417 lines
18 KiB
HTML
417 lines
18 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>qdelaunay Qu -- furthest-site Delaunay triangulation</title>
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</head>
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<body>
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<!-- Navigation links -->
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<a name="TOP"><b>Up</b></a><b>:</b>
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<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><br>
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<b>To:</b> <a href="#synopsis">sy</a>nopsis
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• <a href="#input">in</a>put • <a href="#outputs">ou</a>tputs
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• <a href="#controls">co</a>ntrols • <a href="#graphics">gr</a>aphics
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• <a href="#notes">no</a>tes • <a href="#conventions">co</a>nventions
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• <a href="#options">op</a>tions
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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>qdelaunay Qu -- furthest-site Delaunay triangulation</h1>
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<p>The furthest-site Delaunay triangulation corresponds to the upper facets of the <a href="qdelaun.htm">Delaunay construction</a>.
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Its vertices are the
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extreme points of the input sites.
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It is the dual of the <a
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href="qvoron_f.htm">furthest-site Voronoi diagram</a>.
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<blockquote>
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<dl>
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<dt><b>Example:</b> rbox 10 D2 | qdelaunay <a
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href="qh-optq.htm#Qu">Qu</a> <a
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href="qh-optq.htm#Qt">Qt</a> <a href="qh-opto.htm#s">s</a>
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<a href="qh-opto.htm#i">i</a> <a href="qh-optt.htm#TO">TO
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result</a></dt>
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<dd>Compute the 2-d, furthest-site Delaunay triangulation of 10 random
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points. Triangulate the output.
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Write a summary to the console and the regions to
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'result'.</dd>
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<dt> </dt>
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<dt><b>Example:</b> rbox 10 D2 | qdelaunay <a
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href="qh-optq.htm#Qu">Qu</a> <a
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href="qh-optq.htm#QJn">QJ</a> <a href="qh-opto.htm#s">s</a>
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<a href="qh-opto.htm#i">i</a> <a href="qh-optt.htm#TO">TO
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result</a></dt>
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<dd>Compute the 2-d, furthest-site Delaunay triangulation of 10 random
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points. Joggle the input to guarantee triangular output.
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Write a summary to the console and the regions to
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'result'.</dd>
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<dt> </dt>
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<dt><b>Example:</b> rbox r y c G1 D2 | qdelaunay <a
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href="qh-optq.htm#Qu">Qu</a> <a href="qh-opto.htm#s">s</a>
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<a href="qh-optf.htm#Fv">Fv</a> <a href="qh-optt.htm#TO">TO
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result</a></dt>
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<dd>Compute the 2-d, furthest-site Delaunay triangulation of a triangle inside
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a square.
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Write a summary to the console and unoriented regions to 'result'.
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Merge regions for cocircular input sites (e.g., the square).
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The square is the only furthest-site
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Delaunay region.</dd>
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</dl>
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</blockquote>
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<p>As with the Delaunay triangulation, Qhull computes the
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furthest-site Delaunay triangulation by lifting the input sites to a
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paraboloid. The lower facets correspond to the Delaunay
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triangulation while the upper facets correspond to the
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furthest-site triangulation. Neither triangulation includes
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"vertical" facets (i.e., facets whose last hyperplane
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coefficient is nearly zero). Vertical facets correspond to input
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sites that are coplanar to the convex hull of the input. An
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example is points on the boundary of a lattice.</p>
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<p>By default, qdelaunay merges cocircular and cospherical regions.
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For example, the furthest-site Delaunay triangulation of a square inside a diamond
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('rbox D2 c d G4 | qdelaunay Qu') consists of one region (the diamond).
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<p>If you use '<a href="qh-optq.htm#Qt">Qt</a>' (triangulated output),
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all furthest-site Delaunay regions will be simplicial (e.g., triangles in 2-d).
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Some regions may be
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degenerate and have zero area.
