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diff --git a/src/qhull/html/qhalf.htm b/src/qhull/html/qhalf.htm new file mode 100644 index 000000000..c87fe719e --- /dev/null +++ b/src/qhull/html/qhalf.htm @@ -0,0 +1,626 @@ +<!DOCTYPE HTML PUBLIC "-//IETF//DTD HTML//EN"> +<html> + +<head> +<title>qhalf -- halfspace intersection about a point</title> +</head> + +<body> +<!-- Navigation links --> +<p><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/half.html"><img +src="qh--half.gif" alt="[halfspace]" align="middle" width="100" +height="100"></a>qhalf -- halfspace intersection about a point</h1> + +<p>The intersection of a set of halfspaces is a polytope. The +polytope may be unbounded. See Preparata & Shamos [<a +href="index.htm#pre-sha85">'85</a>] for a discussion. In low +dimensions, halfspace intersection may be used for linear +programming. + +<blockquote> +<dl compact> + <dt><p><b>Example:</b> rbox c | qconvex <a href="qh-optf.htm#FQ">FQ</a> <a href="qh-optf.htm#FV">FV</a> + <a href="qh-opto.htm#n">n</a> | qhalf <a + href="qh-optf.htm#Fp">Fp</a></dt> + <dd>Print the intersection of the facets of a cube. <tt>rbox c</tt> + generates the vertices of a cube. <tt>qconvex FV n</tt> returns of average + of the cube's vertices (in this case, the origin) and the halfspaces + that define the cube. <tt>qhalf Fp</tt> computes the intersection of + the halfspaces about the origin. The intersection is the vertices + of the original cube.</dd> + + <dt><p><b>Example:</b> rbox c d G0.55 | qconvex <a href="qh-optf.htm#FQ">FQ</a> <a href="qh-optf.htm#FV">FV</a> + <a href="qh-opto.htm#n">n</a> | qhalf <a + href="qh-optf.htm#Fp">Fp</a></dt> + <dd>Print the intersection of the facets of a cube and a diamond. There + are 24 facets and 14 intersection points. Four facets define each diamond + vertex. Six facets define each cube vertex. + </dd> + + <dt><p><b>Example:</b> rbox c d G0.55 | qconvex <a href="qh-optf.htm#FQ">FQ</a> <a href="qh-optf.htm#FV">FV</a> + <a href="qh-opto.htm#n">n</a> | qhalf <a + href="qh-optf.htm#Fp">Fp</a> + <a href="qh-optq.htm#Qt">Qt</a></dt> + <dd>Same as above except triangulate before computing + the intersection points. Three facets define each intersection + point. There are two duplicates of the diamond and four duplicates of the cube. + </dd> + + <dt><p><b>Example:</b> rbox 10 s t10 | qconvex <a href="qh-optf.htm#FQ">FQ</a> <a href="qh-optf.htm#FV">FV</a> + <a href="qh-opto.htm#n">n</a> | qhalf <a + href="qh-optf.htm#Fp">Fp</a> <a + href="qh-optf.htm#Fn">Fn</a></dt> + <dd>Print the intersection of the facets of the convex hull of 10 cospherical points. + Include the intersection points and the neighboring intersections. + As in the previous examples, the intersection points are nearly the same as the + original input points. + </dd> +</dl> +</blockquote> + +<p>In Qhull, a <i>halfspace</i> is defined by the points on or below a hyperplane. +The distance of each point to the hyperplane is less than or equal to zero. + +<p>Qhull computes a halfspace intersection by the geometric +duality between points and halfspaces. +See <a href="qh-eg.htm#half">halfspace examples</a>, +<a href="#notes">qhalf notes</a>, and +option 'p' of <a href="#outputs">qhalf outputs</a>. </p> + +<p>Qhalf's <a href="#outputs">outputs</a> are the intersection +points (<a href="qh-optf.htm#Fp">Fp</a>) and +the neighboring intersection points (<a href="qh-optf.htm#Fn">Fn</a>). +For random inputs, halfspace +intersections are usually defined by more than <i>d</i> halfspaces. See the sphere example. + +<p>You can try triangulated output ('<a href="qh-optq.htm#Qt">Qt</a>') and joggled input ('<a href="qh-optq.htm#QJn">QJ</a>'). +It demonstrates that triangulated output is more accurate than joggled input. + +<p>If you use '<a href="qh-optq.htm#Qt">Qt</a>' (triangulated output), all +halfspace intersections are simplicial (e.g., three halfspaces per +intersection in 3-d). In 3-d, if more than three halfspaces intersect +at the same point, triangulated output will produce +duplicate intersections, one for each additional halfspace. See the third example, or +add 'Qt' to the sphere example.