938893903907228e-018
			-0.1 -0.1 0.1 -6.938893903907228e-018
			-0.1 0.1 -0.1 -6.938893903907228e-018
			-0.1 0.1 0.1 -6.938893903907228e-018
			0.1 -0.1 -0.1 -6.938893903907228e-018
			0.1 -0.1 0.1 -6.938893903907228e-018
			0.1 0.1 -0.1 -6.938893903907228e-018
			0.1 0.1 0.1 -6.938893903907228e-018
			4 2 11 1 0
			4 10 1 0 3
			4 11 10 1 0
			4 2 9 0 3
			4 9 11 2 0
			4 7 2 1 3
			4 11 7 2 1
			4 8 10 0 3
			4 9 8 0 3
			5 8 9 10 11 0
			4 10 6 1 3
			4 6 7 1 3
			5 6 8 10 4 3
			5 6 7 10 11 1
			4 5 9 2 3
			4 7 5 2 3
			5 5 8 9 4 3
			5 5 6 7 4 3
			8 5 6 8 7 9 10 11 4
			5 5 7 9 11 2
			</pre>
			</blockquote>

			<p>If you want simplicial output use options
			'<A href="qh-optq.htm#Qt">Qt</A> <A
			href="qh-optf.htm#Ft" >i</A>' or
			'<A href="qh-optq.htm#QJn">QJ</A> <A
			href="qh-optf.htm#Ft" >i</A>', e.g.,
			</p>

			<blockquote>
			<pre>
			rbox r y c G0.1 \| qdelaunay Qt i
			31
			2 11 1 0
			11 10 1 0
			9 11 2 0
			11 7 2 1
			8 10 0 3
			9 8 0 3
			10 6 1 3
			6 7 1 3
			5 9 2 3
			7 5 2 3
			9 8 10 11
			8 10 11 0
			9 8 11 0
			6 8 10 4
			8 6 10 3
			6 8 4 3
			6 7 10 11
			10 6 11 1
			6 7 11 1
			8 5 4 3
			5 8 9 3
			5 6 4 3
			6 5 7 3
			5 9 10 11
			8 5 9 10
			7 5 10 11
			5 6 7 10
			8 5 10 4
			5 6 10 4
			5 9 11 2
			7 5 11 2
			</pre>
			</blockquote>

			</blockquote><h4><A href="#TOC">»</A><a name="2d">How</a> do I get the
			triangles for a 2-d Delaunay triangulation and the vertices of
			its Voronoi diagram?</h4><blockquote>

			<p>To compute the Delaunay triangles indexed by the indices of
			the input sites, use </p>

			<blockquote>
			<p>rbox 10 D2 \| qdelaunay Qt i </p>
			</blockquote>

			<p>To compute the Voronoi vertices and the Voronoi region for
			each input site, use </p>

			<blockquote>
			<p>rbox 10 D2 \| qvoronoi o </p>
			</blockquote>

			<p>To compute each edge ("ridge") of the Voronoi
			diagram for each pair of adjacent input sites, use</p>

			<blockquote>
			<p>rbox 10 D2 \| qvoronoi Fv </p>
			</blockquote>

			<p>To compute the area and volume of the Voronoi region for input site 5 (site 0 is the first one),
			use </p>

			<blockquote>
			<p>rbox 10 D2 \| qvoronoi QV5 p \| qconvex s FS </p>
			</blockquote>

			<p>To compute the lines ("hyperplanes") that define the
			Voronoi region for input site 5, use </p>

			<blockquote>
			<p>rbox 10 D2 \| qvoronoi QV5 p \| qconvex n </p>
			</blockquote>
			or
			<blockquote>
			<p>rbox 10 D2 \| qvoronoi QV5 Fi Fo</p>
			</blockquote>

			<p>To list the extreme points of the input sites use </p>

			<blockquote>
			<p>rbox 10 D2 \| qdelaunay Fx </p>
			</blockquote>

			<p>You will get the same point ids with </p>

			<blockquote>
			<p>rbox 10 D2 \| qconvex Fx </p>
			</blockquote>

			</blockquote><h4><A href="#TOC">»</A><a name="big">Can </a>Qhull triangulate
			a hundred 16-d points?</h4><blockquote>

