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693 lines
30 KiB
HTML
693 lines
30 KiB
HTML
<!DOCTYPE HTML PUBLIC "-//IETF//DTD HTML//EN">
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<html>
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<head>
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<title>Examples of Qhull</title>
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</head>
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<body>
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<!-- Navigation links -->
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<p><a name="TOP"><b>Up:</b></a> <a href="http://www.qhull.org">Home
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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><br>
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<b>To: </b><a href="#TOC">Qhull examples: Table of Contents</a> (please wait
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while loading)<br>
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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/half.html"><img
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src="qh--half.gif" alt="[halfspace]" align="middle" width="100"
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height="100"></a> Examples of Qhull</h1>
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<p>This section of the Qhull manual will introduce you to Qhull
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and its options. Each example is a file for viewing with <a
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href="index.htm#geomview">Geomview</a>. You will need to
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use a Unix computer with a copy of Geomview.
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<p>
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If you are not running Unix, you can view <a
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href="http://www.geom.uiuc.edu/graphics/pix/Special_Topics/Computational_Geometry/welcome.html">pictures</a>
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for some of the examples. To understand Qhull without Geomview, try the
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examples in <a href="qh-quick.htm#programs">Programs</a> and
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<a href="qh-quick.htm#programs">Programs/input</a>. You can also try small
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examples that you compute by hand. Use <a href="rbox.htm">rbox</a>
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to generate examples.
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<p>
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To generate the Geomview examples, execute the shell script <tt>eg/q_eg</tt>.
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It uses <tt>rbox</tt>. The shell script <tt>eg/q_egtest</tt> generates
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test examples, and <tt>eg/q_test</tt> exercises the code. If you
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find yourself viewing the inside of a 3-d example, use Geomview's
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normalization option on the 'obscure' menu.</p>
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<p><b>Copyright © 1995-2015 C.B. Barber</b></p>
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<hr>
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<h2><a href="#TOP">»</a><a name="TOC">Qhull examples: Table of
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Contents </a></h2>
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<ul>
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<li><a href="#2d">2-d and 3-d examples</a></li>
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<li><a href="#how">How Qhull adds a point</a></li>
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<li><a href="#joggle">Triangulated output or joggled input</a></li>
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<li><a href="#delaunay">Delaunay and Voronoi diagrams</a></li>
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<li><a href="#merge">Facet merging for imprecision</a></li>
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<li><a href="#4d">4-d objects</a></li>
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<li><a href="#half">Halfspace intersections</a></li>
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</ul>
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<hr>
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<ul>
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<li><a href="#TOC">»</a><a name="2d">2-d and 3-d examples</a><ul>
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<li><a href="#01">eg.01.cube</a></li>
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<li><a href="#02">eg.02.diamond.cube</a></li>
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<li><a href="#03">eg.03.sphere</a></li>
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<li><a href="#04">eg.04.circle</a></li>
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<li><a href="#05">eg.05.spiral</a></li>
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<li><a href="#06">eg.06.merge.square</a></li>
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<li><a href="#07">eg.07.box</a></li>
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<li><a href="#08a">eg.08a.cube.sphere</a></li>
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<li><a href="#08b">eg.08b.diamond.sphere</a></li>
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<li><a href="#09">eg.09.lens</a></li>
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</ul>
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</li>
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<li><a href="#TOC">»</a><a name="how">How Qhull adds a point</a><ul>
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<li><a href="#10a">eg.10a.sphere.visible</a></li>
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<li><a href="#10b">eg.10b.sphere.beyond</a></li>
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<li><a href="#10c">eg.10c.sphere.horizon</a></li>
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<li><a href="#10d">eg.10d.sphere.cone</a></li>
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<li><a href="#10e">eg.10e.sphere.new</a></li>
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<li><a href="#14">eg.14.sphere.corner</a></li>
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</ul>
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</li>
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<li><a href="#TOC">»</a> <a name="joggle">Triangulated output or joggled input</a>
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<ul>
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<li><a href="#15a">eg.15a.surface</a></li>
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<li><a href="#15b">eg.15b.triangle</a></li>
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<li><a href="#15c">eg.15c.joggle</a></li>
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</ul>
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<li><a href="#TOC">»</a><a name="delaunay"> Delaunay and
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Voronoi diagrams</a><ul>
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<li><a href="#17a">eg.17a.delaunay.2</a></li>
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<li><a href="#17b">eg.17b.delaunay.2i</a></li>
