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			<h1><a
			href="http://www.geom.uiuc.edu/graphics/pix/Special_Topics/Computational_Geometry/4dcube.html"><img
			src="qh--4d.gif" alt="[4-d cube]" align="middle" width="100"
			height="100"></a> Imprecision in Qhull</h1>

			<p>This section of the Qhull manual discusses the problems caused
			by coplanar points and why Qhull uses options '<a
			href="qh-optc.htm#C0">C-0</a>' or '<a href="qh-optq.htm#Qx">Qx</a>'
			by default. If you ignore precision issues with option '<a
			href="qh-optq.htm#Q0">Q0</a>', the output from Qhull can be
			arbitrarily bad. Qhull avoids precision problems if you merge facets (the default) or joggle
			the input ('<a
			href="qh-optq.htm#QJn">QJ</a>'). </p>

			<p>Use option '<a href="qh-optt.htm#Tv">Tv</a>' to verify the
			output from Qhull. It verifies that adjacent facets are clearly
			convex. It verifies that all points are on or below all facets. </p>

			<p>Qhull automatically tests for convexity if it detects
			precision errors while constructing the hull. </p>

			<p><b>Copyright © 1995-2015 C.B. Barber</b></p>

			<hr>

			<h2><a href="#TOP">»</a><a name="TOC">Qhull
			imprecision: Table of Contents </a></h2>

			<ul>
			<li><a href="#prec">Precision problems</a></li>
			<li><a href="#joggle">Merged facets or joggled input</a></li>
			<li><a href="#delaunay">Delaunay triangulations</a></li>
			<li><a href="#halfspace">Halfspace intersection/a></li>
			<li><a href="#imprecise">Merged facets</a></li>
			<li><a href="#how">How Qhull merges facets</a></li>
			<li><a href="#limit">Limitations of merged facets</a></li>
			<li><a href="#injoggle">Joggled input</a></li>
			<li><a href="#exact">Exact arithmetic</a></li>
			<li><a href="#approximate">Approximating a convex hull</a></li>
			</ul>

			<hr>

			<h2><a href="#TOC">»</a><a name="prec">Precision problems</a></h2>

			<p>Since Qhull uses floating point arithmetic, roundoff error
			occurs with each calculation. This causes problems for
			geometric algorithms. Other floating point codes for convex
			hulls, Delaunay triangulations, and Voronoi diagrams also suffer
			from these problems. Qhull handles most of them.</p>

			<p>There are several kinds of precision errors:</p>

			<ul>
			<li>Representation error occurs when there are not enough
			digits to represent a number, e.g., 1/3.</li>
			<li>Measurement error occurs when the input coordinates are
			from measurements. </li>
			<li>Roundoff error occurs when a calculation is rounded to a
			fixed number of digits, e.g., a floating point
			calculation.</li>
			<li>Approximation error occurs when the user wants an
			approximate result because the exact result contains too
			much detail.</li>
			</ul>

			<p>Under imprecision, calculations may return erroneous results.
			For example, roundoff error can turn a small, positive number
			into a small, negative number. See Milenkovic [<a
			href="index.htm#mile93">'93</a>] for a discussion of <em>strict
			robust geometry</em>. Qhull does not meet Milenkovic's criterion
			for accuracy. Qhull's error bound is empirical instead of
			theoretical.</p>

			<p>Qhull 1.0 checked for precision errors but did not handle
			them. The output could contain concave facets, facets with
			inverted orientation ("flipped" facets), more than two
			facets adjacent to a ridge, and two facets with exactly the same
			set of vertices.</p>

			<p>Qhull 2.4 and later automatically handles errors due to
			machine round-off. Option '<a href="qh-optc.htm#C0">C-0</a>' or '<a
			href="qh-optq.htm#Qx">Qx</a>' is set by default. In 5-d and
			higher, the output is clearly convex but an input point could be
			outside of the hull. This may be corrected by using option '<a
			href="qh-optc.htm#C0">C-0</a>', but then the output may contain
			wide facets.</p>

			<p>Qhull 2.5 and later provides option '<a href="qh-optq.htm#QJn">QJ</a>'
			to joggled input. Each input coordinate is modified by a
			small, random quantity. If a precision error occurs, a larger
			modification is tried. When no precision errors occur, Qhull is
			done. </p>

			<p>Qhull 3.1 and later provides option '<a href="qh-optq.htm#Qt">Qt</a>'
			for triangulated output. This removes the need for
			joggled input ('<a href="qh-optq.htm#QJn">QJ</a>').
			Non-simplicial facets are triangulated.
			The facets may have zero area.
			Triangulated output is particularly useful for Delaunay triangulations.</p>