			<p>No. This is an immense structure. A triangulation of 19, 16-d
			points has 43 simplices. If you add one point at a time, the
			triangulation increased as follows: 43, 189, 523, 1289, 2830,
			6071, 11410, 20487. The last triangulation for 26 points used 13
			megabytes of memory. When Qhull uses virtual memory, it becomes
			too slow to use. </p>

			</blockquote>
			<h2><A href="#TOC">»</A><a name="voronoi">Voronoi
			diagram questions</a></h2>

			<h4><A href="#TOC">»</A><a name="volume">How</a> do I compute the volume of a Voronoi region?</h4><blockquote>

			<p>For each Voronoi region, compute the convex hull of the region's Voronoi vertices. The volume of each convex hull is the volume
			of the corresponding Vornoi region.</p>

			<p>For example, to compute the volume of the bounded Voronoi region about [0,0,0]: output the origin's Voronoi vertices and
			compute the volume of their convex hull. The last number from option '<A href="qh-optf.htm#FS">FS</A>' is the volume.</p>
			<blockquote><pre>
			rbox P0 10 \| qvoronoi QV0 p \| qhull FS
			0
			2 1.448134756744281 0.1067973560800857
			</pre></blockquote>

			<p>For another example, see <A href="#2d">How</A> do I get the triangles for a 2-d Delaunay
			triangulation and the vertices of its Voronoi diagram?</p>

			<p>This approach is slow if you are using the command line. A faster approcach is to call Qhull from
			a program. The fastest method is Clarkson's <a href="http://www.netlib.org/voronoi/hull.html">hull</a> program.
			It computes the volume for all Voronoi regions.</p>

			<p>An unbounded Voronoi region does not have a volume.</p>

			</blockquote><h4><A href="#TOC">»</A><a name="maxsphere">How</a> do I get the radii of the empty
			spheres for each Voronoi vertex?</h4><blockquote>

			Use option '<A href="qh-optf.htm#Fi">Fi</A>' to list each bisector (i.e. Delaunay ridge). Then compute the
			minimum distance for each Voronoi vertex.

			<p>There's other ways to get the same information. Let me know if you
			find a better method.

			</blockquote><h4><A href="#TOC">»</A><a name="square">What</a> is the Voronoi diagram
			of a square?</h4><blockquote>

			<p>
			Consider a square,
			<blockquote><pre>
			C:\qhull>rbox c D2
			2 RBOX c D2
			4
			-0.5 -0.5
			-0.5 0.5
			0.5 -0.5
			0.5 0.5
			</pre></blockquote>

			<p>There's two ways to compute the Voronoi diagram: with facet merging
			or with joggle. With facet merging, the
			result is:

			<blockquote><pre>
			C:\qhull>rbox c D2 \| qvoronoi Qz

			Voronoi diagram by the convex hull of 5 points in 3-d:

			Number of Voronoi regions and at-infinity: 5
			Number of Voronoi vertices: 1
			Number of facets in hull: 5

			Statistics for: RBOX c D2 \| QVORONOI Qz

			Number of points processed: 5
			Number of hyperplanes created: 7
			Number of distance tests for qhull: 8
			Number of merged facets: 1
			Number of distance tests for merging: 29
			CPU seconds to compute hull (after input): 0

			C:\qhull>rbox c D2 \| qvoronoi Qz o
			2
			2 5 1
			-10.101 -10.101
			0 0
			2 0 1
			2 0 1
			2 0 1
			2 0 1
			0

			C:\qhull>rbox c D2 \| qvoronoi Qz Fv
			4
			4 0 1 0 1
			4 0 2 0 1
			4 1 3 0 1
			4 2 3 0 1
			</pre></blockquote>

			<p>There is one Voronoi vertex at the origin and rays from the origin
			along each of the coordinate axes.
			The last line '4 2 3 0 1' means that there is
			a ray that bisects input points #2 and #3 from infinity (vertex 0) to
			the origin (vertex 1).
			Option 'Qz' adds an artificial point since the input is cocircular.
			Coordinates -10.101 indicate the
			vertex at infinity.

