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diff --git a/src/qhull/html/qh-optf.htm b/src/qhull/html/qh-optf.htm new file mode 100644 index 000000000..3c7a2b1db --- /dev/null +++ b/src/qhull/html/qh-optf.htm @@ -0,0 +1,736 @@ +<!DOCTYPE HTML PUBLIC "-//IETF//DTD HTML//EN"> +<html> + +<head> +<title>Qhull format options (F)</title> +</head> + +<body><!-- Navigation links --> +<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></p> +<hr> +<!-- Main text of document --> +<h1><a + href="http://www.geom.uiuc.edu/graphics/pix/Special_Topics/Computational_Geometry/delaunay.html"><IMG + align=middle alt=[delaunay] height=100 + src="qh--dt.gif" width=100 ></a> Qhull format options (F)</h1> + +<p>This section lists the format options for Qhull. These options +are indicated by 'F' followed by a letter. See <A + href="qh-opto.htm#output" >Output</a>, <a href="qh-optp.htm#print">Print</a>, +and <a href="qh-optg.htm#geomview">Geomview</a> for other output +options. </p> + +<p><b>Copyright © 1995-2015 C.B. Barber</b></p> + +<hr> + +<p><a href="index.htm#TOC">»</a> <a href="qh-quick.htm#programs">Programs</a> +<a name="format">•</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></p> + +<h2>Additional input & output formats</h2> + +<p>These options allow for automatic processing of Qhull output. +Options '<a href="qh-opto.htm#i">i</a>', '<a href="qh-opto.htm#o">o</a>', +'<a href="qh-opto.htm#n">n</a>', and '<a href="qh-opto.htm#p">p</a>' +may also be used.</p> + +<dl compact> + <dt> + <dd><b>Summary and control</b> + <dt><a href="#FA">FA</a> + <dd>compute total area and volume for option '<A + href="qh-opto.htm#s">s</a>' + + <dt><a href="#FV">FV</a> + <dd>print average vertex (interior point for '<A + href="qhalf.htm">qhalf</a>') + <dt><a href="#FQ">FQ</a> + <dd>print command for qhull and input + <dt><a href="#FO">FO</a> + <dd>print options to stderr or stdout + <dt><a href="#FS">FS</a> + <dd>print sizes: total area and volume + <dt><a href="#Fs">Fs</a> + <dd>print summary: dim, #points, total vertices and + facets, #vertices, #facets, max outer and inner plane + <dt><a href="#Fd">Fd</a> + <dd>use format for input (offset first) + <dt><a href="#FD">FD</a> + <dd>use cdd format for normals (offset first) + <dt><a href="#FM">FM</a> + <dd>print Maple output (2-d and 3-d) + <dt> + <dt> + <dd><b>Facets, points, and vertices</b> + <dt><a href="#Fa">Fa</a> + <dd>print area for each facet + <dt><a href="#FC">FC</a> + <dd>print centrum for each facet + <dt><a href="#Fc">Fc</a> + <dd>print coplanar points for each facet + <dt><a href="#Fx">Fx</a> + <dd>print extreme points (i.e., vertices) of convex hull. + + <dt><a href="#FF">FF</a> + <dd>print facets w/o ridges + <dt><a href="#FI">FI</a> + <dd>print ID for each facet + <dt><a href="#Fi">Fi</a> + <dd>print inner planes for each facet + <dt><a href="#Fm">Fm</a> + <dd>print merge count for each facet (511 max) + <dt><a href="#FP">FP</a> + <dd>print nearest vertex for coplanar points + <dt><a href="#Fn">Fn</a> + <dd>print neighboring facets for each facet + <dt><a href="#FN">FN</a> + <dd>print neighboring facets for each point + <dt><a href="#Fo">Fo</a> + <dd>print outer planes for each facet + <dt><a href="#Ft">Ft</a> + <dd>print triangulation with added points + <dt><a href="#Fv">Fv</a> + <dd>print vertices for each facet + <dt> + <dt> + <dd><b>Delaunay, Voronoi, and halfspace</b> + <dt><a href="#Fx">Fx</a> + <dd>print extreme input sites of Delaunay triangulation + or Voronoi diagram. + <dt><a href="#Fp">Fp</a> + <dd>print points at halfspace intersections + <dt><a href="#Fi2">Fi</a> + <dd>print separating hyperplanes for inner, bounded + Voronoi regions + <dt><a href="#Fo2">Fo</a> + <dd>print separating hyperplanes for outer, unbounded + Voronoi regions + <dt><a href="#Fv2">Fv</a> + <dd>print Voronoi diagram as ridges for each input pair + <dt><a href="#FC">FC</a> + <dd>print Voronoi vertex ("center") for each facet</dd> +</dl> + +<hr> + +<h3><a href="#format">»</a><a name="Fa">Fa - print area for each +facet </a></h3> + +<p>The first line is the number of facets. The remaining lines +are the area for each facet, one facet per line. See '<A + href="#FA" >FA</a>' and '<a href="#FS">FS</a>' for computing the total area and volume.