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<li><a href="#17c">eg.17c.delaunay.2-3</a></li>
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<li><a href="#17d">eg.17d.voronoi.2</a></li>
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<li><a href="#17e">eg.17e.voronoi.2i</a></li>
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<li><a href="#17f">eg.17f.delaunay.3</a></li>
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<li><a href="#18a">eg.18a.furthest.2-3</a></li>
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<li><a href="#18b">eg.18b.furthest-up.2-3</a></li>
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<li><a href="#18c">eg.18c.furthest.2</a></li>
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<li><a href="#19">eg.19.voronoi.region.3</a></li>
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</ul>
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</li>
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<li><a href="#TOC">»</a><a name="merge">Facet merging for
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imprecision </a><ul>
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<li><a href="#20">eg.20.cone</a></li>
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<li><a href="#21a">eg.21a.roundoff.errors</a></li>
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<li><a href="#21b">eg.21b.roundoff.fixed</a></li>
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<li><a href="#22a">eg.22a.merge.sphere.01</a></li>
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<li><a href="#22b">eg.22b.merge.sphere.-01</a></li>
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<li><a href="#22c">eg.22c.merge.sphere.05</a></li>
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<li><a href="#22d">eg.22d.merge.sphere.-05</a></li>
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<li><a href="#23">eg.23.merge.cube</a></li>
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</ul>
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</li>
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<li><a href="#TOC">»</a><a name="4d">4-d objects</a><ul>
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<li><a href="#24">eg.24.merge.cube.4d-in-3d</a></li>
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<li><a href="#30">eg.30.4d.merge.cube</a></li>
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<li><a href="#31">eg.31.4d.delaunay</a></li>
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<li><a href="#32">eg.32.4d.octant</a></li>
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</ul>
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</li>
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<li><a href="#TOC">»</a><a name="half">Halfspace
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intersections</a><ul>
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<li><a href="#33a">eg.33a.cone</a></li>
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<li><a href="#33b">eg.33b.cone.dual</a></li>
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<li><a href="#33c">eg.33c.cone.halfspace</a></li>
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</ul>
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</li>
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</ul>
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<hr>
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<h2><a href="#TOC">»</a>2-d and 3-d examples</h2>
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<h3><a href="#2d">»</a><a name="01">rbox c D3 | qconvex G
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>eg.01.cube </a></h3>
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<p>The first example is a cube in 3-d. The color of each facet
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indicates its normal. For example, normal [0,0,1] along the Z
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axis is (r=0.5, g=0.5, b=1.0). With the 'Dn' option in <tt>rbox</tt>,
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you can generate hypercubes in any dimension. Above 7-d the
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number of intermediate facets grows rapidly. Use '<a
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href="qh-optt.htm#TFn">TFn</a>' to track qconvex's progress. Note
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that each facet is a square that qconvex merged from coplanar
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triangles.</p>
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<h3><a href="#2d">»</a><a name="02">rbox c d G3.0 | qconvex G
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>eg.02.diamond.cube </a></h3>
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<p>The second example is a cube plus a diamond ('d') scaled by <tt>rbox</tt>'s
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'G' option. In higher dimensions, diamonds are much simpler than
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hypercubes. </p>
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<h3><a href="#2d">»</a><a name="03">rbox s 100 D3 | qconvex G
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>eg.03.sphere </a></h3>
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<p>The <tt>rbox s</tt> option generates random points and
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projects them to the d-sphere. All points should be on the convex
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hull. Notice that random points look more clustered than you
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might expect. You can get a smoother distribution by merging
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facets and printing the vertices, e.g.,<i> rbox 1000 s | qconvex
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A-0.95 p | qconvex G >eg.99</i>.</p>
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<h3><a href="#2d">»</a><a name="04">rbox s 100 D2 | qconvex G
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>eg.04.circle </a></h3>
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<p>In 2-d, there are many ways to generate a convex hull. One of
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the earliest algorithms, and one of the fastest, is the 2-d
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Quickhull algorithm [c.f., Preparata & Shamos <a
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href="index.htm#pre-sha85">'85</a>]. It was the model for
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Qhull.</p>
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<h3><a href="#2d">»</a><a name="05">rbox 10 l | qconvex G
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>eg.05.spiral </a></h3>
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<p>One rotation of a spiral.</p>
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<h3><a href="#2d">»</a><a name="06">rbox 1000 D2 | qconvex C-0.03
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Qc Gapcv >eg.06.merge.square</a></h3>
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<p>This demonstrates how Qhull handles precision errors. Option '<a
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href="qh-optc.htm#Cn">C-0.03</a>' requires a clearly convex angle
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between adjacent facets. Otherwise, Qhull merges the facets. </p>
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<p>This is the convex hull of random points in a square. The
