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			<title>qhull -- convex hull and related structures</title>
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			<!-- Navigation links -->
			<p><b><a name="TOP">Up</a></b><b>:</b> <a href="http://www.geom.umn.edu/software/qhull">Home page</a> for Qhull<br>
			<b>Up:</b> <a href="index.htm#TOC">Qhull manual</a>: Table of 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><br>
			<b>To:</b> <a href="#synopsis">sy</a>nopsis <a href="#input">in</a>put
			<a href="#outputs">ou</a>tputs <a href="#controls">co</a>ntrols
			<a href="#options">op</a>tions
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			<h1><a
			href="http://www.geom.umn.edu/graphics/pix/Special_Topics/Computational_Geometry/cone.html"><img
			src="qh--cone.gif" alt="[cone]" align="middle" width="100"
			height="100"></a>qhull -- convex hull and related structures</h1>

			<p>The convex hull of a set of points is the smallest convex set
			containing the points. The Delaunay triangulation and furthest-site
			Delaunay triangulation are equivalent to a convex hull in one
			higher dimension. Halfspace intersection about a point is
			equivalent to a convex hull by polar duality.

			<p>The <tt>qhull</tt> program provides options to build these
			structures and to experiment with the process. Use the
			<a href=qconvex.htm>qconvex</a>,
			<a href=qdelaun.htm>qdelaunay</a>, <a href=qhalf.htm>qhalf</a>,
			and <a href=qvoronoi.htm>qvoronoi</a> programs
			to build specific structures. You may use <tt>qhull</tt> instead.
			It takes the same options and uses the same code.
			<blockquote>
			<dl>
			<dt><b>Example:</b> rbox 1000 D3 \| qhull
			<a href="qh-optc.htm#Cn">C-1e-4</a>
			<a href="qh-optf.htm#FO">FO</a>
			<a href="qh-optt.htm#Ts">Ts</a>
			</dt>
			<dd>Compute the 3-d convex hull of 1000 random
			points.
			Centrums must be 10^-4 below neighboring
			hyperplanes. Print the options and precision constants.
			When done, print statistics. These options may be
			used with any of the Qhull programs.</dd>
			<dt> </dt>
			<dt><b>Example:</b> rbox 1000 D3 \| qhull <a href=qhull.htm#outputs>d</a>
			<a href="qh-optc.htm#Rn">R1e-4</a>
			<a href="qh-optq.htm#Q0">Q0</a></dt>
			<dd>Compute the 3-d Delaunay triangulation of 1000 random
			points. Randomly perturb all calculations by
			[0.9999,1.0001]. Do not correct precision problems.
			This leads to serious precision errors.</dd>
			</dl>
			</blockquote>
			<p>Use the following equivalences when calling <tt>qhull</tt> in 2-d to 4-d (a 3-d
			Delaunay triangulation is a 4-d convex hull):
			<blockquote>
			<ul>
			<li>
			<a href="qconvex.htm">qconvex</a> == qhull
			<li>
			<a href=qdelaun.htm>qdelaunay</a> == qhull d <a href="qh-optq.htm#Qbb">Qbb</a>
			<li>
			<a href=qhalf.htm>qhalf</a> == qhull H
			<li>
			<a href=qvoronoi.htm>qvoronoi</a> == qhull v <a href="qh-optq.htm#Qbb">Qbb</a>
			</ul>
			</blockquote>

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


			<p>If you use '<a href="qh-optq.htm#Qt">Qt</a>' (triangulated output),
			all furthest-site Delaunay regions
			will be simplicial (e.g., triangles in 2-d). Some regions may be
			degenerate and have zero area. Qhull can identify coincident
			points. With facet merging, cocircular and cospherical extreme points will lead
			to non-simplicial output (e.g., a square).

