732 lines
32 KiB
HTML
732 lines
32 KiB
HTML
|
<!DOCTYPE HTML PUBLIC "-//IETF//DTD HTML//EN">
|
||
|
|
<html>
|
||
|
|
|
||
|
|
<head>
|
||
|
|
<title>Qhull control options (Q)</title>
|
||
|
|
</head>
|
||
|
|
|
||
|
|
<body>
|
||
|
|
<!-- Navigation links -->
|
||
|
|
<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></p>
|
||
|
|
|
||
|
|
<hr>
|
||
|
|
<!-- Main text of document -->
|
||
|
|
<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> Qhull control options (Q)</h1>
|
||
|
|
|
||
|
|
<p>This section lists the control options for Qhull. These
|
||
|
|
options are indicated by 'Q' followed by a letter. </p>
|
||
|
|
|
||
|
|
<p><b>Copyright © 1995-2015 C.B. Barber</b></p>
|
||
|
|
|
||
|
|
<hr>
|
||
|
|
|
||
|
|
<p><a href="index.htm#TOC">»</a> <a href="qh-quick.htm#programs">Programs</a>
|
||
|
|
<a name="qhull">•</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></p>
|
||
|
|
|
||
|
|
<h2>Qhull control options</h2>
|
||
|
|
|
||
|
|
<dl compact>
|
||
|
|
<dt> </dt>
|
||
|
|
<dd><b>General</b></dd>
|
||
|
|
<dt><a href="#Qu">Qu</a></dt>
|
||
|
|
<dd>compute upper hull for furthest-site Delaunay
|
||
|
|
triangulation </dd>
|
||
|
|
<dt><a href="#Qc">Qc</a></dt>
|
||
|
|
<dd>keep coplanar points with nearest facet</dd>
|
||
|
|
<dt><a href="#Qi">Qi</a></dt>
|
||
|
|
<dd>keep interior points with nearest facet</dd>
|
||
|
|
<dt><a href="#QJn">QJ</a></dt>
|
||
|
|
<dd>joggled input to avoid precision problems</dd>
|
||
|
|
<dt><a href="#Qt">Qt</a></dt>
|
||
|
|
<dd>triangulated output</dd>
|
||
|
|
<dt> </dt>
|
||
|
|
<dt> </dt>
|
||
|
|
<dd><b>Precision handling</b></dd>
|
||
|
|
<dt><a href="#Qz">Qz</a></dt>
|
||
|
|
<dd>add a point-at-infinity for Delaunay triangulations</dd>
|
||
|
|
<dt><a href="#Qx">Qx</a></dt>
|
||
|
|
<dd>exact pre-merges (allows coplanar facets)</dd>
|
||
|
|
<dt><a href="#Qs">Qs</a></dt>
|
||
|
|
<dd>search all points for the initial simplex</dd>
|
||
|
|
<dt><a href="#Qbb">Qbb</a></dt>
|
||
|
|
<dd>scale last coordinate to [0,m] for Delaunay</dd>
|
||
|
|
<dt><a href="#Qv">Qv</a></dt>
|
||
|
|
<dd>test vertex neighbors for convexity</dd>
|
||
|
|
<dt> </dt>
|
||
|
|
<dt> </dt>
|
||
|
|
<dd><b>Transform input</b></dd>
|
||
|
|
<dt><a href="#Qb0">Qbk:0Bk:0</a></dt>
|
||
|
|
<dd>drop dimension k from input</dd>
|
||
|
|
<dt><a href="#QRn">QRn</a></dt>
|
||
|
|
<dd>random rotation (n=seed, n=0 time, n=-1 time/no rotate)</dd>
|
||
|
|
<dt><a href="#Qbk">Qbk:n</a></dt>
|
||
|
|
<dd>scale coord[k] to low bound of n (default -0.5)</dd>
|
||
|
|
<dt><a href="#QBk">QBk:n</a></dt>
|
||
|
|
<dd>scale coord[k] to upper bound of n (default 0.5)</dd>
|
||
|
|
<dt><a href="#QbB">QbB</a></dt>
|
||
|
|
<dd>scale input to fit the unit cube</dd>
|
||
|
|
<dt> </dt>
|
||
|
|
<dt> </dt>
|
||
|
|
<dd><b>Select facets</b></dd>
|
||
|
|
<dt><a href="#QVn">QVn</a></dt>
|
||
|
|
<dd>good facet if it includes point n, -n if not</dd>
|
||
|
|
<dt><a href="#QGn">QGn</a></dt>
|
||
|
|
<dd>good facet if visible from point n, -n for not visible</dd>
|
||
|
|
<dt><a href="#Qg">Qg</a></dt>
|
||
|
|
<dd>only build good facets (needs '<a href="#QGn">QGn</a>', '<a
|
||
|
|
href="#QVn">QVn </a>', or '<a href="qh-optp.htm#Pdk">Pdk</a>')</dd>
|
||
|
|
<dt> </dt>
|
||
|
|
<dt> </dt>
|
||
|
|
<dd><b>Experimental</b></dd>
|
||
|
|
<dt><a href="#Q4">Q4</a></dt>
|
||
|
|
<dd>avoid merging old facets into new facets</dd>
|
||
|
|
<dt><a href="#Q5">Q5</a></dt>
|
||
|
|
<dd>do not correct outer planes at end of qhull</dd>
|
||
|
|
<dt><a href="#Q3">Q3</a></dt>
|
||
|
|
<dd>do not merge redundant vertices</dd>
|
||
|
|
<dt><a href="#Q6">Q6</a></dt>
|
||
|
|
<dd>do not pre-merge concave or coplanar facets</dd>
|
||
|
|
<dt><a href="#Q0">Q0</a></dt>
|
||
|
|
<dd>do not pre-merge facets with 'C-0' or 'Qx'</dd>
|
||
|
|
<dt><a href="#Q8">Q8</a></dt>
|
||
|
|
<dd>ignore near-interior points</dd>
