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	<title>qdelaunay -- Delaunay triangulation</title>
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	<h1><a
	href="http://www.geom.uiuc.edu/graphics/pix/Special_Topics/Computational_Geometry/delaunay.html"><img
	src="qh--dt.gif" alt="[delaunay]" align="middle" width="100"
	height="100"></a>qdelaunay -- Delaunay triangulation</h1>

	<p>The Delaunay triangulation is the triangulation with empty
	circumspheres. It has many useful properties and applications.
	See the survey article by Aurenhammer [<a
	href="index.htm#aure91">'91</a>] and the detailed introduction
	by O'Rourke [<a href="index.htm#orou94">'94</a>]. </p>

	<blockquote>
	<dl>
	<dt><b>Example:</b> rbox r y c G0.1 D2 \| qdelaunay <a href="qh-opto.htm#s">s</a>
	<a href="qh-optf.htm#Fv">Fv</a> <a href="qh-optt.htm#TO">TO
	result</a></dt>
	<dd>Compute the 2-d Delaunay triangulation of a triangle and
	a small square.
	Write a summary to the console and unoriented regions to 'result'.
	Merge regions for cocircular input sites (i.e., the
	square).</dd>
	<dt> </dt>
	<dt><b>Example:</b> rbox r y c G0.1 D2 \| qdelaunay <a href="qh-opto.htm#s">s</a>
	<a href="qh-optf.htm#Fv">Fv</a> <a href="qh-optq.htm#Qt">Qt</a></dt>
	<dd>Compute the 2-d Delaunay triangulation of a triangle and
	a small square. Write a summary and unoriented
	regions to the console. Produce triangulated output.</dd>
	<dt> </dt>
	<dt><b>Example:</b> rbox 10 D2 \| qdelaunay <a
	href="qh-optq.htm#QJn">QJ</a> <a href="qh-opto.htm#s">s</a>
	<a href="qh-opto.htm#i">i</a> <a href="qh-optt.htm#TO">TO
	result</a></dt>
	<dd>Compute the 2-d Delaunay triangulation of 10 random
	points. Joggle the input to guarantee triangular output.
	Write a summary to the console and the regions to
	'result'.</dd>
	</dl>
	</blockquote>

	<p>Qhull computes the Delaunay triangulation by computing a
	convex hull. It lifts the input sites to a paraboloid by adding
	the sum of the squares of the coordinates. It scales the height
	of the paraboloid to improve numeric precision ('<a href=qh-optq.htm#Qbb>Qbb</a>').
	It computes the convex
	hull of the lifted sites, and projects the lower convex hull to
	the input.

	<p>Each region of the Delaunay triangulation
	corresponds to a facet of the lower half of the convex hull.
	Facets of the upper half of the convex hull correspond to the <a
	href="qdelau_f.htm">furthest-site Delaunay triangulation</a>.
	See the examples, <a href="qh-eg.htm#delaunay">Delaunay and
	Voronoi diagrams</a>.</p>

	<p>See <a href="http://www.qhull.org/html/qh-faq.htm#TOC">Qhull FAQ</a> - Delaunay and
	Voronoi diagram questions.</p>

	<p>By default, qdelaunay merges cocircular and cospherical regions.
	For example, the Delaunay triangulation of a square inside a diamond
	('rbox D2 c d G4 \| qdelaunay') contains one region for the square.

	<p>Use option '<a href="qh-optq.htm#Qz">Qz</a>' if the input is circular, cospherical, or
	nearly so. It improves precision by adding a point "at infinity," above the corresponding paraboloid.

	<p>If you use '<a href="qh-optq.htm#Qt">Qt</a>' (triangulated output),
	all Delaunay regions will be simplicial (e.g., triangles in 2-d).
	Some regions may be
	degenerate and have zero area. Triangulated output identifies coincident
	points.

