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			<p><a name="TOP"><b>Up:</b></a> <a
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			<h1><a
			href="http://www.geom.umn.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> Qhull internals</h1>

			<p>This section discusses the internals of Qhull. </p>

			<p><b>Copyright © 1995-2002 The Geometry Center, Minneapolis MN</b></p>

			<hr>

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

			<ul>
			<li><a href="#performance">Performance</a> of Qhull</li>
			<li><a href="#library">Calling Qhull</a> from your program
			<ul>
			<li><a href="#constrained">Constrained Delaunay</a>
			triangulation</li>
			<li><a href="#dids">Delaunay triangulations</a> and point indices</li>
			<li><a href="#findfacet">Locate facet</a> with
			qh_findbestfacet()</li>
			<li><a href="#inc">On-line construction</a> with
			qh_addpoint()</li>
			<li><a href="#mem">Sets and quick memory</a> allocation</li>
			<li><a href="#tricoplanar">Tricoplanar facets</a> and option 'Qt'</li>
			<li><a href="#vneighbor">Vertex neighbors</a> of a vertex</li>
			<li><a href="#vertices">Voronoi vertices</a> of a region</li>
			<li><a href="#ridge">Voronoi vertices</a> of a ridge</li>
			</ul>
			</li>
			<li><a href="#enhance">Enhancements</a> to Qhull</li>
			<li><a href="../src/index.htm">Qhull functions</a>, macros, and data
			structures </li>
			</ul>

			<hr>

			<h2><a href="#TOC">ｻ</a><a name="performance">Performance of
			Qhull </a></h2>

			<p>Empirically, Qhull's performance is balanced in the sense that
			the average case happens on average. This may always be true if
			the precision of the input is limited to at most <i>O(log n)</i>
			bits. Empirically, the maximum number of vertices occurs at the
			end of constructing the hull. </p>

			<p>Let <i>n</i> be the number of input points, <i>v</i> be the
			number of output vertices, and <i>f_v </i>be the maximum number
			of facets for a convex hull of <i>v</i> vertices. If both
			conditions hold, Qhull runs in <i>O(n log v)</i> in 2-d and 3-d
			and <i>O(n f_v/v)</i> otherwise. The function <i>f_v</i>
			increases rapidly with dimension. It is <em>O(v^floor(d/2) /
			floor(d/2)!)</em>.</p>

			<p>The time complexity for merging is unknown. Options '<a
			href="qh-optc.htm#C0">C-0</a>' and '<a href="qh-optq.htm#Qx">Qx</a>'
			(defaults) handle precision problems due to floating-point
			arithmetic. They are optimized for simplicial outputs. </p>

			<p>When running large data sets, you should monitor Qhull's
			performance with the '<a href="qh-optt.htm#TFn">TFn</a>' option.
			The time per facet is approximately constant. In high-d with many
			merged facets, the size of the ridge sets grows rapidly. For
			example the product of 8-d simplices contains 18 facets and
			500,000 ridges. This will increase the time needed per facet. </p>

			<p>As dimension increases, the number of facets and ridges in a
			convex hull grows rapidly for the same number of vertices. For
			example, the convex hull of 300 cospherical points in 6-d has
			30,000 facets. </p>

			<p>If Qhull appears to stop processing facets, check the memory
			usage of Qhull. If more than 5-10% of Qhull is in virtual memory,
			its performance will degrade rapidly. </p>

			<p>When building hulls in 20-d and higher, you can follow the
			progress of Qhull with option '<a href="qh-optt.htm#Tn">T1</a>'.
			It reports each major event in processing a point. </p>

			<p>To reduce memory requirements, recompile Qhull for
			single-precision reals (REALfloat in <tt>user.h</tt>).
			Single-precision does not work with joggle ('<a
			href="qh-optq.htm#QJn">QJ</a>'). Check qh_MEMalign in <tt>user.h</tt>
			and the match between free list sizes and data structure sizes
			(see the end of the statistics report from '<a
			href="qh-optt.htm#Ts">Ts</a>'). If free list sizes do not match,
			you may be able to use a smaller qh_MEMalign. Setting
			qh_COMPUTEfurthest saves a small amount of memory, as does
			clearing qh_MAXoutside (both in <tt>user.h</tt>).</p>

