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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> Qhull code</h1>

			<p>This section discusses the code for Qhull. </p>

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

			<hr>

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

			<ul>
			<li><a href="#reentrant">Reentrant</a> Qhull
			<li><a href="#64bit">Qhull</a> on 64-bit computers
			<li><a href="#cpp">Calling</a> Qhull from C++ programs
			<ul>
			<li><a href="#questions-cpp">Cpp questions for Qhull</a></li>
			<li><a href="#coordinate-cpp">CoordinateIterator</a></li>
			<li><a href="#qhull-cpp">Qhull</a></li>
			<li><a href="#error-cpp">QhullError</a></li>
			<li><a href="#facet-cpp">QhullFacet</a></li>
			<li><a href="#facetlist-cpp">QhullFacetList</a></li>
			<li><a href="#facetset-cpp">QhullFacetSet</a></li>
			<li><a href="#iterator-cpp">QhullIterator</a></li>
			<li><a href="#linkedlist-cpp">QhullLinkedList</a></li>
			<li><a href="#point-cpp">QhullPoint</a></li>
			<li><a href="#qh-cpp">QhullQh</a></li>
			<li><a href="#pointset-cpp">QhullPointSet</a></li>
			<li><a href="#ridge-cpp">QhullRidge</a></li>
			<li><a href="#ridgeset-cpp">QhullRidgeSet</a></li>
			<li><a href="#set-cpp">QhullSet</a></li>
			<li><a href="#vertex-cpp">QhullVertex</a></li>
			<li><a href="#vertexlist-cpp">QhullVertexList</a></li>
			<li><a href="#vertexset-cpp">QhullVertexSet</a></li>
			<li><a href="#rbox-cpp">RboxPoints</a></li>
			</ul>
			<li><a href="#library">Calling</a> Qhull from C programs
			<ul>
			<li><a href="#exit">How to avoid</a> exit(), fprintf(), stderr, and stdout</li>
			<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="#performance">Performance</a> of Qhull</li>
			<li><a href="#enhance">Enhancements</a> to Qhull</li>
			<li><a href="../src/libqhull_r/index.htm">Qhull</a> functions, macros, and data
			structures </li>
			</ul>

			<hr>

			<h2><a href="#TOC">»</a><a name="reentrant">Reentrant Qhull</a></h2>

			<p>Qhull-2015 introduces reentrant Qhull (libqhull_r). Reentrant Qhull uses a qhT* argument instead of global data structures.
			The qhT* pointer is the first argument to most Qhull routines. It allows multiple instances of Qhull to run at the same time.
			It simplifies the C++ interface to Qhull.

			<p>New code should be written with libqhull_r. Existing users of libqhull should consider converting to libqhull_r.
			Although libqhull will be supported indefinitely, improvements may not be implemented.
			Reentrant qhull is 1-2% slower than non-reentrant qhull.

			<p>C++ users need to convert to libqhull_r.
			The new C++ interface does a better, but not perfect, job of hiding Qhull's C data structures.
			The previous C++ interface was unusual
			due to Qhull's global data structures (e.g., UsingLibQhull.cpp).

			<b>Note:</b> Reentrant Qhull is <i>not</i> thread safe. Do not invoke Qhull routines with the same qhT* pointer from multiple threads.

			<h2><a href="#TOC">»</a><a name="64bit">Qhull on 64-bit computers</a></h2>

			<p>Qhull compiles for 64-bit hosts. Since the size of a pointer on a 64-bit host is double the size on a 32-bit host,
			memory consumption increases about 50% for simplicial facets and up-to 100% for non-simplicial facets.

			You can check memory consumption with option <a href="qh-optt.htm#Ts">Ts</a>. It includes the size of
			each data structure:
			<ul>
			<li>32-bit -- merge 24 ridge 20 vertex 28 facet 88 normal 24 ridge vertices 16 facet vertices or neighbors 20
			<li>64-bit -- merge 32 ridge 32 vertex 48 facet 120 normal 32 ridge vertices 40 facet vertices or neighbors 48
			</ul>

			<p>A custom version of Qhull for 3-D Delaunay triangulations may reduce space requirements substantially.

			<p>On 64-bit hosts, reentrant qhull is nearly as fast as non-reentrant qhull.