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facets have thickness because they must be outside all points and
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must include their vertices. The colored lines represent the
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original points and the spheres represent the vertices. Floating
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in the middle of each facet is the centrum. Each centrum is at
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least 0.03 below the planes of its neighbors. This guarantees
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that the facets are convex.</p>
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<h3><a href="#2d">»</a><a name="07">rbox 1000 D3 | qconvex G
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>eg.07.box </a></h3>
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<p>Here's the same distribution but in 3-d with Qhull handling
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machine roundoff errors. Note the large number of facets. </p>
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<h3><a href="#2d">»</a><a name="08a">rbox c G0.4 s 500 | qconvex G
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>eg.08a.cube.sphere </a></h3>
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<p>The sphere is just barely poking out of the cube. Try the same
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distribution with randomization turned on ('<a
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href="qh-optq.htm#Qr">Qr</a>'). This turns Qhull into a
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randomized incremental algorithm. To compare Qhull and
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randomization, look at the number of hyperplanes created and the
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number of points partitioned. Don't compare CPU times since Qhull's
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implementation of randomization is inefficient. The number of
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hyperplanes and partitionings indicate the dominant costs for
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Qhull. With randomization, you'll notice that the number of
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facets created is larger than before. This is especially true as
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you increase the number of points. It is because the randomized
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algorithm builds most of the sphere before it adds the cube's
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vertices.</p>
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<h3><a href="#2d">»</a><a name="08b">rbox d G0.6 s 500 | qconvex G
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>eg.08b.diamond.sphere </a></h3>
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<p>This is a combination of the diamond distribution and the
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sphere.</p>
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<h3><a href="#2d">»</a><a name="09">rbox 100 L3 G0.5 s | qconvex
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G >eg.09.lens </a></h3>
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<p>Each half of the lens distribution lies on a sphere of radius
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three. A directed search for the furthest facet below a point
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(e.g., qh_findbest in <tt>geom.c</tt>) may fail if started from
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an arbitrary facet. For example, if the first facet is on the
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opposite side of the lens, a directed search will report that the
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point is inside the convex hull even though it is outside. This
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problem occurs whenever the curvature of the convex hull is less
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than a sphere centered at the test point. </p>
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<p>To prevent this problem, Qhull does not use directed search
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all the time. When Qhull processes a point on the edge of the
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lens, it partitions the remaining points with an exhaustive
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search instead of a directed search (see qh_findbestnew in <tt>geom2.c</tt>).
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</p>
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<h2><a href="#TOC">»</a>How Qhull adds a point</h2>
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<h3><a href="#how">»</a><a name="10a">rbox 100 s P0.5,0.5,0.5 |
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qconvex Ga QG0 >eg.10a.sphere.visible</a></h3>
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<p>The next 4 examples show how Qhull adds a point. The point
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[0.5,0.5,0.5] is at one corner of the bounding box. Qhull adds a
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point using the beneath-beyond algorithm. First Qhull finds all
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of the facets that are visible from the point. Qhull will replace
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these facets with new facets.</p>
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<h3><a href="#how">»</a><a name="10b">rbox 100 s
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P0.5,0.5,0.5|qconvex Ga QG-0 >eg.10b.sphere.beyond </a></h3>
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<p>These are the facets that are not visible from the point.
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Qhull will keep these facets.</p>
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<h3><a href="#how">»</a><a name="10c">rbox 100 s P0.5,0.5,0.5 |
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qconvex PG Ga QG0 >eg.10c.sphere.horizon </a></h3>
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<p>These facets are the horizon facets; they border the visible
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facets. The inside edges are the horizon ridges. Each horizon
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ridge will form the base for a new facet.</p>
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<h3><a href="#how">»</a><a name="10d">rbox 100 s P0.5,0.5,0.5 |
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qconvex Ga QV0 PgG >eg.10d.sphere.cone </a></h3>
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<p>This is the cone of points from the new point to the horizon
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facets. Try combining this image with <tt>eg.10c.sphere.horizon</tt>
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and <tt>eg.10a.sphere.visible</tt>.
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</p>
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<h3><a href="#how">»</a><a name="10e">rbox 100 s P0.5,0.5,0.5 |
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qconvex Ga >eg.10e.sphere.new</a></h3>
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<p>This is the convex hull after [0.5,0.5,0.5] has been added.
