293 lines
12 KiB
HTML
293 lines
12 KiB
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<!DOCTYPE HTML PUBLIC "-//IETF//DTD HTML//EN">
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<html>
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<head>
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<title>Qhull precision options</title>
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</head>
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<body>
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<!-- Navigation links -->
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<p><b>Up:</b> <a href="http://www.qhull.org">Home page</a> for Qhull<br>
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<b>Up:</b> <a href="index.htm#TOC">Qhull manual</a>: Table of Contents<br>
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<b>To:</b> <a href="qh-quick.htm#programs">Programs</a>
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• <a href="qh-quick.htm#options">Options</a>
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• <a href="qh-opto.htm#output">Output</a>
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• <a href="qh-optf.htm#format">Formats</a>
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• <a href="qh-optg.htm#geomview">Geomview</a>
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• <a href="qh-optp.htm#print">Print</a>
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• <a href="qh-optq.htm#qhull">Qhull</a>
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• <a href="qh-optc.htm#prec">Precision</a>
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• <a href="qh-optt.htm#trace">Trace</a>
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• <a href="../src/libqhull_r/index.htm">Functions</a></p>
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<hr>
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<!-- Main text of document -->
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<h1><a
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href="http://www.geom.uiuc.edu/graphics/pix/Special_Topics/Computational_Geometry/delaunay.html"><img
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src="qh--dt.gif" alt="[delaunay]" align="middle" width="100"
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height="100"></a> Qhull precision options</h1>
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This section lists the precision options for Qhull. These options are
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indicated by an upper-case letter followed by a number.
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<p><b>Copyright © 1995-2015 C.B. Barber</b></p>
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<hr>
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<p><a href="index.htm#TOC">»</a> <a href="qh-quick.htm#programs">Programs</a>
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<a name="prec">•</a> <a href="qh-quick.htm#options">Options</a>
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• <a href="qh-opto.htm#output">Output</a>
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• <a href="qh-optf.htm#format">Formats</a>
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• <a href="qh-optg.htm#geomview">Geomview</a>
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• <a href="qh-optp.htm#print">Print</a>
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• <a href="qh-optq.htm#qhull">Qhull</a>
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• <a href="qh-optc.htm#prec">Precision</a>
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• <a href="qh-optt.htm#trace">Trace</a>
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• <a href="../src/libqhull_r/index.htm">Functions</a></p>
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<h2>Precision options</h2>
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<p>Most users will not need to set these options. They are best
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used for <a href="qh-impre.htm#approximate">approximating</a> a
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convex hull. They may also be used for testing Qhull's handling
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of precision errors.</p>
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<p>By default, Qhull uses options '<a href="#C0">C-0</a>' for
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2-d, 3-d and 4-d, and '<a href="qh-optq.htm#Qx">Qx</a>' for 5-d
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and higher. These options use facet merging to handle precision
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errors. You may also use joggled input '<a href="qh-optq.htm#QJn">QJ</a>'
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to avoid precision problems.
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For more information see <a
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href="qh-impre.htm">Imprecision in Qhull</a>.</p>
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<dl compact>
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<dt> </dt>
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<dd><b>General</b></dd>
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<dt><a href="#Cn2">Cn</a></dt>
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<dd>centrum radius for post-merging</dd>
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<dt><a href="#Cn">C-n</a></dt>
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<dd>centrum radius for pre-merging</dd>
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<dt><a href="#An2">An</a></dt>
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<dd>cosine of maximum angle for post-merging</dd>
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<dt><a href="#An">A-n</a></dt>
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<dd>cosine of maximum angle for pre-merging</dd>
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<dt><a href="qh-optq.htm#Qx">Qx</a></dt>
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<dd>exact pre-merges (allows coplanar facets)</dd>
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<dt><a href="#C0">C-0</a></dt>
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<dd>handle all precision errors</dd>
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<dt><a href="#Wn">Wn</a></dt>
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<dd>min distance above plane for outside points</dd>
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</dl>
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<dl compact>
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<dt> </dt>
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<dd><b>Experimental</b></dd>
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<dt><a href="#Un">Un</a></dt>
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<dd>max distance below plane for a new, coplanar point</dd>
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<dt><a href="#En">En</a></dt>
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<dd>max roundoff error for distance computation</dd>
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<dt><a href="#Vn">Vn</a></dt>
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<dd>min distance above plane for a visible facet</dd>
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<dt><a href="#Rn">Rn</a></dt>
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<dd>randomly perturb computations by a factor of [1-n,1+n]</dd>
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</dl>
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<dl compact>
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</dl>
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<hr>
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<h3><a href="#prec">»</a><a name="An">A-n - cosine of maximum
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angle for pre-merging.</a></h3>
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<p>Pre-merging occurs while Qhull constructs the hull. It is
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indicated by '<a href="#Cn">C-n</a>', 'A-n', or '<a
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href="qh-optq.htm#Qx">Qx</a>'.</p>
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<p>If the angle between a pair of facet normals is greater than <i>n</i>,
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Qhull merges one of the facets into a neighbor. It selects the
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facet that is closest to a neighboring facet.</p>
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<p>For example, option 'A-0.99' merges facets during the
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construction of the hull. If the cosine of the angle between
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facets is greater than 0.99, one or the other facet is merged.