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<p>If you use '<a href="qh-optq.htm#QJn">QJ</a>' (joggled input), all furthest-site
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Delaunay regions
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will be simplicial (e.g., triangles in 2-d). Joggled input
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is less accurate than triangulated output ('Qt'). See <a
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href="qh-impre.htm#joggle">Merged facets or joggled input</a>. </p>
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<p>The output for 3-d, furthest-site Delaunay triangulations may be confusing if the
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input contains cospherical data. See the FAQ item
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<a href=qh-faq.htm#extra>Why
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are there extra points in a 4-d or higher convex hull?</a>
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Avoid these problems with triangulated output ('<a href="qh-optq.htm#Qt">Qt</a>') or
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joggled input ('<a href="qh-optq.htm#QJn">QJ</a>').
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</p>
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<p>The 'qdelaunay' program is equivalent to
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'<a href=qhull.htm#outputs>qhull d</a> <a href=qh-optq.htm#Qbb>Qbb</a>' in 2-d to 3-d, and
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'<a href=qhull.htm#outputs>qhull d</a> <a href=qh-optq.htm#Qbb>Qbb</a> <a href=qh-optq.htm#Qx>Qx</a>'
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in 4-d and higher. It disables the following Qhull
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<a href=qh-quick.htm#options>options</a>: <i>d n v H U Qb QB Qc Qf Qg Qi
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Qm Qr QR Qv Qx TR E V FC Fi Fo Fp FV Q0,etc</i>.
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<p><b>Copyright © 1995-2015 C.B. Barber</b></p>
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<hr>
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<h3><a href="#TOP">»</a><a name="synopsis">furthest-site qdelaunay synopsis</a></h3>
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<blockquote>
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See <a href="qdelaun.htm#synopsis">qdelaunay synopsis</a>. The same
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program is used for both constructions. Use option '<a href="qh-optq.htm#Qu">Qu</a>'
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for furthest-site Delaunay triangulations.
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</blockquote>
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<h3><a href="#TOP">»</a><a name="input">furthest-site qdelaunay
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input</a></h3>
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<blockquote>
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<p>The input data on <tt>stdin</tt> consists of:</p>
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<ul>
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<li>dimension
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<li>number of points</li>
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<li>point coordinates</li>
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</ul>
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<p>Use I/O redirection (e.g., qdelaunay Qu < data.txt), a pipe (e.g., rbox 10 | qdelaunay Qu),
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or the '<a href=qh-optt.htm#TI>TI</a>' option (e.g., qdelaunay Qu TI data.txt).
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<p>For example, this is a square containing four random points.
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Its furthest-site Delaunay
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triangulation contains one square.
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<p>
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<blockquote>
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<tt>rbox c 4 D2 > data</tt>
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<blockquote><pre>
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2 RBOX c 4 D2
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8
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-0.4999921736307369 -0.3684622117955817
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0.2556053225468894 -0.0413498678629751
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0.0327672376602583 -0.2810408135699488
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-0.452955383763607 0.17886471718444
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-0.5 -0.5
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-0.5 0.5
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0.5 -0.5
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0.5 0.5
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</pre></blockquote>
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<p><tt>qdelaunay Qu i < data</tt>
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<blockquote><pre>
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Furthest-site Delaunay triangulation by the convex hull of 8 points in 3-d:
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Number of input sites: 8
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Number of Delaunay regions: 1
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Number of non-simplicial Delaunay regions: 1
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Statistics for: RBOX c 4 D2 | QDELAUNAY s Qu i
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Number of points processed: 8
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Number of hyperplanes created: 20
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Number of facets in hull: 11
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Number of distance tests for qhull: 34
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Number of merged facets: 1
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Number of distance tests for merging: 107
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CPU seconds to compute hull (after input): 0.02
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1
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7 6 4 5
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</pre></blockquote>
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</blockquote>
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</blockquote>
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<h3><a href="#TOP">»</a><a name="outputs">furthest-site qdelaunay
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outputs</a></h3>
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<blockquote>
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<p>These options control the output of furthest-site Delaunay triangulations:</p>
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<blockquote>
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<dl compact>
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<dd><b>furthest-site Delaunay regions</b></dd>
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<dt><a href="qh-opto.htm#i">i</a></dt>
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<dd>list input sites for each furthest-site Delaunay region. The first line is the number of regions. The
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remaining lines list the input sites for each region. The regions are
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oriented. In 3-d and
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higher, report cospherical sites by adding extra points. For the points-in-square example,
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the square is the only furthest-site Delaunay region.</dd>
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<dt><a href="qh-optf.htm#Fv">Fv</a></dt>
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<dd>list input sites for each furthest-site Delaunay region. The first line is the number of regions.