			<p>With triangulated output, the Voronoi vertex is
			duplicated:

			<blockquote><pre>
			C:\qhull3.1>rbox c D2 \| qvoronoi Qt Qz

			Voronoi diagram by the convex hull of 5 points in 3-d:

			Number of Voronoi regions and at-infinity: 5
			Number of Voronoi vertices: 2
			Number of triangulated facets: 1

			Statistics for: RBOX c D2 \| QVORONOI Qt Qz

			Number of points processed: 5
			Number of hyperplanes created: 7
			Number of facets in hull: 6
			Number of distance tests for qhull: 8
			Number of distance tests for merging: 33
			Number of distance tests for checking: 30
			Number of merged facets: 1
			CPU seconds to compute hull (after input): 0.05

			C:\qhull3.1>rbox c D2 \| qvoronoi Qt Qz o
			2
			3 5 1
			-10.101 -10.101
			0 0
			0 0
			3 2 0 1
			2 1 0
			2 2 0
			3 2 0 1
			0

			C:\qhull3.1>rbox c D2 \| qvoronoi Qt Qz Fv
			4
			4 0 2 0 2
			4 0 1 0 1
			4 1 3 0 1
			4 2 3 0 2
			</pre></blockquote>


			<p>With joggle, the input is no longer cocircular and the Voronoi vertex is
			split into two:

			<blockquote><pre>
			C:\qhull>rbox c D2 \| qvoronoi Qt Qz

			C:\qhull>rbox c D2 \| qvoronoi QJ o
			2
			3 4 1
			-10.101 -10.101
			-4.71511718558304e-012 -1.775812830118184e-011
			9.020340030474472e-012 -4.02267108512433e-012
			2 0 1
			3 2 1 0
			3 2 0 1
			2 2 0

			C:\qhull>rbox c D2 \| qvoronoi QJ Fv
			5
			4 0 2 0 1
			4 0 1 0 1
			4 1 2 1 2
			4 1 3 0 2
			4 2 3 0 2
			</pre></blockquote>

			<p>Note that the Voronoi diagram includes the same rays as
			before plus a short edge between the two vertices.</p>


			</blockquote><h4><A href="#TOC">»</A><a name="vsphere">How</a> do I construct
			the Voronoi diagram of cospherical points?</h4><blockquote>

			<p>Three-d terrain data can be approximated with cospherical
			points. The Delaunay triangulation of cospherical points is the
			same as their convex hull. If the points lie on the unit sphere,
			the facet normals are the Voronoi vertices [via S. Fortune]. </p>

			<p>For example, consider the points {[1,0,0], [-1,0,0], [0,1,0],
			...}. Their convex hull is: </p>

			<pre>
			rbox d G1 \| qconvex o
			3
			6 8 12
			0 0 -1
			0 0 1
			0 -1 0
			0 1 0
			-1 0 0
			1 0 0
			3 3 1 4
			3 1 3 5
			3 0 3 4
			3 3 0 5
			3 2 1 5
			3 1 2 4
			3 2 0 4
			3 0 2 5
			</pre>

			<p>The facet normals are: </p>

			<pre>
			rbox d G1 \| qconvex n
			4
			8
			-0.5773502691896258 0.5773502691896258 0.5773502691896258 -0.5773502691896258
			0.5773502691896258 0.5773502691896258 0.5773502691896258 -0.5773502691896258
			-0.5773502691896258 0.5773502691896258 -0.5773502691896258 -0.5773502691896258
			0.5773502691896258 0.5773502691896258 -0.5773502691896258 -0.5773502691896258
			0.5773502691896258 -0.5773502691896258 0.5773502691896258 -0.5773502691896258
			-0.5773502691896258 -0.5773502691896258 0.5773502691896258 -0.5773502691896258
			-0.5773502691896258 -0.5773502691896258 -0.5773502691896258 -0.5773502691896258
			0.5773502691896258 -0.5773502691896258 -0.5773502691896258 -0.5773502691896258
			</pre>

			<p>If you drop the offset from each line (the last number), each
			line is the Voronoi vertex for the corresponding facet. The
			neighboring facets for each point define the Voronoi region for
			each point. For example: </p>