			<p>By handling round-off errors, Qhull can provide a variety of
			output formats. For example, it can return the halfspace that
			defines each facet ('<a href="qh-opto.htm#n">n</a>'). The
			halfspaces include roundoff error. If the halfspaces were exact,
			their intersection would return the original extreme points. With
			imprecise halfspaces and exact arithmetic, nearly incident points
			may be returned for an original extreme point. By handling
			roundoff error, Qhull returns one intersection point for each of
			the original extreme points. Qhull may split or merge an extreme
			point, but this appears to be unlikely.</p>

			<p>The following pipe implements the identity function for
			extreme points (with roundoff):
			<blockquote>
			qconvex <a href="qh-optf.htm#FV">FV</a> <a
			href="qh-opto.htm#n">n</a> \| qhalf <a href="qh-optf.htm#Fp">Fp</a>
			</blockquote>

			<p>Bernd Gartner published his
			<a href=http://www.inf.ethz.ch/personal/gaertner/miniball.html>Miniball</a>
			algorithm ["Fast and robust smallest enclosing balls", <i>Algorithms - ESA '99</i>, LNCS 1643].
			It uses floating point arithmetic and a carefully designed primitive operation.
			It is practical to 20-D or higher, and identifies at least two points on the
			convex hull of the input set. Like Qhull, it is an incremental algorithm that
			processes points furthest from the intermediate result and ignores
			points that are close to the intermediate result.

			<h2><a href="#TOC">»</a><a name="joggle">Merged facets or joggled input</a></h2>

			<p>This section discusses the choice between merged facets and joggled input.
			By default, Qhull uses merged facets to handle
			precision problems. With option '<a href="qh-optq.htm#QJn">QJ</a>',
			the input is joggled. See <a href="qh-eg.htm#joggle">examples</a>
			of joggled input and triangulated output.
			<ul>
			<li>Use merged facets (the default)
			when you want non-simplicial output (e.g., the faces of a cube).
			<li>Use merged facets and triangulated output ('<a href="qh-optq.htm#Qt">Qt</a>') when
			you want simplicial output and coplanar facets (e.g., triangles for a Delaunay triangulation).
			<li>Use joggled input ('<a href="qh-optq.htm#QJn">QJ</a>') when you need clearly-convex,
			simplicial output.
			</ul>

			<p>The choice between merged facets and joggled input depends on
			the application. Both run about the same speed. Joggled input may
			be faster if the initial joggle is sufficiently large to avoid
			precision errors.

			<p>Most applications should used merged facets
			with triangulated output. </p>

			<p>Use merged facets (the
			default, '<a href="qh-optc.htm#C0">C-0</a>')
			or triangulated output ('<a href="qh-optq.htm#Qt">Qt</a>') if </p>

			<ul>
			<li>Your application supports non-simplicial facets, or
			it allows degenerate, simplicial facets (option '<a href="qh-optq.htm#Qt">Qt</a>'). </li>
			<li>You do not want the input modified. </li>
			<li>You want to set additional options for approximating the
			hull. </li>
			<li>You use single precision arithmetic (<a href="../src/libqhull/user.h#realT">realT</a>).
			</li>
			</ul>

			<p>Use joggled input ('<a href="qh-optq.htm#QJn">QJ</a>') if </p>

			<ul>
			<li>Your application needs clearly convex, simplicial output</li>
			<li>Your application supports perturbed input points and narrow triangles.</li>
			<li>Seven significant digits is sufficient accuracy.</li>
			</ul>

			<p>You may use both techniques or combine joggle with
			post-merging ('<a href="qh-optc.htm#Cn2">Cn</a>'). </p>

			<p>Other researchers have used techniques similar to joggled
			input. Sullivan and Beichel [ref?] randomly perturb the input
			before computing the Delaunay triangulation. Corkum and Wyllie
			[news://comp.graphics, 1990] randomly rotate a polytope before
			testing point inclusion. Edelsbrunner and Mucke [Symp. Comp.
			Geo., 1988] and Yap [J. Comp. Sys. Sci., 1990] symbolically
			perturb the input to remove singularities. </p>

			<p>Merged facets ('<a href="qh-optc.htm#C0">C-0</a>') handles
			precision problems directly. If a precision problem occurs, Qhull
			merges one of the offending facets into one of its neighbors.
			Since all precision problems in Qhull are associated with one or
			more facets, Qhull will either fix the problem or attempt to merge the
			last remaining facets. </p>

			<h2><a href="#TOC">»</a><a name="delaunay">Delaunay
			triangulations </a></h2>

			<p>Programs that use Delaunay triangulations traditionally assume
			a triangulated input. By default, <a href=qdelaun.htm>qdelaunay</a>
			merges regions with cocircular or cospherical input sites.
			If you want a simplicial triangulation
			use triangulated output ('<a href="qh-optq.htm#Qt">Qt</a>') or joggled
			input ('<a href="qh-optq.htm#QJn">QJ</a>').