	<p>If you use '<a href="qh-optq.htm#QJn">QJ</a>' (joggled input), all Delaunay regions
	will be simplicial (e.g., triangles in 2-d). Coincident points will
	create small regions since the points are joggled apart. Joggled input
	is less accurate than triangulated output ('Qt'). See <a
	href="qh-impre.htm#joggle">Merged facets or joggled input</a>. </p>

	<p>The output for 3-d Delaunay triangulations may be confusing if the
	input contains cospherical data. See the FAQ item
	<a href=qh-faq.htm#extra>Why
	are there extra points in a 4-d or higher convex hull?</a>
	Avoid these problems with triangulated output ('<a href="qh-optq.htm#Qt">Qt</a>') or
	joggled input ('<a href="qh-optq.htm#QJn">QJ</a>').
	</p>

	<p>The 'qdelaunay' program is equivalent to
	'<a href=qhull.htm#outputs>qhull d</a> <a href=qh-optq.htm#Qbb>Qbb</a>' in 2-d to 3-d, and
	'<a href=qhull.htm#outputs>qhull d</a> <a href=qh-optq.htm#Qbb>Qbb</a> <a href=qh-optq.htm#Qx>Qx</a>'
	in 4-d and higher. It disables the following Qhull
	<a href=qh-quick.htm#options>options</a>: <i>d n v H U Qb QB Qc Qf Qg Qi
	Qm Qr QR Qv Qx TR E V FC Fi Fo Fp Ft FV Q0,etc</i>.


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

	<hr>

	<h3><a href="#TOP">»</a><a name="synopsis">qdelaunay synopsis</a></h3>

	<pre>
	qdelaunay- compute the Delaunay triangulation.
	input (stdin): dimension, number of points, point coordinates
	comments start with a non-numeric character

	options (qdelaun.htm):
	Qt - triangulated output
	QJ - joggle input instead of merging facets
	Qu - furthest-site Delaunay triangulation
	Tv - verify result: structure, convexity, and in-circle test
	. - concise list of all options
	- - one-line description of all options

	output options (subset):
	s - summary of results (default)
	i - vertices incident to each Delaunay region
	Fx - extreme points (vertices of the convex hull)
	o - OFF format (shows the points lifted to a paraboloid)
	G - Geomview output (2-d and 3-d points lifted to a paraboloid)
	m - Mathematica output (2-d inputs lifted to a paraboloid)
	QVn - print Delaunay regions that include point n, -n if not
	TO file- output results to file, may be enclosed in single quotes

	examples:
	rbox c P0 D2 \| qdelaunay s o rbox c P0 D2 \| qdelaunay i
	rbox c P0 D3 \| qdelaunay Fv Qt rbox c P0 D2 \| qdelaunay s Qu Fv
	rbox c G1 d D2 \| qdelaunay s i rbox c G1 d D2 \| qdelaunay s i Qt
	rbox M3,4 z 100 D2 \| qdelaunay s rbox M3,4 z 100 D2 \| qdelaunay s Qt
	</pre>


	<h3><a href="#TOP">»</a><a name="input">qdelaunay
	input</a></h3>

	<blockquote>
	<p>The input data on <tt>stdin</tt> consists of:</p>
	<ul>
	<li>dimension
	<li>number of points</li>
	<li>point coordinates</li>
	</ul>

	<p>Use I/O redirection (e.g., qdelaunay < data.txt), a pipe (e.g., rbox 10 \| qdelaunay),
	or the '<a href=qh-optt.htm#TI>TI</a>' option (e.g., qdelaunay TI data.txt).