			<pre>
			rbox d G1 \| qconvex FN
			6
			4 7 3 2 6
			4 5 0 1 4
			4 7 4 5 6
			4 3 1 0 2
			4 6 2 0 5
			4 7 3 1 4
			</pre>

			<p>The Voronoi vertices {7, 3, 2, 6} define the Voronoi region
			for point 0. Point 0 is [0,0,-1]. Its Voronoi vertices are </p>

			<pre>
			-0.5773502691896258 0.5773502691896258 -0.5773502691896258
			0.5773502691896258 0.5773502691896258 -0.5773502691896258
			-0.5773502691896258 -0.5773502691896258 -0.5773502691896258
			0.5773502691896258 -0.5773502691896258 -0.5773502691896258
			</pre>

			<p>In this case, the Voronoi vertices are oriented, but in
			general they are unordered. </p>

			<p>By taking the dual of the Delaunay triangulation, you can
			construct the Voronoi diagram. For cospherical points, the convex
			hull vertices for each facet, define the input sites for each
			Voronoi vertex. In 3-d, the input sites are oriented. For
			example: </p>

			<pre>
			rbox d G1 \| qconvex i
			8
			3 1 4
			1 3 5
			0 3 4
			3 0 5
			2 1 5
			1 2 4
			2 0 4
			0 2 5
			</pre>

			<p>The convex hull vertices for facet 0 are {3, 1, 4}. So Voronoi
			vertex 0 (i.e., [-0.577, 0.577, 0.577]) is the Voronoi vertex for
			input sites {3, 1, 4} (i.e., {[0,1,0], [0,0,1], [-1,0,0]}). </p>

			</blockquote><h4><A href="#TOC">»</A><a name="rays">Can</a> Qhull compute the
			unbounded rays of the Voronoi diagram?</h4><blockquote>

			<p>Use '<A href="qh-optf.htm#Fo2">Fo</A>' to compute the separating
			hyperplanes for unbounded Voronoi regions. The corresponding ray
			goes to infinity from the Voronoi vertices. If you enclose the
			input sites in a large enough box, the outermost bounded regions
			will represent the unbounded regions of the original points. </p>

			<p>If you do not box the input sites, you can identify the
			unbounded regions. They list '0' as a vertex. Vertex 0 represents
			"infinity". Each unbounded ray includes vertex 0 in
			option '<A href="qh-optf.htm#Fv2">Fv</A>. See <A
			href="qvoronoi.htm#graphics" >Voronoi graphics</A> and <A
			href="qvoronoi.htm#notes" >Voronoi notes</A>. </p>

			</blockquote>
			<h2><A href="#TOC">»</A>Approximation questions</h2>

			<h4><A href="#TOC">»</A><a name="simplex">How</a> do I
			approximate data with a simplex</h4><blockquote>

			<p>Qhull may be used to help select a simplex that approximates a
			data set. It will take experimentation. Geomview will help to
			visualize the results. This task may be difficult to do in 5-d
			and higher. Use rbox options 'x' and 'y' to produce random
			distributions within a simplex. Your methods work if you can
			recover the simplex. </p>

			<p>Use Qhull's precision options to get a first approximation to
			the hull, say with 10 to 50 facets. For example, try 'C0.05' to
			remove small facets after constructing the hull. Use 'W0.05' to
			ignore points within 0.05 of a facet. Use 'PA5' to print the five
			largest facets by area. </p>

			<p>Then use other methods to fit a simplex to this data. Remove
			outlying vertices with few nearby points. Look for large facets
			in different quadrants. You can use option 'Pd0d1d2' to print all
			the facets in a quadrant. </p>

			<p>In 4-d and higher, use the outer planes (option 'Fo' or
			'facet->maxoutside') since the hyperplane of an approximate
			facet may be below many of the input points. </p>

			<p>For example, consider fitting a cube to 1000 uniformly random
			points in the unit cube. In this case, the first try was good: </p>

