Gedichte Musik Bilder Quellcodebibliothek Diashow Normaldarstellung
Quellcodebibliothek Statistik Leitseite products/Sources/formale Sprachen/GAP/pkg/nconvex/doc/   (Algebra von RWTH Aachen Version 4.15.1©)  Datei vom 28.11.2024 mit Größe 29 kB image not shown  

Quelle  chap7.html   Sprache: HTML

 
 products/Sources/formale Sprachen/GAP/pkg/nconvex/doc/chap7.html


<?xml version="1.0" encoding="UTF-8"?>

<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
         "http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">

<html xmlns="http://www.w3.org/1999/xhtml" xml:lang="en">
<head>
<title>GAP (NConvex) - Chapter 7: Polytopes</title>
<meta http-equiv="content-type" content="text/html; charset=UTF-8" />
<meta name="generator" content="GAPDoc2HTML" />
<link rel="stylesheet" type="text/css" href="manual.css" />
<script src="manual.js" type="text/javascript"></script>
<script type="text/javascript">overwriteStyle();</script>
</head>
<body class="chap7"  onload="jscontent()">


<div class="chlinktop"><span class="chlink1">Goto Chapter: </span><a href="chap0.html">Top</a>  <a href="chap1.html">1</a>  <a href="chap2.html">2</a>  <a href="chap3.html">3</a>  <a href="chap4.html">4</a>  <a href="chap5.html">5</a>  <a href="chap6.html">6</a>  <a href="chap7.html">7</a>  <a href="chapInd.html">Ind</a>  </div>

<div class="chlinkprevnexttop"> <a href="chap0.html">[Top of Book]</a>   <a href="chap0.html#contents">[Contents]</a>    <a href="chap6.html">[Previous Chapter]</a>    <a href="chapInd.html">[Next Chapter]</a>   </div>

<p id="mathjaxlink" class="pcenter"><a href="chap7_mj.html">[MathJax on]</a></p>
<p><a id="X7B933C4686727183" name="X7B933C4686727183"></a></p>
<div class="ChapSects"><a href="chap7.html#X7B933C4686727183">7 <span class="Heading">Polytopes</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap7.html#X801887DC7E5F6AB7">7.1 <span class="Heading">Creating polytopes</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7.html#X83FB00B383A19F03">7.1-1 PolytopeByInequalities</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7.html#X81DC4EE8849B08DA">7.1-2 Polytope</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap7.html#X7C701DBF7BAE649A">7.2 <span class="Heading">Attributes</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7.html#X7D8727917B72396A">7.2-1 ExternalCddPolytope</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7.html#X83B1CBC67B34BE04">7.2-2 LatticePoints</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7.html#X7FB9E36C7B6C616F">7.2-3 RelativeInteriorLatticePoints</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7.html#X816ACBEF786C23F2">7.2-4 VerticesOfPolytope</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7.html#X805FCCD583DACF8F">7.2-5 Vertices</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7.html#X7B39989A7F321147">7.2-6 DefiningInequalities</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7.html#X7DAF68C08043105D">7.2-7 EqualitiesOfPolytope</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7.html#X7FDABF3D84E67F2C">7.2-8 FacetInequalities</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7.html#X7EC0B0818106EA50">7.2-9 VerticesInFacets</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7.html#X7DE7E52D800A7258">7.2-10 NormalFan</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7.html#X8453091478A4C0F1">7.2-11 FaceFan</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7.html#X85EC550B7FCAECB5">7.2-12 AffineCone</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7.html#X7ACC98FD78ACDC57">7.2-13 PolarPolytope</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7.html#X8349F8DA81418C35">7.2-14 DualPolytope</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap7.html#X871597447BB998A1">7.3 <span class="Heading">Properties</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7.html#X863F64947BDFC654">7.3-1 IsEmpty</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7.html#X85181A5986107254">7.3-2 IsLatticePolytope</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7.html#X7918DAE87B159619">7.3-3 IsVeryAmple</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7.html#X851AD9E684CA7434">7.3-4 IsNormalPolytope</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7.html#X7F39F2BF835AA337">7.3-5 IsSimplicial</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7.html#X797F6A0778C70DD0">7.3-6 IsSimplexPolytope</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7.html#X815A80E785E52474">7.3-7 IsSimplePolytope</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7.html#X85BE2236780D0957">7.3-8 IsReflexive</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7.html#X85DEC39081FD04E9">7.3-9 IsFanoPolytope</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7.html#X780D5BDE7C177166">7.3-10 IsCanonicalFanoPolytope</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7.html#X78B2B9747E9A6FEA">7.3-11 IsTerminalFanoPolytope</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7.html#X7946CEA886A48F96">7.3-12 IsSmoothFanoPolytope</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap7.html#X7B71F2A2834E84CE">7.4 <span class="Heading">Operations on polytopes</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7.html#X7F9303847F938C87"><code>7.4-1 \+</code></a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7.html#X7AEF19987E29612E"><code>7.4-2 \*</code></a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7.html#X8079AF527BA681D1">7.4-3 IntersectionOfPolytopes</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7.html#X7F2AE1197AAFF273">7.4-4 RandomInteriorPoint</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7.html#X7E36517B808D2E44">7.4-5 IsInteriorPoint</a></span>
</div></div>
</div>