</p> + +<p>Use '<a href="qh-optp.htm#PAn">PAn</a>' for printing the n +largest facets. Use option '<a href="qh-optp.htm#PFn">PFn</a>' +for printing facets larger than <i>n</i>.</p> + +<p>For Delaunay triangulations, the area is the area of each +Delaunay triangle. For Voronoi vertices, the area is the area of +the dual facet to each vertex. </p> + +<p>Qhull uses the centrum and ridges to triangulate +non-simplicial facets. The area for non-simplicial facets is the +sum of the areas for each triangle. It is an approximation of the +actual area. The ridge's vertices are projected to the facet's +hyperplane. If a vertex is far below a facet (qh_WIDEcoplanar in <tt>user.h</tt>), +the corresponding triangles are ignored.</p> + +<p>For non-simplicial facets, vertices are often below the +facet's hyperplane. If so, the approximation is less than the +actual value and it may be significantly less. </p> + +<h3><a href="#format">»</a><a name="FA">FA - compute total area +and volume for option 's' </a></h3> + +<p>With option 'FA', Qhull includes the total area and volume in +the summary ('<a href="qh-opto.htm#s">s</a>'). Option '<a href="#FS">FS</a>' also includes the total area and volume. +If facets are +merged, the area and volume are approximations. Option 'FA' is +automatically set for options '<a href="#Fa">Fa</a>', '<A + href="qh-optp.htm#PAn" >PAn</a>', and '<a href="qh-optp.htm#PFn">PFn</a>'. +</p> + +<p>With '<a href="qdelaun.htm">qdelaunay</a> <A + href="qh-opto.htm#s" >s</a> FA', Qhull computes the total area of +the Delaunay triangulation. This equals the volume of the convex +hull of the data points. With options '<a href="qdelau_f.htm">qdelaunay Qu</a> +<a href="qh-opto.htm#s">s</a> FA', Qhull computes the +total area of the furthest-site Delaunay triangulation. This +equals of the total area of the Delaunay triangulation. </p> + +<p>See '<a href="#Fa">Fa</a>' for further details. Option '<a href="#FS">FS</a>' also computes the total area and volume.</p> + +<h3><a href="#format">»</a><a name="Fc">Fc - print coplanar +points for each facet </a></h3> + +<p>The output starts with the number of facets. Then each facet +is printed one per line. Each line is the number of coplanar +points followed by the point ids. </p> + +<p>By default, option 'Fc' reports coplanar points +('<a href="qh-optq.htm#Qc">Qc</a>'). You may also use +option '<a href="qh-optq.htm#Qi">Qi</a>'. Options 'Qi Fc' prints +interior points while 'Qci Fc' prints both coplanar and interior +points. + +<p>Each coplanar point or interior point is assigned to the +facet it is furthest above (resp., least below). </p> + +<p>Use 'Qc <a href="qh-opto.htm#p">p</a>' to print vertex and +coplanar point coordinates. Use '<a href="qh-optf.htm#Fv">Fv</a>' +to print vertices. </p> + +<h3><a href="#format">»</a><a name="FC">FC - print centrum or +Voronoi vertex for each facet </a></h3> + +<p>The output starts with the dimension followed by the number of +facets. Then each facet centrum is printed, one per line. For +<a href="qvoronoi.htm">qvoronoi</a>, Voronoi vertices are +printed instead. </p> + +<h3><a href="#format">»</a><a name="Fd">Fd - use cdd format for +input </a></h3> + +<p>The input starts with comments. The first comment is reported +in the summary. Data starts after a "begin" line. The +next line is the number of points followed by the dimension plus +one and "real" or "integer". Then the points +are listed with a leading "1" or "1.0". The +data ends with an "end" line.