			<blockquote>
			<pre>
			rbox 1000 \| qconvex W0.05 C0.05 PA6 Fo
			4
			6
			0.35715408374381 0.08706467018177928 -0.9299788727015564 -0.5985514741284483
			0.995841591359023 -0.02512604712761577 0.08756829720435189 -0.5258834069202866
			0.02448099521570909 -0.02685210459017302 0.9993396046151313 -0.5158104982631999
			-0.9990223929415094 -0.01261133513150079 0.04236994958247349 -0.509218270408407
			-0.0128069014364698 -0.9998380680115362 0.01264203427283151 -0.5002512653670584
			0.01120895057872914 0.01803671994177704 -0.9997744926535512 -0.5056824072956361
			</pre>
			</blockquote>

			</blockquote>
			<h2><A href="#TOC">»</A>Halfspace questions</h2>

			<h4><A href="#TOC">»</A><a name="halfspace">How</a> do I compute the
			intersection of halfspaces with Qhull?</h4><blockquote>

			<p>Qhull computes the halfspace intersection about a point. The
			point must be inside all of the halfspaces. Given a point, a
			duality turns a halfspace intersection problem into a convex
			hull problem.

			<p>Use linear programming if you
			do not know a point in the interior of the halfspaces.
			See the <A href="qhalf.htm#notes">notes</A> for qhalf. You will need
			a linear programming code. This may require a fair amount of work to
			implement.</p>



			</blockquote>
			<h2><A href="#TOC">»</A><a name="library">Qhull library
			questions</a></h2>

			<h4><A href="#TOC">»</A><a name="math">Is</a> Qhull available for Mathematica, Matlab, or Maple?</h4><blockquote>

			<p><b>MATLAB</b>

			<p>Z. You of <a href="http://www.mathworks.com">MathWorks</a> added qhull to MATLAB 6.
			See 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/griddata3.html"
			>griddata3</a>,
			<a href="http://www.mathworks.com/help/techdoc/ref/griddatan.html"
			>griddatan</a>,
			<a href="http://www.mathworks.com/help/techdoc/ref/tsearch.html"
			>tsearch</a>,
			<a href="http://www.mathworks.com/help/techdoc/ref/tsearchn.html"
			>tsearchn</a>, and
			<a href="http://www.mathworks.com/help/techdoc/ref/voronoin.html"
			>voronoin</a>. V. Brumberg update MATLAB R14 for Qhull 2003.1 and triangulated output.

			<p>Engwirda wrote <a href="http://www.mathworks.com/matlabcentral/fileexchange/loadFile.do?objectId=10307&objectType=file">mesh2d</a> for unstructured mesh generation in MATLAB.
			It is based on the iterative method of Persson and generally results in better quality meshes than delaunay refinement.


			<p><b>Mathematica and Maple</b>

			<p>See <a href="http://library.wolfram.com/infocenter/MathSource/1160/"
			>qh-math</a>
			for a Delaunay interface to Mathematica. It includes projects for CodeWarrior
			on the Macintosh and Visual C++ on Win32 PCs.

			<p>See Mathematica ('<a
			href="qh-opto.htm#m">m</a>') and Maple ('<a
			href="qh-optf.htm#FM">FM</a>') output options.

			<p></p>
			</blockquote><h4><A href="#TOC">»</A><a name="ridges">Why</a> are there too few ridges?</h4><blockquote>

			The following sample code may produce fewer ridges than expected:

			<blockquote><pre>
			facetT *facetp;
			ridgeT ridge, *ridgep;

			FORALLfacets {
			printf("facet f%d\n", facet->id);
			FOREACHridge_(facet->ridges) {
			printf(" ridge r%d between f%d and f%d\n", ridge->id, ridge->top->id, ridge->bottom->id);
			}
			}
			</pre></blockquote>

			<p> Qhull does not create ridges for simplicial facets.
			Instead it computes ridges from facet->neighbors. To make ridges for a
			simplicial facet, use qh_makeridges() in merge.c. Usefacet->visit_id to visit
			each ridge once (instead of twice). For example,