<h3>7 <span class="Heading">Polytopes</span></h3>

<p><a id="X801887DC7E5F6AB7" name="X801887DC7E5F6AB7"></a></p>

<h4>7.1 <span class="Heading">Creating polytopes</span></h4>

<p><a id="X83FB00B383A19F03" name="X83FB00B383A19F03"></a></p>

<h5>7.1-1 PolytopeByInequalities</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ PolytopeByInequalities</code>( <var class="Arg">L</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: a Polytope Object</p>

<p>The operation takes a list <span class="Math">L</span> of lists <span class="Math">[L_1, L_2, ...]</span> where each <span class="Math">L_j</span> represents an inequality and returns the polytope defined by them (if they define a polytope). For example the <span class="Math">j</span>'th entry L_j = [c_j,a_{j1},a_{j2},...,a_{jn}] corresponds to the inequality c_j+\sum_{i=1}^n a_{ji}x_i \geq 0.



<p><a id="X81DC4EE8849B08DA" name="X81DC4EE8849B08DA"></a></p>

<h5>7.1-2 Polytope</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Polytope</code>( <var class="Arg">L</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: a Polytope Object</p>

<p>The operation takes the list of the vertices and returns the polytope defined by them.</p>

<p><a id="X7C701DBF7BAE649A" name="X7C701DBF7BAE649A"></a></p>

<h4>7.2 <span class="Heading">Attributes</span></h4>

<p><a id="X7D8727917B72396A" name="X7D8727917B72396A"></a></p>

<h5>7.2-1 ExternalCddPolytope</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ExternalCddPolytope</code>( <var class="Arg">P</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: a CddPolyhedron</p>

<p>Converts the polytope to a CddPolyhedron. The operations of CddInterface can then be applied on this polyhedron.</p>

<p><a id="X83B1CBC67B34BE04" name="X83B1CBC67B34BE04"></a></p>

<h5>7.2-2 LatticePoints</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ LatticePoints</code>( <var class="Arg">P</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: a List</p>

<p>The operation returns the list of integer points inside the polytope.</p>

<p><a id="X7FB9E36C7B6C616F" name="X7FB9E36C7B6C616F"></a></p>

<h5>7.2-3 RelativeInteriorLatticePoints</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ RelativeInteriorLatticePoints</code>( <var class="Arg">P</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: a List</p>

<p>The operation returns the interior lattice points inside the polytope.</p>

<p><a id="X816ACBEF786C23F2" name="X816ACBEF786C23F2"></a></p>

<h5>7.2-4 VerticesOfPolytope</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ VerticesOfPolytope</code>( <var class="Arg">P</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: a list of lists</p>

<p>The operation returns the vertices of the polytope</p>

<p><a id="X805FCCD583DACF8F" name="X805FCCD583DACF8F"></a></p>

<h5>7.2-5 Vertices</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Vertices</code>( <var class="Arg">P</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: a list of lists</p>

<p>The same output as <code class="code">VerticesOfPolytope</code>.</p>

<p><a id="X7B39989A7F321147" name="X7B39989A7F321147"></a></p>

<h5>7.2-6 DefiningInequalities</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ DefiningInequalities</code>( <var class="Arg">P</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: a list of lists</p>