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Each remaining line starts with the number of input sites. The regions
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are unoriented. For the points-in-square example,
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the square is the only furthest-site Delaunay region.</dd>
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<dt><a href="qh-optf.htm#Ft">Ft</a></dt>
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<dd>print a triangulation of the furthest-site Delaunay regions in OFF format. The first line
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is the dimension. The second line is the number of input sites and added points,
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followed by the number of simplices and the number of ridges.
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The input coordinates are next, followed by the centrum coordinates. There is
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one centrum for each non-simplicial furthest-site Delaunay region. Each remaining line starts
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with dimension+1. The
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simplices are oriented.
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For the points-in-square example, the square has a centrum at the
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origin. It splits the square into four triangular regions.</dd>
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<dt><a href="qh-optf.htm#Fn">Fn</a></dt>
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<dd>list neighboring regions for each furthest-site Delaunay region. The first line is the
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number of regions. Each remaining line starts with the number of
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neighboring regions. Negative indices (e.g., <em>-1</em>) indicate regions
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outside of the furthest-site Delaunay triangulation.
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For the points-in-square example, the four neighboring regions
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are outside of the triangulation. They belong to the regular
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Delaunay triangulation.</dd>
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<dt><a href="qh-optf.htm#FN">FN</a></dt>
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<dd>list the furthest-site Delaunay regions for each input site. The first line is the
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total number of input sites. Each remaining line starts with the number of
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furthest-site Delaunay regions. Negative indices (e.g., <em>-1</em>) indicate regions
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outside of the furthest-site Delaunay triangulation.
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For the points-in-square example, the four random points belong to no region
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while the square's vertices belong to region <em>0</em> and three
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regions outside of the furthest-site Delaunay triangulation.</dd>
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<dt><a href="qh-optf.htm#Fa">Fa</a></dt>
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<dd>print area for each furthest-site Delaunay region. The first line is the number of regions.
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The areas follow, one line per region. For the points-in-square example, the
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square has unit area. </dd>
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<dt> </dt>
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<dt> </dt>
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<dd><b>Input sites</b></dd>
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<dt><a href="qh-optf.htm#Fx">Fx</a></dt>
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<dd>list extreme points of the input sites. These points are vertices of the furthest-point
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Delaunay triangulation. They are on the
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boundary of the convex hull. The first line is the number of
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extreme points. Each point is listed, one per line. The points-in-square example
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has four extreme points.</dd>
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<dt> </dt>
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<dt> </dt>
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<dd><b>General</b></dd>
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<dt><a href="qh-optf.htm#FA">FA</a></dt>
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<dd>compute total area for '<a href="qh-opto.htm#s">s</a>'
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and '<a href="qh-optf.htm#FS">FS</a>'. This is the