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Note that in actual practice, the above sequence would never
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happen. Unlike the randomized algorithms, Qhull always processes
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a point that is furthest in an outside set. A point like
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[0.5,0.5,0.5] would be one of the first points processed.</p>
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<h3><a href="#how">»</a><a name="14">rbox 100 s P0.5,0.5,0.5 |
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qhull Ga QV0g Q0 >eg.14.sphere.corner</a></h3>
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<p>The '<a href="qh-optq.htm#QVn">QVn</a>', '<a
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href="qh-optq.htm#QGn">QGn </a>' and '<a href="qh-optp.htm#Pdk">Pdk</a>'
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options define good facets for Qhull. In this case '<a
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href="qh-optq.htm#QVn">QV0</a>' defines the 0'th point
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[0.5,0.5,0.5] as the good vertex, and '<a href="qh-optq.htm#Qg">Qg</a>'
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tells Qhull to only build facets that might be part of a good
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facet. This technique reduces output size in low dimensions. It
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does not work with facet merging.</p>
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<h2><a href="#TOC">»</a>Triangulated output or joggled input</h2>
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<h3><a href="#joggle">»</a><a name="15a">rbox 500 W0 | qconvex QR0 Qc Gvp >eg.15a.surface</a></h3>
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<p>This is the convex hull of 500 points on the surface of
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a cube. Note the large, non-simplicial facet for each face.
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Qhull merges non-convex facets.
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<p>If the facets were not merged, Qhull
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would report precision problems. For example, turn off facet merging
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with option '<a href="qh-optq.htm#Q0">Q0</a>'. Qhull may report concave
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facets, flipped facets, or other precision errors:
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<blockquote>
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rbox 500 W0 | qhull QR0 Q0
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</blockquote>
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<p>
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<h3><a href="#joggle">»</a><a name="15b">rbox 500 W0 | qconvex QR0 Qt Qc Gvp >eg.15b.triangle</a></h3>
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<p>Like the previous examples, this is the convex hull of 500 points on the
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surface of a cube. Option '<a href="qh-optq.htm#Qt">Qt</a>' triangulates the
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non-simplicial facets. Triangulated output is
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particularly helpful for Delaunay triangulations.
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<p>
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<h3><a href="#joggle">»</a><a name="15c">rbox 500 W0 | qconvex QR0 QJ5e-2 Qc Gvp >eg.15c.joggle</a></h3>
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<p>This is the convex hull of 500 joggled points on the surface of
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a cube. The option '<a href="qh-optq.htm#QJn">QJ5e-2</a>'
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sets a very large joggle to make the effect visible. Notice
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that all of the facets are triangles. If you rotate the cube,
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you'll see red-yellow lines for coplanar points.
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<p>
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With option '<a href="qh-optq.htm#QJn">QJ</a>', Qhull joggles the
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input to avoid precision problems. It adds a small random number
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to each input coordinate. If a precision
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error occurs, it increases the joggle and tries again. It repeats
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this process until no precision problems occur.
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<p>
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Joggled input is a simple solution to precision problems in
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computational geometry. Qhull can also merge facets to handle
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precision problems. See <a href="qh-impre.htm#joggle">Merged facets or joggled input</a>.
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<h2><a href="#TOC">»</a>Delaunay and Voronoi diagrams</h2>
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<h3><a href="#delaunay">»</a><a name="17a">qdelaunay Qt
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<eg.data.17 GnraD2 >eg.17a.delaunay.2</a></h3>
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<p>
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The input file, <tt>eg.data.17</tt>, consists of a square, 15 random
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points within the outside half of the square, and 6 co-circular
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points centered on the square.
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<p>The Delaunay triangulation is the triangulation with empty
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circumcircles. The input for this example is unusual because it
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includes six co-circular points. Every triangular subset of these
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points has the same circumcircle. Option '<a href="qh-optq.htm#Qt">Qt</a>'
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triangulates the co-circular facet.</p>
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<h3><a href="#delaunay">»</a><a name="17b">qdelaunay <eg.data.17
|
|
GnraD2 >eg.17b.delaunay.2i</a></h3>
|
|
|
|
<p>This is the same example without triangulated output ('<a href="qh-optq.htm#Qt">Qt</a>'). qdelaunay
|
|
merges the non-unique Delaunay triangles into a hexagon.</p>
|
|
|
|
<h3><a href="#delaunay">»</a><a name="17c">qdelaunay <eg.data.17
|
|
Ga >eg.17c.delaunay.2-3 </a></h3>
|
|
|
|
<p>This is how Qhull generated both diagrams. Use Geomview's
|
|
'obscure' menu to turn off normalization, and Geomview's
|
|
'cameras' menu to turn off perspective. Then load this <a
|
|
href="http://www.geom.uiuc.edu/graphics/pix/Special_Topics/Computational_Geometry/delaunay.html">object</a>
|
|
with one of the previous diagrams.</p>
|
|
|
|
<p>The points are lifted to a paraboloid by summing the squares
|
|
of each coordinate. These are the light blue points. Then the
|
|
convex hull is taken. That's what you see here. If you look up
|
|
the Z-axis, you'll see that points and edges coincide.</p>
|
|
|
|
<h3><a href="#delaunay">»</a><a name="17d">qvoronoi QJ
|
|
<eg.data.17 Gna >eg.17d.voronoi.2</a></h3>
|
|
|
|
<p>The Voronoi diagram is the dual of the Delaunay triangulation.
