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diff --git a/src/qhull/html/qvoronoi.htm b/src/qhull/html/qvoronoi.htm new file mode 100644 index 000000000..6d81d48c1 --- /dev/null +++ b/src/qhull/html/qvoronoi.htm @@ -0,0 +1,667 @@ +<!DOCTYPE HTML PUBLIC "-//IETF//DTD HTML//EN"> +<html> + +<head> +<title>qvoronoi -- Voronoi diagram</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.archinect.com/gallery/displayimage.php?pos=-4658"><img +src="normal_voronoi_knauss_oesterle.jpg" alt="[voronoi]" align="middle" +height="100"></a>qvoronoi -- Voronoi diagram</h1> + +<p>The Voronoi diagram is the nearest-neighbor map for a set of +points. Each region contains those points that are nearer +one input site than any other input site. 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>]. The Voronoi diagram is the +dual of the <a href=qdelaun.htm>Delaunay triangulation</a>. </p> + +<blockquote> +<dl> + <dt><b>Example:</b> rbox 10 D3 | qvoronoi <a href="qh-opto.htm#s">s</a> + <a href="qh-opto.htm#o">o</a> <a href="qh-optt.htm#TO">TO + result</a></dt> + <dd>Compute the 3-d Voronoi diagram of 10 random points. Write a + summary to the console and the Voronoi vertices and + regions to 'result'. The first vertex of the result + indicates unbounded regions.</dd> + + <dt> </dt> + <dt><b>Example:</b> rbox r y c G0.1 D2 | qvoronoi + <a href="qh-opto.htm#s">s</a> + <a href="qh-opto.htm#o">o</a> <a href="qh-optt.htm#TO">TO + result</a></dt> + <dd>Compute the 2-d Voronoi diagram of a triangle and a small + square. Write a + summary to the console and Voronoi vertices and regions + to 'result'. Report a single Voronoi vertex for + cocircular input sites. The first vertex of the result + indicates unbounded regions. The origin is the Voronoi + vertex for the square.</dd> + + <dt> </dt> + <dt><b>Example:</b> rbox r y c G0.1 D2 | qvoronoi <a href="qh-optf.htm#Fv2">Fv</a> + <a href="qh-optt.htm#TO">TO result</a></dt> + <dd>Compute the 2-d Voronoi diagram of a triangle and a small + square. Write a + summary to the console and the Voronoi ridges to + 'result'. Each ridge is the perpendicular bisector of a + pair of input sites. Vertex "0" indicates + unbounded ridges. Vertex "8" is the Voronoi + vertex for the square.</dd> + + <dt> </dt> + <dt><b>Example:</b> rbox r y c G0.1 D2 | qvoronoi <a href="qh-optf.htm#Fi2">Fi</a></dt> + <dd>Print the bounded, separating hyperplanes for the 2-d Voronoi diagram of a + triangle and a small + square. Note the four hyperplanes (i.e., lines) for Voronoi vertex + "8". It is at the origin. + </dd> +</dl> +</blockquote> + +<p>Qhull computes the Voronoi diagram via the <a href="qdelaun.htm">Delaunay +triangulation</a>. Each Voronoi +vertex is the circumcenter of a facet of the Delaunay +triangulation. Each Voronoi region corresponds to a vertex (i.e., input site) of the +Delaunay triangulation. </p> + +<p>Qhull outputs the Voronoi vertices for each Voronoi region. With +option '<a href="qh-optf.htm#Fv2">Fv</a>', +it lists all ridges of the Voronoi diagram with the corresponding +pairs of input sites. With +options '<a href="qh-optf.htm#Fi2">Fi</a>' and '<a href="qh-optf.htm#Fo2">Fo</a>', +it lists the bounded and unbounded separating hyperplanes. +You can also output a single Voronoi region +for further processing [see <a href="#graphics">graphics</a>].</p> + +<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>See <a href="http://www.qhull.org/html/qh-faq.htm#TOC">Qhull FAQ</a> - Delaunay and +Voronoi diagram questions.