			<p>Shewchuk is working on a 3-d version of his triangle
			program. It is optimized for 3-d simplicial Delaunay triangulation
			and uses less memory than Qhull.</p>

			<p>To reduce the size of the Qhull executable, consider
			qh_NOtrace and qh_KEEPstatistics 0 in <tt>user.h</tt>. By
			changing <tt>user.c </tt>you can also remove the input/output
			code in <tt>io.c</tt>. If you don't need facet merging, then
			version 1.01 of Qhull is much smaller. It contains some bugs that
			prevent Qhull from initializing in simple test cases. It is
			slower in high dimensions.</p>

			<p>The precision options, '<a href="qh-optc.htm#Vn">Vn</a>', '<a
			href="qh-optc.htm#Wn">Wn</a>', '<a href="qh-optc.htm#Un">Un</a>'.
			'<a href="qh-optc.htm#An">A-n</a>', '<a href="qh-optc.htm#Cn">C-n</a>',
			'<a href="qh-optc.htm#An2">An</a>', '<a href="qh-optc.htm#Cn2">Cn</a>',
			and '<a href="qh-optq.htm#Qx">Qx</a>', may have large effects on
			Qhull performance. You will need to experiment to find the best
			combination for your application. </p>

			<p>The verify option ('<a href="qh-optt.htm#Tv">Tv</a>') checks
			every point after the hull is complete. If facet merging is used,
			it checks that every point is inside every facet. This can take a
			very long time if there are many points and many facets. You can
			interrupt the verify without losing your output. If facet merging
			is not used and there are many points and facets, Qhull uses a
			directed search instead of an exhaustive search. This should be
			fast enough for most point sets. Directed search is not used for
			facet merging because directed search was already used for
			updating the facets' outer planes.</p>

			<p>The check-frequently option ('<a href="qh-optt.htm#Tc">Tc</a>')
			becomes expensive as the dimension increases. The verify option
			('<a href="qh-optt.htm#Tv">Tv</a>') performs many of the same
			checks before outputting the results.</p>

			<p>Options '<a href="qh-optq.htm#Q0">Q0</a>' (no pre-merging), '<a
			href="qh-optq.htm#Q3">Q3</a>' (no checks for redundant vertices),
			'<a href="qh-optq.htm#Q5">Q5</a>' (no updates for outer planes),
			and '<a href="qh-optq.htm#Q8">Q8</a>' (no near-interior points)
			increase Qhull's speed. The corresponding operations may not be
			needed in your application.</p>

			<p>In 2-d and 3-d, a partial hull may be faster to produce.
			Option '<a href="qh-optq.htm#QGn">QgGn</a>' only builds facets
			visible to point n. Option '<a href="qh-optq.htm#QVn">QgVn</a>'
			only builds facets that contain point n. In higher-dimensions,
			this does not reduce the number of facets.</p>

			<p><tt>User.h</tt> includes a number of performance-related
			constants. Changes may improve Qhull performance on your data
			sets. To understand their effect on performance, you will need to
			read the corresponding code. </p>

			<p>GNU <tt>gprof</tt> reports that the dominate cost for 3-d
			convex hull of cosperical points is qh_distplane(), mainly called
			from qh_findbestnew(). The dominate cost for 3-d Delaunay triangulation
			is creating new facets in qh_addpoint(), while qh_distplane() remains
			the most expensive function.

			</p>
			<h2><a href="#TOC">ｻ</a><a name="library">Calling Qhull from
			your program</a></h2>

			<p>The <tt>Makefile</tt> produces a library, <tt>libqhull.a</tt>,
			for inclusion in your programs. First review <tt>qhull.h</tt>.
			This defines the data structures used by Qhull and provides
			prototypes for the top-level functions. </p>

			<p>Most users will only need qhull.h in their programs. For
			example, the Qhull program is defined with <tt>qhull.h</tt> and <tt>unix.c</tt>.
			To access all functions, use <tt>qhull_a.h</tt>. Include the file
			with "<tt>#include <qhull/qhull_a.h>".</tt> This
			avoids potential name conflicts.</p>