</p> + +<p>If you use '<a href="qh-optq.htm#QJn">QJ</a>' (joggled input), all halfspace +intersections are simplicial. This may lead to nearly identical +intersections. For example, either replace 'Qt' with 'QJ' above, or add +'QJ' to the sphere example. +See <a +href="qh-impre.htm#joggle">Merged facets or joggled input</a>. </p> + +<p>The 'qhalf' program is equivalent to +'<a href=qhull.htm#outputs>qhull H</a>' in 2-d to 4-d, and +'<a href=qhull.htm#outputs>qhull H</a> <a href=qh-optq.htm#Qx>Qx</a>' +in 5-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> +<hr> + +<h3><a href="#TOP">»</a><a name="synopsis">qhalf synopsis</a></h3> +<pre> +qhalf- halfspace intersection about a point. + input (stdin): [dim, 1, interior point] + dim+1, n + halfspace coefficients + offset + comments start with a non-numeric character + +options (qhalf.htm): + Hn,n - specify coordinates of interior point + Qt - triangulated output + QJ - joggle input instead of merging facets + Tv - verify result: structure, convexity, and redundancy + . - concise list of all options + - - one-line description of all options + +output options (subset): + s - summary of results (default) + Fp - intersection coordinates + Fv - non-redundant halfspaces incident to each intersection + Fx - non-redundant halfspaces + o - OFF file format (dual convex hull) + G - Geomview output (dual convex hull) + m - Mathematica output (dual convex hull) + QVn - print intersections for halfspace n, -n if not + TO file - output results to file, may be enclosed in single quotes + +examples: + rbox d | qconvex n | qhalf s H0,0,0 Fp + rbox c | qconvex FV n | qhalf s i + rbox c | qconvex FV n | qhalf s o +</pre> + +<h3><a href="#TOP">»</a><a name="input">qhalf input</a></h3> + +<blockquote> +<p>The input data on <tt>stdin</tt> consists of:</p> +<ul> + <li>[optional] interior point + <ul> + <li>dimension + <li>1 + <li>coordinates of interior point + </ul> + <li>dimension + 1 + <li>number of halfspaces</li> + <li>halfspace coefficients followed by offset</li> +</ul> + +<p>Use I/O redirection (e.g., qhalf < data.txt), a pipe (e.g., rbox c | qconvex FV n | qhalf), +or the '<a href=qh-optt.htm#TI>TI</a>' option (e.g., qhalf TI data.txt). + +<p>Qhull needs an interior point to compute the halfspace +intersection. An interior point is clearly inside all of the halfspaces. +A point is <i>inside</i> a halfspace if its distance to the corresponding hyperplane is negative. + +<p>The interior point may be listed at the beginning of the input (as shown above). +If not, option +'Hn,n' defines the interior point as +[n,n,0,...] where <em>0</em> is the default coordinate (e.g., +'H0' is the origin). Use linear programming if you do not know +the interior point (see <a href="#notes">halfspace notes</a>),</p> + +<p>The input to qhalf is a set of halfspaces that are defined by their hyperplanes. +Each halfspace is defined by +<em>d</em> coefficients followed by a signed offset. This defines +a linear inequality. The coefficients define a vector that is +normal to the halfspace. +The vector may have any length. If it +has length one, the offset is the distance from the origin to the +halfspace's boundary. Points in the halfspace have a negative distance to the hyperplane. +The distance from the interior point to each +halfspace is likewise negative.