</p> + +<p>The 'qvonoroi' program is equivalent to +'<a href=qhull.htm#outputs>qhull v</a> <a href=qh-optq.htm#Qbb>Qbb</a>' in 2-d to 3-d, and +'<a href=qhull.htm#outputs>qhull v</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 Qbb QbB Qf Qg Qm +Qr QR Qv Qx Qz TR E V Fa FA FC FD FS Ft FV Gt Q0,etc</i>. + +<p><b>Copyright © 1995-2015 C.B. Barber</b></p> + +<p>Voronoi image by KOOK Architecture, Silvan Oesterle and Michael Knauss. + +<hr> +<h3><a href="#TOP">»</a><a name="synopsis">qvoronoi synopsis</a></h3> + +<pre> +qvoronoi- compute the Voronoi diagram. + input (stdin): dimension, number of points, point coordinates + comments start with a non-numeric character + +options (qh-voron.htm): + Qu - compute furthest-site Voronoi diagram + 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) + p - Voronoi vertices + o - OFF file format (dim, Voronoi vertices, and Voronoi regions) + FN - count and Voronoi vertices for each Voronoi region + Fv - Voronoi diagram as Voronoi vertices between adjacent input sites + Fi - separating hyperplanes for bounded regions, 'Fo' for unbounded + G - Geomview output (2-d only) + QVn - Voronoi vertices for input point n, -n if not + TO file- output results to file, may be enclosed in single quotes + +examples: +rbox c P0 D2 | qvoronoi s o rbox c P0 D2 | qvoronoi Fi +rbox c P0 D2 | qvoronoi Fo rbox c P0 D2 | qvoronoi Fv +rbox c P0 D2 | qvoronoi s Qu Fv rbox c P0 D2 | qvoronoi Qu Fo +rbox c G1 d D2 | qvoronoi s p rbox c P0 D2 | qvoronoi s Fv QV0 +</pre> + +<h3><a href="#TOP">»</a><a name="input">qvoronoi input</a></h3> +<blockquote> +The input data on <tt>stdin</tt> consists of: +<ul> + <li>dimension + <li>number of points</li> + <li>point coordinates</li> +</ul> + +<p>Use I/O redirection (e.g., qvoronoi < data.txt), a pipe (e.g., rbox 10 | qvoronoi), +or the '<a href=qh-optt.htm#TI>TI</a>' option (e.g., qvoronoi TI data.txt). + +<p>For example, this is four cocircular points inside a square. Their Voronoi +diagram has nine vertices and eight regions. Notice the Voronoi vertex +at the origin, and the Voronoi vertices (on each axis) for the four +sides of the square. +<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>qvoronoi s p < data</tt> +<blockquote><pre> + +Voronoi diagram by the convex hull of 8 points in 3-d: + + Number of Voronoi regions: 8 + Number of Voronoi vertices: 9 + Number of non-simplicial Voronoi vertices: 1 + +Statistics for: RBOX s 4 W0 c D2 | QVORONOI s p + + 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): 0.094 + +2 +9 +4.217546450968612e-17 1.735507986399734 +-8.402566836762659e-17 -1.364368854147395 +0.3447488772716865 -0.6395484723719818 +1.719446929853986 2.136555906154247e-17 +0.4967882915039657 0.68662371396699 +-1.729928876283549 1.343733067524222e-17 +-0.8906163241424728 -0.4594150543829102 +-0.6656840313875723 0.5003013793414868 +-7.318364664277155e-19 -1.188217818408333e-16 +</pre></blockquote> +</blockquote> + +</blockquote> +<h3><a href="#TOP">»</a> <a name="outputs">qvoronoi +outputs</a></h3> +<blockquote> + +<p>These options control the output of Voronoi diagrams.</p> +<blockquote> + +<dl compact> + <dt> </dt> + <dd><b>Voronoi vertices</b></dd> + <dt><a href="qh-opto.htm#p">p</a></dt> + <dd>print the coordinates of the Voronoi vertices. The first line + is the dimension. The second line is the number of vertices. Each + remaining line is a Voronoi vertex.</dd> + <dt><a href="qh-optf.htm#Fn">Fn</a></dt> + <dd>list the neighboring Voronoi vertices for each Voronoi + vertex. The first line is the number of Voronoi vertices. Each + remaining line starts with the number of neighboring vertices. + Negative vertices (e.g., <em>-1</em>) indicate vertices + outside of the Voronoi diagram. + In the circle-in-box example, the + Voronoi vertex at the origin has four neighbors.</dd> + <dt><a href="qh-optf.htm#FN">FN</a></dt> + <dd>list the Voronoi vertices for each Voronoi region. The first line is + the number of Voronoi regions. Each remaining line starts with the + number of Voronoi vertices. Negative indices (e.g., <em>-1</em>) indicate vertices + outside of the Voronoi diagram. + In the circle-in-box example, the four bounded regions are defined by four + Voronoi vertices.