			<p>If you use '<a href="qh-optq.htm#QJn">QJ</a>' (joggle), all furthest-site
			Delaunay regions will be simplicial (e.g., triangles in 2-d). Duplicate points will
			create small regions since the points are joggled apart. See <a
			href="qh-impre.htm#joggle">Joggled input or merged facets</a>. </p>

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

			<p><b>Copyright © 1995-2002 The Geometry Center, Minneapolis MN</b></p>

			<hr>

			<h3><a href="#TOP">ｻ</a><a name="synopsis">qhull synopsis</a></h3>
			<pre>
			qhull- compute convex hulls and related structures.
			input (stdin): dimension, n, point coordinates
			comments start with a non-numeric character
			halfspace: use dim+1 and put offsets after coefficients

			options (qh-quick.htm):
			d - Delaunay triangulation by lifting points to a paraboloid
			d Qu - furthest-site Delaunay triangulation (upper convex hull)
			v - Voronoi diagram as the dual of the Delaunay triangulation
			v Qu - furthest-site Voronoi diagram
			H1,1 - Halfspace intersection about [1,1,0,...] via polar duality
			Qt - triangulated output
			QJ - joggle input instead of merging facets
			Tv - verify result: structure, convexity, and point inclusion
			. - concise list of all options
			- - one-line description of all options

			Output options (subset):
			s - summary of results (default)
			i - vertices incident to each facet
			n - normals with offsets
			p - vertex coordinates (if 'Qc', includes coplanar points)
			if 'v', Voronoi vertices
			Fp - halfspace intersections
			Fx - extreme points (convex hull vertices)
			FA - compute total area and volume
			o - OFF format (if 'v', outputs Voronoi regions)
			G - Geomview output (2-d, 3-d and 4-d)
			m - Mathematica output (2-d and 3-d)
			QVn - print facets that include point n, -n if not
			TO file- output results to file, may be enclosed in single quotes

			examples:
			rbox c d D2 \| qhull Qc s f Fx \| more rbox 1000 s \| qhull Tv s FA
			rbox 10 D2 \| qhull d QJ s i TO result rbox 10 D2 \| qhull v Qt p
			rbox 10 D2 \| qhull d Qu QJ m rbox 10 D2 \| qhull v Qu QJ o
			rbox c \| qhull n rbox c \| qhull FV n \| qhull H Fp
			rbox d D12 \| qhull QR0 FA rbox c D7 \| qhull FA TF1000
			rbox y 1000 W0 \| qhull rbox 10 \| qhull v QJ o Fv
			</pre>

			<h3><a href="#TOP">ｻ</a><a name="input">qhull input</a></h3>
			<blockquote>

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

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

			<p>Comments start with a non-numeric character. Error reporting is
			simpler if there is one point per line. Dimension
			and number of points may be reversed. For halfspace intersection,
			an interior point may be prepended (see <a href=qhalf.htm#input>qhalf input</a>).

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

			<blockquote>
			<p>rbox c > data </p>
			<pre>
			3 RBOX c
			8
			-0.5 -0.5 -0.5
			-0.5 -0.5 0.5
			-0.5 0.5 -0.5
			-0.5 0.5 0.5
			0.5 -0.5 -0.5
			0.5 -0.5 0.5
			0.5 0.5 -0.5
			0.5 0.5 0.5
			</pre>
			<p>qhull s n < data</p>
			<pre>

			Convex hull of 8 points in 3-d:

			Number of vertices: 8
			Number of facets: 6
			Number of non-simplicial facets: 6

			Statistics for: RBOX c \| QHULL s n

			Number of points processed: 8
			Number of hyperplanes created: 11
			Number of distance tests for qhull: 35
			Number of merged facets: 6
			Number of distance tests for merging: 84
			CPU seconds to compute hull (after input): 0.081

			4
			6
			0 0 -1 -0.5
			0 -1 0 -0.5
			1 0 0 -0.5
			-1 0 0 -0.5
			0 1 0 -0.5
			0 0 1 -0.5
			</pre>
			</blockquote>

			</blockquote>
			<h3><a href="#TOP">ｻ</a><a name="outputs">qhull outputs</a></h3>
			<blockquote>