			<h2><a href="#TOC">»</a><a name="cpp">Calling Qhull from
			C++ programs</a></h2>

			<p>Qhull 2015 uses reentrant Qhull for its C++ interface. If you used
			the C++ interface from qhull 2012.1, you will need to adjust how you initialize and use
			the Qhull classes. See <a href="../src/Changes.txt">Changes.txt</a> for details.

			<p>
			Qhull's C++ interface allows you to explore the results of running Qhull.
			It provides access to Qhull's data structures.
			Most of the classes derive from the corresponding qhull data structure.
			For example, <a href="#facet-cpp">QhullFacet</a> is an instance of Qhull's <a href="../src/libqhull_r/libqhull_r.h#facetT">facetT</a>.
			</p>

			<p>You can retain most of the data in Qhull and use the C++ interface to explore its results.
			Each object contains a reference the Qhull's data structure (via QhullQh), making the C++ representation less memory efficient.
			</p>

			<p>Besides using the C++ interface, you can also use libqhull_r directly. For example,
			the FOREACHfacet_(...) macro will visit each facet in turn.
			</p>

			<p>The C++ interface to Qhull is incomplete. You may need to extend the interface.
			If so, you will need to understand Qhull's data structures and read the code.

			Example (c.f., <code>user_eg3 eg-100</code>). It prints the facets generated by Qhull.

			<pre>
			RboxPoints rbox;
			rbox.appendRandomPoints("100");
			Qhull qhull;
			qhull.runQhull("", rbox);
			QhullFacetList facets(qhull);
			cout<< facets;
			</pre>

			<p>
			The C++ iterface for RboxPoints redefines the fprintf() calls
			in rboxlib.c. Instead of writing its output to stdout, RboxPoints appends
			the output to a std::vector.
			The same technique may be used for calling Qhull from C++.
			</p>
			<ul><li>
			Run Qhull with option '<a href="qh-optt.htm#Ta">Ta</a>' to annotate the
			output with qh_fprintf() identifiers.
			</li><li>
			Redefine qh_fprintf() for these identifiers.
			</li><li>
			See RboxPoints.cpp for an example.
			</li></ul>
			<p>
			A more flexible approach extends Qhull's classes. For example,
			to access the vertices of a <a href="#facet-cpp">QhullFacet</a>,
			define a constructor of <a href="#vertexset-cpp">QhullVertexSet</a>
			that takes a QhullFacet as a parameter.
			</p>
			<p>
			Since the C++ interface uses reentrant Qhull, multiple threads may run Qhull at the same time. Each thread is
			one run of Qhull. Do <i>not</i> have two threads accessing the same Qhull instance.
			</p>

			<h3><a href="#TOC">»</a><a name="coordinate-cpp">CoordinateIterator</a></h3>
			<p>
			A CoordinateIterator or ConstCoordinateIterator [RboxPoints.cpp] is a <code>std::vector<realT>::iterator</code> for Rbox and Qhull coordinates.
			It is the result type of <a href="#rbox-cpp">RboxPoints</a>.coordinates().
			</p>

			<p>Qhull does not use CoordinateIterator for its data structures. A point in Qhull is an array of reals instead of a std::vector.
			See <a href="#point-cpp">QhullPoint</a>.
			</p>

			<h3><a href="#TOC">»</a><a name="qhull-cpp">Qhull</a></h3>
			<p>
			Qhull is the top-level class for running Qhull.
			It initializes Qhull, runs the computation, and records errors.
			It provides access to the global data structure <a href="#qh-cpp">QhullQh</a>,
			Qhull's <a href="#facet-cpp">facets</a>, and <a href="#vertex-cpp">vertices</a>.
			</p>

			<h3><a href="#TOC">»</a><a name="error-cpp">QhullError</a></h3>
			<p>
			QhullError is derived from <code>std::exception</code>. It reports errors from Qhull and captures the output to stderr.
			</p>

			<p>
			If error handling is not set up, Qhull exits with a code from 1 to 5. The codes are defined by
			qh_ERR* in libqhull_r.h. The exit is via qh_exit() in usermem_r.c.
			The C++ interface does not report the
			captured output in QhullError. Call Qhull::setErrorStream to send output to cerr instead.
			</p>

			<h3><a href="#TOC">»</a><a name="facet-cpp">QhullFacet</a></h3>
			<p>
			A QhullFacet is a facet of the convex hull, a region of the Delaunay triangulation, a vertex of a Voronoi diagram,
			or an intersection of the halfspace intersection about a point.
			A QhullFacet has a set of <a href="#vertex-cpp">QhullVertex</a>, a set of <a href="#ridge-cpp">QhullRidge</a>, and
			a set of neighboring QhullFacets.
			</p>

