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diff --git a/xs/src/qhull/html/qh-optc.htm b/xs/src/qhull/html/qh-optc.htm new file mode 100644 index 000000000..87308180d --- /dev/null +++ b/xs/src/qhull/html/qh-optc.htm @@ -0,0 +1,292 @@ +<!DOCTYPE HTML PUBLIC "-//IETF//DTD HTML//EN"> +<html> + +<head> +<title>Qhull precision options</title> +</head> + +<body> +<!-- Navigation links --> +<p><b>Up:</b> <a href="http://www.qhull.org">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> +• <a href="../src/libqhull_r/index.htm">Functions</a></p> + +<hr> +<!-- Main text of document --> +<h1><a +href="http://www.geom.uiuc.edu/graphics/pix/Special_Topics/Computational_Geometry/delaunay.html"><img +src="qh--dt.gif" alt="[delaunay]" align="middle" width="100" +height="100"></a> Qhull precision options</h1> + +This section lists the precision options for Qhull. These options are +indicated by an upper-case letter followed by a number. + +<p><b>Copyright © 1995-2015 C.B. Barber</b></p> + +<hr> + +<p><a href="index.htm#TOC">»</a> <a href="qh-quick.htm#programs">Programs</a> +<a name="prec">•</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> +• <a href="../src/libqhull_r/index.htm">Functions</a></p> + +<h2>Precision options</h2> + +<p>Most users will not need to set these options. They are best +used for <a href="qh-impre.htm#approximate">approximating</a> a +convex hull. They may also be used for testing Qhull's handling +of precision errors.</p> + +<p>By default, Qhull uses options '<a href="#C0">C-0</a>' for +2-d, 3-d and 4-d, and '<a href="qh-optq.htm#Qx">Qx</a>' for 5-d +and higher. These options use facet merging to handle precision +errors. You may also use joggled input '<a href="qh-optq.htm#QJn">QJ</a>' +to avoid precision problems. +For more information see <a +href="qh-impre.htm">Imprecision in Qhull</a>.</p> + +<dl compact> + <dt> </dt> + <dd><b>General</b></dd> + <dt><a href="#Cn2">Cn</a></dt> + <dd>centrum radius for post-merging</dd> + <dt><a href="#Cn">C-n</a></dt> + <dd>centrum radius for pre-merging</dd> + <dt><a href="#An2">An</a></dt> + <dd>cosine of maximum angle for post-merging</dd> + <dt><a href="#An">A-n</a></dt> + <dd>cosine of maximum angle for pre-merging</dd> + <dt><a href="qh-optq.htm#Qx">Qx</a></dt> + <dd>exact pre-merges (allows coplanar facets)</dd> + <dt><a href="#C0">C-0</a></dt> + <dd>handle all precision errors</dd> + <dt><a href="#Wn">Wn</a></dt> + <dd>min distance above plane for outside points</dd> +</dl> + +<dl compact> + <dt> </dt> + <dd><b>Experimental</b></dd> + <dt><a href="#Un">Un</a></dt> + <dd>max distance below plane for a new, coplanar point</dd> + <dt><a href="#En">En</a></dt> + <dd>max roundoff error for distance computation</dd> + <dt><a href="#Vn">Vn</a></dt> + <dd>min distance above plane for a visible facet</dd> + <dt><a href="#Rn">Rn</a></dt> + <dd>randomly perturb computations by a factor of [1-n,1+n]</dd> +</dl> + +<dl compact> +</dl> + +<hr> + +<h3><a href="#prec">»</a><a name="An">A-n - cosine of maximum +angle for pre-merging.</a></h3> + +<p>Pre-merging occurs while Qhull constructs the hull. It is +indicated by '<a href="#Cn">C-n</a>', 'A-n', or '<a +href="qh-optq.htm#Qx">Qx</a>'.</p> + +<p>If the angle between a pair of facet normals is greater than <i>n</i>, +Qhull merges one of the facets into a neighbor. It selects the +facet that is closest to a neighboring facet.</p> + +<p>For example, option 'A-0.99' merges facets during the +construction of the hull. If the cosine of the angle between +facets is greater than 0.99, one or the other facet is merged. +Qhull accounts for the maximum roundoff error.