			<p>See <a href="../src/index.htm">Qhull functions, macros, and data
			structures</a> for internal documentation of Qhull. The
			documentation provides an overview and index. To use the library
			you will need to read and understand the code. For most users, it
			is better to write data to a file, call the qhull program, and
			read the results from the output file.</p>

			<p>When you read the code, be aware of the macros "qh"
			and "qhstat", e.g., "qh hull_dim". They are
			defined in <tt>qhull.h</tt>. They allow the global data
			structures to be pre-allocated (faster access) or dynamically
			allocated (allows multiple copies). </p>

			<p><tt>User.c</tt> and <tt>qhull_interface.cpp</tt> include
			an application template. You can use <tt>user_eg.c</tt>
			as an example. It also shows how to reduce memory requirements by
			not loading <tt>io.o</tt>. If you use joggled input ('<a
			href="qh-optq.htm#QJn">QJ</a>'), you may also remove <tt>merge.o</tt>
			with <a href="../src/user.h#NOmerge">qh_NOmerge</a>'. </p>

			<p>The <a href=http://www.boost.org/libs/graph/doc/table_of_contents.html>BGL</a>
			Boost Graph Library [aka GGCL] provides C++ classes for graph data structures
			and algorithms [Dr. Dobb's 9/00 p. 29-38; OOPSLA '99 p. 399-414]. It is modelled after the
			Standard Template Library. It would provide a good interface to Qhull.
			If you are interested in adapting BGL to Qhull, please contact
			<a href="mailto:bradb@geom.umn.edu">bradb@geom.umn.edu</a>.

			<p>If you use the Qhull library, you are on your own as far as
			bugs go. Start with small examples for which you know the output.
			If you get a bug, try to duplicate it with the Qhull program. The
			'<a href="qh-optt.htm#Tc">Tc</a>' option will catch many problems
			as they occur. When an error occurs, use '<a
			href="qh-optt.htm#Tn">T4</a> <a href="qh-optt.htm#TPn">TPn</a>'
			to trace from the last point added to the hull. Compare your
			trace with the trace output from the Qhull program.</p>

			<p>Errors in the Qhull library are more likely than errors in the
			Qhull program. These are usually due to feature interactions that
			do not occur in the Qhull program. Please report all errors that
			you find in the Qhull library. Please include suggestions for
			improvement. </p>

			<h3><a href="#TOC">ｻ</a><a name="mem">sets and quick memory
			allocation</a></h3>

			<p>You can use <tt>mem.c</tt> and<tt> qset.c</tt> individually. <tt>Mem.c
			</tt>implements quick-fit memory allocation. It is faster than
			malloc/free in applications that allocate and deallocate lots of
			memory. </p>

			<p><tt>Qset.c</tt> implements sets and related collections. It's
			the inner loop of Qhull, so speed is more important than
			abstraction. Set iteration is particularly fast. <tt>qset.c</tt>
			just includes the functions needed for Qhull. </p>

			<h3><a href="#TOC">ｻ</a><a name="dids">Delaunay triangulations
			and point indices</a></h3>

			<p>Here some unchecked code to print the point indices of each
			Delaunay triangle. Use option 'QJ' if you want to avoid
			non-simplicial facets. Note that upper Delaunay regions are
			skipped. These facets correspond to the furthest-site Delaunay
			triangulation. </p>

			<blockquote>
			<pre>
			facetT *facet;
			vertexT vertex, *vertexp;

			FORALLfacets {
			if (!facet->upperdelaunay) {
			printf ("%d", qh_setsize (facet->vertices);
			FOREACHvertex_(facet->vertices)
			printf (" %d", qh_pointid (vertex->point));
			printf ("\n");
			}
			}

			</pre>
			</blockquote>

			<h3><a href="#TOC">ｻ</a><a name="findfacet">locate a facet with
			qh_findbestfacet()</a></h3>

			<p>The routine qh_findbestfacet in <tt>poly2.c</tt> is
			particularly useful. It uses a directed search to locate the
			facet that is furthest below a point. For Delaunay
			triangulations, this facet is the Delaunay triangle that contains
			the lifted point. For convex hulls, the distance of a point to
			the convex hull is either the distance to this facet or the
			distance to a subface of the facet.</p>