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same as the area of the convex hull.</dd>
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<dt><a href="qh-opto.htm#o">o</a></dt>
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<dd>print upper facets of the corresponding convex hull (a
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paraboloid)</dd>
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<dt><a href="qh-opto.htm#m">m</a></dt>
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<dd>Mathematica output for the upper facets of the paraboloid (2-d triangulations).</dd>
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<dt><a href="qh-optf.htm#FM">FM</a></dt>
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<dd>Maple output for the upper facets of the paraboloid (2-d triangulations).</dd>
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<dt><a href="qh-optg.htm#G">G</a></dt>
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<dd>Geomview output for the paraboloid (2-d or 3-d triangulations).</dd>
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<dt><a href="qh-opto.htm#s">s</a></dt>
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<dd>print summary for the furthest-site Delaunay triangulation. Use '<a
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href="qh-optf.htm#Fs">Fs</a>' and '<a
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href="qh-optf.htm#FS">FS</a>' for numeric data.</dd>
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</dl>
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</blockquote>
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</blockquote>
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<h3><a href="#TOP">»</a><a name="controls">furthest-site qdelaunay
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controls</a></h3>
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<blockquote>
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<p>These options provide additional control:</p>
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<blockquote>
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<dl compact>
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<dt><a href="qh-optq.htm#Qu">Qu</a></dt>
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<dd>must be used for furthest-site Delaunay triangulation.</dd>
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<dt><a href="qh-optq.htm#Qt">Qt</a></dt>
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<dd>triangulated output. Qhull triangulates non-simplicial facets. It may produce
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degenerate facets of zero area.</dd>
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<dt><a href="qh-optq.htm#QJn">QJ</a></dt>
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<dd>joggle the input to avoid cospherical and coincident
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sites. It is less accurate than triangulated output ('Qt').</dd>
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<dt><a href="qh-optq.htm#QVn">QVn</a></dt>
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<dd>select facets adjacent to input site <em>n</em> (marked
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'good').</dd>
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<dt><a href="qh-optt.htm#Tv">Tv</a></dt>
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<dd>verify result.</dd>
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<dt><a href="qh-optt.htm#TO">TI file</a></dt>
|
||
|
|
<dd>input data from file. The filename may not use spaces or quotes.</dd>
|
||
|
|
<dt><a href="qh-optt.htm#TO">TO file</a></dt>
|
||
|
|
<dd>output results to file. Use single quotes if the filename
|
||
|
|
contains spaces (e.g., <tt>TO 'file with spaces.txt'</tt></dd>
|
||
|
|
<dt><a href="qh-optt.htm#TFn">TFn</a></dt>
|
||
|
|
<dd>report progress after constructing <em>n</em> facets</dd>
|
||
|
|
<dt><a href="qh-optp.htm#PDk">PDk:1</a></dt>
|
||
|
|
<dd>include upper and lower facets in the output. Set <em>k</em>
|
||
|
|
to the last dimension (e.g., 'PD2:1' for 2-d inputs). </dd>
|
||
|
|
<dt><a href="qh-opto.htm#f">f</a></dt>
|
||
|
|
<dd>facet dump. Print the data structure for each facet (i.e., furthest-site Delaunay region).</dd>
|
||
|
|
</dl>
|
||
|
|
</blockquote>
|
||
|
|
|
||
|
|
</blockquote>
|
||
|
|
<h3><a href="#TOP">»</a><a name="graphics">furthest-site qdelaunay
|
||
|
|
graphics</a></h3>
|
||
|
|
<blockquote>
|
||
|
|
|
||
|
|
See <a href="qdelaun.htm#graphics">Delaunay graphics</a>.
|
||
|
|
They are the same except for Mathematica and Maple output.
|
||
|
|
|
||
|
|
</blockquote>
|
||
|
|
<h3><a href="#TOP">»</a><a name="notes">furthest-site
|
||
|
|
qdelaunay notes</a></h3>
|
||
|
|
<blockquote>
|
||
|
|
|
||
|
|
<p>The furthest-site Delaunay triangulation does not
|
||
|
|
record coincident input sites. Use <tt>qdelaunay</tt> instead.
|
||
|
|
|
||
|
|
<p><tt>qdelaunay Qu</tt> does not work for purely cocircular
|
||
|
|
or cospherical points (e.g., rbox c | qdelaunay Qu). Instead,
|
||
|
|
use <tt>qdelaunay Qz</tt> -- when all points are vertices of the convex
|
||
|
|
hull of the input sites, the Delaunay triangulation is the same
|
||
|
|
as the furthest-site Delaunay triangulation.