|
|
Here you see the original sites and the Voronoi vertices.
|
|
Notice the each
|
|
vertex is equidistant from three sites. The edges indicate the
|
|
Voronoi region for a site. Qhull does not draw the unbounded
|
|
edges. Instead, it draws extra edges to close the unbounded
|
|
Voronoi regions. You may find it helpful to enclose the input
|
|
points in a square. You can compute the unbounded
|
|
rays from option '<a href="qh-optf.htm#Fo2">Fo</a>'.
|
|
</p>
|
|
|
|
<p>Instead
|
|
of triangulated output ('<a href="qh-optq.htm#Qt">Qt</a>'), this
|
|
example uses joggled input ('<a href="qh-optq.htm#QJn">QJ</a>').
|
|
Normally, you should use neither 'QJ' nor 'Qt' for Voronoi diagrams.
|
|
|
|
<h3><a href="#delaunay">»</a><a name="17e">qvoronoi <eg.data.17
|
|
Gna >eg.17e.voronoi.2i </a></h3>
|
|
|
|
<p>This looks the same as the previous diagrams, but take a look
|
|
at the data. Run 'qvoronoi p <eg/eg.data.17'. This prints
|
|
the Voronoi vertices.
|
|
|
|
<p>With 'QJ', there are four nearly identical Voronoi vertices
|
|
within 10^-11 of the origin. Option 'QJ' joggled the input. After the joggle,
|
|
the cocircular
|
|
input sites are no longer cocircular. The corresponding Voronoi vertices are
|
|
similar but not identical.
|
|
|
|
<p>This example does not use options 'Qt' or 'QJ'. The cocircular
|
|
input sites define one Voronoi vertex near the origin. </p>
|
|
|
|
<p>Option 'Qt' would triangulate the corresponding Delaunay region into
|
|
four triangles. Each triangle is assigned the same Voronoi vertex.</p>
|
|
|
|
<h3><a href="#delaunay">»</a><a name="17f"> rbox c G0.1 d |
|
|
qdelaunay Gt Qz <eg.17f.delaunay.3 </a></h3>
|
|
|
|
<p>This is the 3-d Delaunay triangulation of a small cube inside
|
|
a prism. Since the outside ridges are transparent, it shows the
|
|
interior of the outermost facets. If you slice open the
|
|
triangulation with Geomview's ginsu, you will see that the innermost
|
|
facet is a cube. Note the use of '<a href="qh-optq.htm#Qz">Qz</a>'
|
|
to add a point "at infinity". This avoids a degenerate
|
|
input due to cospherical points.</p>
|
|
|
|
<h3><a href="#delaunay">»</a><a name="18a">rbox 10 D2 d | qdelaunay
|
|
Qu G >eg.18a.furthest.2-3 </a></h3>
|
|
|
|
<p>The furthest-site Voronoi diagram contains Voronoi regions for
|
|
points that are <i>furthest </i>from an input site. It is the
|
|
dual of the furthest-site Delaunay triangulation. You can
|
|
determine the furthest-site Delaunay triangulation from the
|
|
convex hull of the lifted points (<a href="#17c">eg.17c.delaunay.2-3</a>).
