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<head><title>affineImage(Matrix,Polyhedron,Matrix) -- computes the affine image of a polyhedron</title>
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<div><h1>affineImage(Matrix,Polyhedron,Matrix) -- computes the affine image of 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> P1 = affineImage(A,P,v) </tt><br/><tt>P1 = affineImage(A,P) </tt><br/><tt>P1 = affineImage(P,v)</tt></div>
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<li><span>Function: <a href="_affine__Image.html" title="computes the affine image of a cone or polyhedron">affineImage</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>P</tt>, <span>a <a href="___Polyhedron.html">convex polyhedron</a></span></span></li>
<li><span><tt>v</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>P1</tt>, <span>a <a href="___Polyhedron.html">convex polyhedron</a></span></span></li>
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<div class="single"><h2>Description</h2>
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<tt>A</tt> must be a matrix from the ambient space of the polyhedron <tt>P</tt> to some
other target space and <tt>v</tt> must be a vector in that target space, i.e. the number of
columns of <tt>A</tt> must equal the ambient dimension of <tt>P</tt> and <tt>A</tt> and <tt>v</tt>
must have the same number of rows. Then <tt>affineImage</tt> computes the
polyhedron <tt>{(A*p)+v | p in P}</tt> where <tt>v</tt> is set to 0 if omitted and <tt>A</tt> is the
identity if omitted.<p/>
For example, consider the following two dimensional polytope:<table class="examples"><tr><td><pre>i1 : P = convexHull matrix {{-2,0,2,4},{-8,-2,2,8}}
o1 = {ambient dimension => 2 }
dimension of lineality space => 0
dimension of polyhedron => 2
number of facets => 4
number of rays => 0
number of vertices => 4
o1 : Polyhedron</pre>
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This polytope is the affine image of the square:<table class="examples"><tr><td><pre>i2 : A = matrix {{-5,2},{3,-1}}
o2 = | -5 2 |
| 3 -1 |
2 2
o2 : Matrix ZZ <--- ZZ</pre>
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<tr><td><pre>i3 : v = matrix {{5},{-3}}
o3 = | 5 |
| -3 |
2 1
o3 : Matrix ZZ <--- ZZ</pre>
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<tr><td><pre>i4 : Q = affineImage(A,P,v)
o4 = {ambient dimension => 2 }
dimension of lineality space => 0
dimension of polyhedron => 2
number of facets => 4
number of rays => 0
number of vertices => 4
o4 : Polyhedron</pre>
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<tr><td><pre>i5 : vertices Q
o5 = | -1 1 -1 1 |
| -1 -1 1 1 |
2 4
o5 : Matrix QQ <--- QQ</pre>
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