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<head><title>minkowskiSum -- computes the Minkowski sum of two convex objects</title>
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<div><h1>minkowskiSum -- computes the Minkowski sum of two convex objects</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> Q = minkowskiSum(X,Y)</tt></div>
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<li><div class="single">Inputs:<ul><li><span><tt>X</tt>, <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>
<li><span><tt>Y</tt>, <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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<li><div class="single">Outputs:<ul><li><span><tt>Q</tt>, <span>a <a href="___Polyhedron.html">convex polyhedron</a></span>, <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>
<div><p/>
The Minkowski sum of <tt>X</tt> and <tt>Y</tt> is the polyhedron
<tt>X + Y = {x + y | x in X, y in Y}</tt>. If <tt>X</tt> and <tt>Y</tt> are both
cones, then their Minkowski sum is their positive hull, which is a cone, so the
output is a <a href="___Cone.html" title="the class of all rational convex polyhedral cones">Cone</a>. Otherwise the output is a <a href="___Polyhedron.html" title="the class of all convex polyhedra">Polyhedron</a>. <tt>X</tt>
and <tt>Y</tt> have to lie in the same ambient space.<table class="examples"><tr><td><pre>i1 : P1 = convexHull matrix {{0,1,-1},{0,-1,-1}}
o1 = {ambient dimension => 2 }
dimension of lineality space => 0
dimension of polyhedron => 2
number of facets => 3
number of rays => 0
number of vertices => 3
o1 : Polyhedron</pre>
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<tr><td><pre>i2 : P2 = convexHull matrix {{0,1,-1},{0,1,1}}
o2 = {ambient dimension => 2 }
dimension of lineality space => 0
dimension of polyhedron => 2
number of facets => 3
number of rays => 0
number of vertices => 3
o2 : Polyhedron</pre>
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<tr><td><pre>i3 : Q = minkowskiSum(P1,P2)
o3 = {ambient dimension => 2 }
dimension of lineality space => 0
dimension of polyhedron => 2
number of facets => 6
number of rays => 0
number of vertices => 6
o3 : Polyhedron</pre>
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<tr><td><pre>i4 : vertices Q
o4 = | -2 2 -1 1 -1 1 |
| 0 0 -1 -1 1 1 |
2 6
o4 : Matrix QQ <--- QQ</pre>
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See also <a href="___Cone_sp_pl_sp__Cone.html" title="computes the Minkowski sum of two cones">Cone + Cone</a>, <a href="___Cone_sp_pl_sp__Polyhedron.html" title="computes the Minkowski sum of a cone and a polyhedron">Cone + Polyhedron</a>, <a href="___Polyhedron_sp_pl_sp__Cone.html" title="computes the Minkowski sum of a polyhedron and a cone">Polyhedron + Cone</a>, and <a href="___Polyhedron_sp_pl_sp__Cone.html" title="computes the Minkowski sum of a polyhedron and a cone">Polyhedron + Cone</a>.</div>
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<div class="waystouse"><h2>Ways to use <tt>minkowskiSum</tt> :</h2>
<ul><li>minkowskiSum(Cone,Cone)</li>
<li>minkowskiSum(Cone,Polyhedron)</li>
<li>minkowskiSum(Polyhedron,Cone)</li>
<li>minkowskiSum(Polyhedron,Polyhedron)</li>
</ul>
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