</p> + +<p>For halfspaces ('<a href="qhalf.htm">qhalf</a> Fd'), +the input format is the same. Each halfspace starts with its +offset. The signs of the offset and coefficients are the +opposite of Qhull's +convention. The first two lines of the input may be an interior +point in '<a href="#FV">FV</a>' format.</p> + +<h3><a href="#format">»</a><a name="FD">FD - use cdd format for +normals </a></h3> + +<p>Option 'FD' prints normals ('<a href="qh-opto.htm#n">n</a>', '<A + href="#Fo" >Fo</a>', '<a href="#Fi">Fi</a>') or points ('<A + href="qh-opto.htm#p" >p</a>') in cdd format. The first line is the +command line that invoked Qhull. Data starts with a +"begin" line. The next line is the number of normals or +points followed by the dimension plus one and "real". +Then the normals or points are listed with the offset before the +coefficients. The offset for points is 1.0. For normals, +the offset and coefficients use the opposite sign from Qhull. +The data ends with an "end" line.</p> + +<h3><a href="#format">»</a><a name="FF">FF - print facets w/o +ridges </a></h3> + +<p>Option 'FF' prints all fields of all facets (as in '<A + href="qh-opto.htm#f" >f</a>') without printing the ridges. This is +useful in higher dimensions where a facet may have many ridges. +For simplicial facets, options 'FF' and '<a href="qh-opto.htm#f">f +</a>' are equivalent.</p> + +<h3><a href="#format">»</a><a name="Fi">Fi - print inner planes +for each facet </a></h3> + +<p>The first line is the dimension plus one. The second line is +the number of facets. The remainder is one inner plane per line. +The format is the same as option '<a href="qh-opto.htm#n">n</a>'.</p> + +<p>The inner plane is a plane that is below the facet's vertices. +It is an offset from the facet's hyperplane. It includes a +roundoff error for computing the vertex distance.</p> + +<p>Note that the inner planes for Geomview output ('<A + href="qh-optg.htm#Gi" >Gi</a>') include an additional offset for +vertex visualization and roundoff error. </p> + +<h3><a href="#format">»</a><a name="Fi2">Fi - print separating +hyperplanes for inner, bounded Voronoi regions</a></h3> + +<p>With <a href="qvoronoi.htm" >qvoronoi</a>, 'Fi' prints the +separating hyperplanes for inner, bounded regions of the Voronoi +diagram. The first line is the number of ridges. Then each +hyperplane is printed, one per line. A line starts with the +number of indices and floats. The first pair of indices indicates +an adjacent pair of input sites. The next <i>d</i> floats are the +normalized coefficients for the hyperplane, and the last float is +the offset. The hyperplane is oriented toward '<A + href="qh-optq.htm#QVn" >QVn</a>' (if defined), or the first input +site of the pair. </p> + +<p>Use '<a href="qh-optf.htm#Fo2">Fo</a>' for unbounded regions, +and '<a href="qh-optf.htm#Fv2">Fv</a>' for the corresponding +Voronoi vertices. </p> + +<p>Use '<a href="qh-optt.htm#Tv">Tv</a>' to verify that the +hyperplanes are perpendicular bisectors. It will list relevant +statistics to stderr. The hyperplane is a perpendicular bisector +if the midpoint of the input sites lies on the plane, all Voronoi +vertices in the ridge lie on the plane, and the angle between the +input sites and the plane is ninety degrees. This is true if all +statistics are zero. Roundoff and computation errors make these +non-zero. The deviations appear to be largest when the +corresponding Delaunay triangles are large and thin; for example, +the Voronoi diagram of nearly cospherical