	<p>For example, this is four cocircular points inside a square. Its Delaunay
	triangulation contains 8 triangles and one four-sided
	figure.
	<p>
	<blockquote>
	<tt>rbox s 4 W0 c G1 D2 > data</tt>
	<blockquote><pre>
	2 RBOX s 4 W0 c D2
	8
	-0.4941988586954018 -0.07594397977563715
	-0.06448037284989526 0.4958248496365813
	0.4911154367094632 0.09383830681375946
	-0.348353580869097 -0.3586778257652367
	-1 -1
	-1 1
	1 -1
	1 1
	</pre></blockquote>

	<p><tt>qdelaunay s i < data</tt>
	<blockquote><pre>

	Delaunay triangulation by the convex hull of 8 points in 3-d

	Number of input sites: 8
	Number of Delaunay regions: 9
	Number of non-simplicial Delaunay regions: 1

	Statistics for: RBOX s 4 W0 c D2 \| QDELAUNAY s i

	Number of points processed: 8
	Number of hyperplanes created: 18
	Number of facets in hull: 10
	Number of distance tests for qhull: 33
	Number of merged facets: 2
	Number of distance tests for merging: 102
	CPU seconds to compute hull (after input): 0.028

	9
	1 7 5
	6 3 4
	2 3 6
	7 2 6
	2 7 1
	0 5 4
	3 0 4
	0 1 5
	1 0 3 2
	</pre></blockquote>
	</blockquote>

	</blockquote>
	<h3><a href="#TOP">»</a><a name="outputs">qdelaunay
	outputs</a></h3>
	<blockquote>

	<p>These options control the output of Delaunay triangulations:</p>
	<blockquote>

	<dl compact>
	<dd><b>Delaunay regions</b></dd>
	<dt><a href="qh-opto.htm#i">i</a></dt>
	<dd>list input sites for each Delaunay region. The first line is the number of regions. The
	remaining lines list the input sites for each region. The regions are
	oriented. In 3-d and
	higher, report cospherical sites by adding extra points. Use triangulated
	output ('<a href="qh-optq.htm#Qt">Qt</a>') to avoid non-simpicial regions. For the circle-in-square example,
	eight Delaunay regions are triangular and the ninth has four input sites.</dd>
	<dt><a href="qh-optf.htm#Fv">Fv</a></dt>
	<dd>list input sites for each Delaunay region. The first line is the number of regions.
	Each remaining line starts with the number of input sites. The regions
	are unoriented. For the circle-in-square example,
	eight Delaunay regions are triangular and the ninth has four input sites.</dd>
	<dt><a href="qh-optf.htm#Fn">Fn</a></dt>
	<dd>list neighboring regions for each Delaunay region. The first line is the
	number of regions. Each remaining line starts with the number of
	neighboring regions. Negative indices (e.g., <em>-1</em>) indicate regions
	outside of the Delaunay triangulation.
	For the circle-in-square example, the four regions on the square are neighbors to
	the region-at-infinity.</dd>
	<dt><a href="qh-optf.htm#FN">FN</a></dt>
	<dd>list the Delaunay regions for each input site. The first line is the
	total number of input sites. Each remaining line starts with the number of
	Delaunay regions. Negative indices (e.g., <em>-1</em>) indicate regions
	outside of the Delaunay triangulation.
	For the circle-in-square example, each point on the circle belongs to four
	Delaunay regions. Use '<a href="qh-optq.htm#Qc">Qc</a> FN'
	to include coincident input sites and deleted vertices. </dd>
	<dt><a href="qh-optf.htm#Fa">Fa</a></dt>
	<dd>print area for each Delaunay region. The first line is the number of regions.
	The areas follow, one line per region. For the circle-in-square example, the
	cocircular region has area 0.4. </dd>
	<dt> </dt>
	<dt> </dt>
	<dd><b>Input sites</b></dd>
	<dt><a href="qh-optf.htm#Fc">Fc</a></dt>
	<dd>list coincident input sites for each Delaunay region.
	The first line is the number of regions. The remaining lines start with
	the number of coincident sites and deleted vertices. Deleted vertices
	indicate highly degenerate input (see'<a href="qh-optf.htm#Fs">Fs</a>').
	A coincident site is assigned to one Delaunay
	region. Do not use '<a href="qh-optq.htm#QJn">QJ</a>' with 'Fc'; the joggle will separate
	coincident sites.</dd>
	<dt><a href="qh-optf.htm#FP">FP</a></dt>
	<dd>print coincident input sites with distance to
	nearest site (i.e., vertex). The first line is the
	number of coincident sites. Each remaining line starts with the point ID of
	an input site, followed by the point ID of a coincident point, its region, and distance.
	Includes deleted vertices which
	indicate highly degenerate input (see'<a href="qh-optf.htm#Fs">Fs</a>').
	Do not use '<a href="qh-optq.htm#QJn">QJ</a>' with 'FP'; the joggle will separate
	coincident sites.</dd>
	<dt><a href="qh-optf.htm#Fx">Fx</a></dt>
	<dd>list extreme points of the input sites. These points are on the
	boundary of the convex hull. The first line is the number of
	extreme points. Each point is listed, one per line. The circle-in-square example
	has four extreme points.</dd>
	<dt> </dt>
	<dt> </dt>
	<dd><b>General</b></dd>
	<dt><a href="qh-optf.htm#FA">FA</a></dt>
	<dd>compute total area for '<a href="qh-opto.htm#s">s</a>'
	and '<a href="qh-optf.htm#FS">FS</a>'</dd>
	<dt><a href="qh-opto.htm#o">o</a></dt>
	<dd>print lower facets of the corresponding convex hull (a
	paraboloid)</dd>
	<dt><a href="qh-opto.htm#m">m</a></dt>
	<dd>Mathematica output for the lower facets of the paraboloid (2-d triangulations).</dd>
	<dt><a href="qh-optf.htm#FM">FM</a></dt>
	<dd>Maple output for the lower facets of the paraboloid (2-d triangulations).</dd>
	<dt><a href="qh-optg.htm#G">G</a></dt>
	<dd>Geomview output for the paraboloid (2-d or 3-d triangulations).</dd>
	<dt><a href="qh-opto.htm#s">s</a></dt>
	<dd>print summary for the Delaunay triangulation. Use '<a
	href="qh-optf.htm#Fs">Fs</a>' and '<a
	href="qh-optf.htm#FS">FS</a>' for numeric data.</dd>
	</dl>
	</blockquote>