			<p>For Delaunay triangulations, triangulated
			output should produce good results. All points are within roundoff error of
			a paraboloid. If two points are nearly incident, one will be a
			coplanar point. So all points are clearly separated and convex.
			If qhull reports deleted vertices, the triangulation
			may contain serious precision faults. Deleted vertices are reported
			in the summary ('<a href="qh-opto.htm#s">s</a>', '<a href="qh-optf.htm#Fs">Fs</a>'</p>

			<p>You should use option '<a href="qh-optq.htm#Qbb">Qbb</a>' with Delaunay
			triangulations. It scales the last coordinate and may reduce
			roundoff error. It is automatically set for <a href=qdelaun.htm>qdelaunay</a>,
			<a href=qvoronoi.htm>qvoronoi</a>, and option '<a
			href="qh-optq.htm#QJn">QJ</a>'.</p>

			<p>Edelsbrunner, H, <i>Geometry and Topology for Mesh Generation</i>, Cambridge University Press, 2001.
			Good mathematical treatise on Delaunay triangulation and mesh generation for 2-d
			and 3-d surfaces. The chapter on surface simplification is
			particularly interesting. It is similar to facet merging in Qhull.

			<p>Veron and Leon published an algorithm for shape preserving polyhedral
			simplification with bounded error [<i>Computers and Graphics</i>, 22.5:565-585, 1998].
			It remove nodes using front propagation and multiple remeshing.

			<h2><a href="#TOC">»</a><a name="halfspace">Halfspace intersection</a></h2>

			<p>
			The identity pipe for Qhull reveals some precision questions for
			halfspace intersections. The identity pipe creates the convex hull of
			a set of points and intersects the facets' hyperplanes. It should return the input
			points, but narrow distributions may drop points while offset distributions may add
			points. It may be better to normalize the input set about the origin.
			For example, compare the first results with the later two results: [T. Abraham]
			<blockquote>
			rbox 100 s t \| tee r \| qconvex FV n \| qhalf Fp \| cat - r \| /bin/sort -n \| tail
			<br>
			rbox 100 L1e5 t \| tee r \| qconvex FV n \| qhalf Fp \| cat - r \| /bin/sort -n \| tail
			<br>
			rbox 100 s O10 t \| tee r \| qconvex FV n \| qhalf Fp \| cat - r \| /bin/sort -n \| tail
			</blockquote>


			<h2><a href="#TOC">»</a><a name="imprecise">Merged facets </a></h2>

			<p>Qhull detects precision
			problems when computing distances. A precision problem occurs if
			the distance computation is less than the maximum roundoff error.
			Qhull treats the result of a hyperplane computation as if it
			were exact.</p>

			<p>Qhull handles precision problems by merging non-convex facets.
			The result of merging two facets is a thick facet defined by an <i>inner
			plane</i> and an <i>outer plane</i>. The inner and outer planes
			are offsets from the facet's hyperplane. The inner plane is
			clearly below the facet's vertices. At the end of Qhull, the
			outer planes are clearly above all input points. Any exact convex
			hull must lie between the inner and outer planes.</p>

			<p>Qhull tests for convexity by computing a point for each facet.
			This point is called the facet's <i>centrum</i>. It is the
			arithmetic center of the facet's vertices projected to the
			facet's hyperplane. For simplicial facets with <em>d</em>
			vertices, the centrum is equivalent to the centroid or center of
			gravity. </p>

			<p>Two neighboring facets are convex if each centrum is clearly
			below the other hyperplane. The '<a href="qh-optc.htm#Cn2">Cn</a>'
			or '<a href="qh-optc.htm#Cn">C-n</a>' options sets the centrum's
			radius to <i>n </i>. A centrum is clearly below a hyperplane if
			the computed distance from the centrum to the hyperplane is
			greater than the centrum's radius plus two maximum roundoff
			errors. Two are required because the centrum can be the maximum
			roundoff error above its hyperplane and the distance computation
			can be high by the maximum roundoff error.</p>