</dd> + + <dt> </dt> + <dt> </dt> + <dd><b>Voronoi regions</b></dd> + <dt><a href="qh-opto.htm#o">o</a></dt> + <dd>print the Voronoi regions in OFF format. The first line is the + dimension. The second line is the number of vertices, the number + of input sites, and "1". The third line represents the vertex-at-infinity. + Its coordinates are "-10.101". The next lines are the coordinates + of the Voronoi vertices. Each remaining line starts with the number + of Voronoi vertices in a Voronoi region. In 2-d, the vertices are +listed in adjacency order (unoriented). In 3-d and higher, the +vertices are listed in numeric order. In the circle-in-square + example, each bounded region includes the Voronoi vertex at + the origin. Lines consisting of <em>0</em> indicate + coplanar input sites or '<a href="qh-optq.htm#Qz">Qz</a>'. </dd> + <dt><a href="qh-optf.htm#Fi2">Fi</a></dt> + <dd>print separating hyperplanes for inner, bounded Voronoi + regions. The first number is the number of separating + hyperplanes. Each remaining line starts with <i>3+dim</i>. The + next two numbers are adjacent input sites. The next <i>dim</i> + numbers are the coefficients of the separating hyperplane. The + last number is its offset. Use '<a href="qh-optt.htm#Tv">Tv</a>' to verify that the +hyperplanes are perpendicular bisectors. It will list relevant +statistics to stderr. </dd> + <dt><a href="qh-optf.htm#Fo2">Fo</a></dt> + <dd>print separating hyperplanes for outer, unbounded Voronoi + regions. The first number is the number of separating + hyperplanes. Each remaining line starts with <i>3+dim</i>. The + next two numbers are adjacent input sites on the convex hull. The + next <i>dim</i> + numbers are the coefficients of the separating hyperplane. The + last number is its offset. Use '<a href="qh-optt.htm#Tv">Tv</a>' to verify that the +hyperplanes are perpendicular bisectors. It will list relevant +statistics to stderr,</dd> + <dt> </dt> + <dt> </dt> + <dd><b>Input sites</b></dd> + <dt><a href="qh-optf.htm#Fv2">Fv</a></dt> + <dd>list ridges of Voronoi vertices for pairs of input sites. The + first line is the number of ridges. Each remaining line starts with + two plus the number of Voronoi vertices in the ridge. The next + two numbers are two adjacent input sites. The remaining numbers list + the Voronoi vertices. As with option 'o', a <em>0</em> indicates + the vertex-at-infinity + and an unbounded, separating hyperplane. + The perpendicular bisector (separating hyperplane) + of the input sites is a flat through these vertices. + In the circle-in-square example, the ridge for each edge of the square + is unbounded.</dd> + <dt><a href="qh-optf.htm#Fc">Fc</a></dt> + <dd>list coincident input sites for each Voronoi vertex. + The first line is the number of vertices. 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 Voronoi + vertex. 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 vertex, 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> </dt> + <dt> </dt> + <dd><b>General</b></dd> + <dt><a href="qh-opto.htm#s">s</a></dt> + <dd>print summary of the Voronoi diagram. Use '<a + href="qh-optf.htm#Fs">Fs</a>' for numeric data.</dd> + <dt><a href="qh-opto.htm#i">i</a></dt> + <dd>list input sites for each <a href=qdelaun.htm>Delaunay region</a>. Use option '<a href="qh-optp.htm#Pp">Pp</a>' + to avoid the warning. The first line is the number of regions. The + remaining lines list the input sites for each region. The regions are + oriented. In the circle-in-square example, the cocircular region has four + edges. In 3-d and higher, report cospherical sites by adding extra points. + </dd> + <dt><a href="qh-optg.htm#G">G</a></dt> + <dd>Geomview output for 2-d Voronoi diagrams.</dd> + </dl> +</blockquote> +</blockquote> +<h3><a href="#TOP">»</a> <a name="controls">qvoronoi +controls</a></h3> +<blockquote> + +<p>These options provide additional control:</p> +<blockquote> + +<dl compact> + <dt><a href="qh-optq.htm#Qu">Qu</a></dt> + <dd>compute the <a href="qvoron_f.htm">furthest-site Voronoi diagram</a>.</dd> + <dt><a href="qh-optq.htm#QVn">QVn</a></dt> + <dd>select Voronoi vertices for input site <em>n</em> </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 | qvoronoi Qz').