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diff --git a/src/qhull/html/qdelaun.htm b/src/qhull/html/qdelaun.htm new file mode 100644 index 000000000..a42223c66 --- /dev/null +++ b/src/qhull/html/qdelaun.htm @@ -0,0 +1,628 @@ +<!DOCTYPE HTML PUBLIC "-//IETF//DTD HTML//EN"> +<html> + +<head> +<title>qdelaunay -- Delaunay triangulation</title> +</head> + +<body> +<!-- Navigation links --> +<a name="TOP"><b>Up</b></a><b>:</b> +<a href="http://www.qhull.org">Home page</a> for Qhull<br> +<b>Up:</b> <a href="index.htm#TOC">Qhull manual</a>: Table of Contents<br> +<b>To:</b> <a href="qh-quick.htm#programs">Programs</a> +• <a href="qh-quick.htm#options">Options</a> +• <a href="qh-opto.htm#output">Output</a> +• <a href="qh-optf.htm#format">Formats</a> +• <a href="qh-optg.htm#geomview">Geomview</a> +• <a href="qh-optp.htm#print">Print</a> +• <a href="qh-optq.htm#qhull">Qhull</a> +• <a href="qh-optc.htm#prec">Precision</a> +• <a href="qh-optt.htm#trace">Trace</a> +• <a href="../src/libqhull_r/index.htm">Functions</a><br> +<b>To:</b> <a href="#synopsis">sy</a>nopsis +• <a href="#input">in</a>put • <a href="#outputs">ou</a>tputs +• <a href="#controls">co</a>ntrols • <a href="#graphics">gr</a>aphics +• <a href="#notes">no</a>tes • <a href="#conventions">co</a>nventions +• <a href="#options">op</a>tions + +<hr> +<!-- Main text of document --> +<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> + +<!-- Navigation links --> +<hr> + +<p><b>Up:</b> <a href="http://www.qhull.org">Home page</a> for Qhull<br> +<b>Up:</b> <a href="index.htm#TOC">Qhull manual</a>: Table of Contents<br> +<b>To:</b> <a href="qh-quick.htm#programs">Programs</a> +• <a href="qh-quick.htm#options">Options</a> +• <a href="qh-opto.htm#output">Output</a> +• <a href="qh-optf.htm#format">Formats</a> +• <a href="qh-optg.htm#geomview">Geomview</a> +• <a href="qh-optp.htm#print">Print</a> +• <a href="qh-optq.htm#qhull">Qhull</a> +• <a href="qh-optc.htm#prec">Precision</a> +• <a href="qh-optt.htm#trace">Trace</a> +• <a href="../src/libqhull_r/index.htm">Functions</a><br> +<b>To:</b> <a href="#synopsis">sy</a>nopsis +• <a href="#input">in</a>put • <a href="#outputs">ou</a>tputs +• <a href="#controls">co</a>ntrols • <a href="#graphics">gr</a>aphics +• <a href="#notes">no</a>tes • <a href="#conventions">co</a>nventions +• <a href="#options">op</a>tions +<!-- GC common information --> +<hr> + +<p><a href="http://www.geom.uiuc.edu/"><img src="qh--geom.gif" +align="middle" width="40" height="40"></a><i>The Geometry Center +Home Page </i></p> + +<p>Comments to: <a href=mailto:qhull@qhull.org>qhull@qhull.org</a> +</a><br> +Created: Sept. 25, 1995 --- <!-- hhmts start --> Last modified: see top <!-- hhmts end --> </p> +</body> +</html> |