			<blockquote><pre>
			facetT facet, neighbor;
			ridgeT ridge, *ridgep;

			qh visit_id++;
			FORALLfacets {
			printf("facet f%d\n", facet->id);
			qh_makeridges(facet);
			facet->visitId= qh visit_id;
			FOREACHridge_(facet->ridges) {
			neighbor= otherfacet_(ridge, visible);
			if (neighbor->visitid != qh visit_id)
			printf(" ridge r%d between f%d and f%d\n", ridge->id, ridge->top->id, ridge->bottom->id);
			}
			}
			</pre></blockquote>

			</blockquote><h4><A href="#TOC">»</A><a name="call">Can</a> Qhull use coordinates without placing
			them in a data file?</h4><blockquote>

			<p>You may call Qhull from a program. Please use the reentrant Qhull library (libqhullstatic_r.a, libqhull_r.so, or qhull_r.dll).

			See user_eg.c and "Qhull-template" in user_r.c for examples..

			See <a href="qh-code.htm">Qhull internals</a> for an introduction to Qhull's reentrant library and its C++ interface.

			<p>Hint: Start with a small example for which you know the
			answer.</p>

			</blockquote><h4><A href="#TOC">»</A><a name="size">How</a> large are Qhull's data structures?</h4><blockquote>

			<p>Qhull uses a general-dimension data structure.
			The size depends on the dimension. Use option 'Ts' to print
			out the memory statistics [e.g., 'rbox D2 10 \| qconvex Ts'].


			</blockquote><h4><A href="#TOC">»</A><a name="inc">Can</a> Qhull construct
			convex hulls and Delaunay triangulations one point at a time?</h4><blockquote>

			<p>The Qhull library may be used to construct convex hulls and
			Delaunay triangulations one point at a time. It may not be used
			for deleting points or moving points. </p>

			<p>Qhull is designed for batch processing. Neither Clarkson's
			randomized incremental algorithm nor Qhull are designed for
			on-line operation. For many applications, it is better to
			reconstruct the convex hull or Delaunay triangulation from
			scratch for each new point. </p>

			<p>With random point sets and on-line processing, Clarkson's
			algorithm should run faster than Qhull. Clarkson uses the
			intermediate facets to reject new, interior points, while Qhull,
			when used on-line, visits every facet to reject such points. If
			used on-line for n points, Clarkson may take O(n) times as much
			memory as the average off-line case, while Qhull's space
			requirement does not change. </p>

			<p>If you triangulate the output before adding all the points
			(option 'Qt' and procedure qh_triangulate), you must set
			option '<A href="qh-optq.htm#Q11">Q11</A>'. It duplicates the
			normals of triangulated facets and recomputes the centrums.
			This should be avoided for regular use since triangulated facets
			are not clearly convex with their neighbors. It appears to
			work most of the time, but fails for cases that Qhull normally
			handles well [see the test call to qh_triangulate in qh_addpoint].

			</blockquote><h4><A href="#TOC">»</A><a name="ridges2">How</a> do I visit the
			ridges of a Delaunay triangulation?</h4><blockquote>

			<p>To visit the ridges of a Delaunay triangulation, visit each
			facet. Each ridge will appear twice since it belongs to two
			facets. In pseudo-code: </p>

			<pre>
			for each facet of the triangulation
			if the facet is Delaunay (i.e., part of the lower convex hull)
			for each ridge of the facet
			if the ridge's neighboring facet has not been visited
			... process a ridge of the Delaunay triangulation ...
			</pre>

			<p>In undebugged, C code: </p>

			<pre>
			qh visit_id++;
			FORALLfacets_(facetlist)
			if (!facet->upperdelaunay) {
			facet->visitid= qh visit_id;
			qh_makeridges(facet);
			FOREACHridge_(facet->ridges) {
			neighbor= otherfacet_(ridge, facet);
			if (neighbor->visitid != qh visit_id) {
			/* Print ridge here with facet-id and neighbor-id */
			/fprintf(fp, "f%d\tf%d\t",facet->id,neighbor->ID);/
			FOREACHvertex_(ridge->vertices)
			fprintf(fp,"%d ",qh_pointid (vertex->point) );
			qh_printfacetNvertex_simplicial (fp, facet, format);
			fprintf(fp," ");
			if(neighbor->upperdelaunay)
			fprintf(fp," -1 -1 -1 -1 ");
			else
			qh_printfacetNvertex_simplicial (fp, neighbor, format);
			fprintf(fp,"\n");
			}
			}
			}
			}
			</pre>