</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., + Voronoi vertex).</dd> +</dl> + +</blockquote> +</blockquote> +<h3><a href="#TOP">»</a> <a name="graphics">qvoronoi +graphics</a></h3> +<blockquote> + +<p>In 2-d, Geomview output ('<a href="qh-optg.htm#G">G</a>') +displays a Voronoi diagram with extra edges to close the +unbounded Voronoi regions. To view the unbounded rays, enclose +the input points in a square.</p> + +<p>You can also view <i>individual</i> Voronoi regions in 3-d. To +view the Voronoi region for site 3 in Geomview, execute</p> + +<blockquote> + <p>qvoronoi <data <a href="qh-optq.htm#QVn">QV3</a> <a + href="qh-opto.htm#p">p</a> | qconvex s G >output</p> +</blockquote> + +<p>The <tt>qvoronoi</tt> command returns the Voronoi vertices +for input site 3. The <tt>qconvex</tt> command computes their convex hull. +This is the Voronoi region for input site 3. Its +hyperplane normals (qconvex 'n') are the same as the separating hyperplanes +from options '<a href="qh-optf.htm#Fi">Fi</a>' +and '<a href="qh-optf.htm#Fo">Fo</a>' (up to roundoff error). + +<p>See the <a href="qh-eg.htm#delaunay">Delaunay and Voronoi +examples</a> for 2-d and 3-d examples. Turn off normalization (on +Geomview's 'obscure' menu) when comparing the Voronoi diagram +with the corresponding Delaunay triangulation. </p> + +</blockquote> +<h3><a href="#TOP">»</a><a name="notes">qvoronoi +notes</a></h3> +<blockquote> + +<p>You can simplify the Voronoi diagram by enclosing the input +sites in a large square or cube. This is particularly recommended +for cocircular or cospherical input data.</p> + +<p>See <a href="#graphics">Voronoi graphics</a> for computing +the convex hull of a Voronoi region. </p> + +<p>Voronoi diagrams 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/user.h#ZEROdelaunay">qh_ZEROdelaunay</a>). + +<p>Unbounded regions can be confusing. For example, '<tt>rbox c | +qvoronoi Qz o</tt>' produces the Voronoi regions for the vertices +of a cube centered at the origin. All regions are unbounded. The +output is </p> + +<blockquote> + <pre>3 +2 9 1 +-10.101 -10.101 -10.101 + 0 0 0 +2 0 1 +2 0 1 +2 0 1 +2 0 1 +2 0 1 +2 0 1 +2 0 1 +2 0 1 +0 +</pre> +</blockquote> + +<p>The first line is the dimension. The second line is the number +of vertices and the number of regions. There is one region per +input point plus a region for the point-at-infinity added by +option '<a href="qh-optq.htm#Qz">Qz</a>'. The next two lines +lists the Voronoi vertices. The first vertex is the infinity +vertex. It is indicate by the coordinates <em>-10.101</em>. The +second vertex is the origin. The next nine lines list the +regions. Each region lists two vertices -- the infinity vertex +and the origin. The last line is "0" because no region +is associated with the point-at-infinity. A "0" would +also be listed for nearly incident input sites. </p> + +<p>To use option '<a href="qh-optf.htm#Fv">Fv</a>', add an +interior point. For example, </p> + +<blockquote> + <pre> +rbox c P0 | qvoronoi Fv +20 +5 0 7 1 3 5 +5 0 3 1 4 5 +5 0 5 1 2 3 +5 0 1 1 2 4 +5 0 6 2 3 6 +5 0 2 2 4 6 +5 0 4 4 5 6 +5 0 8 5 3 6 +5 1 2 0 2 4 +5 1 3 0 1 4 +5 1 5 0 1 2 +5 2 4 0 4 6 +5 2 6 0 2 6 +5 3 4 0 4 5 +5 3 7 0 1 5 +5 4 8 0 6 5 +5 5 6 0 2 3 +5 5 7 0 1 3 +5 6 8 0 6 3 +5 7 8 0 3 5 +</pre> +</blockquote> + +<p>The output consists of 20 ridges and each ridge lists a pair +of input sites and a triplet of Voronoi vertices. The first eight +ridges connect the origin ('P0'). The remainder list the edges of +the cube. Each edge generates an unbounded ray through the +midpoint. The corresponding separating planes ('Fo') follow each +pair of coordinate axes. </p> + +<p>Options '<a href="qh-optq.htm#Qt">Qt</a>' (triangulated output) +and '<a href="qh-optq.htm#QJn">QJ</a>' (joggled input) are deprecated. They may produce +unexpected results. If you use these options, cocircular and cospherical input sites will +produce duplicate or