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<head><title>halfspaces -- computes the defining half-spaces of a Cone or a Polyhedron</title>
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<div><h1>halfspaces -- computes the defining half-spaces of a Cone or a Polyhedron</h1>
<div class="single"><h2>Synopsis</h2>
<ul><li><div class="list"><dl class="element"><dt class="heading">Usage: </dt><dd class="value"><div><tt> M = halfspaces C </tt><br/><tt>(M,v) = halfspaces P</tt></div>
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<li><div class="single">Inputs:<ul><li><span><tt>C</tt>, <span>a <a href="___Cone.html">convex rational cone</a></span></span></li>
<li><span><tt>P</tt>, <span>a <a href="___Polyhedron.html">convex polyhedron</a></span></span></li>
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<li><div class="single">Outputs:<ul><li><span><tt>M</tt>, <span>a <a href="../../Macaulay2Doc/html/___Matrix.html">matrix</a></span>, with entries over <a href="../../Macaulay2Doc/html/___Q__Q.html" title="the class of all rational numbers">QQ</a></span></li>
<li><span><tt>v</tt>, <span>a <a href="../../Macaulay2Doc/html/___Matrix.html">matrix</a></span>, with entries over <a href="../../Macaulay2Doc/html/___Q__Q.html" title="the class of all rational numbers">QQ</a> and only one column</span></li>
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<div class="single"><h2>Description</h2>
<div><p/>
<tt>halfspaces</tt> returns the defining affine half-spaces. For a
polyhedron <tt>P</tt> the output is <tt>(M,v)</tt>, where the source
of <tt>M</tt> has the dimension of the ambient space of <tt>P</tt>
and <tt>v</tt> is a one column matrix in the target space
of <tt>M</tt> such that <tt>P = {p in H | M*p =< v}</tt> where
<tt>H</tt> is the intersection of the defining affine hyperplanes.<p/>
For a cone <tt>C</tt> the output is the matrix<tt>M</tt> that is the
same matrix as before but <tt>v</tt> is omitted since it is 0,
so <tt>C = {c in H | M*c =< 0}</tt> and <tt>H</tt> is the intersection
of the defining linear hyperplanes.<table class="examples"><tr><td><pre>i1 : R = matrix {{1,1,2,2},{2,3,1,3},{3,2,3,1}};
3 4
o1 : Matrix ZZ <--- ZZ</pre>
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<tr><td><pre>i2 : V = matrix {{1,-1},{0,0},{0,0}};
3 2
o2 : Matrix ZZ <--- ZZ</pre>
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<tr><td><pre>i3 : C = posHull R
o3 = {ambient dimension => 3 }
dimension of lineality space => 0
dimension of the cone => 3
number of facets => 4
number of rays => 4
o3 : Cone</pre>
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<tr><td><pre>i4 : halfspaces C
o4 = | -2 1 1 |
| 1 -1 1 |
| 1 1 -1 |
| 5 -1 -1 |
4 3
o4 : Matrix QQ <--- QQ</pre>
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Now we take this cone over a line and get a polyhedron.<table class="examples"><tr><td><pre>i5 : P = convexHull(V,R)
o5 = {ambient dimension => 3 }
dimension of lineality space => 0
dimension of polyhedron => 3
number of facets => 6
number of rays => 4
number of vertices => 2
o5 : Polyhedron</pre>
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<tr><td><pre>i6 : halfspaces P
o6 = (| 0 1 -3 |, | 0 |)
| 2 -1 -1 | | 2 |
| -1 1 -1 | | 1 |
| 0 -3 1 | | 0 |
| -1 -1 1 | | 1 |
| -5 1 1 | | 5 |
o6 : Sequence</pre>
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<div class="waystouse"><h2>Ways to use <tt>halfspaces</tt> :</h2>
<ul><li>halfspaces(Cone)</li>
<li>halfspaces(Polyhedron)</li>
</ul>
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