|
||
|
|
<dt><a href="#Q2">Q2</a></dt>
|
||
|
|
<dd>merge all non-convex at once instead of independent sets</dd>
|
||
|
|
<dt><a href="#Qf">Qf</a></dt>
|
||
|
|
<dd>partition point to furthest outside facet</dd>
|
||
|
|
<dt><a href="#Q7">Q7</a></dt>
|
||
|
|
<dd>process facets depth-first instead of breadth-first</dd>
|
||
|
|
<dt><a href="#Q9">Q9</a></dt>
|
||
|
|
<dd>process furthest of furthest points</dd>
|
||
|
|
<dt><a href="#Q10">Q10</a></dt>
|
||
|
|
<dd>no special processing for narrow distributions</dd>
|
||
|
|
<dt><a href="#Q11">Q11</a></dt>
|
||
|
|
<dd>copy normals and recompute centrums for tricoplanar facets</dd>
|
||
|
|
<dt><a href="#Q12">Q12</a></dt>
|
||
|
|
<dd>do not error on wide merge due to duplicate ridge and nearly coincident points</dd>
|
||
|
|
<dt><a href="#Qm">Qm</a></dt>
|
||
|
|
<dd>process points only if they would increase the max. outer
|
||
|
|
plane </dd>
|
||
|
|
<dt><a href="#Qr">Qr</a></dt>
|
||
|
|
<dd>process random outside points instead of furthest one</dd>
|
||
|
|
<dt><a href="#Q1">Q1</a></dt>
|
||
|
|
<dd>sort merges by type instead of angle</dd>
|
||
|
|
</dl>
|
||
|
|
|
||
|
|
<hr>
|
||
|
|
|
||
|
|
<h3><a href="#qhull">»</a><a name="Qbb">Qbb - scale the last
|
||
|
|
coordinate to [0,m] for Delaunay</a></h3>
|
||
|
|
|
||
|
|
<p>After scaling with option 'Qbb', the lower bound of the last
|
||
|
|
coordinate will be 0 and the upper bound will be the maximum
|
||
|
|
width of the other coordinates. Scaling happens after projecting
|
||
|
|
the points to a paraboloid and scaling other coordinates. </p>
|
||
|
|
|
||
|
|
<p>Option 'Qbb' is automatically set for <a href=qdelaun.htm>qdelaunay</a>
|
||
|
|
and <a href=qvoronoi.htm>qvoronoi</a>. Option 'Qbb' is automatically set for joggled input '<a
|
||
|
|
href="qh-optq.htm#QJn">QJ</a>'. </p>
|
||
|
|
|
||
|
|
<p>Option 'Qbb' should be used for Delaunay triangulations with
|
||
|
|
integer coordinates. Since the last coordinate is the sum of
|
||
|
|
squares, it may be much larger than the other coordinates. For
|
||
|
|
example, <tt>rbox 10000 D2 B1e8 | qhull d</tt> has precision
|
||
|
|
problems while <tt>rbox 10000 D2 B1e8 | qhull d Qbb</tt> is OK. </p>
|
||
|
|
|
||
|
|
<h3><a href="#qhull">»</a><a name="QbB">QbB - scale the input to
|
||
|
|
fit the unit cube</a></h3>
|
||
|
|
|
||
|
|
<p>After scaling with option 'QbB', the lower bound will be -0.5
|
||
|
|
and the upper bound +0.5 in all dimensions. For different bounds
|
||
|
|
change qh_DEFAULTbox in <tt>user.h</tt> (0.5 is best for <a
|
||
|
|
href="index.htm#geomview">Geomview</a>).</p>
|
||
|
|
|
||
|
|
<p>For Delaunay and Voronoi diagrams, scaling happens after
|
||
|
|
projection to the paraboloid. Under precise arithmetic, scaling
|
||
|
|
does not change the topology of the convex hull. Scaling may
|
||
|
|
reduce precision errors if coordinate values vary widely.</p>
|
||
|
|
|
||
|
|
<h3><a href="#qhull">»</a><a name="Qbk">Qbk:n - scale coord[k]
|
||
|
|
to low bound</a></h3>
|
||
|
|
|
||
|
|
<p>After scaling, the lower bound for dimension k of the input
|
||
|
|
points will be n. 'Qbk' scales coord[k] to -0.5. </p>
|
||
|
|
|
||
|
|
<h3><a href="#qhull">»</a><a name="QBk">QBk:n - scale coord[k]
|
||
|
|
to upper bound </a></h3>
|
||
|
|
|
||
|
|
<p>After scaling, the upper bound for dimension k of the input
|
||
|
|
points will be n. 'QBk' scales coord[k] to 0.5. </p>
|
||
|
|
|
||
|
|
<h3><a href="#qhull">»</a><a name="Qb0">Qbk:0Bk:0 - drop
|
||
|
|
dimension k from the input points</a></h3>
|
||
|
|
|
||
|
|
<p>Drop dimension<em> k </em>from the input points. For example,
|
||
|
|
'Qb1:0B1:0' deletes the y-coordinate from all input points. This
|
||
|
|
allows the user to take convex hulls of sub-dimensional objects.
|
||
|
|
It happens before the Delaunay and Voronoi transformation.