			<p>These options control the output of qhull. They may be used
			individually or together.</p>
			<blockquote>
			<dl compact>
			<dt> </dt>
			<dd><b>General</b></dd>
			<dt><a name="d">qhull d</a></dt>
			<dd>compute the Delaunay triangulation by lifting the points
			to a paraboloid. See <a href=qdelaun.htm>qdelaunay</a>.</dd>
			<dt><a name="v">qhull v</a></dt>
			<dd>compute the Voronoi diagram by computing the Delaunay
			triangulation. See <a href=qvoronoi.htm>qvoronoi</a>.</dd>
			<dt><a name="H">qhull H</a></dt>
			<dd>compute the halfspace intersection about a point via polar
			duality. See <a href=qhalf.htm>qhalf</a>.</dd>
			</dl>
			</blockquote>

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

			</blockquote>
			<h3><a href="#TOP">ｻ</a><a name="controls">qhull controls</a></h3>
			<blockquote>

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

			</blockquote>
			<h3><a href="#TOP">ｻ</a><a name="options">qhull options</a></h3>

			<pre>
			qhull- compute convex hulls and related structures.
			http://www.geom.umn.edu/software/qhull

			input (stdin):
			first lines: dimension and number of points (or vice-versa).
			other lines: point coordinates, best if one point per line
			comments: start with a non-numeric character
			halfspaces: use dim plus one and put offset after coefficients.
			May be preceded by a single interior point ('H').

			options:
			d - Delaunay triangulation by lifting points to a paraboloid
			d Qu - furthest-site Delaunay triangulation (upper convex hull)
			v - Voronoi diagram (dual of the Delaunay triangulation)
			v Qu - furthest-site Voronoi diagram
			Hn,n,... - halfspace intersection about point [n,n,0,...]
			Qt - triangulated output
			QJ - joggle input instead of merging facets
			Qc - keep coplanar points with nearest facet
			Qi - keep interior points with nearest facet

			Qhull control options:
			Qbk:n - scale coord k so that low bound is n
			QBk:n - scale coord k so that upper bound is n (QBk is 0.5)
			QbB - scale input to unit cube centered at the origin
			Qbb - scale last coordinate to [0,m] for Delaunay triangulations
			Qbk:0Bk:0 - remove k-th coordinate from input
			QJn - randomly joggle input in range [-n,n]
			QRn - random rotation (n=seed, n=0 time, n=-1 time/no rotate)
			Qf - partition point to furthest outside facet
			Qg - only build good facets (needs 'QGn', 'QVn', or 'PdD')
			Qm - only process points that would increase max_outside
			Qr - process random outside points instead of furthest ones
			Qs - search all points for the initial simplex
			Qu - for 'd' or 'v', compute upper hull without point at-infinity
			returns furthest-site Delaunay triangulation
			Qv - test vertex neighbors for convexity
			Qx - exact pre-merges (skips coplanar and anglomaniacs facets)
			Qz - add point-at-infinity to Delaunay triangulation
			QGn - good facet if visible from point n, -n for not visible
			QVn - good facet if it includes point n, -n if not
			Q0 - turn off default p remerge with 'C-0'/'Qx'
			Q1 - sort merges by type instead of angle
			Q2 - merge all non-convex at once instead of independent sets
			Q3 - do not merge redundant vertices
			Q4 - avoid old>new merges
			Q5 - do not correct outer planes at end of qhull
			Q6 - do not pre-merge concave or coplanar facets
			Q7 - depth-first processing instead of breadth-first
			Q8 - do not process near-inside points
			Q9 - process furthest of furthest points
			Q10 - no special processing for narrow distributions
			Q11 - copy normals and recompute centrums for tricoplanar facets

			Towpaths Trace options:
			T4 - trace at level n, 4=all, 5=mem/gauss, -1= events
			Tc - check frequently during execution
			Ts - print statistics
			Tv - verify result: structure, convexity, and point inclusion
			Tz - send all output to stdout
			TFn - report summary when n or more facets created
			TI file - input data from file, no spaces or single quotes
			TO file - output results to file, may be enclosed in single quotes
			TPn - turn on tracing when point n added to hull
			TMn - turn on tracing at merge n
			TWn - trace merge facets when width > n
			TRn - rerun qhull n times. Use with 'QJn'
			TVn - stop qhull after adding point n, -n for before (see TCn)
			TCn - stop qhull after building cone for point n (see TVn)