			<h3><a href="#TOC">»</a><a name="facetlist-cpp">QhullFacetList</a></h3>
			<p>
			A QhullFacetList is a linked list of <a href="#facet-cpp">QhullFacet</a>. The result of <code>Qhull.runQhull</code> is a QhullFacetList stored
			in <a href="#qh-cpp">QhullQh</a>.
			</p>

			<h3><a href="#TOC">»</a><a name="facetset-cpp">QhullFacetSet</a></h3>
			<p>
			A QhullFacetSet is a <a href="#set-cpp">QhullSet</a> of <a href="#facet-cpp">QhullFacet</a>. QhullFacetSet may be ordered or unordered. The neighboring facets of a QhullFacet is a QhullFacetSet.
			The neighbors of a <a href="#facet-cpp">QhullFacet</a> is a QhullFacetSet.
			The neighbors are ordered for simplicial facets, matching the opposite vertex of the facet.
			</p>

			<h3><a href="#TOC">»</a><a name="iterator-cpp">QhullIterator</a></h3>
			<p>
			QhullIterator contains macros for defining Java-style iterator templates from a STL-style iterator template.
			</p>

			<h3><a href="#TOC">»</a><a name="linkedlist-cpp">QhullLinkedList</a></h3>
			<p>
			A QhullLinkedLIst is a template for linked lists with next and previous pointers.
			<a href="#facetlist-cpp">QhullFacetList</a> and <a href="#facetlist-cpp">QhullVertexList</a> are QhullLinkedLists.
			</p>

			<h3><a href="#TOC">»</a><a name="point-cpp">QhullPoint</a></h3>
			<p>
			A QhullPoint is an array of point coordinates, typically doubles. The length of the array is <a href="#qh-cpp">QhullQh</a>.hull_dim.
			The identifier of a QhullPoint is its 0-based index from QhullQh.first_point followed by QhullQh.other_points.
			</p>

			<h3><a href="#TOC">»</a><a name="pointset-cpp">QhullPointSet</a></h3>
			<p>
			A QhullPointSet is a <a href="#set-cpp">QhullSet</a> of <a href="#point-cpp">QhullPoint</a>. The QhullPointSet of a <a href="#facet-cpp">QhullFacet</a> is its coplanar points.
			</p>

			<h3><a href="#TOC">»</a><a name="qh-cpp">QhullQh</a></h3>
			<p>
			QhullQh is the root of Qhull's data structure.
			It contains initialized constants, sets, buffers, and variables.
			It contains an array and a set of <a href="#point-cpp">QhullPoint</a>,
			a list of <a href="#facet-cpp">QhullFacet</a>, and a list of <a href="#vertex-cpp">QhullVertex</a>.
			The points are the input to Qhull. The facets and vertices are the result of running Qhull.
			</p>

			<p>
			Qhull's functions access QhullQh through the global variable, <code>qh_qh</code>.
			The global data structures, qh_stat and qh_mem, record statistics and manage memory respectively.
			</p>

			<h3><a href="#TOC">»</a><a name="ridge-cpp">QhullRidge</a></h3>

			<p>
			A QhullRidge represents the edge between two <a href="#facet-cpp">QhullFacet</a>'s.
			It is always simplicial with qh.hull_dim-1 <a href="#vertex-cpp">QhullVertex</a>)'s.
			</p>

			<h3><a href="#TOC">»</a><a name="ridgeset-cpp">QhullRidgeSet</a></h3>

			<p>
			A QhullRidgeSet is a <a href="#set-cpp">QhullSet</a> of <a href="#ridge-cpp">QhullRidge</a>. Each <a href="#facet-cpp">QhullFacet</a> contains a QhullRidgeSet.
			</p>

			<h3><a href="#TOC">»</a><a name="set-cpp">QhullSet</a></h3>

			<p>
			A QhullSet is a set of pointers to objects. QhullSets may be ordered or unordered. They are the core data structure for Qhull.
			</p>

			<h3><a href="#TOC">»</a><a name="vertex-cpp">QhullVertex</a></h3>

			<p>
			A QhullVertex is a vertex of the convex hull. A simplicial <a href="#facet-cpp">QhullFacet</a> has qh.hull_dim-1 vertices. A QhullVertex contains a <a href="#point-cpp">QhullPoint</a>.
			It may list its neighboring <a href="#facet-cpp">QhullFacet</a>'s.
			</p>

