📄 qhalf.htm
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<title>qhalf -- halfspace intersection about a point</title>
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<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. </p>
<blockquote>
<dl compact>
<dt><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
vertext. 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>
</dl>
</blockquote>
<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>By default, halfspace intersections may be defined by more than
<i>d</i> halfspaces. See the previous cube and diamond example.
This is the expected output for halfspace intersection.
<p>You can try triangulated output and joggled input. 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 previous
cube and diamond 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, replace 'Qt' with 'QJ' above and
compare the duplicated intersections.
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 © 1995-2003 The Geometry Center, Minneapolis MN</b></p>
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