<p>The operation returns the defining inequalities of the polytope. I.e., a list of lists <span class="Math">[L_1, L_2, ...]</span> where each <span class="Math">L_j=[c_j,a_{j1},a_{j2},...,a_{jn}]</span> represents the inequality <span class="Math">c_j+\sum_{i=1}^n a_{ji}x_i \geq 0</span>. If <span class="Math">L</span> and <span class="Math">-L</span> occur in the output then <span class="Math">L</span> is called a defining-equality of the polytope.</p>

<p><a id="X7DAF68C08043105D" name="X7DAF68C08043105D"></a></p>

<h5>7.2-7 EqualitiesOfPolytope</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ EqualitiesOfPolytope</code>( <var class="Arg">P</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: a list of lists</p>

<p>The operation returns the defining-equalities of the polytope.</p>

<p><a id="X7FDABF3D84E67F2C" name="X7FDABF3D84E67F2C"></a></p>

<h5>7.2-8 FacetInequalities</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ FacetInequalities</code>( <var class="Arg">P</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: a list of lists</p>

<p>The operation returns the list of the inequalities of the facets. Each defining inequality that is not defining-equality of the polytope is a facet inequality.</p>

<p><a id="X7EC0B0818106EA50" name="X7EC0B0818106EA50"></a></p>

<h5>7.2-9 VerticesInFacets</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ VerticesInFacets</code>( <var class="Arg">P</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: a list of lists</p>

<p>The operation returns list of lists <span class="Math">L</span>. The entries of each <span class="Math">L_j</span> in <span class="Math">L</span> consists of <span class="Math">0</span>'s or 1's. For instance, if <span class="Math">L_j=[1,0,0,1,0,1]</span>, then The polytope has <span class="Math">6</span> vertices and the vertices of the <span class="Math">j</span>'th facet are \{V_1,V_4,V_6\}.



<p><a id="X7DE7E52D800A7258" name="X7DE7E52D800A7258"></a></p>

<h5>7.2-10 NormalFan</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ NormalFan</code>( <var class="Arg">P</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: a fan</p>

<p>The operation returns the normal fan of the given polytope.</p>

<p><a id="X8453091478A4C0F1" name="X8453091478A4C0F1"></a></p>

<h5>7.2-11 FaceFan</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ FaceFan</code>( <var class="Arg">P</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: a fan</p>

<p>The operation returns the face fan of the given polytope. Remember that the face fan of a polytope is isomorphic to the normal fan of its polar polytope.</p>

<p><a id="X85EC550B7FCAECB5" name="X85EC550B7FCAECB5"></a></p>

<h5>7.2-12 AffineCone</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ AffineCone</code>( <var class="Arg">P</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: a cone</p>

<p>If the ambient space of the polytope is <span class="Math">\mathrm{R}^n</span>, then the output is a cone in <span class="Math">\mathrm{R}^{n+1}</span>. The defining rays of the cone are <span class="Math">{[a_{j1},a_{j2},...,a_{jn},1]}_j</span> such that <span class="Math">V_j=[a_{j1},a_{j2},...,a_{jn}]</span> is a vertex in the polytope.</p>

<p><a id="X7ACC98FD78ACDC57" name="X7ACC98FD78ACDC57"></a></p>

<h5>7.2-13 PolarPolytope</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ PolarPolytope</code>( <var class="Arg">P</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: a Polytope</p>

<p>The operation returns the polar polytope of the given polytope.</p>

<p><a id="X8349F8DA81418C35" name="X8349F8DA81418C35"></a></p>

<h5>7.2-14 DualPolytope</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ DualPolytope</code>( <var class="Arg">P</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: a Polytope</p>

<p>The operation returns the dual polytope of the given polytope.</p>

<p><a id="X871597447BB998A1" name="X871597447BB998A1"></a></p>

<h4>7.3 <span class="Heading">Properties</span></h4>

<p><a id="X863F64947BDFC654" name="X863F64947BDFC654"></a></p>

<h5>7.3-1 IsEmpty</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsEmpty</code>( <var class="Arg">P</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>Returns: a true or false</p>

<p>Returns whether the polytope empty or not</p>

<p><a id="X85181A5986107254" name="X85181A5986107254"></a></p>

<h5>7.3-2 IsLatticePolytope</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsLatticePolytope</code>( <var class="Arg">P</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>Returns: a true or false</p>

