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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 © 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>.

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	<p><a href="http://www.geom.uiuc.edu/"><img src="qh--geom.gif"
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	<p>Comments to: <a href=mailto:qhull@qhull.org>qhull@qhull.org</a>
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