			<p>With the '<a href="qh-optc.htm#Cn">C-n</a>' or '<a
			href="qh-optc.htm#An">A-n </a>' options, Qhull merges non-convex
			facets while constructing the hull. The remaining facets are
			clearly convex. With the '<a href="qh-optq.htm#Qx">Qx </a>'
			option, Qhull merges coplanar facets after constructing the hull.
			While constructing the hull, it merges coplanar horizon facets,
			flipped facets, concave facets and duplicated ridges. With '<a
			href="qh-optq.htm#Qx">Qx</a>', coplanar points may be missed, but
			it appears to be unlikely.</p>

			<p>If the user sets the '<a href="qh-optc.htm#An2">An</a>' or '<a
			href="qh-optc.htm#An">A-n</a>' option, then all neighboring
			facets are clearly convex and the maximum (exact) cosine of an
			angle is <i>n</i>.</p>

			<p>If '<a href="qh-optc.htm#C0">C-0</a>' or '<a
			href="qh-optq.htm#Qx">Qx</a>' is used without other precision
			options (default), Qhull tests vertices instead of centrums for
			adjacent simplices. In 3-d, if simplex <i>abc</i> is adjacent to
			simplex <i>bcd</i>, Qhull tests that vertex <i>a</i> is clearly
			below simplex <i>bcd </i>, and vertex <i>d</i> is clearly below
			simplex <i>abc</i>. When building the hull, Qhull tests vertices
			if the horizon is simplicial and no merges occur. </p>

			<h2><a href="#TOC">»</a><a name="how">How Qhull merges facets</a></h2>

			<p>If two facets are not clearly convex, then Qhull removes one
			or the other facet by merging the facet into a neighbor. It
			selects the merge which minimizes the distance from the
			neighboring hyperplane to the facet's vertices. Qhull also
			performs merges when a facet has fewer than <i>d</i> neighbors (called a
			degenerate facet), when a facet's vertices are included in a
			neighboring facet's vertices (called a redundant facet), when a
			facet's orientation is flipped, or when a ridge occurs between
			more than two facets.</p>

			<p>Qhull performs merges in a series of passes sorted by merge
			angle. Each pass merges those facets which haven't already been
			merged in that pass. After a pass, Qhull checks for redundant
			vertices. For example, if a vertex has only two neighbors in 3-d,
			the vertex is redundant and Qhull merges it into an adjacent
			vertex.</p>

			<p>Merging two simplicial facets creates a non-simplicial facet
			of <em>d+1</em> vertices. Additional merges create larger facets.
			When merging facet A into facet B, Qhull retains facet B's
			hyperplane. It merges the vertices, neighbors, and ridges of both
			facets. It recomputes the centrum if a wide merge has not
			occurred (qh_WIDEcoplanar) and the number of extra vertices is
			smaller than a constant (qh_MAXnewcentrum).</p>


			<h2><a href="#TOC">»</a><a name="limit">Limitations of merged
			facets</a></h2>

			<ul>
			<li><b>Uneven dimensions</b> --
			If one coordinate has a larger absolute value than other
			coordinates, it may dominate the effect of roundoff errors on
			distance computations. You may use option '<a
			href="qh-optq.htm#QbB">QbB</a>' to scale points to the unit cube.
			For Delaunay triangulations and Voronoi diagrams, <a href=qdelaun.htm>qdelaunay</a>
			and <a href=qvoronoi.htm>qvoronoi</a> always set
			option '<a href="qh-optq.htm#Qbb">Qbb</a>'. It scales the last
			coordinate to [0,m] where <i>m</i> is the maximum width of the
			other coordinates. Option '<a href="qh-optq.htm#Qbb">Qbb</a>' is
			needed for Delaunay triangulations of integer coordinates
			and nearly cocircular points.

			<p>For example, compare
			<pre>
			rbox 1000 W0 t \| qconvex Qb2:-1e-14B2:1e-14
			</pre>
			with
			<pre>
			rbox 1000 W0 t \| qconvex
			</pre>
			The distributions are the same but the first is compressed to a 2e-14 slab.
			<p>
			<li><b>Post-merging of coplanar facets</b> -- In 5-d and higher, option '<a href="qh-optq.htm#Qx">Qx</a>'
			(default) delays merging of coplanar facets until post-merging.
			This may allow "dents" to occur in the intermediate
			convex hulls. A point may be poorly partitioned and force a poor
			approximation. See option '<a href="qh-optq.htm#Qx">Qx</a>' for
			further discussion.</p>