points. </p> + +<h3><a href="#format">»</a><a name="FI">FI - print ID for each +facet </a></h3> + +<p>Print facet identifiers. These are used internally and listed +with options '<a href="qh-opto.htm#f">f</a>' and '<a href="#FF">FF</a>'. +Options '<a href="#Fn">Fn </a>' and '<a href="#FN">FN</a>' use +facet identifiers for negative indices. </p> + +<h3><a href="#format">»</a><a name="Fm">Fm - print merge count +for each facet </a></h3> + +<p>The first line is the number of facets. The remainder is the +number of merges for each facet, one per line. At most 511 merges +are reported for a facet. See '<a href="qh-optp.htm#PMn">PMn</a>' +for printing the facets with the most merges. </p> + +<h3><a href="#format">»</a><a name="FM">FM - print Maple +output </a></h3> + +<p>Qhull writes a Maple file for 2-d and 3-d convex hulls, +2-d and 3-d halfspace intersections, +and 2-d Delaunay triangulations. Qhull produces a 2-d +or 3-d plot. + +<p><i>Warning</i>: This option has not been tested in Maple. + +<p>[From T. K. Abraham with help from M. R. Feinberg and N. Platinova.] +The following steps apply while working within the +Maple worksheet environment : +<ol> +<li>Generate the data and store it as an array . For example, in 3-d, data generated +in Maple is of the form : x[i],y[i],z[i] +<p> +<li>Create a single variable and assign the entire array of data points to this variable. +Use the "seq" command within square brackets as shown in the following example. +(The square brackets are essential for the rest of the steps to work.) +<p> +>data:=[seq([x[i],y[i],z[i]],i=1..n)]:# here n is the number of data points + +<li>Next we need to write the data to a file to be read by qhull. Before +writing the data to a file, make sure that the qhull executable files and +the data file lie in the same subdirectory. If the executable files are +stored in the "C:\qhull3.1\" subdirectory, then save the file in the same +subdirectory, say "C:\qhull3.1\datafile.txt". For the sake of integrity of +the data file , it is best to first ensure that the data file does not +exist before writing into the data file. This can be done by running a +delete command first . To write the data to the file, use the "writedata" +and the "writedata[APPEND]" commands as illustrated in the following example : +<p> +>system("del c:\\qhull3.1\\datafile.txt");#To erase any previous versions of the file +<br>>writedata("c:\\qhull3.1\\datafile.txt ",[3, nops(data)]);#writing in qhull format +<br>>writedata[APPEND]("c:\\ qhull3.1\\datafile.txt ", data);#writing the data points +<li> +Use the 'FM' option to produce Maple output. Store the output as a ".mpl" file. +For example, using the file we created above, we type the following (in DOS environment) +<p> +qconvex s FM <datafile.txt >dataplot.mpl + +<li> +To read 3-d output in Maple, we use the 'read' command followed by +a 'display3d' command. For example (in Maple environment): +<p> +>with (plots): +<br>>read `c:\\qhull3.1\\dataplot.mpl`:#IMPORTANT - Note that the punctuation mark used is ' and NOT '. The correct punctuation mark is the one next to the key for "1" (not the punctuation mark near the enter key) +<br>> qhullplot:=%: +<br>> display3d(qhullplot); +</ol> + +<p>For Delaunay triangulation orthogonal projection is better. + +<p>For halfspace intersections, Qhull produces the dual +convex hull. + +<p>See <a href="qh-faq.htm#math">Is Qhull available for Maple?</a> +for other URLs. + +<h3><a href="#format">»</a><a name="Fn">Fn - print neighboring +facets for each facet </a></h3> + +<p>The output starts with the number of facets. Then each facet +is printed one per line. Each line is the number of neighbors +followed by an index for each neighbor. The indices match the +other facet output formats.