			<h3><a href="#TOC">»</a><a name="vertexlist-cpp">QhullVertexList</a></h3>

			<p>
			A QhullVertexList is a <a href="#linkedlist-cpp">QhullLinkedList</a> of <a href="#vertex-cpp">QhullVertex</a>.
			The global data structure, <a href="#qh-cpp">QhullQh</a> contains a QhullVertexList of all
			the vertices.
			</p>

			<h3><a href="#TOC">»</a><a name="vertexset-cpp">QhullVertexSet</a></h3>

			<p>
			A QhullVertexSet is a <a href="#set-cpp">QhullSet</a> of <a href="#vertex-cpp">QhullVertex</a>.
			The QhullVertexSet of a <a href="#facet-cpp">QhullFacet</a> is the vertices of the facet. It is
			ordered for simplicial facets and unordered for non-simplicial facets.
			</p>

			<h3><a href="#TOC">»</a><a name="rbox-cpp">RboxPoints</a></h3>

			<p>
			RboxPoints is a std::vector of point coordinates (<a href="#point-cpp">QhullPoint</a>).
			It's iterator is <a href="#coordinate-cpp">CoordinateIterator</a>.
			</p>
			<p>
			<code>RboxPoints.appendRandomPoints()</code> appends points from a variety of distributions such as uniformly distributed within a cube and random points on a sphere.
			It can also append a cube's vertices or specific points.
			</p>

			<h3><a href="#TOC">»</a><a name="questions-cpp">Cpp questions for Qhull</a></h3>

			Developing C++ code requires many conventions, idioms, and technical details.
			The following questions have either
			mystified the author or do not have a clear answer. See also
			<a href="http://www.qhull.org/road/road-faq/xml/cpp-guideline.xml">C++ and Perl Guidelines</a>.
			and FIXUP notes in the code.
			Please add notes to <a href="http://github.com/qhull/qhull/wiki">Qhull Wiki</a>.

			<ul>
			<li>FIXUP QH11028 Should return reference, but get reference to temporary
			<pre>iterator Coordinates::operator++() { return iterator(++i); }</pre>
			<li>size() as size_t, size_type, or int
			<li>Should all containers have a reserve()?
			<li>Qhull.feasiblePoint interface
			<li>How to avoid copy constructor while logging, maybeThrowQhullMessage()
			<li>How to configure Qhull output. Trace and results should go to stdout/stderr
			<li>Qhull and RboxPoints messaging. e.g., ~Qhull, hasQhullMessage(). Rename them as QhullErrorMessage?
			<li>How to add additional output to an error message, e.g., qh_setprint
			<li>Is idx the best name for an index? It's rather cryptic, but BSD strings.h defines index().
			<li>Qhull::feasiblePoint Qhull::useOutputStream as field or getter?
			<li>Define virtual functions for user customization of Qhull (e.g., qh_fprintf, qh_memfree,etc.)
			<li>Figure out RoadError::global_log. clearQhullMessage currently clearGlobalLog
			<li>Should the false QhullFacet be NULL or empty? e.g., QhullFacet::tricoplanarOwner() and QhullFacetSet::end()
			<li>Should output format for floats be predefined (qh_REAL_1, 2.2g, 10.7g) or as currently set for stream
			<li>Should cout << !point.defined() be blank or 'undefined'
			<li>Infinite point as !defined()
			<li>qlist and qlinkedlist define pointer, reference, size_type, difference_type, const_pointer, const_reference for the class but not for iterator and const_iterator
			vector.h -- <pre>reference operator[](difference_type _Off) const</pre>
			<li>When forwarding an implementation is base() an approriate name (e.g., Coordinates::iterator::base() as std::vector<coordT>::iterator).
			<li>When forwarding an implementation, does not work "returning address of temporary"
			<li>Also --, +=, and -=
			<pre>iterator &operator++() { return iterator(i++); }</pre>
			<li>if vector<coordT> inheritance is bad, is QhullVertexSet OK?
			<li>Should QhullPointSet define pointer and reference data types?
			</ul>

			<h2><a href="#TOC">»</a><a name="library">Calling Qhull from
			C programs</a></h2>

			<p><b>Warning:</b> Qhull was not designed for calling from C
			programs. You may find the <a href="#cpp">C++ interface</a> easier to use.
			You will need to understand the data structures and read the code.
			Most users will find it easier to call Qhull as an external
			command.