			<p>qh_findbestfacet performs an exhaustive search if its directed
			search returns a facet that is above the point. This occurs when
			the point is inside the hull or if the curvature of the convex
			hull is less than the curvature of a sphere centered at the point
			(e.g., a point near a lens-shaped convex hull). When the later
			occurs, the distance function is bimodal and a directed search
			may return a facet on the far side of the convex hull. </p>

			<p>Algorithms that retain the previously constructed hulls
			usually avoid an exhaustive search for the best facet. You may
			use a hierarchical decomposition of the convex hull [Dobkin and
			Kirkpatrick <a href="index.htm#dob-kir90">'90</a>]. </p>

			<p>To use qh_findbestfacet with Delaunay triangulations, lift the
			point to a paraboloid by summing the squares of its coordinates
			(see qh_setdelaunay in geom2.c). Do not scale the input with
			options 'Qbk', 'QBk', 'QbB' or 'Qbb'. See Mucke, et al [<a
			href="index.htm#muck96">'96</a>] for a good point location
			algorithm.</p>

			<p>The intersection of a ray with the convex hull may be found by
			locating the facet closest to a distant point on the ray.
			Intersecting the ray with the facet's hyperplane gives a new
			point to test. </p>

			<h3><a href="#TOC">ｻ</a><a name="inc">on-line construction with
			qh_addpoint()</a></h3>

			<p>The Qhull library may be used for the on-line construction of
			convex hulls, Delaunay triangulations, and halfspace
			intersections about a point. It may be slower than implementations that retain
			intermediate convex hulls (e.g., Clarkson's <a
			href="http://netlib.bell-labs.com/netlib/voronoi/hull.html">hull
			program</a>). These implementations always use a directed search.
			For the on-line construction of convex hulls and halfspace
			intersections, Qhull may use an exhaustive search
			(qh_findbestfacet). </p>

			<p>You may use qh_findbestfacet and qh_addpoint (<tt>qhull.c</tt>) to add a point to
			a convex hull. Do not modify the point's coordinates since
			qh_addpoint does not make a copy of the coordinates. For Delaunay
			triangulations, you need to lift the point to a paraboloid by
			summing the squares of the coordinates (see qh_setdelaunay in
			geom2.c). Do not scale the input with options 'Qbk', 'QBk', 'QbB'
			or 'Qbb'. Do not deallocate the point's coordinates. You need to
			provide a facet that is below the point (<a href="#findfacet">qh_findbestfacet</a>).
			</p>

			<p>You can not delete points. Another limitation is that Qhull
			uses the initial set of points to determine the maximum roundoff
			error (via the upper and lower bounds for each coordinate). </p>

			<p>For many applications, it is better to rebuild the hull from
			scratch for each new point. This is especially true if the point
			set is small or if many points are added at a time.</p>

			<p>Calling qh_addpoint from your program may be slower than
			recomputing the convex hull with qh_qhull. This is especially
			true if the added points are not appended to the qh first_point
			array. In this case, Qhull must search a set to determine a
			point's ID. [R. Weber] </p>

			<p>See user_eg.c for examples of the on-line construction of
			convex hulls, Delaunay triangulations, and halfspace
			intersections. The outline is: </p>

			<blockquote>
			<pre>
			initialize qhull with an initial set of points
			qh_qhull();

			for each additional point p
			append p to the end of the point array or allocate p separately
			lift p to the paraboloid by calling qh_setdelaunay
			facet= qh_findbestfacet (p, !qh_ALL, &bestdist, &isoutside);
			if (isoutside)
			if (!qh_addpoint (point, facet, False))
			break; /* user requested an early exit with 'TVn' or 'TCn' */

			call qh_check_maxout() to compute outer planes
			terminate qhull</pre>
			</blockquote>

			<h3><a href="#TOC">ｻ</a><a name="constrained">Constrained
			Delaunay triangulation </a></h3>

			<p>With a fair amount of work, Qhull is suitable for constrained
			Delaunay triangulation. See Shewchuk, ACM Symposium on
			Computational Geometry, Minneapolis 1998.</p>

			<p>Here's a quick way to add a constraint to a Delaunay
			triangulation: subdivide the constraint into pieces shorter than
			the minimum feature separation. You will need an independent
			check of the constraint in the output since the minimum feature
			separation may be incorrect. [H. Geron] </p>

			<h3><A href="#TOC">ｻ</A><a name="tricoplanar">Tricoplanar facets and option 'Qt'</h3>

			<p>Option '<a href=qh_optq.htm#Qt>Qt</a>' triangulates non-simplicial
			facets (e.g., a square facet in 3-d or a cubical facet in 4-d).
			All facets share the same apex (i.e., the first vertex in facet->vertices).
			For each triangulated facet, Qhull
			sets facet->tricoplanar true and copies facet->center, facet->normal, facet->offset, and facet->maxoutside. One of
			the facets owns facet->normal; its facet->keepcentrum is true.
			If facet->isarea is false, facet->triowner points to the owning
			facet.