</p> + +<p>If 'A-n' is set without '<a href="#Cn">C-n</a>', then '<a +href="#C0">C-0</a>' is automatically set. </p> + +<p>In 5-d and higher, you should set '<a href="qh-optq.htm#Qx">Qx</a>' +along with 'A-n'. It skips merges of coplanar facets until after +the hull is constructed and before '<a href="#An2">An</a>' and '<a +href="#Cn2">Cn</a>' are checked. </p> + +<h3><a href="#prec">»</a><a name="An2">An - cosine of maximum angle for +post-merging.</a></h3> + +<p>Post merging occurs after the hull is constructed. For +example, option 'A0.99' merges a facet if the cosine of the angle +between facets is greater than 0.99. Qhull accounts for the +maximum roundoff error.</p> + +<p>If 'An' is set without '<a href="#Cn2">Cn</a>', then '<a +href="#Cn2">C0</a>' is automatically set. </p> + +<h3><a href="#prec">»</a><a name="C0">C-0 - handle all precision +errors </a></h3> + +<p>Qhull handles precision errors by merging facets. The 'C-0' +option handles all precision errors in 2-d, 3-d, and 4-d. It is +set by default. It may be used in higher dimensions, but +sometimes the facet width grows rapidly. It is usually better to +use '<a href="qh-optq.htm#Qx">Qx</a>' in 5-d and higher. +Use '<a href="qh-optq.htm#QJn">QJ</a>' to joggle the input +instead of merging facets. +Use '<a +href="qh-optq.htm#Q0">Q0</a>' to turn both options off.</p> + +<p>Qhull optimizes 'C-0' ("_zero-centrum") by testing +vertices instead of centrums for adjacent simplices. This may be +slower in higher dimensions if merges decrease the number of +processed points. The optimization may be turned off by setting a +small value such as 'C-1e-30'. See <a href="qh-impre.htm">How +Qhull handles imprecision</a>.</p> + +<h3><a href="#prec">»</a><a name="Cn">C-n - centrum radius for +pre-merging</a></h3> + +<p>Pre-merging occurs while Qhull constructs the hull. It is +indicated by 'C-n', '<a href="#An">A-n</a>', or '<a +href="qh-optq.htm#Qx">Qx</a>'.</p> + +<p>The <i>centrum</i> of a facet is a point on the facet for +testing facet convexity. It is the average of the vertices +projected to the facet's hyperplane. Two adjacent facets are +convex if each centrum is clearly below the other facet. </p> + +<p>If adjacent facets are non-convex, one of the facets is merged +into a neighboring facet. Qhull merges the facet that is closest +to a neighboring facet. </p> + +<p>For option 'C-n', <i>n</i> is the centrum radius. For example, +'C-0.001' merges facets whenever the centrum is less than 0.001 +from a neighboring hyperplane. Qhull accounts for roundoff error +when testing the centrum.</p> + +<p>In 5-d and higher, you should set '<a href="qh-optq.htm#Qx">Qx</a>' +along with 'C-n'. It skips merges of coplanar facets until after +the hull is constructed and before '<a href="#An2">An</a>' and '<a +href="#Cn2">Cn</a>' are checked. </p> + +<h3><a href="#prec">»</a><a name="Cn2">Cn - centrum radius for +post-merging</a></h3> + +<p>Post-merging occurs after Qhull constructs the hull. It is +indicated by '<a href="#Cn2">Cn</a>' or '<a href="#An2">An</a>'. </p> + +<p>For option '<a href="#Cn2">Cn</a>', <i>n</i> is the centrum +radius. For example, 'C0.001' merges facets when the centrum is +less than 0.001 from a neighboring hyperplane. Qhull accounts for +roundoff error when testing the centrum.</p> + +<p>Both pre-merging and post-merging may be defined. If only +post-merging is used ('<a href="qh-optq.htm#Q0">Q0</a>' with +'Cn'), Qhull may fail to produce a hull due to precision errors +during the hull's construction.