			<p>For examples of calling Qhull, see GNU Octave's
			<a href=http://www.gnu.org/software/octave/doc/interpreter/Geometry.html>computational geometry code</a>,
			and Qhull's
			<a href=../src/user_eg/user_eg_r.c>user_eg_r.c</a>,
			<a href=../src/user_eg2/user_eg2_r.c>user_eg2_r.c</a>, and
			<a href=../src/libqhull_r/user_r.c>user_r.c</a>. To see how Qhull calls its library, read
			<a href=../src/qhull/unix_r.c>unix_r.c</a>,
			<a href=../src/qconvex/qconvex.c>qconvex.c</a>,
			<a href=../src/qdelaunay/qdelaun.c>qdelaun.c</a>,
			<a href=../src/qhalf/qhalf.c>qhalf.c</a>, and
			<a href=../src/qvoronoi/qvoronoi.c>qvoronoi.c</a>. The '*_r.c' files are reentrant, otherwise they are non-reentrant.
			Either version may be used. New code should use reentrant Qhull.

			<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@shore.net">bradb@shore.net</a>.

			<p>See <a href="../src/libqhull_r/index.htm">Reentrant 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>If you use non-reentrant Qhull, be aware of the macros "qh"
			and "qhstat", e.g., "qh hull_dim". They are
			defined in <tt>libqhull.h</tt>. They allow the global data
			structures to be pre-allocated (faster access) or dynamically
			allocated (allows multiple copies). </p>

			<p>Qhull's <tt>Makefile</tt> produces a library, <tt>libqhull_r.a</tt>,
			for inclusion in your programs. First review <tt>libqhull_r.h</tt>.
			This defines the data structures used by Qhull and provides
			prototypes for the top-level functions.
			Most users will only need libqhull_r.h in their programs. For
			example, the Qhull program is defined with <tt>libqhull_r.h</tt> and <tt>unix_r.c</tt>.
			To access all functions, use <tt>qhull_ra.h</tt>. Include the file
			with "<tt>#include <libqhull_r/qhull_ra.h></tt>". This
			avoids potential name conflicts.</p>

			<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">»</a><a name="exit">How to avoid exit(), fprintf(), stderr, and stdout</a></h3>

			<p>Qhull sends output to qh.fout and errors, log messages, and summaries to qh.ferr. qh.fout is normally
			stdout and qh.ferr is stderr. qh.fout may be redefined by option '<a
			href="qh-optt.htm#TO">TO</a>' or the caller. qh.ferr may be redirected to qh.fout by option '<a
			href="qh-optt.htm#Tz">Tz</a>'.</p>

			<p>Qhull does not use stderr, stdout, fprintf(), or exit() directly.</p>

			<p>Qhull reports errors via qh_errexit() by writting a message to qh.ferr and invoking longjmp().
			This returns the caller to the corresponding setjmp() (c.f., QH_TRY_ in QhullQh.h). If
			qh_errexit() is not available, Qhull functions call qh_exit(). qh_exit() normally calls exit(),
			but may be redefined by the user. An example is
			libqhullcpp/usermem_r-cpp.cpp. It redefines qh_exit() as a 'throw'.</p>

			<p>If qh_meminit() or qh_new_qhull() is called with ferr==NULL, then they set ferr to stderr.
			Otherwise the Qhull libraries use qh->ferr and qh->qhmem.ferr for error output.</p>

			<p>If an error occurs before qh->ferr is initialized, Qhull invokes qh_fprintf_stderr(). The user
			may redefine this function along with qh_exit(), qh_malloc(), and qh_free().

			<p>The Qhull libraries write output via qh_fprintf() [userprintf_r.c]. Otherwise, the Qhull
			libraries do not use stdout, fprintf(), or printf(). Like qh_exit(), the user may redefine
			qh_fprintf().</p>

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

			<p>You can use <tt>mem_r.c</tt> and <tt>qset_r.c</tt> individually. <tt>Mem_r.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_r.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_r.c</tt>
			just includes the functions needed for Qhull. </p>

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

			<p>The routine qh_findbestfacet in <tt>poly2_r.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>