			<p>This is difficult to produce in 5-d and higher. Option '<a href="qh-optq.htm#Q6">Q6</a>' turns off merging of concave
			facets. This is similar to 'Qx'. It may lead to serious precision errors,
			for example,
			<pre>
			rbox 10000 W1e-13 \| qhull Q6 Tv
			</pre>

			<p>
			<li><b>Maximum facet width</b> --
			Qhull reports the maximum outer plane and inner planes (if
			more than roundoff error apart). There is no upper bound
			for either figure. This is an area for further research. Qhull
			does a good job of post-merging in all dimensions. Qhull does a
			good job of pre-merging in 2-d, 3-d, and 4-d. With the '<a
			href="qh-optq.htm#Qx">Qx</a>' option, it does a good job in
			higher dimensions. In 5-d and higher, Qhull does poorly at
			detecting redundant vertices. </p>

			<p>In the summary ('<a href="qh-opto.htm#s">s</a>'), look at the
			ratio between the maximum facet width and the maximum width of a
			single merge, e.g., "(3.4x)". Qhull usually reports a
			ratio of four or lower in 3-d and six or lower in 4-d. If it
			reports a ratio greater than 10, this may indicate an
			implementation error. Narrow distributions (see following) may
			produce wide facets.

			<p>For example, if special processing for narrow distributions is
			turned off ('<a href="qh-optq.htm#Q10">Q10</a>'), qhull may produce
			a wide facet:</p>
			<pre>
			rbox 1000 L100000 s G1e-16 t1002074964 \| qhull Tv Q10
			</pre>

			<p>
			<li><b>Narrow distribution</b> -- In 3-d, a narrow distribution may result in a poor
			approximation. For example, if you do not use qdelaunay nor option
			'<a href="qh-optq.htm#Qbb">Qbb</a>', the furthest-site
			Delaunay triangulation of nearly cocircular points may produce a poor
			approximation:
			<pre>
			rbox s 5000 W1e-13 D2 t1002151341 \| qhull d Qt
			rbox 1000 s W1e-13 t1002231672 \| qhull d Tv
			</pre>

			<p>During
			construction of the hull, a point may be above two
			facets with opposite orientations that span the input
			set. Even though the point may be nearly coplanar with both
			facets, and can be distant from the precise convex
			hull of the input sites. Additional facets leave the point distant from
			a facet. To fix this problem, add option '<a href="qh-optq.htm#Qbb">Qbb</a>'
			(it scales the last coordinate). Option '<a href="qh-optq.htm#Qbb">Qbb</a>'
			is automatically set for <a href=qdelaun.htm>qdelaunay</a> and <a href=qvoronoi.htm>qvoronoi</a>.

			<p>Qhull generates a warning if the initial simplex is narrow.
			For narrow distributions, Qhull changes how it processes coplanar
			points -- it does not make a point coplanar until the hull is
			finished.
			Use option '<a href="qh-optq.htm#Q10">Q10</a>' to try Qhull without
			special processing for narrow distributions.
			For example, special processing is needed for:
			<pre>
			rbox 1000 L100000 s G1e-16 t1002074964 \| qhull Tv Q10
			</pre>

			<p>You may turn off the warning message by reducing
			qh_WARNnarrow in <tt>user.h</tt> or by setting option
			'<a href="qh-optp.htm#Pp">Pp</a>'. </p>

			<p>Similar problems occur for distributions with a large flat facet surrounded
			with many small facet at a sharp angle to the large facet.
			Qhull 3.1 fixes most of these problems, but a poor approximation can occur.
			A point may be left outside of the convex hull ('<a href="qh-optt.htm#Tv">Tv</a>').
			Examples include
			the furthest-site Delaunay triangulation of nearly cocircular points plus the origin, and the convex hull of a cone of nearly cocircular points. The width of the band is 10^-13.
			<pre>
			rbox s 1000 W1e-13 P0 D2 t996799242 \| qhull d Tv
			rbox 1000 s Z1 G1e-13 t1002152123 \| qhull Tv
			rbox 1000 s Z1 G1e-13 t1002231668 \| qhull Tv
			</pre>

			<p>
			<li><b>Quadratic running time</b> -- If the output contains large, non-simplicial
			facets, the running time for Qhull may be quadratic in the size of the triangulated
			output. For example, <tt>rbox 1000 s W1e-13 c G2 \| qhull d</tt> is 4 times
			faster for 500 points. The convex hull contains two large nearly spherical facets and
			many nearly coplanar facets. Each new point retriangulates the spherical facet and repartitions the remaining points into all of the nearly coplanar facets.
			In this case, quadratic running time is avoided if you use qdelaunay,
			add option '<a href="qh-optq.htm#Qbb">Qbb</a>',
			or add the origin ('P0') to the input.
			<p>
			<li><b>Nearly coincident points within 1e-13</b> --
			Multiple, nearly coincident points within a 1e-13 ball of points in the unit cube
			may lead to wide facets or quadratic running time.
			For example, the convex hull a 1000 coincident, cospherical points in 4-D,
			or the 3-D Delaunay triangulation of nearly coincident points, may lead to very
			wide facets (e.g., 2267021951.3x).

