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<p><b><a name="TOP">Up</a></b><b>:</b> <a href="http://www.geom.umn.edu/locate/qhull">Home page</a> for Qhull<br>
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href="http://www.geom.umn.edu/graphics/pix/Special_Topics/Computational_Geometry/cone.html"><img
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href="http://www.geom.uiuc.edu/graphics/pix/Special_Topics/Computational_Geometry/cone.html"><img
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src="qh--cone.gif" alt="[cone]" align="middle" width="100"
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height="100"></a>qhull -- convex hull and related structures</h1>
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used with any of the Qhull programs.</dd>
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<dt><b>Example:</b> rbox 1000 D3 | qhull <a href=qhull.htm#outputs>d</a>
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<a href="qh-optq.htm#Qbb">Qbb</a>
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<a href="qh-optc.htm#Rn">R1e-4</a>
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<a href="qh-optq.htm#Q0">Q0</a></dt>
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<dd>Compute the 3-d Delaunay triangulation of 1000 random
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<p>By default, Qhull merges coplanar facets. For example, the convex
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hull of a cube's vertices has six facets.
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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
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will be simplicial (e.g., triangles in 2-d). Some regions may be
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degenerate and have zero area. Qhull can identify coincident
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points. With facet merging, cocircular and cospherical extreme points will lead
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to non-simplicial output (e.g., a square).
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all facets will be simplicial (e.g., triangles in 2-d). For the cube
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example, it will have 12 facets. Some facets 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>' (joggle), all furthest-site
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Delaunay regions will be simplicial (e.g., triangles in 2-d). Duplicate points will
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create small regions since the points are joggled apart. See <a
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href="qh-impre.htm#joggle">Joggled input or merged facets</a>. </p>
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<p>If you use '<a href="qh-optq.htm#QJn">QJ</a>' (joggled input),
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all facets will be simplicial. The corresponding vertices will be
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slightly perturbed. Joggled input is less accurate that triangulated
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href="qh-impre.htm#joggle">Merged facets or joggled input</a>. </p>
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<p>The output for 4-d convex hulls may be confusing if the convex
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hull contains non-simplicial facets (e.g., a hypercube). See
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are there extra points in a 4-d or higher convex hull?</a><br>
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<p><b>Copyright © 1995-2001 The Geometry Center, Minneapolis MN</b></p>
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<p><b>Copyright © 1995-2003 The Geometry Center, Minneapolis MN</b></p>
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rbox c d D2 | qhull Qc s f Fx | more rbox 1000 s | qhull Tv s FA
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rbox 10 D2 | qhull d QJ s i TO result rbox 10 D2 | qhull v Qt p
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rbox 10 D2 | qhull d QJ s i TO result rbox 10 D2 | qhull v Qbb Qt p
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rbox 10 D2 | qhull d Qu QJ m rbox 10 D2 | qhull v Qu QJ o
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rbox c | qhull n rbox c | qhull FV n | qhull H Fp
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rbox d D12 | qhull QR0 FA rbox c D7 | qhull FA TF1000
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<li>point coordinates</li>
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<p>Use I/O redirection (e.g., qhull < data.txt), a pipe (e.g., rbox 10 | qhull),
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or the '<a href=qh-optt.htm#TI>TI</a>' option (e.g., qhull TI data.txt).
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<p>Comments start with a non-numeric character. Error reporting is
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simpler if there is one point per line. Dimension
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and number of points may be reversed. For halfspace intersection,
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<dd><b>General</b></dd>
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<dt><a name="d">qhull d</a></dt>
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<dd>compute the convex hull of the input points.
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See <a href=qconvex.htm>qconvex</a>.</dd>
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<dt><a name="d">qhull d Qbb</a></dt>
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<dd>compute the Delaunay triangulation by lifting the points
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to a paraboloid. See <a href=qdelaun.htm>qdelaunay</a>.</dd>
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<dt><a name="v">qhull v</a></dt>
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to a paraboloid. Use option '<a href="qh-optq.htm#Qbb">Qbb</a>'
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to scale the paraboloid and improve numeric precision.
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See <a href=qdelaun.htm>qdelaunay</a>.</dd>
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<dt><a name="v">qhull v Qbb</a></dt>
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<dd>compute the Voronoi diagram by computing the Delaunay
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triangulation. See <a href=qvoronoi.htm>qvoronoi</a>.</dd>
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triangulation. Use option '<a href="qh-optq.htm#Qbb">Qbb</a>'
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to scale the paraboloid and improve numeric precision.
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See <a href=qvoronoi.htm>qvoronoi</a>.</dd>
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<dt><a name="H">qhull H</a></dt>
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<dd>compute the halfspace intersection about a point via polar
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duality. See <a href=qhalf.htm>qhalf</a>.</dd>
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for 'v', separating hyperplanes for bounded Voronoi regions
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FI - ID of each facet
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Fm - merge count for each facet (511 max)
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FM - Maple output (2-d and 3-d)
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Fn - count plus neighboring facets for each facet
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FN - count plus neighboring facets for each point
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Fo - outer plane (or max_outside) for each facet
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<p><b>Up:</b> <a href="http://www.geom.umn.edu/locate/qhull">Home page</a> for Qhull<br>
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<p><b>Up:</b> <a href="http://www.qhull.org">Home page</a> for Qhull<br>
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<b>Up:</b> <a href="index.htm#TOC">Qhull manual</a>: Table of Contents<br>
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<b>To:</b> <a href="qh-quick.htm#programs">Programs</a>
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• <a href="qh-quick.htm#options">Options</a>
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<!-- GC common information -->
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<p><a href="http://www.geom.umn.edu/"><img src="qh--geom.gif"
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<p><a href="http://www.geom.uiuc.edu/"><img src="qh--geom.gif"
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align="middle" width="40" height="40"></a><i>The Geometry Center
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Home Page </i></p>
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href="http://www.geom.umn.edu/software/qhull/qhull-mail.html">qhull@geom.umn.edu
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<p>Comments to: <a href=mailto:qhull@qhull.org>qhull@qhull.org</a>
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Created: Sept. 25, 1995 --- <!-- hhmts start --> Last modified: see top <!-- hhmts end --> </p>
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