			<blockquote>
			<p><b>Warning:</b> If triangulated output ('<a href=qh-optq.htm#Qt>Qt</a>') and
			the best facet is triangulated, qh_findbestfacet() returns one of
			the corresponding 'tricoplanar' facets. The actual best facet may be a different
			tricoplanar facet.
			<p>
			See qh_nearvertex() in poly2.c for sample code to visit each
			tricoplanar facet. To identify the correct tricoplanar facet,
			see Devillers, et. al., [<a href="index.htm#devi01">'01</a>]
			and Mucke, et al [<a href="index.htm#muck96">'96</a>]. If you
			implement this test in general dimension, please notify
			<a href="mailto:qhull@qhull.org">qhull@qhull.org</a>.
			</blockquote>

			<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_r.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">»</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://www.netlib.org/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>libqhull.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">»</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">»</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/libqhull_r/io_r.c#detvridge>qh_detvridge</a> for an example of ignoring tricoplanar facets.</p>

			<h3><a href="#TOC">»</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">»</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">»</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">»</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">»</a><a name="enhance">Enhancements to Qhull </a></h2>

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

			<pre>
			[Jan 2010] Suggestions
			- Generate vcproj from qtpro files
			cd qtpro && qmake -spec win32-msvc2005 -tp vc -recursive
			sed -i 's/C\:\/bash\/local\/qhull\/qtpro\///' qhull-all.sln
			Change qhullcpp to libqhull.dll
			Allow both builds on same host (keep /tmp separate)
			- Make distribution -- remove tmp, news, .git, leftovers from project, change CRLF
			search for 2010.1, Dates
			qhulltest --all added to output
			Add md5sum
			Add test of user_eg3, etc.
			- C++ class for access to statistics, accumulate vs. add
			- Add dialog box to RoadError-- a virtual function?
			- Option 'Gt' does not make visible all facets of the mesh example, rbox 32 M1,0,1 \| qhull d Gt
			- Option to select bounded Voronoi regions [A. Uzunovic]
			- Merge small volume boundary cells into unbounded regions [Dominik Szczerba]
			- Postmerge with merge options
			- Add const to C code
			- Add modify operators and MutablePointCoordinateIterator to PointCoordinates
			- Add Qtest::toString() functions for QhullPoint and others. QByteArray and qstrdup()
			- Fix option Qt for conformant triangulations of merged facets
			- Investigate flipped facet -- rbox 100 s D3 t1263080158 \| qhull R1e-3 Tcv Qc
			- Add doc comments to c++ code
			- Measure performance of Qhull, seconds per point by dimension
			- Report potential wraparound of 64-bit ints -- e.g., a large set or points

			Documentation
			- Qhull::addPoint(). Problems with qh_findbestfacet and otherpoints see
			qh-code.htm#inc on-line construction with qh_addpoint()
			- How to handle 64-bit possible loss of data. WARN64, ptr_intT, size_t/int
			- Show custom of qh_fprintf
			- grep 'qh_mem ' x \| sort \| awk '{ print $2; }' \| uniq -c \| grep -vE ' (2\|4\|6\|8\|10\|12\|14\|16\|20\|64\|162)[^0-9]'
			- qtpro/qhulltest contains .pro and Makefile. Remove Makefiles by setting shadow directory to ../../tmp/projectname
			- Rules for use of qh_qh and multi processes
			UsingQhull
			errorIfAnotherUser
			~QhullPoints() needs ownership of qh_qh
			Does !qh_pointer work?
			When is qh_qh required? Minimize the time.
			qhmem, qhstat.ferr
			qhull_inuse==1 when qhull globals active [not useful?]
			rbox_inuse==1 when rbox globals active
			- Multithreaded -- call largest dimension for infinityPoint() and origin()
			- Better documentation for qhmem totshort, freesize, etc.
			- how to change .h, .c, and .cpp to text/html. OK in Opera
			- QhullVertex.dimension() is not quite correct, epensive
			- Check globalAngleEpsilon
			- Deprecate save_qhull()

			[Dec 2003] Here is a partial list:
			- fix finddelaunay() in user_eg.c for tricoplanar facets
			- write a BGL, C++ interface to Qhull
			http://www.boost.org/libs/graph/doc/table_of_contents.html
			- 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:
			- implement Matlab's tsearch() using Qhull
			- 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
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			<p>Comments to: <a href=mailto:qhull@qhull.org>qhull@qhull.org</a>
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			Created: Sept. 25, 1995 --- <!-- hhmts start --> Last modified: see changes.txt <!-- hhmts end --> </p>
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