	</blockquote>
	<h3><a href="#TOP">»</a><a name="controls">qdelaunay
	controls</a></h3>
	<blockquote>

	<p>These options provide additional control:</p>
	<blockquote>

	<dl compact>
	<dt><a href="qh-optq.htm#Qt">Qt</a></dt>
	<dd>triangulated output. Qhull triangulates non-simplicial facets. It may produce
	degenerate facets of zero area.</dd>
	<dt><a href="qh-optq.htm#QJn">QJ</a></dt>
	<dd>joggle the input to avoid cospherical and coincident
	sites. It is less accurate than triangulated output ('Qt').</dd>
	<dt><a href="qh-optq.htm#Qu">Qu</a></dt>
	<dd>compute the <a href="qdelau_f.htm">furthest-site Delaunay triangulation</a>.</dd>
	<dt><a href="qh-optq.htm#Qz">Qz</a></dt>
	<dd>add a point above the paraboloid to reduce precision
	errors. Use it for nearly cocircular/cospherical input
	(e.g., 'rbox c \| qdelaunay Qz'). The point is printed for
	options '<a href="qh-optf.htm#Ft">Ft</a>' and '<a
	href="qh-opto.htm#o">o</a>'.</dd>
	<dt><a href="qh-optq.htm#QVn">QVn</a></dt>
	<dd>select facets adjacent to input site <em>n</em> (marked
	'good').</dd>
	<dt><a href="qh-optt.htm#Tv">Tv</a></dt>
	<dd>verify result.</dd>
	<dt><a href="qh-optt.htm#TO">TI file</a></dt>
	<dd>input data from file. The filename may not use spaces or quotes.</dd>
	<dt><a href="qh-optt.htm#TO">TO file</a></dt>
	<dd>output results to file. Use single quotes if the filename
	contains spaces (e.g., <tt>TO 'file with spaces.txt'</tt></dd>
	<dt><a href="qh-optt.htm#TFn">TFn</a></dt>
	<dd>report progress after constructing <em>n</em> facets</dd>
	<dt><a href="qh-optp.htm#PDk">PDk:1</a></dt>
	<dd>include upper and lower facets in the output. Set <em>k</em>
	to the last dimension (e.g., 'PD2:1' for 2-d inputs). </dd>
	<dt><a href="qh-opto.htm#f">f</a></dt>
	<dd>facet dump. Print the data structure for each facet (i.e., Delaunay region).</dd>
	</dl>
	</blockquote>