|
|
The upper convex hull (blue) generates the furthest-site Delaunay
|
|
triangulation. </p>
|
|
|
|
<h3><a href="#delaunay">»</a><a name="18b">rbox 10 D2 d | qdelaunay
|
|
Qu Pd2 G >eg.18b.furthest-up.2-3</a></h3>
|
|
|
|
<p>This is the upper convex hull of the preceding example. The
|
|
furthest-site Delaunay triangulation is the projection of the
|
|
upper convex hull back to the input points. The furthest-site
|
|
Voronoi vertices are the circumcenters of the furthest-site
|
|
Delaunay triangles. </p>
|
|
|
|
<h3><a href="#delaunay">»</a><a name="18c">rbox 10 D2 d | qvoronoi
|
|
Qu Gv >eg.18c.furthest.2</a></h3>
|
|
|
|
<p>This shows an incomplete furthest-site Voronoi diagram. It
|
|
only shows regions with more than two vertices. The regions are
|
|
artificially truncated. The actual regions are unbounded. You can
|
|
print the regions' vertices with 'qvoronoi Qu <a
|
|
href="qh-opto.htm#o">o</a>'. </p>
|
|
|
|
<p>Use Geomview's 'obscure' menu to turn off normalization, and
|
|
Geomview's 'cameras' menu to turn off perspective. Then load this
|
|
with the upper convex hull.</p>
|
|
|
|
<h3><a href="#delaunay">»</a><a name="19">rbox 10 D3 | qvoronoi QV5
|
|
p | qconvex G >eg.19.voronoi.region.3 </a></h3>
|
|
|
|
<p>This shows the Voronoi region for input site 5 of a 3-d
|
|
Voronoi diagram.</p>
|
|
|
|
<h2><a href="#TOC">»</a>Facet merging for imprecision</h2>
|
|
|
|
<h3><a href="#merge">»</a><a name="20">rbox r s 20 Z1 G0.2 |
|
|
qconvex G >eg.20.cone </a></h3>
|
|
|
|
<p>There are two things unusual about this <a
|
|
href="http://www.geom.uiuc.edu/graphics/pix/Special_Topics/Computational_Geometry/cone.html">cone</a>.
|
|
One is the large flat disk at one end and the other is the
|
|
rectangles about the middle. That's how the points were
|
|
generated, and if those points were exact, this is the correct
|
|
hull. But <tt>rbox</tt> used floating point arithmetic to
|
|
generate the data. So the precise convex hull should have been
|
|
triangles instead of rectangles. By requiring convexity, Qhull
|
|
has recovered the original design.</p>
|
|
|
|
<h3><a href="#merge">»</a><a name="21a">rbox 200 s | qhull Q0
|
|
R0.01 Gav Po >eg.21a.roundoff.errors </a></h3>
|
|
|
|
<p>This is the convex hull of 200 cospherical points with
|
|
precision errors ignored ('<a href="qh-optq.htm#Q0">Q0</a>'). To
|
|
demonstrate the effect of roundoff error, we've added a random
|
|
perturbation ('<a href="qh-optc.htm#Rn">R0.01</a>') to every
|
|
distance and hyperplane calculation. Qhull, like all other convex
|
|
hull algorithms with floating point arithmetic, makes
|
|
inconsistent decisions and generates wildly wrong results. In
|
|
this case, one or more facets are flipped over. These facets have
|
|
the wrong color. You can also turn on 'normals' in Geomview's
|
|
appearances menu and turn off 'facing normals'. There should be
|
|
some white lines pointing in the wrong direction. These
|
|
correspond to flipped facets. </p>
|
|
|
|
<p>Different machines may not produce this picture. If your
|
|
machine generated a long error message, decrease the number of
|
|
points or the random perturbation ('<a href="qh-optc.htm#Rn">R0.01</a>').
|
|
If it did not report flipped facets, increase the number of
|
|
points or perturbation.</p>
|
|
|
|
<h3><a href="#merge">»</a><a name="21b">rbox 200 s | qconvex Qc
|
|
R0.01 Gpav >eg.21b.roundoff.fixed </a></h3>
|
|
|
|
<p>Qhull handles the random perturbations and returns an
|
|
imprecise <a
|
|
href="http://www.geom.uiuc.edu/graphics/pix/Special_Topics/Computational_Geometry/fixed.html">sphere</a>.
|
|
In this case, the output is a weak approximation to the points.
|
|
This is because a random perturbation of '<a
|
|
href="qh-optc.htm#Rn">R0.01 </a>' is equivalent to losing all but
|
|
1.8 digits of precision. The outer planes float above the points
|
|
because Qhull needs to allow for the maximum roundoff error. </p>
|
|
<p>
|
|
If you start with a smaller random perturbation, you
|
|
can use joggle ('<a href="qh-optq.htm#QJn">QJn</a>') to avoid
|
|
precision problems. You need to set <i>n</i> significantly
|
|
larger than the random perturbation. For example, try
|
|
'rbox 200 s | qconvex Qc R1e-4 QJ1e-1'.