</p> + +<p>For simplicial facets, each neighbor is opposite +the corresponding vertex (option '<a href="#Fv">Fv</a>'). +Do not compare to option '<a href="qh-opto.htm#i">i</a>'. Option 'i' +orients facets by reversing the order of two vertices. For non-simplicial facets, +the neighbors are unordered. + +<p>A negative index indicates an unprinted facet due to printing +only good facets ('<a href="qh-optp.htm#Pg">Pg</a>', <a href="qdelaun.htm" >qdelaunay</a>, +<a href="qvoronoi.htm" >qvoronoi</a>). It +is the negation of the facet's ID (option '<a href="#FI">FI</a>'). +For example, negative indices are used for facets "at +infinity" in the Delaunay triangulation.</p> + +<h3><a href="#format">»</a><a name="FN">FN - print neighboring +facets for each point </a></h3> + +<p>The first line is the number of points. Then each point is +printed, one per line. For unassigned points (either interior or +coplanar), the line is "0". For assigned coplanar +points ('<a href="qh-optq.htm#Qc">Qc</a>'), the line is +"1" followed by the index of the facet that is furthest +below the point. For assigned interior points ('<A + href="qh-optq.htm#Qi" >Qi</a>'), the line is "1" +followed by the index of the facet that is least above the point. +For vertices that do not belong to good facet, the line is +"0"</p> + +<p>For vertices of good facets, the line is the number of +neighboring facets followed by the facet indices. The indices +correspond to the other '<a href="#format">F</a>' formats. In 4-d +and higher, the facets are sorted by index. In 3-d, the facets +are in adjacency order (not oriented).</p> + +<p>A negative index indicates an unprinted facet due to printing +only good facets (<a href="qdelaun.htm" >qdelaunay</a>, +<a href="qvoronoi.htm" >qvoronoi</a>, '<a href="qh-optp.htm#Pdk">Pdk</a>', +'<a href="qh-optp.htm#Pg">Pg</a>'). It is the negation of the +facet's ID ('<a href="#FI"> FI</a>'). For example, negative +indices are used for facets "at infinity" in the +Delaunay triangulation.</p> + +<p>For Voronoi vertices, option 'FN' lists the vertices of the +Voronoi region for each input site. Option 'FN' lists the regions +in site ID order. Option 'FN' corresponds to the second half of +option '<a href="qh-opto.htm#o">o</a>'. To convert from 'FN' to '<A + href="qh-opto.htm#o" >o</a>', replace negative indices with zero +and increment non-negative indices by one. </p> + +<p>If you are using the <a href="qh-code.htm#library">Qhull +library</a> or <a href="qh-code.htm#cpp">C++ interface</a>, option 'FN' has the side effect of reordering the +neighbors for a vertex </p> + +<h3><a href="#format">»</a><a name="Fo">Fo - print outer planes +for each facet </a></h3> + +<p>The first line is the dimension plus one. The second line is +the number of facets. The remainder is one outer plane per line. +The format is the same as option '<a href="qh-opto.htm#n">n</a>'.</p> + +<p>The outer plane is a plane that is above all points. It is an +offset from the facet's hyperplane. It includes a roundoff error +for computing the point distance. When testing the outer plane +(e.g., '<a href="qh-optt.htm#Tv">Tv</a>'), another roundoff error +should be added for the tested point.</p> + +<p>If outer planes are not checked ('<a href="qh-optq.htm#Q5">Q5</a>') +or not computed (!qh_MAXoutside), the maximum, computed outside +distance is used instead. This can be much larger than the actual +outer planes.</p> + +<p>Note that the outer planes for Geomview output ('<A + href="qh-optg.htm#G" >G</a>') include an additional offset for +vertex/point visualization, 'lines closer,' and roundoff error.