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<title>Qhull control options (Q)</title>
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<p><b>Up:</b> <a href="http://www.qhull.org">Home page</a> for Qhull (<a href="../index.htm">local</a>)<br>
<b>Up:</b> <a href="index.htm#TOC">Qhull manual</a>: contents<br>
<b>To:</b> <a href="qh-quick.htm#programs">Programs</a>
&#149; <a href="qh-quick.htm#options">Options</a>
&#149; <a href="qh-opto.htm#output">Output</a>
&#149; <a href="qh-optf.htm#format">Formats</a>
&#149; <a href="qh-optg.htm#geomview">Geomview</a>
&#149; <a href="qh-optp.htm#print">Print</a>
&#149; <a href="qh-optq.htm#qhull">Qhull</a>
&#149; <a href="qh-optc.htm#prec">Precision</a>
&#149; <a href="qh-optt.htm#trace">Trace</a>
&#149; <a href="http://www.qhull.org/src/libqhull_r/index.htm">Functions</a> (<a href="../src/libqhull_r/index.htm">local</a>)</p>
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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> 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 &copy; 1995-2020 C.B. Barber</b></p>
<hr>
<p><a href="index.htm#TOC">&#187;</a> <a href="qh-quick.htm#programs">Programs</a>
<a name="qhull">&#149;</a> <a href="qh-quick.htm#options">Options</a>
&#149; <a href="qh-opto.htm#output">Output</a>
&#149; <a href="qh-optf.htm#format">Formats</a>
&#149; <a href="qh-optg.htm#geomview">Geomview</a>
&#149; <a href="qh-optp.htm#print">Print</a>
&#149; <a href="qh-optq.htm#qhull">Qhull</a>
&#149; <a href="qh-optc.htm#prec">Precision</a>
&#149; <a href="qh-optt.htm#trace">Trace</a>
&#149; <a href="http://www.qhull.org/src/libqhull_r/index.htm">Functions</a> (<a href="../src/libqhull_r/index.htm">local</a>)</p>
<h2>Qhull control options</h2>
<dl compact>
<dt><a href="#Qa">Qa</a></dt>
<dd>allow input with fewer or more points than coordinates</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><a href="#Qu">Qu</a></dt>
<dd>compute upper hull for furthest-site Delaunay
triangulation </dd>
<dt><a href="#Qw">Qw</a></dt>
<dd>allow warnings about Qhull options</dd>
<dt>&nbsp;</dt>
<dt>&nbsp;</dt>
<dd><b>Precision handling</b></dd>
<dt><a href="#Qbb">Qbb</a></dt>
<dd>scale last coordinate to [0,m] for Delaunay</dd>
<dt><a href="#Qs">Qs</a></dt>
<dd>search all points for the initial simplex</dd>
<dt><a href="#Qv">Qv</a></dt>
<dd>test vertex neighbors for convexity</dd>
<dt><a href="#Qx">Qx</a></dt>
<dd>exact pre-merges (allows coplanar facets)</dd>
<dt><a href="#Qz">Qz</a></dt>
<dd>add a point-at-infinity for Delaunay triangulations</dd>
<dt>&nbsp;</dt>
<dt>&nbsp;</dt>
<dd><b>Transform input</b></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="#Qb0">Qbk:0Bk:0</a></dt>
<dd>drop dimension k from input</dd>
<dt><a href="#QbB">QbB</a></dt>
<dd>scale input to fit the unit cube</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="#QRn">QRn</a></dt>
<dd>random rotation (n=seed, n=0 time, n=-1 time/no rotate)</dd>
<dt>&nbsp;</dt>
<dt>&nbsp;</dt>
<dd><b>Select facets</b></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><a href="#QGn">QGn</a></dt>
<dd>good facet if visible from point n, -n for not visible</dd>
<dt><a href="#QVn">QVn</a></dt>
<dd>good facet if it includes point n, -n if not</dd>
<dt>&nbsp;</dt>
<dt>&nbsp;</dt>
<dd><b>Experimental</b></dd>
<dt><a href="#Qf">Qf</a></dt>
<dd>partition point to furthest outside facet</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="#Q0">Q0</a></dt>
<dd>do not pre-merge facets with 'C-0' or 'Qx'</dd>
<dt><a href="#Q1">Q1</a></dt>
<dd>merge by mergetype/angle instead of mergetype/distance</dd>
<dt><a href="#Q2">Q2</a></dt>
<dd>merge all non-convex at once instead of independent sets</dd>
<dt><a href="#Q3">Q3</a></dt>
<dd>do not merge redundant vertices</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="#Q6">Q6</a></dt>
<dd>do not pre-merge concave or coplanar facets</dd>
<dt><a href="#Q7">Q7</a></dt>
<dd>process facets depth-first instead of breadth-first</dd>
<dt><a href="#Q8">Q8</a></dt>
<dd>ignore near-interior points</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>allow wide facets and wide dupridge</dd>
<dt><a href="#Q14">Q14</a></dt>
<dd>merge pinched vertices that create a dupridge</dd>
<dt><a href="#Q15">Q15</a></dt>
<dd>check for duplicate ridges with the same vertices</dd>
</dl>
<hr>
<h3><a href="#qhull">&#187;</a><a name="Qa">Qa - allow input with fewer or more points than coordinates</a></h3>
<p>Option 'Qa' (qh.ALLOWshort) allows input with fewer or more points than coordinates. Qhull's input
starts with the number of points and the dimension, d. The coordinates follow, d coordinates per point.