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Qhull accounts for the maximum roundoff error.</p>
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<p>If 'A-n' is set without '<a href="#Cn">C-n</a>', then '<a
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href="#C0">C-0</a>' is automatically set. </p>
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<p>In 5-d and higher, you should set '<a href="qh-optq.htm#Qx">Qx</a>'
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along with 'A-n'. It skips merges of coplanar facets until after
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the hull is constructed and before '<a href="#An2">An</a>' and '<a
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href="#Cn2">Cn</a>' are checked. </p>
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<h3><a href="#prec">»</a><a name="An2">An - cosine of maximum angle for
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post-merging.</a></h3>
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<p>Post merging occurs after the hull is constructed. For
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example, option 'A0.99' merges a facet if the cosine of the angle
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between facets is greater than 0.99. Qhull accounts for the
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maximum roundoff error.</p>
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<p>If 'An' is set without '<a href="#Cn2">Cn</a>', then '<a
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href="#Cn2">C0</a>' is automatically set. </p>
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<h3><a href="#prec">»</a><a name="C0">C-0 - handle all precision
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errors </a></h3>
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<p>Qhull handles precision errors by merging facets. The 'C-0'
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option handles all precision errors in 2-d, 3-d, and 4-d. It is
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set by default. It may be used in higher dimensions, but
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sometimes the facet width grows rapidly. It is usually better to
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use '<a href="qh-optq.htm#Qx">Qx</a>' in 5-d and higher.
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Use '<a href="qh-optq.htm#QJn">QJ</a>' to joggle the input
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instead of merging facets.
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Use '<a
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href="qh-optq.htm#Q0">Q0</a>' to turn both options off.</p>
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<p>Qhull optimizes 'C-0' ("_zero-centrum") by testing
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vertices instead of centrums for adjacent simplices. This may be
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slower in higher dimensions if merges decrease the number of
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processed points. The optimization may be turned off by setting a
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small value such as 'C-1e-30'. See <a href="qh-impre.htm">How
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Qhull handles imprecision</a>.</p>
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<h3><a href="#prec">»</a><a name="Cn">C-n - centrum radius for
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pre-merging</a></h3>
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<p>Pre-merging occurs while Qhull constructs the hull. It is
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indicated by 'C-n', '<a href="#An">A-n</a>', or '<a
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href="qh-optq.htm#Qx">Qx</a>'.</p>
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<p>The <i>centrum</i> of a facet is a point on the facet for
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testing facet convexity. It is the average of the vertices
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projected to the facet's hyperplane. Two adjacent facets are
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convex if each centrum is clearly below the other facet. </p>
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<p>If adjacent facets are non-convex, one of the facets is merged
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into a neighboring facet. Qhull merges the facet that is closest
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to a neighboring facet. </p>
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<p>For option 'C-n', <i>n</i> is the centrum radius. For example,
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'C-0.001' merges facets whenever the centrum is less than 0.001
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from a neighboring hyperplane. Qhull accounts for roundoff error
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when testing the centrum.</p>
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<p>In 5-d and higher, you should set '<a href="qh-optq.htm#Qx">Qx</a>'
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along with 'C-n'. It skips merges of coplanar facets until after
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the hull is constructed and before '<a href="#An2">An</a>' and '<a
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href="#Cn2">Cn</a>' are checked. </p>
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<h3><a href="#prec">»</a><a name="Cn2">Cn - centrum radius for
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post-merging</a></h3>
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<p>Post-merging occurs after Qhull constructs the hull. It is
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indicated by '<a href="#Cn2">Cn</a>' or '<a href="#An2">An</a>'. </p>
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<p>For option '<a href="#Cn2">Cn</a>', <i>n</i> is the centrum
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radius. For example, 'C0.001' merges facets when the centrum is
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less than 0.001 from a neighboring hyperplane. Qhull accounts for
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roundoff error when testing the centrum.</p>
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<p>Both pre-merging and post-merging may be defined. If only
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post-merging is used ('<a href="qh-optq.htm#Q0">Q0</a>' with
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'Cn'), Qhull may fail to produce a hull due to precision errors
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during the hull's construction.</p>
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<h3><a href="#prec">»</a><a name="En">En - max roundoff error
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for distance computations</a></h3>
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<p>This allows the user to change the maximum roundoff error
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computed by Qhull. The value computed by Qhull may be overly
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pessimistic. If 'En' is set too small, then the output may not be
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convex. The statistic "max. distance of a new vertex to a
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facet" (from option '<a href="qh-optt.htm#Ts">Ts</a>') is a