<p>Returns whether the polytope is lattice polytope or not.</p>

<p><a id="X7918DAE87B159619" name="X7918DAE87B159619"></a></p>

<h5>7.3-3 IsVeryAmple</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsVeryAmple</code>( <var class="Arg">P</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>Returns: a true or false</p>

<p>Returns whether the polytope is very ample or not.</p>

<p><a id="X851AD9E684CA7434" name="X851AD9E684CA7434"></a></p>

<h5>7.3-4 IsNormalPolytope</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsNormalPolytope</code>( <var class="Arg">P</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>Returns: a true or false</p>

<p>Returns whether the polytope is normal or not.</p>

<p><a id="X7F39F2BF835AA337" name="X7F39F2BF835AA337"></a></p>

<h5>7.3-5 IsSimplicial</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsSimplicial</code>( <var class="Arg">P</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>Returns: a true or false</p>

<p>Returns whether the polytope is simplicial or not.</p>

<p><a id="X797F6A0778C70DD0" name="X797F6A0778C70DD0"></a></p>

<h5>7.3-6 IsSimplexPolytope</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsSimplexPolytope</code>( <var class="Arg">P</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>Returns: a true or false</p>

<p>Returns whether the polytope is simplex polytope or not.</p>

<p><a id="X815A80E785E52474" name="X815A80E785E52474"></a></p>

<h5>7.3-7 IsSimplePolytope</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsSimplePolytope</code>( <var class="Arg">P</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>Returns: a true or false</p>

<p>Returns whether the polytope is simple or not.</p>

<p><a id="X85BE2236780D0957" name="X85BE2236780D0957"></a></p>

<h5>7.3-8 IsReflexive</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsReflexive</code>( <var class="Arg">P</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>Returns: a true or false</p>

<p>Returns whether the polytope is reflexive or not, i.e., if its dual polytope is lattice polytope.</p>

<p><a id="X85DEC39081FD04E9" name="X85DEC39081FD04E9"></a></p>

<h5>7.3-9 IsFanoPolytope</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsFanoPolytope</code>( <var class="Arg">P</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>Returns: a true or false</p>

<p>returns whether the polytope is Fano or not. Fano polytope is a full dimensional lattice polytope whose vertices are primitive elements in the containing lattice, i.e., each vertex is not a positive integer multiple of any other lattice element.</p>

<p><a id="X780D5BDE7C177166" name="X780D5BDE7C177166"></a></p>

<h5>7.3-10 IsCanonicalFanoPolytope</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsCanonicalFanoPolytope</code>( <var class="Arg">P</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>Returns: a true or false</p>

<p>returns whether the polytope is canonical Fano or not. A canonical Fano polytope is a full dimensional lattice polytope whose relative interior contains only one lattice point, namely the origin.</p>

<p><a id="X78B2B9747E9A6FEA" name="X78B2B9747E9A6FEA"></a></p>

<h5>7.3-11 IsTerminalFanoPolytope</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsTerminalFanoPolytope</code>( <var class="Arg">P</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>Returns: a true or false</p>

<p>returns whether the polytope is terminal Fano or not. A terminal Fano polytope is a full dimensional lattice polytope whose lattice points are its vertices and the origin.</p>

<p><a id="X7946CEA886A48F96" name="X7946CEA886A48F96"></a></p>

<h5>7.3-12 IsSmoothFanoPolytope</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsSmoothFanoPolytope</code>( <var class="Arg">P</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>Returns: a true or false</p>

<p>Returns whether the polytope is smooth fano polytope or not, i.e, if the vertices in each facet form a basis for the containing lattice or not. polytope.</p>

<p><a id="X7B71F2A2834E84CE" name="X7B71F2A2834E84CE"></a></p>

<h4>7.4 <span class="Heading">Operations on polytopes</span></h4>

<p><a id="X7F9303847F938C87" name="X7F9303847F938C87"></a></p>

<h5><code>7.4-1 \+</code></h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ \+</code>( <var class="Arg">P1</var>, <var class="Arg">P2</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: a polytope</p>