			<p>For Delaunay triangulations, the problem typically occurs for extreme points of the input
			set (i.e., on the edge between the upper and lower convex hull). After multiple facet merges, four
			facets may share the same, duplicate ridge and must be merged.
			Some of these facets may be long and narrow, leading to a very wide merged facet.
			If so, error QH6271 is reported. It may be overriden with option '<a href="qh-optq.htm#Q12">Q12</a>'.

			<p>Duplicate ridges occur when the horizon facets for a new point is "pinched".
			In a duplicate ridge, a subridge (e.g., a line segment in 3-d) is shared by two horizon facets.
			At least two of its vertices are nearly coincident. It is easy to generate coincident points with
			option 'Cn,r,m' of rbox. It generates n points within an r ball for each of m input sites. For example,
			every point of the following distributions has a nearly coincident point within a 1e-13 ball.
			Substantially smaller or larger balls do not lead to pinched horizons.
			<pre>
			rbox 1000 C1,1e-13 D4 s t \| qhull
			rbox 75 C1,1e-13 t \| qhull d
			</pre>
			For Delaunay triangulations, a bounding box may alleviate this error (e.g., <tt>rbox 500 C1,1E-13 t c G1 \| qhull d</tt>).
			A later release of qhull will avoid pinched horizons by merging duplicate subridges. A subridge is
			merged by merging adjacent vertices.
			<p>
			<li><b>Facet with zero-area</b> --
			It is possible for a zero-area facet to be convex with its
			neighbors. This can occur if the hyperplanes of neighboring
			facets are above the facet's centrum, and the facet's hyperplane
			is above the neighboring centrums. Qhull computes the facet's
			hyperplane so that it passes through the facet's vertices. The
			vertices can be collinear. </p>

			<p>
			<li><b>No more facets</b> -- Qhull reports an error if there are <em>d+1</em> facets left
			and two of the facets are not clearly convex. This typically
			occurs when the convexity constraints are too strong or the input
			points are degenerate. The former is more likely in 5-d and
			higher -- especially with option '<a href="qh-optc.htm#Cn">C-n</a>'.</p>

			<p>
			<li><b>Deleted cone</b> -- Lots of merging can end up deleting all
			of the new facets for a point. This is a rare event that has
			only been seen while debugging the code.

			<p>
			<li><b>Triangulated output leads to precision problems</b> -- With sufficient
			merging, the ridges of a non-simplicial facet may have serious topological
			and geometric problems. A ridge may be between more than two
			neighboring facets. If so, their triangulation ('<a href="qh-optq.htm#Qt">Qt</a>')
			will fail since two facets have the same vertex set. Furthermore,
			a triangulated facet may have flipped orientation compared to its
			neighbors.</li>

			<p>The triangulation process detects degenerate facets with
			only two neighbors. These are marked degenerate. They have
			zero area.

			<p>
			<li><b>Coplanar points</b> --
			Option '<a href="qh-optq.htm#Qc">Qc</a>' is determined by
			qh_check_maxout() after constructing the hull. Qhull needs to
			retain all possible coplanar points in the facets' coplanar sets.
			This depends on qh_RATIOnearInside in <tt>user.h.</tt>
			Furthermore, the cutoff for a coplanar point is arbitrarily set
			at the minimum vertex. If coplanar points are important to your
			application, remove the interior points by hand (set '<a
			href="qh-optq.htm#Qc">Qc</a> <a href="qh-optq.htm#Qi">Qi</a>') or
			make qh_RATIOnearInside sufficiently large.</p>

			<p>
			<li><b>Maximum roundoff error</b> -- Qhull computes the maximum roundoff error from the maximum
			coordinates of the point set. Usually the maximum roundoff error
			is a reasonable choice for all distance computations. The maximum
			roundoff error could be computed separately for each point or for
			each distance computation. This is expensive and it conflicts
			with option '<a href="qh-optc.htm#Cn">C-n</a>'.