</p> + +<p>The halfspace format is the same as Qhull's output options '<a +href="qh-opto.htm#n">n</a>', '<a href="qh-optf.htm#Fo">Fo</a>', +and '<a href="qh-optf.htm#Fi">Fi</a>'. Use option '<a +href="qh-optf.htm#Fd">Fd</a>' to use cdd format for the +halfspaces.</p> + +<p>For example, here is the input for computing the intersection +of halfplanes that form a cube.</p> + +<blockquote> + <p>rbox c | qconvex FQ FV n TO data </p> + <pre> +RBOX c | QCONVEX FQ FV n +3 1 + 0 0 0 +4 +6 + 0 0 -1 -0.5 + 0 -1 0 -0.5 + 1 0 0 -0.5 + -1 0 0 -0.5 + 0 1 0 -0.5 + 0 0 1 -0.5 +</pre> + <p>qhalf s Fp < data </p> + <pre> + +Halfspace intersection by the convex hull of 6 points in 3-d: + + Number of halfspaces: 6 + Number of non-redundant halfspaces: 6 + Number of intersection points: 8 + +Statistics for: RBOX c | QCONVEX FQ FV n | QHALF s Fp + + Number of points processed: 6 + Number of hyperplanes created: 11 + Number of distance tests for qhull: 11 + Number of merged facets: 1 + Number of distance tests for merging: 45 + CPU seconds to compute hull (after input): 0 + +3 +3 +8 + -0.5 0.5 0.5 + 0.5 0.5 0.5 + -0.5 0.5 -0.5 + 0.5 0.5 -0.5 + 0.5 -0.5 0.5 + -0.5 -0.5 0.5 + -0.5 -0.5 -0.5 + 0.5 -0.5 -0.5 +</pre> +</blockquote> + +</blockquote> +<h3><a href="#TOP">»</a><a name="outputs">qhalf outputs</a></h3> +<blockquote> + +<p>The following options control the output for halfspace +intersection.</p> +<blockquote> +<dl compact> + <dt> </dt> + <dd><b>Intersections</b></dd> + <dt><a href="qh-optf.htm#FN">FN</a></dt> + <dd>list intersection points for each non-redundant + halfspace. The first line + is the number of non-redundant halfspaces. Each remaining + lines starts with the number of intersection points. For the cube + example, each halfspace has four intersection points.</dd> + <dt><a href="qh-optf.htm#Fn">Fn</a></dt> + <dd>list neighboring intersections for each intersection point. The first line + is the number of intersection points. Each remaining line + starts with the number of neighboring intersections. For the cube + example, each intersection point has three neighboring intersections. + <p> + In 3-d, a non-simplicial intersection has more than three neighboring + intersections. For random data (e.g., the sphere example), non-simplicial intersections are the norm. + Option '<a href="qh-optq.htm#Qt">Qt</a>' produces three + neighboring intersections per intersection by duplicating the intersection + points. Option <a href="qh-optq.htm#QJn">QJ</a>' produces three + neighboring intersections per intersection by joggling the hyperplanes and + hence their intersections. + </dd> + <dt><a href="qh-optf.htm#Fp">Fp</a></dt> + <dd>print intersection coordinates. The first line is the dimension and the + second line is the number of intersection points. The following lines are the + coordinates of each intersection.</dd> + <dt><a href="qh-optf.htm#FI">FI</a></dt> + <dd>list intersection IDs. The first line is the number of + intersections. The IDs follow, one per line.</dd> + <dt> </dt> + <dt> </dt> + <dd><b>Halfspaces</b></dd> + <dt><a href="qh-optf.htm#Fx">Fx</a></dt> + <dd>list non-redundant halfspaces. The first line is the number of + non-redundant halfspaces. The other lines list one halfspace per line. + A halfspace is <i>non-redundant</i> if it + defines a facet of the intersection. Redundant halfspaces are ignored. For + the cube example, all of the halfspaces are non-redundant. + </dd> + <dt><a href="qh-optf.htm#Fv">Fv</a></dt> + <dd>list non-redundant halfspaces incident to each intersection point. + The first line is the number of + non-redundant halfspaces. Each remaining line starts with the number + of non-redundant halfspaces. For the + cube example, each intersection is incident to three halfspaces.