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<title>qhull -- convex hull and related structures</title>
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<p><b><a name="TOP">Up</a></b><b>:</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><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.uiuc.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-optq.htm#Qbb">Qbb</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>By default, Qhull merges coplanar facets. For example, the convex
hull of a cube's vertices has six facets.
<p>If you use '<a href="qh-optq.htm#Qt">Qt</a>' (triangulated output),
all facets will be simplicial (e.g., triangles in 2-d). For the cube
example, it will have 12 facets. Some facets may be
degenerate and have zero area.
<p>If you use '<a href="qh-optq.htm#QJn">QJ</a>' (joggled input),
all facets will be simplicial. The corresponding vertices will be
slightly perturbed. Joggled input is less accurate that triangulated
output.See <a
href="qh-impre.htm#joggle">Merged facets or joggled input</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 © 1995-2003 The Geometry Center, Minneapolis MN</b></p>
<hr>
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