|
|
|
|
<h3><a href="#merge">»</a><a name="22a">rbox 1000 s| qconvex C0.01
|
|
Qc Gcrp >eg.22a.merge.sphere.01</a></h3>
|
|
|
|
<h3><a href="#merge">»</a><a name="22b">rbox 1000 s| qconvex
|
|
C-0.01 Qc Gcrp >eg.22b.merge.sphere.-01</a></h3>
|
|
|
|
<h3><a href="#merge">»</a><a name="22c">rbox 1000 s| qconvex C0.05
|
|
Qc Gcrpv >eg.22c.merge.sphere.05</a></h3>
|
|
|
|
<h3><a href="#merge">»</a><a name="22d">rbox 1000 s| qconvex
|
|
C-0.05 Qc Gcrpv >eg.22d.merge.sphere.-05</a></h3>
|
|
|
|
<p>The next four examples compare post-merging and pre-merging ('<a
|
|
href="qh-optc.htm#Cn2">Cn</a>' vs. '<a href="qh-optc.htm#Cn">C-n</a>').
|
|
Qhull uses '-' as a flag to indicate pre-merging.</p>
|
|
|
|
<p>Post-merging happens after the convex hull is built. During
|
|
post-merging, Qhull repeatedly merges an independent set of
|
|
non-convex facets. For a given set of parameters, the result is
|
|
about as good as one can hope for.</p>
|
|
|
|
<p>Pre-merging does the same thing as post-merging, except that
|
|
it happens after adding each point to the convex hull. With
|
|
pre-merging, Qhull guarantees a convex hull, but the facets are
|
|
wider than those from post-merging. If a pre-merge option is not
|
|
specified, Qhull handles machine round-off errors.</p>
|
|
|
|
<p>You may see coplanar points appearing slightly outside
|
|
the facets of the last example. This is becomes Geomview moves
|
|
line segments forward toward the viewer. You can avoid the
|
|
effect by setting 'lines closer' to '0' in Geomview's camera menu.
|
|
|
|
<h3><a href="#merge">»</a><a name="23">rbox 1000 | qconvex s
|
|
Gcprvah C0.1 Qc >eg.23.merge.cube</a></h3>
|
|
|
|
<p>Here's the 3-d imprecise cube with all of the Geomview
|
|
options. There's spheres for the vertices, radii for the coplanar
|
|
points, dots for the interior points, hyperplane intersections,
|
|
centrums, and inner and outer planes. The radii are shorter than
|
|
the spheres because this uses post-merging ('<a href="qh-optc.htm#Cn2">C0.1</a>')
|
|
instead of pre-merging.
|
|
|
|
<h2><a href="#TOC">»</a>4-d objects</h2>
|
|
|
|
<p>With Qhull and Geomview you can develop an intuitive sense of
|
|
4-d surfaces. When you get into trouble, think of viewing the
|
|
surface of a 3-d sphere in a 2-d plane.</p>
|
|
|
|
<h3><a href="#4d">»</a><a name="24">rbox 5000 D4 | qconvex s GD0v
|
|
Pd0:0.5 C-0.02 C0.1 >eg.24.merge.cube.4d-in-3d</a></h3>
|
|
|
|
<p>Here's one facet of the imprecise cube in 4-d. It's projected
|
|
into 3-d (the '<a href="qh-optg.htm#GDn">GDn</a>' option drops
|
|
dimension n). Each ridge consists of two triangles between this
|
|
facet and the neighboring facet. In this case, Geomview displays
|
|
the topological ridges, i.e., as triangles between three
|
|
vertices. That is why the cube looks lopsided.</p>
|
|
|
|
<h3><a href="#4d">»</a><a name="30">rbox 5000 D4 | qconvex s
|
|
C-0.02 C0.1 Gh >eg.30.4d.merge.cube </a></h3>
|
|
|
|
<p><a
|
|
href="http://www.geom.uiuc.edu/graphics/pix/Special_Topics/Computational_Geometry/4dcube.html">Here</a>
|
|
is the equivalent in 4-d of the imprecise <a href="#06">square</a>
|
|
and imprecise <a href="#23">cube</a>. It's the imprecise convex
|
|
hull of 5000 random points in a hypercube. It's a full 4-d object
|
|
so Geomview's <tt>ginsu </tt>does not work. If you view it in
|
|
Geomview, you'll be inside the hypercube. To view 4-d objects
|
|
directly, either load the <tt>4dview</tt> module or the <tt>ndview
|
|
</tt>module. For <tt>4dview</tt>, you must have started Geomview
|
|
in the same directory as the object. For <tt>ndview</tt>,
|
|
initialize a set of windows with the prefab menu, and load the
|
|
object through Geomview. The <tt>4dview</tt> module includes an
|
|
option for slicing along any hyperplane. If you do this in the x,
|
|
y, or z plane, you'll see the inside of a hypercube.</p>
|
|
|
|
<p>The '<a href="qh-optg.htm#Gh">Gh</a>' option prints the
|
|
geometric intersections between adjacent facets. Note the strong
|
|
convexity constraint for post-merging ('<a href="qh-optc.htm#Cn2">C0.1</a>').
