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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>&nbsp;</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>&nbsp;</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 &copy; 1995-2012 C.B. Barber</b></p>
<hr>
<h3><a href="#TOP">&#187;</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">&#187;</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 &lt; 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 &gt; 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 &lt; 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">&#187;</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>&nbsp;</dt>
<dt>&nbsp;</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>&nbsp;</dt>
<dt>&nbsp;</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">&#187;</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">&#187;</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">&#187;</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 &lt;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">&#187;</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">&#187;</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] &lt;= 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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