</p> + +<h3><a href="#format">»</a><a name="Fo2">Fo - print separating +hyperplanes for outer, unbounded Voronoi regions</a></h3> + +<p>With <a href="qvoronoi.htm" >qvoronoi</a>, 'Fo' prints the +separating hyperplanes for outer, unbounded regions of the +Voronoi diagram. The first line is the number of ridges. Then +each hyperplane is printed, one per line. A line starts with the +number of indices and floats. The first pair of indices indicates +an adjacent pair of input sites. The next <i>d</i> floats are the +normalized coefficients for the hyperplane, and the last float is +the offset. The hyperplane is oriented toward '<A + href="qh-optq.htm#QVn" >QVn</a>' (if defined), or the first input +site of the pair. </p> + +<p>Option 'Fo' gives the hyperplanes for the unbounded rays of +the unbounded regions of the Voronoi diagram. Each hyperplane +goes through the midpoint of the corresponding input sites. The +rays are directed away from the input sites. </p> + +<p>Use '<a href="qh-optf.htm#Fi2">Fi</a>' for bounded regions, +and '<a href="qh-optf.htm#Fv2">Fv</a>' for the corresponding +Voronoi vertices. Use '<a href="qh-optt.htm#Tv">Tv</a>' to verify +that the corresponding Voronoi vertices lie on the hyperplane. </p> + +<h3><a href="#format">»</a><a name="FO">FO - print list of +selected options </a></h3> + +<p>Lists selected options and default values to stderr. +Additional 'FO's are printed to stdout. </p> + +<h3><a href="#format">»</a><a name="Fp">Fp - print points at +halfspace intersections</a></h3> + +<p>The first line is the number of intersection points. The +remainder is one intersection point per line. A intersection +point is the intersection of <i>d</i> or more halfspaces from +'<a href="qhalf.htm">qhalf</a>'. It corresponds to a +facet of the dual polytope. The "infinity" point +[-10.101,-10.101,...] indicates an unbounded intersection.</p> + +<p>If [x,y,z] are the dual facet's normal coefficients and <i>b<0</i> +is its offset, the halfspace intersection occurs at +[x/-b,y/-b,z/-b] plus the interior point. If <i>b>=0</i>, the +halfspace intersection is unbounded. </p> + +<h3><a href="#format">»</a><a name="FP">FP - print nearest +vertex for coplanar points </a></h3> + +<p>The output starts with the number of coplanar points. Then +each coplanar point is printed one per line. Each line is the +point ID of the closest vertex, the point ID of the coplanar +point, the corresponding facet ID, and the distance. Sort the +lines to list the coplanar points nearest to each vertex. </p> + +<p>Use options '<a href="qh-optq.htm#Qc">Qc</a>' and/or '<A + href="qh-optq.htm#Qi" >Qi</a>' with 'FP'. Options 'Qc FP' prints +coplanar points while 'Qci FP' prints coplanar and interior +points. Option 'Qc' is automatically selected if 'Qi' is not +selected. + +<p>For Delaunay triangulations (<a href="qdelaun.htm" >qdelaunay</a> +or <a href="qvoronoi.htm" >qvoronoi</a>), a coplanar point is nearly +incident to a vertex. The distance is the distance in the +original point set.</p> + +<p>If imprecision problems are severe, Qhull will delete input +sites when constructing the Delaunay triangulation. Option 'FP' will +list these points along with coincident points.</p> + +<p>If there are many coplanar or coincident points and non-simplicial +facets are triangulated ('<a href="qh-optq.htm#Qt">Qt</a>'), option +'FP' may be inefficient. It redetermines the original vertex set +for each coplanar point.</p> + +<h3><a href="#format">»</a><a name="FQ">FQ - print command for +qhull and input </a></h3> + +<p>Prints qhull and input command, e.g., "rbox 10 s | qhull +FQ". Option 'FQ' may be repeated multiple times.