	</blockquote>
	<h3><a href="#TOP">»</a><a name="graphics">qdelaunay
	graphics</a></h3>
	<blockquote>

	<p>For 2-d and 3-d Delaunay triangulations, Geomview ('qdelaunay <a
	href="qh-optg.htm#G">G</a>') displays the corresponding convex
	hull (a paraboloid). </p>

	<p>To view a 2-d Delaunay triangulation, use 'qdelaunay <a
	href="qh-optg.htm#GDn">GrD2</a>' to drop the last dimension. This
	is the same as viewing the hull without perspective (see
	Geomview's 'cameras' menu). </p>

	<p>To view a 3-d Delaunay triangulation, use 'qdelaunay <a
	href="qh-optg.htm#GDn">GrD3</a>' to drop the last dimension. You
	may see extra edges. These are interior edges that Geomview moves
	towards the viewer (see 'lines closer' in Geomview's camera
	options). Use option '<a href="qh-optg.htm#Gt">Gt</a>' to make
	the outer ridges transparent in 3-d. See <a
	href="qh-eg.htm#delaunay">Delaunay and Voronoi examples</a>.</p>

	<p>For 2-d Delaunay triangulations, Mathematica ('<a
	href="qh-opto.htm#m">m</a>') and Maple ('<a
	href="qh-optf.htm#FM">FM</a>') output displays the lower facets of the corresponding convex
	hull (a paraboloid). </p>

	<p>For 2-d, furthest-site Delaunay triangulations, Maple and Mathematica output ('<a
	href="qh-optq.htm#Qu">Qu</a> <a
	href="qh-opto.htm#m">m</a>') displays the upper facets of the corresponding convex
	hull (a paraboloid). </p>

	</blockquote>
	<h3><a href="#TOP">»</a><a name="notes">qdelaunay
	notes</a></h3>
	<blockquote>

	<p>You can simplify the Delaunay triangulation by enclosing the input
	sites in a large square or cube. This is particularly recommended
	for cocircular or cospherical input data.

	<p>A non-simplicial Delaunay region indicates nearly cocircular or
	cospherical input sites. To avoid non-simplicial regions either triangulate
	the output ('<a href="qh-optq.htm#Qt">Qt</a>') or joggle
	the input ('<a href="qh-optq.htm#QJn">QJ</a>'). Triangulated output
	is more accurate than joggled input. Alternatively, use an <a
	href="qh-impre.htm#exact">exact arithmetic code</a>.</p>

	<p>Delaunay triangulations do not include facets that are
	coplanar with the convex hull of the input sites. A facet is
	coplanar if the last coefficient of its normal is
	nearly zero (see <a href="../src/libqhull/user.h#ZEROdelaunay">qh_ZEROdelaunay</a>).

	<p>See <a href=qh-impre.htm#delaunay>Imprecision issues :: Delaunay triangulations</a>
	for a discussion of precision issues. Deleted vertices indicate
	highly degenerate input. They are listed in the summary output and
	option '<a href="qh-optf.htm#Fs">Fs</a>'.</p>

	<p>To compute the Delaunay triangulation of points on a sphere,
	compute their convex hull. If the sphere is the unit sphere at
	the origin, the facet normals are the Voronoi vertices of the
	input. The points may be restricted to a hemisphere. [S. Fortune]
	</p>

	<p>The 3-d Delaunay triangulation of regular points on a half
	spiral (e.g., 'rbox 100 l \| qdelaunay') has quadratic size, while the Delaunay triangulation
	of random 3-d points is
	approximately linear for reasonably sized point sets.