nearly duplicate Voronoi vertices. See also <a +href="qh-impre.htm#joggle">Merged facets or joggled input</a>. </p> + +</blockquote> +<h3><a href="#TOP">»</a><a name="conventions">qvoronoi conventions</a></h3> +<blockquote> + +<p>The following terminology is used for Voronoi diagrams in +Qhull. The underlying structure is a Delaunay triangulation from +a convex hull in one higher dimension. Facets of the Delaunay +triangulation correspond to vertices of the Voronoi diagram. +Vertices of the Delaunay triangulation correspond to input sites. +They also correspond to regions of the Voronoi diagram. See <a +href="qconvex.htm#conventions">convex hull conventions</a>, <a +href="qdelaun.htm#conventions">Delaunay conventions</a>, and +<a href="index.htm#structure">Qhull's data structures</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>nearly incident</em> + input site</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>Delaunay facet</em> - a lower facet of the + paraboloid. The last coefficient of its normal is + clearly negative.</li> + <li><em>Voronoi vertex</em> - the circumcenter of a Delaunay + facet</li> + <li><em>Voronoi region</em> - the Voronoi vertices for an + input site. The region of Euclidean space nearest to an + input site.</li> + <li><em>Voronoi diagram</em> - the graph of the Voronoi + regions. It includes the ridges (i.e., edges) between the + regions.</li> + <li><em>vertex-at-infinity</em> - the Voronoi vertex that + indicates unbounded Voronoi regions in '<a + href="qh-opto.htm#o">o</a>' output format. Its + coordinates are <em>-10.101</em>.</li> + <li><em>good facet</em> - a Voronoi vertex 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">qvoronoi options</a></h3> + +<pre> +qvoronoi- compute the Voronoi diagram + 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: + Qu - compute furthest-site Voronoi diagram + +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 Voronoi diagram + QGn - Voronoi vertices if visible from point n, -n if not + QVn - Voronoi vertices for input 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 - 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 non-coincident point (before roundoff) + +Output formats (may be combined; if none, produces a summary to stdout): + s - summary to stderr + p - Voronoi vertices + o - OFF format (dim, Voronoi vertices, and Voronoi regions) + i - Delaunay regions (use 'Pp' to avoid warning) + f - facet dump + +More formats: + Fc - count plus coincident points (by Voronoi vertex) + Fd - use cdd format for input (homogeneous with offset first) + FD - use cdd format for output (offset first) + FF - facet dump without ridges + Fi - separating hyperplanes for bounded Voronoi regions + FI - ID for each Voronoi vertex + Fm - merge count for each Voronoi vertex (511 max) + Fn - count plus neighboring Voronoi vertices for each Voronoi vertex + FN - count and Voronoi vertices for each Voronoi region + Fo - separating hyperplanes for unbounded Voronoi regions + FO - options and precision constants + FP - nearest point and distance for each coincident point + FQ - command used for qvoronoi + Fs - summary: #int (8), dimension, #points, tot vertices, tot facets, + for output: #Voronoi regions, #Voronoi vertices, + #coincident points, #non-simplicial regions + #real (2), max outer plane and min vertex + Fv - Voronoi diagram as Voronoi vertices between adjacent input sites + Fx - extreme points of Delaunay triangulation (on convex hull) + +Geomview options (2-d only) + 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 + +Print options: + PAn - keep n largest Voronoi vertices by 'area' + Pdk:n - drop facet if normal[k] <= n (default 0.0) + PDk:n - drop facet if normal[k] >= n + Pg - print good Voronoi vertices (needs 'QGn' or 'QVn') + PFn - keep Voronoi vertices whose 'area' is at least n + PG - print neighbors of good Voronoi vertices + PMn - keep n Voronoi vertices 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> |