|
||
|
|
It happens after the halfspace transformation for both the data
|
||
|
|
and the feasible point.</p>
|
||
|
|
|
||
|
|
<h3><a href="#qhull">»</a><a name="Qc">Qc - keep coplanar points
|
||
|
|
with nearest facet </a></h3>
|
||
|
|
|
||
|
|
<p>During construction of the hull, a point is coplanar if it is
|
||
|
|
between '<a href="qh-optc.htm#Wn">Wn</a>' above and '<a
|
||
|
|
href="qh-optc.htm#Un">Un</a>' below a facet's hyperplane. A
|
||
|
|
different definition is used for output from Qhull. </p>
|
||
|
|
|
||
|
|
<p>For output, a coplanar point is above the minimum vertex
|
||
|
|
(i.e., above the inner plane). With joggle ('<a
|
||
|
|
href="qh-optq.htm#QJn">QJ</a>'), a coplanar point includes points
|
||
|
|
within one joggle of the inner plane. </p>
|
||
|
|
|
||
|
|
<p>With option 'Qc', output formats '<a href="qh-opto.htm#p">p </a>',
|
||
|
|
'<a href="qh-opto.htm#f">f</a>', '<a href="qh-optg.htm#Gp">Gp</a>',
|
||
|
|
'<a href="qh-optf.htm#Fc">Fc</a>', '<a href="qh-optf.htm#FN">FN</a>',
|
||
|
|
and '<a href="qh-optf.htm#FP">FP</a>' will print the coplanar
|
||
|
|
points. With options 'Qc <a href="#Qi">Qi</a>' these outputs
|
||
|
|
include the interior points.</p>
|
||
|
|
|
||
|
|
<p>For Delaunay triangulations (<a href=qdelaun.htm>qdelaunay</a>
|
||
|
|
or <a href=qvoronoi.htm>qvoronoi</a>), a coplanar point is a point
|
||
|
|
that is nearly incident to a vertex. All input points are either
|
||
|
|
vertices of the triangulation or coplanar.</p>
|
||
|
|
|
||
|
|
<p>Qhull stores coplanar points with a facet. While constructing
|
||
|
|
the hull, it retains all points within qh_RATIOnearInside
|
||
|
|
(user.h) of a facet. In qh_check_maxout(), it uses these points
|
||
|
|
to determine the outer plane for each facet. With option 'Qc',
|
||
|
|
qh_check_maxout() retains points above the minimum vertex for the
|
||
|
|
hull. Other points are removed. If qh_RATIOnearInside is wrong or
|
||
|
|
if options '<a href="#Q5">Q5</a> <a href="#Q8">Q8</a>' are set, a
|
||
|
|
coplanar point may be missed in the output (see <a
|
||
|
|
href="qh-impre.htm#limit">Qhull limitations</a>).</p>
|
||
|
|
|
||
|
|
<h3><a href="#qhull">»</a><a name="Qf">Qf - partition point to
|
||
|
|
furthest outside facet </a></h3>
|
||
|
|
|
||
|
|
<p>After adding a new point to the convex hull, Qhull partitions
|
||
|
|
the outside points and coplanar points of the old, visible
|
||
|
|
facets. Without the '<a href="qh-opto.htm#f">f </a>' option and
|
||
|
|
merging, it assigns a point to the first facet that it is outside
|
||
|
|
('<a href="qh-optc.htm#Wn">Wn</a>'). When merging, it assigns a
|
||
|
|
point to the first facet that is more than several times outside
|
||
|
|
(see qh_DISToutside in user.h).</p>
|
||
|
|
|
||
|
|
<p>If option 'Qf' is selected, Qhull performs a directed search
|
||
|
|
(no merging) or an exhaustive search (merging) of new facets.
|
||
|
|
Option 'Qf' may reduce precision errors if pre-merging does not
|
||
|
|
occur.</p>
|
||
|
|
|
||
|
|
<p>Option '<a href="#Q9">Q9</a>' processes the furthest of all
|
||
|
|
furthest points.</p>
|
||
|
|
|
||
|
|
<h3><a href="#qhull">»</a><a name="Qg">Qg - only build good
|
||
|
|
facets (needs 'QGn' 'QVn' or 'Pdk') </a></h3>
|
||
|
|
|
||
|
|
<p>Qhull has several options for defining and printing good
|
||
|
|
facets. With the '<a href="#Qg">Qg</a>' option, Qhull will only
|
||
|
|
build those facets that it needs to determine the good facets in
|
||
|
|
the output. This may speed up Qhull in 2-d and 3-d. It is
|
||
|
|
useful for furthest-site Delaunay
|
||
|
|
triangulations (<a href=qdelau_f.htm>qdelaunay Qu</a>,
|
||
|
|
invoke with 'qhull d Qbb <a href="#Qu">Qu</a> Qg').
|
||
|
|
It is not effective in higher
|
||
|
|
dimensions because many facets see a given point and contain a
|
||
|
|
given vertex. It is not guaranteed to work for all combinations.</p>
|
||
|
|
|
||
|
|
<p>See '<a href="#QGn">QGn</a>', '<a href="#QVn">QVn</a>', and '<a
|
||
|
|
href="qh-optp.htm#Pdk">Pdk</a>' for defining good facets, and '<a
|
||
|
|
href="qh-optp.htm#Pg">Pg</a>' and '<a href="qh-optp.htm#PG">PG</a>'
|
||
|
|
for printing good facets and their neighbors. If pre-merging ('<a
|
||
|
|
href="qh-optc.htm#Cn">C-n</a>') is not used and there are
|
||
|
|
coplanar facets, then 'Qg Pg' may produce a different result than
|
||
|
|
'<a href="qh-optp.htm#Pg">Pg</a>'. </p>
|
||
|
|
|
||
|
|
<h3><a href="#qhull">»</a><a name="QGn">QGn - good facet if
|
||
|
|
visible from point n, -n for not visible </a></h3>
|
||
|
|
|
||
|
|
<p>With option 'QGn', a facet is good (see '<a href="#Qg">Qg</a>'
|
||
|
|
and '<a href="qh-optp.htm#Pg">Pg</a>') if it is visible from
|
||
|
|
point n. If <i>n < 0</i>, a facet is good if it is not visible
|
||
|
|
from point n. Point n is not added to the hull (unless '<a
|
||
|
|