			<p>Qhull sets facet->degenerate if the facet's vertices belong
			to the same ridge of the non-simplicial facet.

			<p>To visit each tricoplanar facet of a non-simplicial facet,
			either visit all neighbors of the apex or recursively visit
			all neighbors of a tricoplanar facet. The tricoplanar facets
			will have the same facet->center.</p>

			<p>See <a href=../src/io.c#detvridge>qh_detvridge</a> for an example of ignoring tricoplanar facets.</p>

			<h3><a href="#TOC">ｻ</a><a name="vertices">Voronoi vertices of a
			region</a></h3>

			<p>The following code iterates over all Voronoi vertices for each
			Voronoi region. Qhull computes Voronoi vertices from the convex
			hull that corresponds to a Delaunay triangulation. An input site
			corresponds to a vertex of the convex hull and a Voronoi vertex
			corresponds to an adjacent facet. A facet is
			"upperdelaunay" if it corresponds to a Voronoi vertex
			"at-infinity". Qhull uses qh_printvoronoi in <tt>io.c</tt>
			for '<a href=qvoronoi.htm>qvoronoi</a> <a href="qh-opto.htm#o">o'</a> </p>

			<blockquote>
			<pre>
			/* please review this code for correctness */
			qh_setvoronoi_all();
			FORALLvertices {
			site_id = qh_pointid (vertex->point);
			if (qh hull_dim == 3)
			qh_order_vertexneighbors(vertex);
			infinity_seen = 0;
			FOREACHneighbor_(vertex) {
			if (neighbor->upperdelaunay) {
			if (!infinity_seen) {
			infinity_seen = 1;
			... process a Voronoi vertex "at infinity" ...
			}
			}else {
			voronoi_vertex = neighbor->center;
			... your code goes here ...
			}
			}
			}
			</pre>
			</blockquote>

			<h3><a href="#TOC">ｻ</a><a name="ridge">Voronoi vertices of a
			ridge</a></h3>

			<p>Qhull uses qh_printvdiagram() in io.c to print the ridges of a
			Voronoi diagram for option '<a href="qh-optf.htm#Fv2">Fv</a>'.
			The helper function qh_eachvoronoi() does the real work. It calls
			the callback 'printvridge' for each ridge of the Voronoi diagram.
			</p>

			<p>You may call qh_printvdiagram2(), qh_eachvoronoi(), or
			qh_eachvoronoi_all() with your own function. If you do not need
			the total number of ridges, you can skip the first call to
			qh_printvdiagram2(). See qh_printvridge() and qh_printvnorm() in
			io.c for examples. </p>

			<h3><a href="#TOC">ｻ</a><a name="vneighbor">vertex neighbors of
			a vertex</a></h3>

			<p>To visit all of the vertices that share an edge with a vertex:
			</p>

			<ul>
			<li>Generate neighbors for each vertex with
			qh_vertexneighbors in <tt>poly2.c</tt>. </li>
			<li>For simplicial facets, visit the vertices of each
			neighbor </li>
			<li>For non-simplicial facets, <ul>
			<li>Generate ridges for neighbors with qh_makeridges
			in <tt>merge.c</tt>. </li>
			<li>Generate ridges for a vertex with qh_vertexridges
			in <tt>merge.c</tt>. </li>
			<li>Visit the vertices of these ridges. </li>
			</ul>
			</li>
			</ul>

			<p>For non-simplicial facets, the ridges form a simplicial
			decomposition of the (d-2)-faces between each pair of facets --
			if you need 1-faces, you probably need to generate the full face
			graph of the convex hull. </p>

