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<title>qdelaunay Qu -- furthest-site Delaunay triangulation</title>
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<h1><a
href="http://www.geom.uiuc.edu/graphics/pix/Special_Topics/Computational_Geometry/delaunay.html"><img
src="qh--dt.gif" alt="[delaunay]" align="middle" width="100"
height="100"></a>qdelaunay Qu -- furthest-site Delaunay triangulation</h1>
<p>The furthest-site Delaunay triangulation corresponds to the upper facets of the <a href="qdelaun.htm">Delaunay construction</a>.
Its vertices are the
extreme points of the input sites.
It is the dual of the <a
href="qvoron_f.htm">furthest-site Voronoi diagram</a>.
<blockquote>
<dl>
<dt><b>Example:</b> rbox 10 D2 | qdelaunay <a
href="qh-optq.htm#Qu">Qu</a> <a
href="qh-optq.htm#Qt">Qt</a> <a href="qh-opto.htm#s">s</a>
<a href="qh-opto.htm#i">i</a> <a href="qh-optt.htm#TO">TO
result</a></dt>
<dd>Compute the 2-d, furthest-site Delaunay triangulation of 10 random
points. Triangulate the output.
Write a summary to the console and the regions to
'result'.</dd>
<dt> </dt>
<dt><b>Example:</b> rbox 10 D2 | qdelaunay <a
href="qh-optq.htm#Qu">Qu</a> <a
href="qh-optq.htm#QJn">QJ</a> <a href="qh-opto.htm#s">s</a>
<a href="qh-opto.htm#i">i</a> <a href="qh-optt.htm#TO">TO
result</a></dt>
<dd>Compute the 2-d, furthest-site Delaunay triangulation of 10 random
points. Joggle the input to guarantee triangular output.
Write a summary to the console and the regions to
'result'.</dd>
<dt> </dt>
<dt><b>Example:</b> rbox r y c G1 D2 | qdelaunay <a
href="qh-optq.htm#Qu">Qu</a> <a href="qh-opto.htm#s">s</a>
<a href="qh-optf.htm#Fv">Fv</a> <a href="qh-optt.htm#TO">TO
result</a></dt>
<dd>Compute the 2-d, furthest-site Delaunay triangulation of a triangle inside
a square.
Write a summary to the console and unoriented regions to 'result'.
Merge regions for cocircular input sites (e.g., the square).
The square is the only furthest-site
Delaunay region.</dd>
</dl>
</blockquote>
<p>As with the Delaunay triangulation, Qhull computes the
furthest-site Delaunay triangulation by lifting the input sites to a
paraboloid. The lower facets correspond to the Delaunay
triangulation while the upper facets correspond to the
furthest-site triangulation. Neither triangulation includes
"vertical" facets (i.e., facets whose last hyperplane
coefficient is nearly zero). Vertical facets correspond to input
sites that are coplanar to the convex hull of the input. An
example is points on the boundary of a lattice.</p>
<p>By default, qdelaunay merges cocircular and cospherical regions.
For example, the furthest-site Delaunay triangulation of a square inside a diamond
('rbox D2 c d G4 | qdelaunay Qu') consists of one region (the diamond).
<p>If you use '<a href="qh-optq.htm#Qt">Qt</a>' (triangulated output),
all furthest-site Delaunay regions will be simplicial (e.g., triangles in 2-d).
Some regions may be
degenerate and have zero area.
<p>If you use '<a href="qh-optq.htm#QJn">QJ</a>' (joggled input), all furthest-site
Delaunay regions
will be simplicial (e.g., triangles in 2-d). Joggled input
is less accurate than triangulated output ('Qt'). See <a
href="qh-impre.htm#joggle">Merged facets or joggled input</a>. </p>
<p>The output for 3-d, furthest-site Delaunay triangulations may be confusing if the
input contains cospherical data. See the FAQ item
<a href=qh-faq.htm#extra>Why
are there extra points in a 4-d or higher convex hull?</a>
Avoid these problems with triangulated output ('<a href="qh-optq.htm#Qt">Qt</a>') or
joggled input ('<a href="qh-optq.htm#QJn">QJ</a>').
</p>
<p>The 'qdelaunay' program is equivalent to
'<a href=qhull.htm#outputs>qhull d</a> <a href=qh-optq.htm#Qbb>Qbb</a>' in 2-d to 3-d, and
'<a href=qhull.htm#outputs>qhull d</a> <a href=qh-optq.htm#Qbb>Qbb</a> <a href=qh-optq.htm#Qx>Qx</a>'
in 4-d and higher. It disables the following Qhull
<a href=qh-quick.htm#options>options</a>: <i>d n v H U Qb QB Qc Qf Qg Qi
Qm Qr QR Qv Qx TR E V FC Fi Fo Fp FV Q0,etc</i>.
<p><b>Copyright © 1995-2003 The Geometry Center, Minneapolis MN</b></p>
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