href="qh-optt.htm#TCn">TCn</a>' or '<a href="qh-optt.htm#TPn">TPn</a>').</p>
|
||
|
|
|
||
|
|
<p>With <a href="rbox.htm">rbox</a>, use the 'Pn,m,r' option
|
||
|
|
to define your point; it will be point 0 ('QG0'). </p>
|
||
|
|
|
||
|
|
<h3><a href="#qhull">»</a><a name="Qi">Qi - keep interior points
|
||
|
|
with nearest facet </a></h3>
|
||
|
|
|
||
|
|
<p>Normally Qhull ignores points that are clearly interior to the
|
||
|
|
convex hull. With option 'Qi', Qhull treats interior points the
|
||
|
|
same as coplanar points. Option 'Qi' does not retain coplanar
|
||
|
|
points. You will probably want '<a href="#Qc">Qc </a>' as well. </p>
|
||
|
|
|
||
|
|
<p>Option 'Qi' is automatically set for '<a href=qdelaun.htm>qdelaunay</a>
|
||
|
|
<a href="#Qc">Qc</a>' and '<a href=qvoronoi.htm>qvoronoi</a>
|
||
|
|
<a href="#Qc">Qc</a>'. If you use
|
||
|
|
'<a href=qdelaun.htm>qdelaunay</a> Qi' or '<a href=qvoronoi.htm>qvoronoi</a>
|
||
|
|
Qi', option '<a href="qh-opto.htm#s">s</a>' reports all nearly
|
||
|
|
incident points while option '<a href="qh-optf.htm#Fs">Fs</a>'
|
||
|
|
reports the number of interior points (should always be zero).</p>
|
||
|
|
|
||
|
|
<p>With option 'Qi', output formats '<a href="qh-opto.htm#p">p</a>',
|
||
|
|
'<a href="qh-opto.htm#f">f</a>','<a href="qh-optg.htm#Gp">Gp</a>',
|
||
|
|
'<a href="qh-optf.htm#Fc">Fc</a>', '<a href="qh-optf.htm#FN">FN</a>',
|
||
|
|
and '<a href="qh-optf.htm#FP">FP</a>' include interior points. </p>
|
||
|
|
|
||
|
|
<h3><a href="#qhull">»</a><a name="QJ">QJ</a> or <a name="QJn">QJn</a> - joggled
|
||
|
|
input to avoid precision errors</a></h3>
|
||
|
|
|
||
|
|
<p>Option 'QJ' or 'QJn' joggles each input coordinate by adding a
|
||
|
|
random number in the range [-n,n]. If a precision error occurs,
|
||
|
|
It tries again. If precision errors still occur, Qhull increases <i>n</i>
|
||
|
|
ten-fold and tries again. The maximum value for increasing <i>n</i>
|
||
|
|
is 0.01 times the maximum width of the input. Option 'QJ' selects
|
||
|
|
a default value for <i>n</i>. <a href="../src/user.h#JOGGLEdefault">User.h</a>
|
||
|
|
defines these parameters and a maximum number of retries. See <a
|
||
|
|
href="qh-impre.htm#joggle">Merged facets or joggled input</a>. </p>
|
||
|
|
|
||
|
|
<p>Users of joggled input should consider converting to
|
||
|
|
triangulated output ('<a href="../html/qh-optq.htm#Qt">Qt</a>'). Triangulated output is
|
||
|
|
approximately 1000 times more accurate than joggled input.
|
||
|
|
|
||
|
|
<p>Option 'QJ' also sets '<a href="qh-optq.htm#Qbb">Qbb</a>' for
|
||
|
|
Delaunay triangulations and Voronoi diagrams. It does not set
|
||
|
|
'Qbb' if '<a href="qh-optq.htm#Qbk">Qbk:n</a>' or '<a
|
||
|
|
href="qh-optq.htm#QBk">QBk:n</a>' are set. </p>
|
||
|
|
|
||
|
|
<p>If 'QJn' is set, Qhull does not merge facets unless requested
|
||
|
|
to. All facets are simplicial (triangular in 2-d). This may be
|
||
|
|
important for your application. You may also use triangulated output
|
||
|
|
('<a href="qh-optq.htm#Qt">Qt</a>') or Option '<a href="qh-optf.htm#Ft">Ft</a>'.
|
||
|
|
|
||
|
|
<p>Qhull adjusts the outer and inner planes for 'QJn' ('<a
|
||
|
|
href="qh-optf.htm#Fs">Fs</a>'). They are increased by <i>sqrt(d)*n</i>
|
||
|
|
to account for the maximum distance between a joggled point and
|
||
|
|
the corresponding input point. Coplanar points ('<a
|
||
|
|
href="qh-optq.htm#Qc">Qc</a>') require an additional <i>sqrt(d)*n</i>
|
||
|
|
since vertices and coplanar points may be joggled in opposite
|
||
|
|
directions. </p>
|
||
|
|
|
||
|
|
<p>For Delaunay triangulations (<a href=qdelaun.htm>qdelaunay</a>), joggle
|
||
|
|
happens before lifting the input sites to a paraboloid. Instead of
|
||
|
|
'QJ', you may use triangulated output ('<a
|
||
|
|
href="qh-optq.htm#Qt">Qt</a>')</p>
|
||
|
|
|
||
|
|
<p>This option is deprecated for Voronoi diagrams (<a href=qvoronoi.htm>qvoronoi</a>).
|
||
|
|
It triangulates cospherical points, leading to duplicated Voronoi vertices.</p>
|
||
|
|
|
||
|
|
<p>By default, 'QJn' uses a fixed random number seed. To use time
|
||
|
|
as the random number seed, select '<a href="qh-optq.htm#QRn">QR-1</a>'.
|
||
|
|
The summary ('<a href="qh-opto.htm#s">s</a>') will show the
|
||
|
|
selected seed as 'QR-n'.
|
||
|
|
|
||
|
|
<p>With 'QJn', Qhull does not error on degenerate hyperplane
|
||
|
|
computations. Except for Delaunay and Voronoi computations, Qhull
|
||
|
|
does not error on coplanar points. </p>
|
||
|
|
|
||
|
|
<p>Use option '<a href="qh-optf.htm#FO">FO</a>' to display the
|
||
|
|
selected options. Option 'FO' displays the joggle and the joggle
|
||
|
|
seed. If Qhull restarts, it will redisplay the options. </p>
|
||
|
|
|
||
|
|
<p>Use option '<a href="qh-optt.htm#TRn">TRn</a>' to estimate the
|
||
|
|
probability that Qhull will fail for a given 'QJn'.