			<p>
			<li><b>All flipped or upper Delaunay</b> -- When a lot of merging occurs for
			Delaunay triangulations, a new point may lead to no good facets. For example,
			try a strong convexity constraint:
			<pre>
			rbox 1000 s t993602376 \| qdelaunay C-1e-3
			</pre>

			</ul>

			<h2><a href="#TOC">»</a><a name="injoggle">Joggled input</a></h2>

			<p>Joggled input is a simple work-around for precision problems
			in computational geometry ["joggle: to shake or jar
			slightly," Amer. Heritage Dictionary]. Other names are
			<i>jostled input</i> or <i>random perturbation</i>.
			Qhull joggles the
			input by modifying each coordinate by a small random quantity. If
			a precision problem occurs, Qhull joggles the input with a larger
			quantity and the algorithm is restarted. This process continues
			until no precision problems occur. Unless all inputs incur
			precision problems, Qhull will terminate. Qhull adjusts the inner
			and outer planes to account for the joggled input. </p>

			<p>Neither joggle nor merged facets has an upper bound for the width of the output
			facets, but both methods work well in practice. Joggled input is
			easier to justify. Precision errors occur when the points are
			nearly singular. For example, four points may be coplanar or
			three points may be collinear. Consider a line and an incident
			point. A precision error occurs if the point is within some
			epsilon of the line. Now joggle the point away from the line by a
			small, uniformly distributed, random quantity. If the point is
			changed by more than epsilon, the precision error is avoided. The
			probability of this event depends on the maximum joggle. Once the
			maximum joggle is larger than epsilon, doubling the maximum
			joggle will halve the probability of a precision error. </p>

			<p>With actual data, an analysis would need to account for each
			point changing independently and other computations. It is easier
			to determine the probabilities empirically ('<a href="qh-optt.htm#TRn">TRn</a>') . For example, consider
			computing the convex hull of the unit cube centered on the
			origin. The arithmetic has 16 significant decimal digits. </p>

			<blockquote>
			<p><b>Convex hull of unit cube</b> </p>
			<table border="1">
			<tr>
			<th align="left">joggle</th>
			<th align="right">error prob. </th>
			</tr>
			<tr>
			<td align="right">1.0e-15</td>
			<td align="center">0.983 </td>
			</tr>
			<tr>
			<td align="right">2.0e-15</td>
			<td align="center">0.830 </td>
			</tr>
			<tr>
			<td align="right">4.0e-15</td>
			<td align="center">0.561 </td>
			</tr>
			<tr>
			<td align="right">8.0e-15</td>
			<td align="center">0.325 </td>
			</tr>
			<tr>
			<td align="right">1.6e-14</td>
			<td align="center">0.185 </td>
			</tr>
			<tr>
			<td align="right">3.2e-14</td>
			<td align="center">0.099 </td>
			</tr>
			<tr>
			<td align="right">6.4e-14</td>
			<td align="center">0.051 </td>
			</tr>
			<tr>
			<td align="right">1.3e-13</td>
			<td align="center">0.025 </td>
			</tr>
			<tr>
			<td align="right">2.6e-13</td>
			<td align="center">0.010 </td>
			</tr>
			<tr>
			<td align="right">5.1e-13</td>
			<td align="center">0.004 </td>
			</tr>
			<tr>
			<td align="right">1.0e-12</td>
			<td align="center">0.002 </td>
			</tr>
			<tr>
			<td align="right">2.0e-12</td>
			<td align="center">0.001 </td>
			</tr>
			</table>
			</blockquote>

			<p>A larger joggle is needed for multiple points. Since the
			number of potential singularities increases, the probability of
			one or more precision errors increases. Here is an example. </p>

			<blockquote>
			<p><b>Convex hull of 1000 points on unit cube</b> </p>
			<table border="1">
			<tr>
			<th align="left">joggle</th>
			<th align="right">error prob. </th>
			</tr>
			<tr>
			<td align="right">1.0e-12</td>
			<td align="center">0.870 </td>
			</tr>
			<tr>
			<td align="right">2.0e-12</td>
			<td align="center">0.700 </td>
			</tr>
			<tr>
			<td align="right">4.0e-12</td>
			<td align="center">0.450 </td>
			</tr>
			<tr>
			<td align="right">8.0e-12</td>
			<td align="center">0.250 </td>
			</tr>
			<tr>
			<td align="right">1.6e-11</td>
			<td align="center">0.110 </td>
			</tr>
			<tr>
			<td align="right">3.2e-11</td>
			<td align="center">0.065 </td>
			</tr>
			<tr>
			<td align="right">6.4e-11</td>
			<td align="center">0.030 </td>
			</tr>
			<tr>
			<td align="right">1.3e-10</td>
			<td align="center">0.010 </td>
			</tr>
			<tr>
			<td align="right">2.6e-10</td>
			<td align="center">0.008 </td>
			</tr>
			<tr>
			<td align="right">5.1e-09</td>
			<td align="center">0.003 </td>
			</tr>
			</table>
			</blockquote>