|
||
|
|
|
||
|
|
<p>A non-simplicial, furthest-site Delaunay region indicates nearly cocircular or
|
||
|
|
cospherical input sites. To avoid non-simplicial regions triangulate
|
||
|
|
the output ('<a href="qh-optq.htm#Qt">Qt</a>') or joggle
|
||
|
|
the input ('<a href="qh-optq.htm#QJn">QJ</a>'). Joggled input
|
||
|
|
is less accurate than triangulated output.
|
||
|
|
You may also triangulate
|
||
|
|
non-simplicial regions with option '<a
|
||
|
|
href="qh-optf.htm#Ft">Ft</a>'. It adds
|
||
|
|
the centrum to non-simplicial regions. Alternatively, use an <a
|
||
|
|
href="qh-impre.htm#exact">exact arithmetic code</a>.</p>
|
||
|
|
|
||
|
|
<p>Furthest-site Delaunay triangulations do not include facets that are
|
||
|
|
coplanar with the convex hull of the input sites. A facet is
|
||
|
|
coplanar if the last coefficient of its normal is
|
||
|
|
nearly zero (see <a href="../src/libqhull/user.h#ZEROdelaunay">qh_ZEROdelaunay</a>).
|
||
|
|
|
||
|
|
</blockquote>
|
||
|
|
<h3><a href="#TOP">»</a><a name="conventions">furthest-site qdelaunay conventions</a></h3>
|
||
|
|
<blockquote>
|
||
|
|
|
||
|
|
<p>The following terminology is used for furthest-site Delaunay
|
||
|
|
triangulations in Qhull. The underlying structure is the upper
|
||
|
|
facets of a convex hull in one higher dimension. See <a
|
||
|
|
href="qconvex.htm#conventions">convex hull conventions</a>, <a
|
||
|
|
href="qdelaun.htm#conventions">Delaunay conventions</a>,
|
||
|
|
and <a href="index.htm#structure">Qhull's data structures</a></p>
|
||
|
|
<blockquote>
|
||
|
|
<ul>
|
||
|
|
<li><em>input site</em> - a point in the input (one dimension
|
||
|
|
lower than a point on the convex hull)</li>
|
||
|
|
<li><em>point</em> - <i>d+1</i> coordinates. The last
|
||
|
|
coordinate is the sum of the squares of the input site's
|
||
|
|
coordinates</li>
|
||
|
|
<li><em>vertex</em> - a point on the paraboloid. It
|
||
|
|
corresponds to a unique input site. </li>
|
||
|
|
<li><em>furthest-site Delaunay facet</em> - an upper facet of the
|
||
|
|
paraboloid. The last coefficient of its normal is
|
||
|
|
clearly positive.</li>
|
||
|
|
<li><em>furthest-site Delaunay region</em> - a furthest-site Delaunay
|
||
|
|
facet projected to the input sites</li>
|
||
|
|
<li><em>non-simplicial facet</em> - more than <em>d</em>
|
||
|
|
points are cocircular or cospherical</li>
|
||
|
|
<li><em>good facet</em> - a furthest-site Delaunay facet with optional
|
||
|
|
restrictions by '<a href="qh-optq.htm#QVn">QVn</a>', etc.</li>
|
||
|
|
</ul>
|
||
|
|
</blockquote>
|
||
|
|
</blockquote>
|
||
|
|
<h3><a href="#TOP">»</a><a name="options">furthest-site qdelaunay options</a></h3>
|
||
|
|
<blockquote>
|
||
|
|
|
||
|
|
See <a href="qdelaun.htm#options">qdelaunay options</a>. The same
|
||
|
|
program is used for both constructions. Use option '<a href="qh-optq.htm#Qu">Qu</a>'
|
||
|
|
for furthest-site Delaunay triangulations.
|
||
|
|
|
||
|
|
</blockquote>
|
||
|
|
<!-- 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><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="#graphics">gr</a>aphics
|
||
|
|
• <a href="#notes">no</a>tes • <a href="#conventions">co</a>nventions
|
||
|
|
• <a href="#options">op</a>tions
|
||
|
|
<!-- 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>
|