|
||
|
|
|
||
|
|
<h3><a href="#qhull">»</a><a name="Qm">Qm - only process points
|
||
|
|
that increase the maximum outer plane </a></h3>
|
||
|
|
|
||
|
|
<p>Qhull reports the maximum outer plane in its summary ('<a
|
||
|
|
href="qh-opto.htm#s">s</a>'). With option 'Qm', Qhull does not
|
||
|
|
process points that are below the current, maximum outer plane.
|
||
|
|
This is equivalent to always adjusting '<a href="qh-optc.htm#Wn">Wn
|
||
|
|
</a>' to the maximum distance of a coplanar point to a facet. It
|
||
|
|
is ignored for points above the upper convex hull of a Delaunay
|
||
|
|
triangulation. Option 'Qm' is no longer important for merging.</p>
|
||
|
|
|
||
|
|
<h3><a href="#qhull">»</a><a name="Qr">Qr - process random
|
||
|
|
outside points instead of furthest ones </a></h3>
|
||
|
|
|
||
|
|
<p>Normally, Qhull processes the furthest point of a facet's
|
||
|
|
outside points. Option 'Qr' instead selects a random outside
|
||
|
|
point for processing. This makes Qhull equivalent to the
|
||
|
|
randomized incremental algorithms.</p>
|
||
|
|
|
||
|
|
<p>The original randomization algorithm by Clarkson and Shor [<a
|
||
|
|
href="index.htm#cla-sho89">'89</a>] used a conflict list which
|
||
|
|
is equivalent to Qhull's outside set. Later randomized algorithms
|
||
|
|
retained the previously constructed facets. </p>
|
||
|
|
|
||
|
|
<p>To compare Qhull to the randomized algorithms with option
|
||
|
|
'Qr', compare "hyperplanes constructed" and
|
||
|
|
"distance tests". Qhull does not report CPU time
|
||
|
|
because the randomization is inefficient. </p>
|
||
|
|
|
||
|
|
<h3><a href="#qhull">»</a><a name="QRn">QRn - random rotation </a></h3>
|
||
|
|
|
||
|
|
<p>Option 'QRn' randomly rotates the input. For Delaunay
|
||
|
|
triangulations (<a href=qdelaun.htm>qdelaunay</a> or <a href=qvoronoi.htm>qvoronoi</a>),
|
||
|
|
it rotates the lifted points about
|
||
|
|
the last axis. </p>
|
||
|
|
|
||
|
|
<p>If <em>n=0</em>, use time as the random number seed. If <em>n>0</em>,
|
||
|
|
use n as the random number seed. If <em>n=-1</em>, don't rotate
|
||
|
|
but use time as the random number seed. If <em>n<-1</em>,
|
||
|
|
don't rotate but use <em>n</em> as the random number seed. </p>
|
||
|
|
|
||
|
|
<h3><a href="#qhull">»</a><a name="Qs">Qs - search all points
|
||
|
|
for the initial simplex </a></h3>
|
||
|
|
|
||
|
|
<p>Qhull constructs an initial simplex from <i>d+1</i> points. It
|
||
|
|
selects points with the maximum and minimum coordinates and
|
||
|
|
non-zero determinants. If this fails, it searches all other
|
||
|
|
points. In 8-d and higher, Qhull selects points with the minimum
|
||
|
|
x or maximum coordinate (see qh_initialvertices in <tt>poly2.c </tt>).
|
||
|
|
It rejects points with nearly zero determinants. This should work
|
||
|
|
for almost all input sets.</p>
|
||
|
|
|
||
|
|
<p>If Qhull can not construct an initial simplex, it reports a
|
||
|
|
descriptive message. Usually, the point set is degenerate and one
|
||
|
|
or more dimensions should be removed ('<a href="#Qb0">Qbk:0Bk:0</a>').
|
||
|
|
If not, use option 'Qs'. It performs an exhaustive search for the
|
||
|
|
best initial simplex. This is expensive is high dimensions. </p>
|
||
|
|
|
||
|
|
<h3><a href="#qhull">»</a><a name="Qt">Qt - triangulated output</a></h3>
|
||
|
|
|
||
|
|
<p>By default, qhull merges facets to handle precision errors. This
|
||
|
|
produces non-simplicial facets (e.g., the convex hull of a cube has 6 square
|
||
|
|
facets). Each facet is non-simplicial because it has four vertices.
|
||
|
|
|
||
|
|
<p>Use option 'Qt' to triangulate all non-simplicial facets before generating
|
||
|
|
results. Alternatively, use joggled input ('<a href="#QJn">QJ</a>') to
|
||
|
|
prevent non-simplical facets. Unless '<a href="qh-optp.htm#Pp">Pp</a>' is set,
|
||
|
|
qhull produces a warning if 'QJ' and 'Qt' are used together.
|
||
|
|
|
||
|
|
<p>For Delaunay triangulations (<a href=qdelaun.htm>qdelaunay</a>), triangulation
|
||
|
|
occurs after lifting the input sites to a paraboloid and computing the convex hull.
|
||
|
|
</p>
|
||
|
|
|
||
|
|
<p>Option 'Qt' is deprecated for Voronoi diagrams (<a href=qvoronoi.htm>qvoronoi</a>).
|
||
|
|
It triangulates cospherical points, leading to duplicated Voronoi vertices.</p>
|
||
|
|
|
||
|
|
<p>Option 'Qt' may produce degenerate facets with zero area.</p>
|
||
|
|
|
||
|
|
<p>Facet area and hull volumes may differ with and without
|
||
|
|
'Qt'. The triangulations are different and different triangles
|
||
|
|
may be ignored due to precision errors.