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reasonable upper bound for the actual roundoff error. </p>
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<h3><a href="#prec">»</a><a name="Rn">Rn - randomly perturb
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computations </a></h3>
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<p>This option perturbs every distance, hyperplane, and angle
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computation by up to <i>(+/- n * max_coord)</i>. It simulates the
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effect of roundoff errors. Unless '<a href="#En">En</a>' is
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explicitly set, it is adjusted for 'Rn'. The command 'qhull Rn'
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will generate a convex hull despite the perturbations. See the <a
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href="qh-eg.htm#merge">Examples </a>section for an example.</p>
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<p>Options 'Rn C-n' have the effect of '<a href="#Wn">W2n</a>'
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and '<a href="#Cn">C-2n</a>'. To use time as the random number
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seed, use option '<a href="qh-optq.htm#QRn">QR-1</a>'.</p>
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<h3><a href="#prec">»</a><a name="Un">Un - max distance for a
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new, coplanar point </a></h3>
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<p>This allows the user to set coplanarity. When pre-merging ('<a
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href="#Cn">C-n </a>', '<a href="#An">A-n</a>' or '<a
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href="qh-optq.htm#Qx">Qx</a>'), Qhull merges a new point into any
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coplanar facets. The default value for 'Un' is '<a href="#Vn">Vn</a>'.</p>
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<h3><a href="#prec">»</a><a name="Vn">Vn - min distance for a
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visible facet </a></h3>
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<p>This allows the user to set facet visibility. When adding a
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point to the convex hull, Qhull determines all facets that are
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visible from the point. A facet is visible if the distance from
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the point to the facet is greater than 'Vn'.</p>
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<p>Without merging, the default value for 'Vn' is the roundoff
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error ('<a href="#En">En</a>'). With merging, the default value
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is the pre-merge centrum ('<a href="#Cn">C-n</a>') in 2-d or 3-d,
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or three times that in other dimensions. If the outside width is
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specified with option '<a href="#Wn">Wn </a>', the maximum,
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default value for 'Vn' is '<a href="#Wn">Wn</a>'.</p>
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<p>Qhull warns if 'Vn' is greater than '<a href="#Wn">Wn</a>' and
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furthest outside ('<a href="qh-optq.htm#Qf">Qf</a>') is not
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selected; this combination usually results in flipped facets
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(i.e., reversed normals).</p>
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<h3><a href="#prec">»</a><a name="Wn">Wn - min distance above
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plane for outside points</a></h3>
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<p>Points are added to the convex hull only if they are clearly
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outside of a facet. A point is outside of a facet if its distance
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to the facet is greater than 'Wn'. Without pre-merging, the
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default value for 'Wn' is '<a href="#En">En </a>'. If the user
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specifies pre-merging and does not set 'Wn', than 'Wn' is set to
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the maximum of '<a href="#Cn">C-n</a>' and <i>maxcoord*(1 - </i><a
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href="#An"><i>A-n</i></a><i>)</i>.</p>
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<p>This option is good for <a href="qh-impre.htm#approximate">approximating</a>
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a convex hull.</p>
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<p>Options '<a href="qh-optq.htm#Qc">Qc</a>' and '<a
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href="qh-optq.htm#Qi">Qi</a>' use the minimum vertex to
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distinguish coplanar points from interior points.</p>
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<!-- Navigation links -->
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<hr>
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<p><b>Up:</b> <a href="http://www.qhull.org">Home page</a> for Qhull<br>
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<b>Up:</b> <a href="index.htm#TOC">Qhull manual</a>: Table of Contents<br>
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<b>To:</b> <a href="qh-quick.htm#programs">Programs</a>
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• <a href="qh-quick.htm#options">Options</a>
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• <a href="qh-opto.htm#output">Output</a>
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• <a href="qh-optf.htm#format">Formats</a>
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• <a href="qh-optg.htm#geomview">Geomview</a>
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• <a href="qh-optp.htm#print">Print</a>
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• <a href="qh-optq.htm#qhull">Qhull</a>
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• <a href="qh-optc.htm#prec">Precision</a>
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• <a href="qh-optt.htm#trace">Trace</a>
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• <a href="../src/libqhull_r/index.htm">Functions</a></p>
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<!-- GC common information -->
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<hr>
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<p><a href="http://www.geom.uiuc.edu/"><img src="qh--geom.gif"
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align="middle" width="40" height="40"></a><i>The Geometry Center
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Home Page </i></p>
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<p>Comments to: <a href=mailto:qhull@qhull.org>qhull@qhull.org</a>
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</a><br>
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Created: Sept. 25, 1995 --- <!-- hhmts start --> Last modified: see top <!-- hhmts end --> </p>
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</body>
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</html>
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