</dd> + <dt><a href="qh-opto.htm#i">i</a></dt> + <dd>list non-redundant halfspaces incident to each intersection point. The first + line is the number of intersection points. Each remaining line + lists the incident, non-redundant halfspaces. For the + cube example, each intersection is incident to three halfspaces. + </dd> + <dt><a href="qh-optf.htm#Fc">Fc</a></dt> + <dd>list coplanar halfspaces for each intersection point. The first line is + the number of intersection points. Each remaining line starts with + the number of coplanar halfspaces. A coplanar halfspace is listed for + one intersection point even though it is coplanar to multiple intersection + points.</dd> + <dt><a href="qh-optq.htm#Qc">Qi</a> <a href="qh-optf.htm#Fc">Fc</a></dt> + <dd>list redundant halfspaces for each intersection point. The first line is + the number of intersection points. Each remaining line starts with + the number of redundant halfspaces. Use options '<a href="qh-optq.htm#Qc">Qc</a> Qi Fc' to list + coplanar and redundant halfspaces.</dd> + + <dt> </dt> + <dt> </dt> + <dd><b>General</b></dd> + <dt><a href="qh-opto.htm#s">s</a></dt> + <dd>print summary for the halfspace intersection. Use '<a + href="qh-optf.htm#Fs">Fs</a>' if you need numeric data.</dd> + <dt><a href="qh-opto.htm#o">o</a></dt> + <dd>print vertices and facets of the dual convex hull. The + first line is the dimension. The second line is the number of + vertices, facets, and ridges. The vertex + coordinates are next, followed by the facets, one per line.</dd> + <dt><a href="qh-opto.htm#p">p</a></dt> + <dd>print vertex coordinates of the dual convex hull. Each vertex corresponds + to a non-redundant halfspace. Its coordinates are the negative of the hyperplane's coefficients + divided by the offset plus the inner product of the coefficients and + the interior point (-c/(b+a.p). + Options 'p <a href="qh-optq.htm#Qc">Qc</a>' includes coplanar halfspaces. + Options 'p <a href="qh-optq.htm#Qi">Qi</a>' includes redundant halfspaces.</dd> + <dt><a href="qh-opto.htm#m">m</a></dt> + <dd>Mathematica output for the dual convex hull in 2-d or 3-d.</dd> + <dt><a href="qh-optf.htm#FM">FM</a></dt> + <dd>Maple output for the dual convex hull in 2-d or 3-d.</dd> + <dt><a href="qh-optg.htm#G">G</a></dt> + <dd>Geomview output for the dual convex hull in 2-d, 3-d, or 4-d.</dd> + </dl> +</blockquote> + +</blockquote> +<h3><a href="#TOP">»</a><a name="controls">qhalf 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. If a 3-d intersection is defined by more than + three hyperplanes, Qhull produces duplicate intersections -- one for + each extra hyperplane.</dd> + <dt><a href="qh-optq.htm#QJn">QJ</a></dt> + <dd>joggle the input instead of merging facets. In 3-d, this guarantees that + each intersection is defined by three hyperplanes.</dd> + <dt><a href="qh-opto.htm#f">f </a></dt> + <dd>facet dump. Print the data structure for each intersection (i.e., + facet)</dd> + <dt><a href="qh-optt.htm#TFn">TFn</a></dt> + <dd>report summary after constructing <em>n</em> + intersections</dd> + <dt><a href="qh-optq.htm#QVn">QVn</a></dt> + <dd>select intersection points for halfspace <em>n</em> + (marked 'good')</dd> + <dt><a href="qh-optq.htm#QGn">QGn</a></dt> + <dd>select intersection points that are visible to halfspace <em>n</em> + (marked 'good'). Use <em>-n</em> for the remainder.</dd> + <dt><a href="qh-optq.htm#Qb0">Qbk:0Bk:0</a></dt> + <dd>remove the k-th coordinate from the input. This computes the + halfspace intersection in one lower dimension.