|
||
|
|
|
||
|
|
<p>With sufficient merging, the ridges of a non-simplicial facet may share more than two neighboring facets. If so, their triangulation ('<a href="#Qt">Qt</a>') will fail since two facets have the same vertex set. </p>
|
||
|
|
|
||
|
|
<h3><a href="#qhull">»</a><a name="Qu">Qu - compute upper hull
|
||
|
|
for furthest-site Delaunay triangulation </a></h3>
|
||
|
|
|
||
|
|
<p>When computing a Delaunay triangulation (<a href=qdelaun.htm>qdelaunay</a>
|
||
|
|
or <a href=qvoronoi.htm>qvoronoi</a>),
|
||
|
|
Qhull computes both the the convex hull of points on a
|
||
|
|
paraboloid. It normally prints facets of the lower hull. These
|
||
|
|
correspond to the Delaunay triangulation. With option 'Qu', Qhull
|
||
|
|
prints facets of the upper hull. These correspond to the <a
|
||
|
|
href="qdelau_f.htm">furthest-site Delaunay triangulation</a>
|
||
|
|
and the <a href="qvoron_f.htm">furthest-site Voronoi diagram</a>.</p>
|
||
|
|
|
||
|
|
<p>Option 'qhull d Qbb Qu <a href="#Qg">Qg</a>' may improve the speed of option
|
||
|
|
'Qu'. If you use the Qhull library, a faster method is 1) use
|
||
|
|
Qhull to compute the convex hull of the input sites; 2) take the
|
||
|
|
extreme points (vertices) of the convex hull; 3) add one interior
|
||
|
|
point (e.g.,
|
||
|
|
'<a href="qh-optf.htm#FV">FV</a>', the average of <em>d</em> extreme points); 4) run
|
||
|
|
'qhull d Qbb Qu' or 'qhull v Qbb Qu' on these points.</p>
|
||
|
|
|
||
|
|
<h3><a href="#qhull">»</a><a name="Qv">Qv - test vertex
|
||
|
|
neighbors for convexity </a></h3>
|
||
|
|
|
||
|
|
<p>Normally, Qhull tests all facet neighbors for convexity.
|
||
|
|
Non-neighboring facets which share a vertex may not satisfy the
|
||
|
|
convexity constraint. This occurs when a facet undercuts the
|
||
|
|
centrum of another facet. They should still be convex. Option
|
||
|
|
'Qv' extends Qhull's convexity testing to all neighboring facets
|
||
|
|
of each vertex. The extra testing occurs after the hull is
|
||
|
|
constructed.. </p>
|
||
|
|
|
||
|
|
<h3><a href="#qhull">»</a><a name="QVn">QVn - good facet if it
|
||
|
|
includes point n, -n if not </a></h3>
|
||
|
|
|
||
|
|
<p>With option 'QVn', a facet is good ('<a href="#Qg">Qg</a>',
|
||
|
|
'<a href="qh-optp.htm#Pg">Pg</a>') if one of its vertices is
|
||
|
|
point n. If <i>n<0</i>, a good facet does not include point n.
|
||
|
|
|
||
|
|
<p>If options '<a href="qh-optp.htm#PG">PG</a>'
|
||
|
|
and '<a href="#Qg">Qg</a>' are not set, option '<a href="qh-optp.htm#Pg">Pg</a>'
|
||
|
|
(print only good)
|
||
|
|
is automatically set.
|
||
|
|
</p>
|
||
|
|
|
||
|
|
<p>Option 'QVn' behaves oddly with options '<a href="qh-optf.htm#Fx">Fx</a>'
|
||
|
|
and '<a href=qvoronoi.htm>qvoronoi</a> <a href="qh-optf.htm#Fv2">Fv</a>'.
|
||
|
|
|
||
|
|
<p>If used with option '<a href="#Qg">Qg</a>' (only process good facets), point n is
|
||
|
|
either in the initial simplex or it is the first
|
||
|
|
point added to the hull. Options 'QVn Qg' require either '<a href="#QJn">QJ</a>' or
|
||
|
|
'<a href="#Q0">Q0</a>' (no merging).</p>
|
||
|
|
|
||
|
|
<h3><a href="#qhull">»</a><a name="Qx">Qx - exact pre-merges
|
||
|
|
(allows coplanar facets) </a></h3>
|
||
|
|
|
||
|
|
<p>Option 'Qx' performs exact merges while building the hull.
|
||
|
|
Option 'Qx' is set by default in 5-d and higher. Use option '<a
|
||
|
|
href="#Q0">Q0</a>' to not use 'Qx' by default. Unless otherwise
|
||
|
|
specified, option 'Qx' sets option '<a href="qh-optc.htm#C0">C-0</a>'.