</p> + +<h3><a href="#format">»</a><a name="Fs">Fs - print summary</a></h3> + +<p>The first line consists of number of integers ("10") +followed by the: +<ul> +<li>dimension +<li>number of points +<li>number of vertices +<li>number of facets +<li>number of vertices selected for output +<li>number of facets selected for output +<li>number of coplanar points for selected facets +<li>number of nonsimplicial or merged facets selected for + output +<LI>number of deleted vertices</LI> +<LI>number of triangulated facets ('<a href="qh-optq.htm#Qt">Qt</a>')</LI> +</ul> + +<p>The second line consists of the number of reals +("2") followed by the: +<ul> +<li>maximum offset to an outer plane +<li>minimum offset to an inner plane.</li> +</ul> +Roundoff and joggle are included. +<P></P> + +<p>For Delaunay triangulations and Voronoi diagrams, the +number of deleted vertices should be zero. If greater than zero, then the +input is highly degenerate and coplanar points are not necessarily coincident +points. For example, <tt>'RBOX 1000 s W1e-13 t995138628 | QHULL d Qbb'</tt> reports +deleted vertices; the input is nearly cospherical.</p> + +<P>Later versions of Qhull may produce additional integers or reals.</P> + +<h3><a href="#format">»</a><a name="FS">FS - print sizes</a></h3> + +<p>The first line consists of the number of integers +("0"). The second line consists of the number of reals +("2"), followed by the total facet area, and the total +volume. Later versions of Qhull may produce additional integers +or reals.</p> + +<p>The total volume measures the volume of the intersection of +the halfspaces defined by each facet. It is computed from the +facet area. Both area and volume are approximations for +non-simplicial facets. See option '<a href="#Fa">Fa </a>' for +further notes. Option '<a href="#FA">FA </a>' also computes the total area and volume. </p> + +<h3><a href="#format">»</a><a name="Ft">Ft - print triangulation</a></h3> + +<p>Prints a triangulation with added points for non-simplicial +facets. The output is </p> + +<ul> + <li>The first line is the dimension + <li>The second line is the number of points, the number + of facets, and the number of ridges. + <li>All of the input points follow, one per line. + <li>The centrums follow, one per non-simplicial facet + <li>Then the facets follow as a list of point indices + preceded by the number of points. The simplices are + oriented. </li> +</ul> + +<p>For convex hulls with simplicial facets, the output is the +same as option '<a href="qh-opto.htm#o">o</a>'.</p> + +<p>The added points are the centrums of the non-simplicial +facets. Except for large facets, the centrum is the average +vertex coordinate projected to the facet's hyperplane. Large +facets may use an old centrum to avoid recomputing the centrum +after each merge. In either case, the centrum is clearly below +neighboring facets. See <a href="qh-impre.htm">Precision issues</a>. +</p> + +<p>The new simplices will not be clearly convex with their +neighbors and they will not satisfy the Delaunay property. They +may even have a flipped orientation. Use triangulated input ('<A + href="qh-optq.htm#Qt">Qt</a>') for Delaunay triangulations. + +<p>For Delaunay triangulations with simplicial facets, the output is the +same as option '<a href="qh-opto.htm#o">o</a>' without the lifted +coordinate. Since 'Ft' is invalid for merged Delaunay facets, option +'Ft' is not available for qdelaunay or qvoronoi. It may be used with +joggled input ('<a href="qh-optq.htm#QJn" >QJ</a>') or triangulated output ('<A + href="qh-optq.htm#Qt" >Qt</a>'), for example, rbox 10 c G 0.01 | qhull d QJ Ft</p> + +<p>If you add a point-at-infinity with '<a href="qh-optq.htm#Qz">Qz</a>', +it is printed after the input sites and before any centrums. It +will not be used in a Delaunay facet.