</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-optq.htm#Qs">Qs</a></dt> + <dd>search all points for the initial simplex. If Qhull can + not construct an initial simplex, it reports a +descriptive message. Usually, the point set is degenerate and one +or more dimensions should be removed ('<a href="qh-optq.htm#Qb0">Qbk:0Bk:0</a>'). +If not, use option 'Qs'. It performs an exhaustive search for the +best initial simplex. This is expensive is high dimensions.</dd> +</dl> +</blockquote> + + +</blockquote> +<h3><a href="#TOP">»</a><a name="graphics">qhalf graphics</a></h3> +<blockquote> + +<p>To view the results with Geomview, compute the convex hull of +the intersection points ('qhull FQ H0 Fp | qhull G'). See <a +href="qh-eg.htm#half">Halfspace examples</a>.</p> + +</blockquote> +<h3><a href="#TOP">»</a><a name="notes">qhalf notes</a></h3> +<blockquote> + +<p>See <a href="qh-impre.htm#halfspace">halfspace intersection</a> for precision issues related to qhalf.</p> + +<p>If you do not know an interior point for the halfspaces, use +linear programming to find one. Assume, <em>n</em> halfspaces +defined by: <em>aj*x1+bj*x2+cj*x3+dj<=0, j=1..n</em>. Perform +the following linear program: </p> + +<blockquote> + <p>max(x5) aj*x1+bj*x2+cj*x3+dj*x4+x5<=0, j=1..n</p> +</blockquote> + +<p>Then, if <em>[x1,x2,x3,x4,x5]</em> is an optimal solution with +<em>x4>0</em> and <em>x5>0</em> we get: </p> + +<blockquote> + <p>aj*(x1/x4)+bj*(x2/x4)+cj*(x3/x4)+dj<=(-x5/x4) j=1..n and (-x5/x4)<0, + </p> +</blockquote> + +<p>and conclude that the point <em>[x1/x4,x2/x4,x3/x4]</em> is in +the interior of all the halfspaces. Since <em>x5</em> is +optimal, this point is "way in" the interior (good +for precision errors).</p> + +<p>After finding an interior point, the rest of the intersection +algorithm is from Preparata & Shamos [<a +href="index.htm#pre-sha85">'85</a>, p. 316, "A simple case +..."]. Translate the halfspaces so that the interior point +is the origin. Calculate the dual polytope. The dual polytope is +the convex hull of the vertices dual to the original faces in +regard to the unit sphere (i.e., halfspaces at distance <em>d</em> +from the origin are dual to vertices at distance <em>1/d</em>). +Then calculate the resulting polytope, which is the dual of the +dual polytope, and translate the origin back to the interior +point [S. Spitz, S. Teller, D. Strawn].</p> + + +</blockquote> +<h3><a href="#TOP">»</a><a name="conventions">qhalf +conventions</a></h3> +<blockquote> + +<p>The following terminology is used for halfspace intersection +in Qhull. This is the hardest structure to understand. The +underlying structure is a convex hull with one vertex per +non-redundant halfspace. See <a href="#conventions">convex hull +conventions</a> and <a href="index.htm#structure">Qhull's data structures</a>.</p> + +<ul> + <li><em>interior point</em> - a point in the intersection of + the halfspaces. Qhull needs an interior point to compute + the intersection. See <a href="#input">halfspace input</a>.</li> + <li><em>halfspace</em> - <em>d</em> coordinates for the + normal and a signed offset. The distance to an interior + point is negative.</li> + <li><em>non-redundant halfspace</em> - a halfspace that + defines an output facet</li> + <li><em>vertex</em> - a dual vertex in the convex hull + corresponding to a non-redundant halfspace</li> + <li><em>coplanar point</em> - the dual point corresponding to + a similar halfspace</li> + <li><em>interior point</em> - the dual point corresponding to + a redundant halfspace</li> + <li><em>intersection point</em>- the intersection of <em>d</em> + or more non-redundant halfspaces</li> + <li><em>facet</em> - a dual facet in the convex hull + corresponding to an intersection point</li> + <li><em>non-simplicial facet</em> - more than <em>d</em> + halfspaces intersect at a point</li> + <li><em>good facet</em> - an intersection point that + satisfies restriction '<a href="qh-optq.htm#QVn">QVn</a>', + etc.