|
||
|
|
</p>
|
||
|
|
|
||
|
|
<p>The "exact" merges are merging a point into a
|
||
|
|
coplanar facet (defined by '<a href="qh-optc.htm#Vn">Vn </a>', '<a
|
||
|
|
href="qh-optc.htm#Un">Un</a>', and '<a href="qh-optc.htm#Cn">C-n</a>'),
|
||
|
|
merging concave facets, merging duplicate ridges, and merging
|
||
|
|
flipped facets. Coplanar merges and angle coplanar merges ('<a
|
||
|
|
href="qh-optc.htm#An">A-n</a>') are not performed. Concavity
|
||
|
|
testing is delayed until a merge occurs.</p>
|
||
|
|
|
||
|
|
<p>After the hull is built, all coplanar merges are performed
|
||
|
|
(defined by '<a href="qh-optc.htm#Cn">C-n</a>' and '<a
|
||
|
|
href="qh-optc.htm#An">A-n</a>'), then post-merges are performed
|
||
|
|
(defined by '<a href="qh-optc.htm#Cn2">Cn</a>' and '<a
|
||
|
|
href="qh-optc.htm#An2">An</a>'). If facet progress is logged ('<a
|
||
|
|
href="qh-optt.htm#TFn">TFn</a>'), Qhull reports each phase and
|
||
|
|
prints intermediate summaries and statistics ('<a
|
||
|
|
href="qh-optt.htm#Ts">Ts</a>'). </p>
|
||
|
|
|
||
|
|
<p>Without 'Qx' in 5-d and higher, options '<a
|
||
|
|
href="qh-optc.htm#Cn">C-n</a>' and '<a href="qh-optc.htm#An">A-n</a>'
|
||
|
|
may merge too many facets. Since redundant vertices are not
|
||
|
|
removed effectively, facets become increasingly wide. </p>
|
||
|
|
|
||
|
|
<p>Option 'Qx' may report a wide facet. With 'Qx', coplanar
|
||
|
|
facets are not merged. This can produce a "dent" in an
|
||
|
|
intermediate hull. If a point is partitioned into a dent and it
|
||
|
|
is below the surrounding facets but above other facets, one or
|
||
|
|
more wide facets will occur. In practice, this is unlikely. To
|
||
|
|
observe this effect, run Qhull with option '<a href="#Q6">Q6</a>'
|
||
|
|
which doesn't pre-merge concave facets. A concave facet makes a
|
||
|
|
large dent in the intermediate hull.</p>
|
||
|
|
|
||
|
|
<p>Option 'Qx' may set an outer plane below one of the input
|
||
|
|
points. A coplanar point may be assigned to the wrong facet
|
||
|
|
because of a "dent" in an intermediate hull. After
|
||
|
|
constructing the hull, Qhull double checks all outer planes with
|
||
|
|
qh_check_maxout in <tt>poly2.c </tt>. If a coplanar point is
|
||
|
|
assigned to the wrong facet, qh_check_maxout may reach a local
|
||
|
|
maximum instead of locating all coplanar facets. This appears to
|
||
|
|
be unlikely.</p>
|
||
|
|
|
||
|
|
<h3><a href="#qhull">»</a><a name="Qz">Qz - add a
|
||
|
|
point-at-infinity for Delaunay triangulations</a></h3>
|
||
|
|
|
||
|
|
<p>Option 'Qz' adds a point above the paraboloid of lifted sites
|
||
|
|
for a Delaunay triangulation. It allows the Delaunay
|
||
|
|
triangulation of cospherical sites. It reduces precision errors
|
||
|
|
for nearly cospherical sites.</p>
|
||
|
|
|
||
|
|
<h3><a href="#qhull">»</a><a name="Q0">Q0 - no merging with C-0
|
||
|
|
and Qx</a></h3>
|
||
|
|
|
||
|
|
<p>Turn off default merge options '<a href="qh-optc.htm#C0">C-0</a>'
|
||
|
|
and '<a href="#Qx">Qx</a>'. </p>
|
||
|
|
|
||
|
|
<p>With 'Q0' and without other pre-merge options, Qhull ignores
|
||
|
|
precision issues while constructing the convex hull. This may
|
||
|
|
lead to precision errors. If so, a descriptive warning is
|
||
|
|
generated. See <a href="qh-impre.htm">Precision issues</a>.</p>
|
||
|
|
|
||
|
|
<h3><a href="#qhull">»</a><a name="Q1">Q1 - sort merges by type
|
||
|
|
instead of angle </a></h3>
|
||
|
|
|
||
|
|
<p>Qhull sorts the coplanar facets before picking a subset of the
|
||
|
|
facets to merge. It merges concave and flipped facets first. Then
|
||
|
|
it merges facets that meet at a steep angle. With 'Q1', Qhull
|
||
|
|
sorts merges by type (coplanar, angle coplanar, concave) instead
|
||
|
|
of by angle. This may make the facets wider. </p>
|
||
|
|
|
||
|
|
<h3><a href="#qhull">»</a><a name="Q2">Q2 - merge all non-convex
|
||
|
|
at once instead of independent sets </a></h3>
|
||
|
|
|
||
|
|
<p>With 'Q2', Qhull merges all facets at once instead of
|
||
|
|
performing merges in independent sets. This may make the facets
|
||
|
|
wider. </p>
|
||
|
|
|
||
|
|
<h3><a href="#qhull">»</a><a name="Q3">Q3 - do not merge
|
||
|
|
redundant vertices </a></h3>
|
||
|
|
|
||
|
|
<p>With 'Q3', Qhull does not remove redundant vertices. In 6-d
|
||
|
|
and higher, Qhull never removes redundant vertices (since
|
||
|
|
vertices are highly interconnected). Option 'Q3' may be faster,
|
||
|
|
but it may result in wider facets. Its effect is easiest to see
|
||
|
|
in 3-d and 4-d.</p>
|
||
|
|
|
||
|
|
<h3><a href="#qhull">»</a><a name="Q4">Q4 - avoid merging </a>old
|
||
|
|
facets into new facets</h3>
|
||
|
|
|
||
|
|
<p>With 'Q4', Qhull avoids merges of an old facet into a new
|
||
|
|
facet. This sometimes improves facet width and sometimes makes it
|
||
|
|
worse. </p>
|
||
|
|
|
||
|
|
<h3><a href="#qhull">»</a><a name="Q5">Q5 - do not correct outer
|
||
|
|
planes at end of qhull </a></h3>
|
||
|
|
|
||
|
|
<p>When merging facets or approximating a hull, Qhull tests
|
||
|
|
coplanar points and outer planes after constructing the hull. It
|
||
|
|
does this by performing a directed search (qh_findbest in <tt>geom.c</tt>).