</p> + +<h3><a href="#format">»</a><a name="Fv">Fv - print vertices for +each facet</a></h3> + +<p>The first line is the number of facets. Then each facet is +printed, one per line. Each line is the number of vertices +followed by the corresponding point ids. Vertices are listed in +the order they were added to the hull (the last one added is the +first listed). +</p> +<p>Option '<a href="qh-opto.htm#i">i</a>' also lists the vertices, +but it orients facets by reversing the order of two +vertices. Option 'i' triangulates non-simplicial, 4-d and higher facets by +adding vertices for the centrums. +</p> + +<h3><a href="#format">»</a><a name="Fv2">Fv - print Voronoi +diagram</a></h3> + +<p>With <a href="qvoronoi.htm" >qvoronoi</a>, 'Fv' prints the +Voronoi diagram or furthest-site Voronoi diagram. The first line +is the number of ridges. Then each ridge is printed, one per +line. The first number is the count of indices. The second pair +of indices indicates a pair of input sites. The remaining indices +list the corresponding ridge of Voronoi vertices. Vertex 0 is the +vertex-at-infinity. It indicates an unbounded ray. </p> + +<p>All vertices of a ridge are coplanar. If the ridge is +unbounded, add the midpoint of the pair of input sites. The +unbounded ray is directed from the Voronoi vertices to infinity. </p> + +<p>Use '<a href="qh-optf.htm#Fo2">Fo</a>' for separating +hyperplanes of outer, unbounded regions. Use '<A + href="qh-optf.htm#Fi2" >Fi</a>' for separating hyperplanes of +inner, bounded regions. </p> + +<p>Option 'Fv' does not list ridges that require more than one +midpoint. For example, the Voronoi diagram of cospherical points +lists zero ridges (e.g., 'rbox 10 s | qvoronoi Fv Qz'). +Other examples are the Voronoi diagrams of a rectangular mesh +(e.g., 'rbox 27 M1,0 | qvoronoi Fv') or a point set with +a rectangular corner (e.g., +'rbox P4,4,4 P4,2,4 P2,4,4 P4,4,2 10 | qvoronoi Fv'). +Both cases miss unbounded rays at the corners. +To determine these ridges, surround the points with a +large cube (e.g., 'rbox 10 s c G2.0 | qvoronoi Fv Qz'). +The cube needs to be large enough to bound all Voronoi regions of the original point set. +Please report any other cases that are missed. If you +can formally describe these cases or +write code to handle them, please send email to <A + href="mailto:qhull@qhull.org" >qhull@qhull.org</a>. </p> + +<h3><a href="#format">»</a><a name="FV">FV - print average +vertex </a></h3> + +<p>The average vertex is the average of all vertex coordinates. +It is an interior point for halfspace intersection. The first +line is the dimension and "1"; the second line is the +coordinates. For example,</p> + +<blockquote> + <p>qconvex FV <A + href="qh-opto.htm#n">n</a> | qhalf <a href="#Fp">Fp</a></p> +</blockquote> + +<p>prints the extreme points of the original point set (roundoff +included). </p> + +<h3><a href="#format">»</a><a name="Fx">Fx - print extreme +points (vertices) of convex hulls and Delaunay triangulations</a></h3> + +<p>The first line is the number of points. The following lines +give the index of the corresponding points. The first point is +'0'. </p> + +<p>In 2-d, the extreme points (vertices) are listed in +counterclockwise order (by qh_ORIENTclock in user.h). </p> + +<p>In 3-d and higher convex hulls, the extreme points (vertices) +are sorted by index. This is the same order as option '<A + href="qh-opto.htm#p" >p</a>' when it doesn't include coplanar or +interior points. </p> + +<p>For Delaunay triangulations, 'Fx' lists the extreme +points of the input sites (i.e., the vertices of their convex hull). The points +are unordered. <!-- Navigation links --> </p> + +<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></p><!-- GC common information --> +<hr> + +<p><a href="http://www.geom.uiuc.edu/"><IMG align=middle + height=40 src="qh--geom.gif" width=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> |