<p>With option 'Qa', there may be fewer or more points than coordinates. This simplifies exploring an input point set by
adjusting the number of points.
<h3><a href="#qhull">&#187;</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">&#187;</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">&#187;</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">&#187;</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">&#187;</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">&#187;</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">&#187;</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">&#187;</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. Outside points that are not above good facets are ignored.
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 may not work for all combinations of options.</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>
<p>Option Qg disables renaming vertices due to duplicate ridges.
See ('<a href="#Q14">Q14</a>' merge-pinched-vertices).</p>
<h3><a href="#qhull">&#187;</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 &lt; 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">&#187;</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">&#187;</a><a name="QJ">QJ</a> or <a name="QJn">QJn</a> - joggled
input to avoid precision errors</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>.
The header file <a href="http://www.qhull.org/src/libqhull_r/user_.h#JOGGLEdefault">user_r.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>On repeated joggles, Qhull errors (QH6010) if the joggle is greater
than 0.1 or one quarter of the maximum coordinate width. </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. </p>
<p>Coplanar points ('<a
href="qh-optq.htm#Qc">Qc</a>', option '_near-inside') 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'.
<p>If your application incrementally adds points with <a
href="http://www.qhull.org/src/libqhull_r/libqhull_r.c#addpoint">qh_addpoint</a>,
facet merging for precision problems is disabled by default.
You can reduce the probability of failure by joggling the point's coordinates before calling qh_addpoint.
See <a href="http://www.qhull.org/src/libqhull_r/geom2_r.c#joggleinput">qh_joggleinput</a>
for sample code.
<h3><a href="#qhull">&#187;</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">&#187;</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 &quot;hyperplanes constructed&quot; and
&quot;distance tests&quot;. Qhull does not report CPU time
because the randomization is inefficient. </p>
<h3><a href="#qhull">&#187;</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&gt;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&lt;-1</em>,
don't rotate but use <em>n</em> as the random number seed. </p>
<p>If set, the summary option ('<a href="qh-opto.htm#s">s</a>') displays
the corresponding 'QRn' option after "Statistics for: ..."
<h3><a href="#qhull">&#187;</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">&#187;</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">&#187;</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">&#187;</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">&#187;</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&lt;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">&#187;</a><a name="Qw">Qw - allow warnings about Qhull options</a></h3>
<p>Option 'Qw' allows warning about unknown, malformed, or incompatible Qhull options.
If 'Qw' is not specified, option warnings produce a 'Qhull option error' and return exit status qh_ERRinput (1). </p>
<h3><a href="#qhull">&#187;</a><a name="Qx">Qx - exact pre-merges (allows coplanar facets)</a></h3>
<p>Option 'Qx' performs exact merges while building the hull.
Options 'Qx' and 'C-0' are set by default in 5-d and higher. To disable this default,
set option '<a href="qh-optc.htm#C0">C-0</a>' or another pre-merge option.
Use option '<a href="#Q0">Q0</a>' to disable all merging, including 'Qx' and 'C-0'.