			<h2><a href="#TOC">ｻ</a><a name="enhance">Enhancements to Qhull </a></h2>

			<p>There are many ways in which Qhull can be improved. </p>

			<pre>Here is a partial list:
			- write a BGL, C++ interface to Qhull
			http://www.boost.org/libs/graph/doc/table_of_contents.html
			- rewrite Qhull in C++ to make it easier to use Qhull in a program
			- change qh_save_qhull to swap the qhT structure instead of using pointers
			- change error handling and tracing to be independent of 'qh ferr'
			- determine the maximum width for a given set of parameters
			- prove that directed search locates all coplanar facets
			- in high-d merging, can a loop of facets become disconnected?
			- find a way to improve inner hulls in 5-d and higher
			- determine the best policy for facet visibility ('<a
			href="qh-optc.htm#Vn">Vn</a>')
			- determine the limitations of '<a href="qh-optq.htm#Qg">Qg</a>'

			Precision improvements:
			- For 'Qt', resolve cross-linked, butterfly ridges.
			May allow retriangulation in qh_addpoint().
			- for Delaunay triangulations ('d' or 'v') under joggled input ('QJ'),
			remove vertical facets whose lowest vertex may be coplanar with convex hull
			- review use of 'Qbb' with 'd QJ'. Is MAXabs_coord better than MAXwidth?
			- check Sugihara and Iri's better in-sphere test [Canadian
			Conf. on Comp. Geo., 1989; Univ. of Tokyo RMI 89-05]
			- replace centrum with center of mass and facet area
			- handle numeric overflow in qh_normalize and elsewhere
			- merge flipped facets into non-flipped neighbors.
			currently they merge into best neighbor (appears ok)
			- determine min norm for Cramer's rule (qh_sethyperplane_det). It looks high.
			- improve facet width for very narrow distributions

			New features:
			- compute volume of Voronoi regions. You need to determine the dual face
			graph in all dimensions [see Clarkson's hull program]
			- compute alpha shapes [see Clarkson's hull program]
			- implement deletion of Delaunay vertices
			see Devillers, ACM Symposium on Computational Geometry, Minneapolis 1999.
			- compute largest empty circle [see O'Rourke, chapter 5.5.3] [Hase]
			- list redundant (i.e., coincident) vertices [Spitz]
			- implement Mucke, et al, ['96] for point location in Delaunay triangulations
			- implement convex hull of moving points
			- implement constrained Delaunay diagrams
			see Shewchuk, ACM Symposium on Computational Geometry, Minneapolis 1998.
			- estimate outer volume of hull
			- automatically determine lower dimensional hulls
			- allow "color" data for input points
			need to insert a coordinate for Delaunay triangulations

			Input/output improvements:
			- Support the VTK Visualization Toolkit, http://www.kitware.com/vtk.html
			- generate output data array for Qhull library [Gautier]
			- need improved DOS window with screen fonts, scrollbar, cut/paste
			- generate Geomview output for Voronoi ridges and unbounded rays
			- generate Geomview output for halfspace intersection
			- generate Geomview display of furthest-site Voronoi diagram
			- use '<a href="qh-optg.htm#GDn">GDn</a>' to view 5-d facets in 4-d
			- convert Geomview output for other 3-d viewers
			- add interactive output option to avoid recomputing a hull
			- orient vertex neighbors for '<a href="qh-optf.htm#Fv">Fv</a>' in 3-d and 2-d
			- track total number of ridges for summary and logging

			Performance improvements:
			- optimize Qhull for 2-d Delaunay triangulations
			- use O'Rourke's <a href="index.htm#orou94">'94</a> vertex->duplicate_edge
			- add bucketing
			- better to specialize all of the code (ca. 2-3x faster w/o merging)
			- use updated LU decomposition to speed up hyperplane construction
			- [Gill et al. 1974, Math. Comp. 28:505-35]
			- construct hyperplanes from the corresponding horizon/visible facets
			- for merging in high d, do not use vertex->neighbors

			</pre>

			<p>Please let us know about your applications and improvements. </p>
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			align="middle" width="40" height="40"></a><i>The Geometry Center
			Home Page </i></p>

			<p>Comments to: <a href=mailto:qhull@geom.umn.edu>qhull@geom.umn.edu</a>
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