|
|
It deletes the small facets.</p>
|
|
|
|
<h3><a href="#4d">»</a><a name="31">rbox 20 D3 | qdelaunay G
|
|
>eg.31.4d.delaunay </a></h3>
|
|
|
|
<p>The Delaunay triangulation of 3-d sites corresponds to a 4-d
|
|
convex hull. You can't see 4-d directly but each facet is a 3-d
|
|
object that you can project to 3-d. This is exactly the same as
|
|
projecting a 2-d facet of a soccer ball onto a plane.</p>
|
|
|
|
<p>Here we see all of the facets together. You can use Geomview's
|
|
<tt>ndview</tt> to look at the object from several directions.
|
|
Try turning on edges in the appearance menu. You'll notice that
|
|
some edges seem to disappear. That's because the object is
|
|
actually two sets of overlapping facets.</p>
|
|
|
|
<p>You can slice the object apart using Geomview's <tt>4dview</tt>.
|
|
With <tt>4dview</tt>, try slicing along the w axis to get a
|
|
single set of facets and then slice along the x axis to look
|
|
inside. Another interesting picture is to slice away the equator
|
|
in the w dimension.</p>
|
|
|
|
<h3><a href="#4d">»</a><a name="32">rbox 30 s D4 | qconvex s G
|
|
Pd0d1d2D3</a></h3>
|
|
|
|
<p>This is the positive octant of the convex hull of 30 4-d
|
|
points. When looking at 4-d, it's easier to look at just a few
|
|
facets at once. If you picked a facet that was directly above
|
|
you, then that facet looks exactly the same in 3-d as it looks in
|
|
4-d. If you pick several facets, then you need to imagine that
|
|
the space of a facet is rotated relative to its neighbors. Try
|
|
Geomview's <tt>ndview</tt> on this example.</p>
|
|
|
|
<h2><a href="#TOC">»</a>Halfspace intersections</h2>
|
|
|
|
<h3><a href="#half">»</a><a name="33a">rbox 10 r s Z1 G0.3 |
|
|
qconvex G >eg.33a.cone </a></h3>
|
|
|
|
<h3><a href="#half">»</a><a name="33b">rbox 10 r s Z1 G0.3 |
|
|
qconvex FV n | qhalf G >eg.33b.cone.dual</a></h3>
|
|
|
|
<h3><a href="#half">»</a><a name="33c">rbox 10 r s Z1 G0.3 |
|
|
qconvex FV n | qhalf Fp | qconvex G >eg.33c.cone.halfspace</a></h3>
|
|
|
|
<p>These examples illustrate halfspace intersection. The first
|
|
picture is the convex hull of two 20-gons plus an apex. The
|
|
second picture is the dual of the first. Try loading <a
|
|
href="http://www.geom.uiuc.edu/graphics/pix/Special_Topics/Computational_Geometry/half.html">both</a>
|
|
at once. Each vertex of the second picture corresponds to a facet
|
|
or halfspace of the first. The vertices with four edges
|
|
correspond to a facet with four neighbors. Similarly the facets
|
|
correspond to vertices. A facet's normal coefficients divided by
|
|
its negative offset is the vertice's coordinates. The coordinates
|
|
are the intersection of the original halfspaces. </p>
|
|
|
|
<p>The third picture shows how Qhull can go back and forth
|
|
between equivalent representations. It starts with a cone,
|
|
generates the halfspaces that define each facet of the cone, and
|
|
then intersects these halfspaces. Except for roundoff error, the
|
|
third picture is a duplicate of the first. </p>
|
|
<!-- Navigation links -->
|
|
<hr>
|
|
|
|
<p><b>Up:</b> <a href="http://www.qhull.org">Home
|
|
page for Qhull</a> <br>
|
|
<b>Up:</b> <a href="index.htm#TOC">Qhull manual: Table of Contents</a><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><br>
|
|
<b>To: </b><a href="#TOC">Qhull examples: Table of Contents</a> (please wait
|
|
while loading)<br>
|
|
<!-- 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>
|
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</html>
|