			<p>Qhull should be redesigned as a class library, or at least as
			an API. It currently provides everything needed, but the
			programmer has to do a lot of work. Hopefully someone will write
			C++ wrapper classes or a Python module for Qhull. </p>

			</blockquote><h4><A href="#TOC">»</A><a name="listd">How</a> do I visit the
			Delaunay regions?</h4><blockquote>

			<p>Qhull constructs a Delaunay triangulation by lifting the
			input sites to a paraboloid. The Delaunay triangulation
			corresponds to the lower convex hull of the lifted points. To
			visit each facet of the lower convex hull, use: </p>

			<pre>
			facetT *facet;

			...
			FORALLfacets {
			if (!facet->upperdelaunay) {
			... only facets for Delaunay regions ...
			}
			}
			</pre>

			</blockquote><h4><A href="#TOC">»</A><a name="outside">When</a> is a point
			outside or inside a facet?</h4><blockquote>

			<p>A point is outside of a facet if it is clearly outside the
			facet's outer plane. The outer plane is defined by an offset
			(facet->maxoutside) from the facet's hyperplane. </p>

			<pre>
			facetT *facet;
			pointT *point;
			realT dist;

			...
			qh_distplane(point, facet, &dist);
			if (dist > facet->maxoutside + 2 * qh DISTround) {
			/* point is clearly outside of facet */
			}
			</pre>

			<p>A point is inside of a facet if it is clearly inside the
			facet's inner plane. The inner plane is computed as the maximum
			distance of a vertex to the facet. It may be computed for an
			individual facet, or you may use the maximum over all facets. For
			example: </p>

			<pre>
			facetT *facet;
			pointT *point;
			realT dist;

			...
			qh_distplane(point, facet, &dist);
			if (dist < qh min_vertex - 2 * qh DISTround) {
			/* point is clearly inside of facet */
			}
			</pre>

			<p>Both tests include two qh.DISTrounds because the computation
			of the furthest point from a facet may be off by qh.DISTround and
			the computation of the current distance to the facet may be off
			by qh.DISTround. </p>

			</blockquote><h4><A href="#TOC">»</A><a name="closest">How</a> do I find the
			facet that is closest to a point?</h4><blockquote>

			<p>Use qh_findbestfacet(). For example, </p>

			<pre>
			coordT point[ DIM ];
			boolT isoutside;
			realT bestdist;
			facetT *facet;

			... set coordinates for point ...

			facet= qh_findbestfacet (point, qh_ALL, &bestdist, &isoutside);

			/* 'facet' is the closest facet to 'point' */
			</pre>

			<p>qh_findbestfacet() performs a directed search for the facet
			furthest below the point. If the point lies inside this facet,
			qh_findbestfacet() performs an exhaustive search of all facets.
			An exhaustive search may be needed because a facet on the far
			side of a lens-shaped distribution may be closer to a point than
			all of the facet's neighbors. The exhaustive search may be
			skipped for spherical distributions. </p>

			<p>Also see, "<A href="#vclosest">How</A> do I find the
			Delaunay triangle that is closest to a
			point?" </p>

			</blockquote><h4><A href="#TOC">»</A><a name="vclosest">How</a> do I find the
			Delaunay triangle or Voronoi region that is closest to a point?</h4><blockquote>

			<p>A Delaunay triangulation subdivides the plane, or in general
			dimension, subdivides space. Given a point, how do you determine
			the subdivision containing the point? Or, given a set of points,
			how do you determine the subdivision containing each point of the set?
			Efficiency is important -- an exhaustive search of the subdivision
			is too slow.