			<p>Other distributions behave similarly. No distribution should
			behave significantly worse. In Euclidean space, the probability
			measure of all singularities is zero. With floating point
			numbers, the probability of a singularity is non-zero. With
			sufficient digits, the probability of a singularity is extremely
			small for random data. For a sufficiently large joggle, all data
			is nearly random data. </p>

			<p>Qhull uses an initial joggle of 30,000 times the maximum
			roundoff error for a distance computation. This avoids most
			potential singularities. If a failure occurs, Qhull retries at
			the initial joggle (in case bad luck occurred). If it occurs
			again, Qhull increases the joggle by ten-fold and tries again.
			This process repeats until the joggle is a hundredth of the width
			of the input points. Qhull reports an error after 100 attempts.
			This should never happen with double-precision arithmetic. Once
			the probability of success is non-zero, the probability of
			success increases about ten-fold at each iteration. The
			probability of repeated failures becomes extremely small. </p>

			<p>Merged facets produces a significantly better approximation.
			Empirically, the maximum separation between inner and outer
			facets is about 30 times the maximum roundoff error for a
			distance computation. This is about 2,000 times better than
			joggled input. Most applications though will not notice the
			difference. </p>

			<h2><a href="#TOC">»</a><a name="exact">Exact arithmetic</a></h2>

			<p>Exact arithmetic may be used instead of floating point.
			Singularities such as coplanar points can either be handled
			directly or the input can be symbolically perturbed. Using exact
			arithmetic is slower than using floating point arithmetic and the
			output may take more space. Chaining a sequence of operations
			increases the time and space required. Some operations are
			difficult to do.</p>

			<p>Clarkson's <a
			href="http://www.netlib.org/voronoi/hull.html">hull
			program</a> and Shewchuk's <a
			href="http://www.cs.cmu.edu/~quake/triangle.html">triangle
			program</a> are practical implementations of exact arithmetic.</p>

			<p>Clarkson limits the input precision to about fifteen digits.
			This reduces the number of nearly singular computations. When a
			determinant is nearly singular, he uses exact arithmetic to
			compute a precise result.</p>

			<h2><a href="#TOC">»</a><a name="approximate">Approximating a
			convex hull</a></h2>

			<p>Qhull may be used for approximating a convex hull. This is
			particularly valuable in 5-d and higher where hulls can be
			immense. You can use '<a href="qh-optq.htm#Qx">Qx</a> <a
			href="qh-optc.htm#Cn">C-n</a>' to merge facets as the hull is
			being constructed. Then use '<a href="qh-optc.htm#Cn2">Cn</a>'
			and/or '<a href="qh-optc.htm#An2">An</a>' to merge small facets
			during post-processing. You can print the <i>n</i> largest facets
			with option '<a href="qh-optp.htm#PAn">PAn</a>'. You can print
			facets whose area is at least <i>n</i> with option '<a
			href="qh-optp.htm#PFn">PFn</a>'. You can output the outer planes
			and an interior point with '<a href="qh-optf.htm#FV">FV</a> <a
			href="qh-optf.htm#Fo">Fo</a>' and then compute their intersection
			with 'qhalf'. </p>

			<p>To approximate a convex hull in 6-d and higher, use
			post-merging with '<a href="qh-optc.htm#Wn">Wn</a>' (e.g., qhull
			W1e-1 C1e-2 TF2000). Pre-merging with a convexity constraint
			(e.g., qhull Qx C-1e-2) often produces a poor approximation or
			terminates with a simplex. Option '<a href="qh-optq.htm#QbB">QbB</a>'
			may help to spread out the data.</p>

			<p>You will need to experiment to determine a satisfactory set of
			options. Use <a href="rbox.htm">rbox</a> to generate test sets
			quickly and <a href="index.htm#geomview">Geomview</a> to view
			the results. You will probably want to write your own driver for
			Qhull using the Qhull library. For example, you could select the
			largest facet in each quadrant.</p>

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			<p><b>Up:</b> <a href="http://www.qhull.org">Home
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			<p><a href="http://www.geom.uiuc.edu/"><img src="qh--geom.gif"
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			Home Page </i></p>

			<p>Comments to: <a href=mailto:qhull@qhull.org>qhull@qhull.org</a>
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