</p> + +<h3><a href="#prec">»</a><a name="En">En - max roundoff error +for distance computations</a></h3> + +<p>This allows the user to change the maximum roundoff error +computed by Qhull. The value computed by Qhull may be overly +pessimistic. If 'En' is set too small, then the output may not be +convex. The statistic "max. distance of a new vertex to a +facet" (from option '<a href="qh-optt.htm#Ts">Ts</a>') is a +reasonable upper bound for the actual roundoff error. </p> + +<h3><a href="#prec">»</a><a name="Rn">Rn - randomly perturb +computations </a></h3> + +<p>This option perturbs every distance, hyperplane, and angle +computation by up to <i>(+/- n * max_coord)</i>. It simulates the +effect of roundoff errors. Unless '<a href="#En">En</a>' is +explicitly set, it is adjusted for 'Rn'. The command 'qhull Rn' +will generate a convex hull despite the perturbations. See the <a +href="qh-eg.htm#merge">Examples </a>section for an example.</p> + +<p>Options 'Rn C-n' have the effect of '<a href="#Wn">W2n</a>' +and '<a href="#Cn">C-2n</a>'. To use time as the random number +seed, use option '<a href="qh-optq.htm#QRn">QR-1</a>'.</p> + +<h3><a href="#prec">»</a><a name="Un">Un - max distance for a +new, coplanar point </a></h3> + +<p>This allows the user to set coplanarity. When pre-merging ('<a +href="#Cn">C-n </a>', '<a href="#An">A-n</a>' or '<a +href="qh-optq.htm#Qx">Qx</a>'), Qhull merges a new point into any +coplanar facets. The default value for 'Un' is '<a href="#Vn">Vn</a>'.</p> + +<h3><a href="#prec">»</a><a name="Vn">Vn - min distance for a +visible facet </a></h3> + +<p>This allows the user to set facet visibility. When adding a +point to the convex hull, Qhull determines all facets that are +visible from the point. A facet is visible if the distance from +the point to the facet is greater than 'Vn'.</p> + +<p>Without merging, the default value for 'Vn' is the roundoff +error ('<a href="#En">En</a>'). With merging, the default value +is the pre-merge centrum ('<a href="#Cn">C-n</a>') in 2-d or 3-d, +or three times that in other dimensions. If the outside width is +specified with option '<a href="#Wn">Wn </a>', the maximum, +default value for 'Vn' is '<a href="#Wn">Wn</a>'.</p> + +<p>Qhull warns if 'Vn' is greater than '<a href="#Wn">Wn</a>' and +furthest outside ('<a href="qh-optq.htm#Qf">Qf</a>') is not +selected; this combination usually results in flipped facets +(i.e., reversed normals).</p> + +<h3><a href="#prec">»</a><a name="Wn">Wn - min distance above +plane for outside points</a></h3> + +<p>Points are added to the convex hull only if they are clearly +outside of a facet. A point is outside of a facet if its distance +to the facet is greater than 'Wn'. Without pre-merging, the +default value for 'Wn' is '<a href="#En">En </a>'. If the user +specifies pre-merging and does not set 'Wn', than 'Wn' is set to +the maximum of '<a href="#Cn">C-n</a>' and <i>maxcoord*(1 - </i><a +href="#An"><i>A-n</i></a><i>)</i>.</p> + +<p>This option is good for <a href="qh-impre.htm#approximate">approximating</a> +a convex hull.</p> + +<p>Options '<a href="qh-optq.htm#Qc">Qc</a>' and '<a +href="qh-optq.htm#Qi">Qi</a>' use the minimum vertex to +distinguish coplanar points from interior points.</p> +<!-- Navigation links --> +<hr> + +<p><b>Up:</b> <a href="http://www.qhull.org">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> +• <a href="../src/libqhull_r/index.htm">Functions</a></p> +<!-- GC common information --> +<hr> + +<p><a href="http://www.geom.uiuc.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@qhull.org>qhull@qhull.org</a> +</a><br> +Created: Sept. 25, 1995 --- <!-- hhmts start --> Last modified: see top <!-- hhmts end --> </p> +</body> +</html> |