</li> +</ul> + +</blockquote> +<h3><a href="#TOP">»</a><a name="options">qhalf options</a></h3> + +<pre> +qhalf- compute the intersection of halfspaces about a point + http://www.qhull.org + +input (stdin): + optional interior point: dimension, 1, coordinates + first lines: dimension+1 and number of halfspaces + other lines: halfspace coefficients followed by offset + comments: start with a non-numeric character + +options: + Hn,n - specify coordinates of interior point + Qt - triangulated ouput + QJ - joggle input instead of merging facets + Qc - keep coplanar halfspaces + Qi - keep other redundant halfspaces + +Qhull control options: + QJn - randomly joggle input in range [-n,n] + Qbk:0Bk:0 - remove k-th coordinate from input + Qs - search all halfspaces for the initial simplex + QGn - print intersection if redundant to halfspace n, -n for not + QVn - print intersections for halfspace 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 redundancy + 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 halfspace n added to intersection + TMn - turn on tracing at merge n + TWn - trace merge facets when width > n + TVn - stop qhull after adding halfspace n, -n for before (see TCn) + TCn - stop qhull after building cone for halfspace 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] + Un - max distance below plane for a new, coplanar halfspace + Wn - min facet width for outside halfspace (before roundoff) + +Output formats (may be combined; if none, produces a summary to stdout): + f - facet dump + G - Geomview output (dual convex hull) + i - non-redundant halfspaces incident to each intersection + m - Mathematica output (dual convex hull) + o - OFF format (dual convex hull: dimension, points, and facets) + p - vertex coordinates of dual convex hull (coplanars if 'Qc' or 'Qi') + s - summary (stderr) + +More formats: + Fc - count plus redundant halfspaces for each intersection + - Qc (default) for coplanar and Qi for other redundant + Fd - use cdd format for input (homogeneous with offset first) + FF - facet dump without ridges + FI - ID of each intersection + Fm - merge count for each intersection (511 max) + FM - Maple output (dual convex hull) + Fn - count plus neighboring intersections for each intersection + FN - count plus intersections for each non-redundant halfspace + FO - options and precision constants + Fp - dim, count, and intersection coordinates + FP - nearest halfspace and distance for each redundant halfspace + FQ - command used for qhalf + Fs - summary: #int (8), dim, #halfspaces, #non-redundant, #intersections + for output: #non-redundant, #intersections, #coplanar + halfspaces, #non-simplicial intersections + #real (2), max outer plane, min vertex + Fv - count plus non-redundant halfspaces for each intersection + Fx - non-redundant halfspaces + +Geomview output (2-d, 3-d and 4-d; dual convex hull) + Ga - all points (i.e., transformed halfspaces) as dots + Gp - coplanar points and vertices as radii + Gv - vertices (i.e., non-redundant halfspaces) as spheres + Gi - inner planes (i.e., halfspace intersections) 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 facets (i.e., intersections) by area + Pdk:n- drop facet if normal[k] <= n (default 0.0) + PDk:n- drop facet if normal[k] >= n + Pg - print good facets (needs 'QGn' or 'QVn') + PFn - keep facets whose area is at least n + PG - print neighbors of good facets + PMn - keep n facets 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> + |