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<head><title>affinePreimage(Matrix,Cone,Matrix) -- computes the affine preimage of a cone</title>
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<div><h1>affinePreimage(Matrix,Cone,Matrix) -- computes the affine preimage of a cone</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> C1 = affinePreimage(A,C,b) </tt><br/><tt>C1 = affinePreimage(A,C) </tt><br/><tt>C1 = affinePreimage(C,b)</tt></div>
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<li><span>Function: <a href="_affine__Preimage.html" title="computes the affine preimage of a cone or polyhedron">affinePreimage</a></span></li>
<li><div class="single">Inputs:<ul><li><span><tt>A</tt>, <span>a <a href="../../Macaulay2Doc/html/___Matrix.html">matrix</a></span>, with entries in <a href="../../Macaulay2Doc/html/___Z__Z.html" title="the class of all integers">ZZ</a> or <a href="../../Macaulay2Doc/html/___Q__Q.html" title="the class of all rational numbers">QQ</a></span></li>
<li><span><tt>C</tt>, <span>a <a href="___Cone.html">convex rational cone</a></span></span></li>
<li><span><tt>b</tt>, <span>a <a href="../../Macaulay2Doc/html/___Matrix.html">matrix</a></span>, with entries in <a href="../../Macaulay2Doc/html/___Z__Z.html" title="the class of all integers">ZZ</a> or <a href="../../Macaulay2Doc/html/___Q__Q.html" title="the class of all rational numbers">QQ</a> and only one column representing a vector</span></li>
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<li><div class="single">Outputs:<ul><li><span><tt>C1</tt>, <span>a <a href="___Polyhedron.html">convex polyhedron</a></span>, of class <a href="___Cone.html" title="the class of all rational convex polyhedral cones">Cone</a> or <a href="___Polyhedron.html" title="the class of all convex polyhedra">Polyhedron</a></span></li>
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<div class="single"><h2>Description</h2>
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<tt>A</tt> must be a matrix from some source space to the ambient space of <tt>C</tt> and <tt>b</tt> must be
a vector in that ambient space, i.e. the number of rows of <tt>A</tt> must equal the ambient dimension of <tt>C</tt>
and the number of rows of <tt>b</tt>. <tt>affinePreimage</tt> then computes the
polyhedron <tt>{q | (A*q)+b in C}</tt> or the cone <tt>{q | (A*q) in C}</tt> if <tt>b</tt> is 0 or omitted.
If <tt>A</tt> is omitted then it is set to identity.<p/>
For example, consider the following three dimensional cone:<table class="examples"><tr><td><pre>i1 : C = posHull matrix {{1,2,3},{3,1,2},{2,3,1}}
o1 = {ambient dimension => 3 }
dimension of lineality space => 0
dimension of the cone => 3
number of facets => 3
number of rays => 3
o1 : Cone</pre>
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We can look at its preimage under the following map:<table class="examples"><tr><td><pre>i2 : A = matrix {{-5,7,1},{1,-5,7},{7,1,-5}}
o2 = | -5 7 1 |
| 1 -5 7 |
| 7 1 -5 |
3 3
o2 : Matrix ZZ <--- ZZ</pre>
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<tr><td><pre>i3 : C1 = affinePreimage(A,C)
o3 = {ambient dimension => 3 }
dimension of lineality space => 0
dimension of the cone => 3
number of facets => 3
number of rays => 3
o3 : Cone</pre>
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<tr><td><pre>i4 : rays C1
o4 = | 13 13 10 |
| 13 10 13 |
| 10 13 13 |
3 3
o4 : Matrix QQ <--- QQ</pre>
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