			Precision options:
			Cn - radius of centrum (roundoff added). Merge facets if non-convex
			An - cosine of maximum angle. Merge facets if cosine > n or non-convex
			C-0 roundoff, A-0.99/C-0.01 pre-merge, A0.99/C0.01 post-merge
			En - max roundoff error for distance computation
			Rn - randomly perturb computations by a factor of [1-n,1+n]
			Vn - min distance above plane for a visible facet (default 3C-n or En)
			Un - max distance below plane for a new, coplanar point (default Vn)
			Wn - min facet width for outside point (before roundoff, default 2Vn)

			Output formats (may be combined; if none, produces a summary to stdout):
			f - facet dump
			G - Geomview output (see below)
			i - vertices incident to each facet
			m - Mathematica output (2-d and 3-d)
			o - OFF format (dim, points and facets; Voronoi regions)
			n - normals with offsets
			p - vertex coordinates or Voronoi vertices (coplanar points if 'Qc')
			s - summary (stderr)

			More formats:
			Fa - area for each facet
			FA - compute total area and volume for option 's'
			Fc - count plus coplanar points for each facet
			use 'Qc' (default) for coplanar and 'Qi' for interior
			FC - centrum or Voronoi center for each facet
			Fd - use cdd format for input (homogeneous with offset first)
			FD - use cdd format for numeric output (offset first)
			FF - facet dump without ridges
			Fi - inner plane for each facet
			for 'v', separating hyperplanes for bounded Voronoi regions
			FI - ID of each facet
			Fm - merge count for each facet (511 max)
			Fn - count plus neighboring facets for each facet
			FN - count plus neighboring facets for each point
			Fo - outer plane (or max_outside) for each facet
			for 'v', separating hyperplanes for unbounded Voronoi regions
			FO - options and precision constants
			Fp - dim, count, and intersection coordinates (halfspace only)
			FP - nearest vertex and distance for each coplanar point
			FQ - command used for qhull
			Fs - summary: #int (8), dimension, #points, tot vertices, tot facets,
			output: #vertices, #facets, #coplanars, #nonsimplicial
			#real (2), max outer plane, min vertex
			FS - sizes: #int (0)
			#real(2) tot area, tot volume
			Ft - triangulation with centrums for non-simplicial facets (OFF format)
			Fv - count plus vertices for each facet
			for 'v', Voronoi diagram as Voronoi vertices for pairs of sites
			FV - average of vertices (a feasible point for 'H')
			Fx - extreme points (in order for 2-d)

			Geomview options (2-d, 3-d, and 4-d; 2-d Voronoi)
			Ga - all points as dots
			Gp - coplanar points and vertices as radii
			Gv - vertices as spheres
			Gi - inner planes only
			Gn - no planes
			Go - outer planes only
			Gc - centrums
			Gh - hyperplane intersections
			Gr - ridges
			GDn - drop dimension n in 3-d and 4-d output
			Gt - for 3-d 'd', transparent outer ridges

			Print options:
			PAn - keep n largest facets by area
			Pdk:n - drop facet if normal[k] <= n (default 0.0)
			PDk:n - drop facet if normal[k] >= n
			Pg - print good facets (needs 'QGn' or 'QVn')
			PFn - keep facets whose area is at least n
			PG - print neighbors of good facets
			PMn - keep n facets with most merges
			Po - force output. If error, output neighborhood of facet
			Pp - do not report precision problems

			. - list of all options
			- - one line descriptions of all options
			</pre>

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			<p><b>Up:</b> <a href="http://www.geom.umn.edu/software/qhull">Home page</a> for Qhull<br>
			<b>Up:</b> <a href="index.htm#TOC">Qhull manual</a>: Table of 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><br>
			<b>To:</b> <a href="#synopsis">sy</a>nopsis <a href="#input">in</a>put
			<a href="#outputs">ou</a>tputs <a href="#controls">co</a>ntrols
			<a href="#options">op</a>tions
			<!-- GC common information -->
			<hr>

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

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