	<p>With the <a href="qh-code.htm#library">Qhull library</a>, you
	can use <tt>qh_findbestfacet</tt> in <tt>poly2.c</tt> to locate the facet
	that contains a point. You should first lift the point to the
	paraboloid (i.e., the last coordinate is the sum of the squares
	of the point's coordinates -- <tt>qh_setdelaunay</tt>). Do not use options
	'<a href="qh-optq.htm#Qbb">Qbb</a>', '<a href="qh-optq.htm#QbB">QbB</a>',
	'<a href="qh-optq.htm#Qbk">Qbk:n</a>', or '<a
	href="qh-optq.htm#QBk">QBk:n</a>' since these scale the last
	coordinate. </p>

	<p>If a point is interior to the convex hull of the input set, it
	is interior to the adjacent vertices of the Delaunay
	triangulation. This is demonstrated by the following pipe for
	point 0:

	<pre>
	qdelaunay <data s FQ QV0 p \| qconvex s Qb3:0B3:0 p
	</pre>

	<p>The first call to qdelaunay returns the neighboring points of
	point 0 in the Delaunay triangulation. The second call to qconvex
	returns the vertices of the convex hull of these points (after
	dropping the lifted coordinate). If point 0 is interior to the
	original point set, it is interior to the reduced point set. </p>

	</blockquote>
	<h3><a href="#TOP">»</a><a name="conventions">qdelaunay conventions</a></h3>
	<blockquote>

	<p>The following terminology is used for Delaunay triangulations
	in Qhull for dimension <i>d</i>. The underlying structure is the
	lower facets of a convex hull in dimension <i>d+1</i>. For
	further information, see <a href="index.htm#structure">data
	structures</a> and <a href="qconvex.htm#conventions">convex hull
	conventions</a>.</p>
	<blockquote>
	<ul>
	<li><em>input site</em> - a point in the input (one dimension
	lower than a point on the convex hull)</li>
	<li><em>point</em> - a point has <i>d+1</i> coordinates. The
	last coordinate is the sum of the squares of the input
	site's coordinates</li>
	<li><em>coplanar point</em> - a <em>coincident</em>
	input site or a deleted vertex. Deleted vertices
	indicate highly degenerate input.</li>
	<li><em>vertex</em> - a point on the paraboloid. It
	corresponds to a unique input site. </li>
	<li><em>point-at-infinity</em> - a point added above the
	paraboloid by option '<a href="qh-optq.htm#Qz">Qz</a>'</li>
	<li><em>lower facet</em> - a facet corresponding to a
	Delaunay region. The last coefficient of its normal is
	clearly negative.</li>
	<li><em>upper facet</em> - a facet corresponding to a
	furthest-site Delaunay region. The last coefficient of
	its normal is clearly positive. </li>
	<li><em>Delaunay region</em> - a
	lower facet projected to the input sites</li>
	<li><em>upper Delaunay region</em> - an upper facet projected
	to the input sites</li>
	<li><em>non-simplicial facet</em> - more than <em>d</em>
	input sites are cocircular or cospherical</li>
	<li><em>good facet</em> - a Delaunay region with optional
	restrictions by '<a href="qh-optq.htm#QVn">QVn</a>', etc.</li>
	</ul>
	</blockquote>
	</blockquote>
	<h3><a href="#TOP">»</a><a name="options">qdelaunay options</a></h3>

	<pre>
	qdelaunay- compute the Delaunay triangulation
	http://www.qhull.org

	input (stdin):
	first lines: dimension and number of points (or vice-versa).
	other lines: point coordinates, best if one point per line
	comments: start with a non-numeric character

	options:
	Qt - triangulated output
	QJ - joggle input instead of merging facets
	Qu - compute furthest-site Delaunay triangulation