|
||
|
|
It includes points that are just inside the hull. </p>
|
||
|
|
|
||
|
|
<p>With options 'Q5' or '<a href="qh-optp.htm#Po">Po</a>', Qhull
|
||
|
|
does not test outer planes. The maximum outer plane is used
|
||
|
|
instead. Coplanar points ('<a href="#Qc">Qc</a>') are defined by
|
||
|
|
'<a href="qh-optc.htm#Un">Un</a>'. An input point may be outside
|
||
|
|
of the maximum outer plane (this appears to be unlikely). An
|
||
|
|
interior point may be above '<a href="qh-optc.htm#Un">Un</a>'
|
||
|
|
from a hyperplane.</p>
|
||
|
|
|
||
|
|
<p>Option 'Q5' may be used if outer planes are not needed. Outer
|
||
|
|
planes are needed for options '<a href="qh-opto.htm#s">s</a>', '<a
|
||
|
|
href="qh-optg.htm#G">G</a>', '<a href="qh-optg.htm#Go">Go </a>',
|
||
|
|
'<a href="qh-optf.htm#Fs">Fs</a>', '<a href="qh-optf.htm#Fo">Fo</a>',
|
||
|
|
'<a href="qh-optf.htm#FF">FF</a>', and '<a href="qh-opto.htm#f">f</a>'.</p>
|
||
|
|
|
||
|
|
<h3><a href="#qhull">»</a><a name="Q6">Q6 - do not pre-merge
|
||
|
|
concave or coplanar facets </a></h3>
|
||
|
|
|
||
|
|
<p>With 'Q6', Qhull does not pre-merge concave or coplanar
|
||
|
|
facets. This demonstrates the effect of "dents" when
|
||
|
|
using '<a href="#Qx">Qx</a>'. </p>
|
||
|
|
|
||
|
|
<h3><a href="#qhull">»</a><a name="Q7">Q7 - depth-first
|
||
|
|
processing instead of breadth-first </a></h3>
|
||
|
|
|
||
|
|
<p>With 'Q7', Qhull processes facets in depth-first order instead
|
||
|
|
of breadth-first order. This may increase the locality of
|
||
|
|
reference in low dimensions. If so, Qhull may be able to use
|
||
|
|
virtual memory effectively. </p>
|
||
|
|
|
||
|
|
<p>In 5-d and higher, many facets are visible from each
|
||
|
|
unprocessed point. So each iteration may access a large
|
||
|
|
proportion of allocated memory. This makes virtual memory
|
||
|
|
ineffectual. Once real memory is used up, Qhull will spend most
|
||
|
|
of its time waiting for I/O.</p>
|
||
|
|
|
||
|
|
<p>Under 'Q7', Qhull runs slower and the facets may be wider. </p>
|
||
|
|
|
||
|
|
<h3><a href="#qhull">»</a><a name="Q8">Q8 - ignore near-interior
|
||
|
|
points </a></h3>
|
||
|
|
|
||
|
|
<p>With 'Q8' and merging, Qhull does not process interior points
|
||
|
|
that are near to a facet (as defined by qh_RATIOnearInside in
|
||
|
|
user.h). This avoids partitioning steps. It may miss a coplanar
|
||
|
|
point when adjusting outer hulls in qh_check_maxout(). The best
|
||
|
|
value for qh_RATIOnearInside is not known. Options 'Q8 <a
|
||
|
|
href="#Qc">Qc</a>' may be sufficient. </p>
|
||
|
|
|
||
|
|
<h3><a href="#qhull">»</a><a name="Q9">Q9 - process furthest of
|
||
|
|
furthest points </a></h3>
|
||
|
|
|
||
|
|
<p>With 'Q9', Qhull processes the furthest point of all outside
|
||
|
|
sets. This may reduce precision problems. The furthest point of
|
||
|
|
all outside sets is not necessarily the furthest point from the
|
||
|
|
convex hull.</p>
|
||
|
|
|
||
|
|
<h3><a href="#qhull">»</a><a name="Q10">Q10 - no special processing
|
||
|
|
for narrow distributions</a></h3>
|
||
|
|
|
||
|
|
<p>With 'Q10', Qhull does not special-case narrow distributions.
|
||
|
|
See <a href=qh-impre.htm#limit>Limitations of merged facets</a> for
|
||
|
|
more information.
|
||
|
|
|
||
|
|
<h3><a href="#qhull">»</a><a name="Q11">Q11 - copy normals and recompute
|
||
|
|
centrums for
|
||
|
|
tricoplanar facets</a></h3>
|
||
|
|
|
||
|
|
Option '<a href="#Qt">Qt</a>' triangulates non-simplicial facets
|
||
|
|
into "tricoplanar" facets.
|
||
|
|
Normally tricoplanar facets share the same normal, centrum, and
|
||
|
|
Voronoi vertex. They can not be merged or replaced. With
|
||
|
|
option 'Q11', Qhull duplicates the normal and Voronoi vertex.
|
||
|
|
It recomputes the centrum.
|
||
|
|
|
||
|
|
<p>Use 'Q11' if you use the Qhull library to add points
|
||
|
|
incrementally and call qh_triangulate() after each point.
|
||
|
|
Otherwise, Qhull will report an error when it tries to
|
||
|
|
merge and replace a tricoplanar facet.
|
||
|
|
|
||
|
|
<p>With sufficient merging and new points, option 'Q11' may
|
||
|
|
lead to precision problems such
|
||
|
|
as duplicate ridges and concave facets. For example, if qh_triangulate()
|
||
|
|
is added to qh_addpoint(), RBOX 1000 s W1e-12 t1001813667 P0 | QHULL d Q11 Tv,
|
||
|
|
reports an error due to a duplicate ridge.
|
||
|
|
|
||
|
|
<h3><a href="#qhull">»</a><a name="Q12">Q12 - do not error
|
||
|
|
on wide merge due to duplicate ridge and nearly coincident points</a></h3>
|
||
|
|
|
||
|
|
<p>In 3-d and higher Delaunay Triangulations or 4-d and higher convex hulls, multiple,
|
||
|
|
nearly coincident points may lead to very wide facets. An error is reported if a
|
||
|
|
merge across a duplicate ridge would increase the facet width by 100x or more.
|
||
|
|
|
||
|
|
<p>Use option 'Q12' to log a warning instead of throwing an error.
|
||
|
|
|
||
|
|
<p>For Delaunay triangulations, a bounding box may alleviate this error (e.g., <tt>rbox 500 C1,1E-13 t c G1 | qhull d</tt>).
|
||
|
|
This avoids the ill-defined edge between upper and lower convex hulls.
|
||
|
|
The problem will be fixed in a future release of Qhull.
|
||
|
|
|
||
|
|
<p>To demonstrate the problem, use rbox option 'Cn,r,m' to generate nearly coincident points.
|
||
|
|
For more information, see "Nearly coincident points on an edge"
|
||
|
|
in <a href="qh-impre.htm#limit"</a>Nearly coincident points on an edge</a>.
|
||
|
|
|
||
|
|
<!-- 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></p>
|
||
|
|
<!-- 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>
|