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<head><title>mConvexHullFacesAndDuals -- Faces and their duals of a convex hull.</title>
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<div><h1>mConvexHullFacesAndDuals -- Faces and their duals of a convex hull.</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>mConvexHullFacesAndDuals(L)</tt></div>
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<li><div class="single">Inputs:<ul><li><span><tt>L</tt>, <span>a <a href="../../Macaulay2Doc/html/___List.html">list</a></span>, of <a href="../../Macaulay2Doc/html/___Vector.html" title="the class of all elements of free modules that are handled by the engine">Vector</a>s</span></li>
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<li><div class="single">Outputs:<ul><li><span><span>a <a href="../../Macaulay2Doc/html/___List.html">list</a></span></span></li>
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<li><div class="single"><a href="../../Macaulay2Doc/html/_using_spfunctions_spwith_spoptional_spinputs.html">Optional inputs</a>:<ul><li><span><a href="_to__File.html">toFile => ...</a>, -- Store result in a file.</span></li>
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<div class="single"><h2>Description</h2>
<div><p>Returns a list of</p>
<p>- a <a href="../../Macaulay2Doc/html/___Matrix.html" title="the class of all matrices">Matrix</a> A with the vertices of the convex hull of L in its rows</p>
<p>- a <a href="../../Macaulay2Doc/html/___Matrix.html" title="the class of all matrices">Matrix</a> Adual with the vertices of the dual in its rows</p>
<p>- a list of lists with the faces of the convex hull sorted by increasing dimension.</p>
<p></p>
<p>Each face is a list with two elements. The first is a list of the vertices of the face, the second a list of the vertices of the dual face.</p>
<p>The vertices of the faces are represented by the indices of the rows of A.</p>
<p>This requires that the convex hull of L contains 0 in its interior.</p>
<p>This uses the Convex functions convHull and faces.</p>
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<table class="examples"><tr><td><pre>i1 : L={vector {1,0,0},vector {-1,0,0},vector {0,1,0},vector {0,-1,0},vector {0,0,1},vector {0,0,-1}}
o1 = {| 1 |, | -1 |, | 0 |, | 0 |, | 0 |, | 0 |}
| 0 | | 0 | | 1 | | -1 | | 0 | | 0 |
| 0 | | 0 | | 0 | | 0 | | 1 | | -1 |
o1 : List</pre>
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<tr><td><pre>i2 : P=mConvexHullFacesAndDuals(L)
o2 = {| 1 0 0 |, | 1 -1 -1 |, {{{{}, {1, 2, 3, 4, 5, 6, 7, 8}}}, {{{1},
| -1 0 0 | | -1 -1 -1 |
| 0 1 0 | | 1 1 -1 |
| 0 -1 0 | | -1 1 -1 |
| 0 0 1 | | 1 -1 1 |
| 0 0 -1 | | -1 -1 1 |
| 1 1 1 |
| -1 1 1 |
------------------------------------------------------------------------
{2, 4, 6, 8}}, {{2}, {1, 3, 5, 7}}, {{3}, {1, 2, 5, 6}}, {{4}, {3, 4, 7,
------------------------------------------------------------------------
8}}, {{5}, {1, 2, 3, 4}}, {{6}, {5, 6, 7, 8}}}, {{{2, 5}, {1, 3}}, {{3,
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5}, {1, 2}}, {{1, 5}, {2, 4}}, {{4, 5}, {3, 4}}, {{3, 6}, {5, 6}}, {{2,
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3}, {1, 5}}, {{1, 3}, {2, 6}}, {{2, 4}, {3, 7}}, {{1, 4}, {4, 8}}, {{2,
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6}, {5, 7}}, {{1, 6}, {6, 8}}, {{4, 6}, {7, 8}}}, {{{2, 3, 5}, {1}},
------------------------------------------------------------------------
{{1, 3, 5}, {2}}, {{2, 4, 5}, {3}}, {{1, 4, 5}, {4}}, {{2, 3, 6}, {5}},
------------------------------------------------------------------------
{{1, 3, 6}, {6}}, {{2, 4, 6}, {7}}, {{1, 4, 6}, {8}}}, {{{1, 2, 3, 4, 5,
------------------------------------------------------------------------
6}, {}}}}}
o2 : List</pre>
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<div class="waystouse"><h2>Ways to use <tt>mConvexHullFacesAndDuals</tt> :</h2>
<ul><li>mConvexHullFacesAndDuals(List)</li>
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