</p>
<p>The &quot;exact&quot; pre-merges are merging a point into a
coplanar horizon 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. Superfluous
vertices are allowed. 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>'). Superfluous
vertices are removed.</p>
<p>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. Facets become increasingly wide. </p>
<p>Option 'Qx' may report a wide facet. With 'Qx', coplanar
facets are not merged. This can produce a &quot;dent&quot; 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 &quot;dent&quot; 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">&#187;</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">&#187;</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">&#187;</a><a name="Q1">Q1 - merge by mergetype/angle instead
of mergetype/distance</a></h3>
<p>Qhull merges coplanar facets first, then concave and flipped facets.
Within a merge-type, it merges coplanar facets by distance first.
</p>
<p>With option 'Q1', Qhull merges coplanar facets by angle instead of by distance.
</p>
<h3><a href="#qhull">&#187;</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">&#187;</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">&#187;</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. See qh_merge_nonconvex. It is not supported for qh_merge_twisted.</p>
<h3><a href="#qhull">&#187;</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">&#187;</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 &quot;dents&quot; when
using '<a href="#Qx">Qx</a>'. </p>
<h3><a href="#qhull">&#187;</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">&#187;</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">&#187;</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">&#187;</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">&#187;</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">&#187;</a><a name="Q12">Q12 - allow wide facets and wide dupridge</a></h3>
<p>Option 'Q12' (qh.ALLOWwide) disables error exits due to wide facets and wide dupridge. With 'Q12', Qhull may produce arbitrarily
wide facets. A facet is 'wide' if there is a wide gap between its outer plane (f.maxoutside above all points) and its
inner plane (below its vertices). A wide facet is typically due to multiple topological and geometric problems that
Qhull cannot resolve. As Qhull merges facets, they become increasingly wide, making further problems more likely.</p>
<p>A 'dupridge' is a ridge with more than two neighboring facets. A dupridge is a loop in the facet topology. A convex hull should never have dupridges, but with imprecise
arithmetic, they may occur. Dupridges may occur in 3-D and higher. They are more likely for
nearly adjacent points in 4-D and higher.</p>
<p>Qhull handles dupridges by merging all but one pair of facets (qh_matchdupridge in poly2_r.c)
or by dropping one of its vertices as a coplanar point (qh_getpinchedmerges in merge_r.c). If
vertices are too far apart and a merge would produce an wide facet, qh_check_dupridge reports an error. The
corresponding error exit is disabled by 'Q12'.</p>
<p>To demonstrate the problem, use rbox option 'Cn,r,m' to generate nearly adjacent points.
For Delaunay triangulations, a bounding box may alleviate this error (e.g., <tt>rbox 500 C1,1E-13 c G1 | qhull d</tt>).
This avoids the ill-defined edge between upper and lower convex hulls.
For more information, see "Nearly adjacent vertices within 1e-13" in <a href="qh-impre.htm#limit">Limitations of merged facets</a>.
</p>
<p>Errors disabled by 'Q12' return an error status of qh_ERRwide (8). Wide facets are defined by constants in user_r.h (qh_WIDE... and qh_RATIO...).
<h3><a href="#qhull">&#187;</a><a name="Q14">Q14 - merge pinched vertices due to a dupridge</a></h3>
<p>A <i>ridge</i> in Qhull is the d-1 simplex between two neighboring facets. For example in 3-d, a ridge is the edge shared by two neighboring polygons.
With non-simplicial facets, Qhull may create a "dupridge", a ridge with four or more neighboring facets. In effect, the surface
of the convex hull forms a loop that crosses itself at the dupridge. Qhull resolves a dupridge by
merging a pair of facets. With experimental option, 'Q14' it may resolve a dupridge by merging a pair of vertices.
<p>In 3-d and higher, nearly adjacent vertices may lead to dupridges (see
"Nearly adjacent vertices within 1e-13" in <a href="qh-impre.htm">Imprecision in Qhull</a>).
With option 'Q14', Qhull merges nearly adjacent vertices when they create a dupridge.
<h3><a href="#qhull">&#187;</a><a name="Q15">Q15 - check for duplicate ridges with the same vertices</a></h3>
<p>Option 'Q15' (qh.CHECKduplicates) checks for duplicate ridges after each merge (qh_maybe_duplicateridges,
qh_checkfacet).
Duplicate ridges have the same vertices. It is a topological error for convex hulls.
Qhull resolves a duplicate ridge by merging vertices and partitioning the deleted vertex as a coplanar
point (qh_renamevertex).
<p>Option 'Q15' does not prevent duplicate ridges for non-neighboring facets, nor does it prevent other topological
errors such as inconsistent facet orientation. Resolving a duplicate ridge may itself lead to topological errors
and wide facets.
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