			<p>First compute the Delaunay triangle with qh_new_qhull() in user_r.c or Qhull::runQhull().
			Lift the point to the paraboloid by summing the squares of the
			coordinates. Use qh_findbestfacet() [poly2.c] to find the closest Delaunay
			triangle. Determine the closest vertex to find the corresponding
			Voronoi region. Do not use options
			'<A href="qh-optq.htm#Qbb">Qbb</A>', '<A href="qh-optq.htm#QbB">QbB</A>',
			'<A href="qh-optq.htm#Qbk">Qbk:n</A>', or '<A
			href="qh-optq.htm#QBk" >QBk:n</A>' since these scale the last
			coordinate. Optimizations of qh_findbestfacet() should
			be possible for Delaunay triangulations.</p>

			<p>You first need to lift the point to the paraboloid (i.e., the
			last coordinate is the sum of the squares of the point's coordinates).
			The
			routine, qh_setdelaunay() [geom2.c], lifts an array of points to the
			paraboloid. The following excerpt is from findclosest() in
			user_eg.c. </p>

			<pre>
			coordT point[ DIM + 1]; /* one extra coordinate for lifting the point */
			boolT isoutside;
			realT bestdist;
			facetT *facet;

			... set coordinates for point[] ...

			qh_setdelaunay (DIM+1, 1, point);
			facet= qh_findbestfacet (point, qh_ALL, &bestdist, &isoutside);
			/* 'facet' is the closest Delaunay triangle to 'point' */
			</pre>

			<p>The returned facet either contains the point or it is the
			closest Delaunay triangle along the convex hull of the input set.

			<p>Point location is an active research area in Computational
			Geometry. For a practical approach, see Mucke, et al, "Fast randomized
			point location without preprocessing in two- and
			three-dimensional Delaunay triangulations," <i>Computational
			Geometry '96</i>, p. 274-283, May 1996.
			For an introduction to planar point location see [O'Rourke '93].
			Also see, "<A
			href="#closest" >How</A> do I find the facet that is closest to a
			point?" </p>

			<p>To locate the closest Voronoi region, determine the closest
			vertex of the closest Delaunay triangle. </p>

			<pre>
			realT dist, bestdist= REALmax;
			vertexT bestvertex= NULL, vertex, **vertexp;

			/* 'facet' is the closest Delaunay triangle to 'point' */

			FOREACHvertex_( facet->vertices ) {
			dist= qh_pointdist( point, vertex->point, DIM );
			if (dist < bestdist) {
			bestdist= dist;
			bestvertex= vertex;
			}
			}
			/* 'bestvertex' represents the Voronoi region closest to 'point'. The corresponding
			input site is 'bestvertex->point' */
			</pre>

			</blockquote><h4><A href="#TOC">»</A><a name="vertices">How</a> do I list the
			vertices?</h4><blockquote>

			<p>To list the vertices (i.e., extreme points) of the convex hull
			use </p>

			<blockquote>
			<pre>
			vertexT *vertex;

			FORALLvertices {
			...
			// vertex->point is the coordinates of the vertex
			// qh_pointid(vertex->point) is the point ID of the vertex
			...
			}
			</pre>
			</blockquote>

			</blockquote><h4><A href="#TOC">»</A><a name="test">How</a> do I test code
			that uses the Qhull library?</h4><blockquote>

			<p>Compare the output from your program with the output from the
			Qhull program. Use option 'T1' or 'T4' to trace what Qhull is
			doing. Prepare a <i>small</i> example for which you know the
			output. Run the example through the Qhull program and your code.
			Compare the trace outputs. If you do everything right, the two
			trace outputs should be almost the same. The trace output will
			also guide you to the functions that you need to review. </p>

			</blockquote><h4><A href="#TOC">»</A><a name="orient">When</a> I compute a
			plane equation from a facet, I sometimes get an outward-pointing
			normal and sometimes an inward-pointing normal</h4><blockquote>

			<p>Qhull orients simplicial facets, and prints oriented output
			for 'i', 'Ft', and other options. The orientation depends on <i>both</i>
			the vertex order and the flag facet->toporient.</p>

			<p>Qhull does not orient
			non-simplicial facets. Instead it orients the facet's ridges. These are
			printed with the 'Qt' and 'Ft' option. The facet's hyperplane is oriented. </p>

			</blockquote>
			<hr><!-- Navigation links -->

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			<p>Comments to: <a href=mailto:qhull@qhull.org>qhull@qhull.org</a>
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