	Qhull control options:
	QJn - randomly joggle input in range [-n,n]
	Qs - search all points for the initial simplex
	Qz - add point-at-infinity to Delaunay triangulation
	QGn - print Delaunay region if visible from point n, -n if not
	QVn - print Delaunay regions that include point n, -n if not

	Trace options:
	T4 - trace at level n, 4=all, 5=mem/gauss, -1= events
	Tc - check frequently during execution
	Ts - print statistics
	Tv - verify result: structure, convexity, and in-circle test
	Tz - send all output to stdout
	TFn - report summary when n or more facets created
	TI file - input data from file, no spaces or single quotes
	TO file - output results to file, may be enclosed in single quotes
	TPn - turn on tracing when point n added to hull
	TMn - turn on tracing at merge n
	TWn - trace merge facets when width > n
	TVn - stop qhull after adding point n, -n for before (see TCn)
	TCn - stop qhull after building cone for point n (see TVn)

	Precision options:
	Cn - radius of centrum (roundoff added). Merge facets if non-convex
	An - cosine of maximum angle. Merge facets if cosine > n or non-convex
	C-0 roundoff, A-0.99/C-0.01 pre-merge, A0.99/C0.01 post-merge
	Rn - randomly perturb computations by a factor of [1-n,1+n]
	Wn - min facet width for outside point (before roundoff)

	Output formats (may be combined; if none, produces a summary to stdout):
	f - facet dump
	G - Geomview output (see below)
	i - vertices incident to each Delaunay region
	m - Mathematica output (2-d only, lifted to a paraboloid)
	o - OFF format (dim, points, and facets as a paraboloid)
	p - point coordinates (lifted to a paraboloid)
	s - summary (stderr)

	More formats:
	Fa - area for each Delaunay region
	FA - compute total area for option 's'
	Fc - count plus coincident points for each Delaunay region
	Fd - use cdd format for input (homogeneous with offset first)
	FD - use cdd format for numeric output (offset first)
	FF - facet dump without ridges
	FI - ID of each Delaunay region
	Fm - merge count for each Delaunay region (511 max)
	FM - Maple output (2-d only, lifted to a paraboloid)
	Fn - count plus neighboring region for each Delaunay region
	FN - count plus neighboring region for each point
	FO - options and precision constants
	FP - nearest point and distance for each coincident point
	FQ - command used for qdelaunay
	Fs - summary: #int (8), dimension, #points, tot vertices, tot facets,
	for output: #vertices, #Delaunay regions,
	#coincident points, #non-simplicial regions
	#real (2), max outer plane, min vertex
	FS - sizes: #int (0)
	#real (2), tot area, 0
	Fv - count plus vertices for each Delaunay region
	Fx - extreme points of Delaunay triangulation (on convex hull)

	Geomview options (2-d and 3-d)
	Ga - all points as dots
	Gp - coplanar points and vertices as radii
	Gv - vertices as spheres
	Gi - inner planes only
	Gn - no planes
	Go - outer planes only
	Gc - centrums
	Gh - hyperplane intersections
	Gr - ridges
	GDn - drop dimension n in 3-d and 4-d output
	Gt - transparent outer ridges to view 3-d Delaunay

	Print options:
	PAn - keep n largest Delaunay regions by area
	Pdk:n - drop facet if normal[k] <= n (default 0.0)
	PDk:n - drop facet if normal[k] >= n
	Pg - print good Delaunay regions (needs 'QGn' or 'QVn')
	PFn - keep Delaunay regions whose area is at least n
	PG - print neighbors of good regions (needs 'QGn' or 'QVn')
	PMn - keep n Delaunay regions with most merges
	Po - force output. If error, output neighborhood of facet
	Pp - do not report precision problems

	. - list of all options
	- - one line descriptions of all options
	</pre>

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