<p>The output is Minkowski sum of the input polytopes.</p>

<p><a id="X7AEF19987E29612E" name="X7AEF19987E29612E"></a></p>

<h5><code>7.4-2 \*</code></h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ \*</code>( <var class="Arg">n</var>, <var class="Arg">P</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: a polytope</p>

<p>The output is Minkowski sum of the input polytope with itself <span class="Math">n</span> times.</p>

<p><a id="X8079AF527BA681D1" name="X8079AF527BA681D1"></a></p>

<h5>7.4-3 IntersectionOfPolytopes</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IntersectionOfPolytopes</code>( <var class="Arg">P1</var>, <var class="Arg">P2</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: a polytope</p>

<p>The output is the intersection of the input polytopes.</p>

<p><a id="X7F2AE1197AAFF273" name="X7F2AE1197AAFF273"></a></p>

<h5>7.4-4 RandomInteriorPoint</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ RandomInteriorPoint</code>( <var class="Arg">P</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: a list</p>

<p>Returns a random interior point in the polytope.</p>

<p><a id="X7E36517B808D2E44" name="X7E36517B808D2E44"></a></p>

<h5>7.4-5 IsInteriorPoint</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsInteriorPoint</code>( <var class="Arg">M</var>, <var class="Arg">P</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: true or false</p>

<p>Checks if the given point is interior point of the polytope.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">P:= Polytope( [ [ 0, 0, 0 ], [ 1, 0, 0 ], [ 0, 1, 0 ], [ 1, 1, 2 ] ] );</span>
<A polytope in |R^3>
<span class="GAPprompt">gap></span> <span class="GAPinput">IsNormalPolytope( P );</span>
false
<span class="GAPprompt">gap></span> <span class="GAPinput">IsVeryAmple( P );</span>
false
<span class="GAPprompt">gap></span> <span class="GAPinput">Q:= Polytope( [ [ 0, 0, 0 ], [ 1, 0, 0 ], [ 0, 1, 0 ], [ 1, 1, 1 ] ] );</span>
<A polytope in |R^3>
<span class="GAPprompt">gap></span> <span class="GAPinput">IsNormalPolytope( Q );</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">IsVeryAmple( Q );</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">Q;</span>
<A normal very ample polytope in |R^3 with 4 vertices>
<span class="GAPprompt">gap></span> <span class="GAPinput">T:= Polytope( [ [ 0, 0, 0 ], [ 1, 0, 0 ], [ 0, 1, 0 ], [ 1, 1, 4 ] ] ); </span>
<A polytope in |R^3>
<span class="GAPprompt">gap></span> <span class="GAPinput">I:= Polytope( [ [ 0, 0, 0 ], [ 0, 0, 1 ] ] );</span>
<A polytope in |R^3>
<span class="GAPprompt">gap></span> <span class="GAPinput">J:= T + I; </span>
<A polytope in |R^3>
<span class="GAPprompt">gap></span> <span class="GAPinput">IsVeryAmple( J );</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">IsNormalPolytope( J );</span>
false
<span class="GAPprompt">gap></span> <span class="GAPinput">J;</span>
<A very ample polytope in |R^3 with 8 vertices>
<span class="GAPprompt">gap></span> <span class="GAPinput"># Example 2.2.20 Cox, Toric Varieties</span>
<span class="GAPprompt">></span> <span class="GAPinput">A:= [ [1,1,1,0,0,0], [1,1,0,1,0,0], [1,0,1,0,1,0], [ 1,0,0,1,0,1], </span>
<span class="GAPprompt">></span> <span class="GAPinput">[ 1,0,0,0,1,1], [ 0,1,1,0,0,1], [0,1,0,1,1,0], [0,1,0,0,1,1], </span>
<span class="GAPprompt">></span> <span class="GAPinput">[0,0,1,1,1,0], [0,0,1,1,0,1] ];</span>
[ [ 1, 1, 1, 0, 0, 0 ], [ 1, 1, 0, 1, 0, 0 ], [ 1, 0, 1, 0, 1, 0 ],
[ 1, 0, 0, 1, 0, 1 ], [ 1, 0, 0, 0, 1, 1 ], [ 0, 1, 1, 0, 0, 1 ], 
 [ 0, 1, 0, 1, 1, 0 ], [ 0, 1, 0, 0, 1, 1 ], [ 0, 0, 1, 1, 1, 0 ], 
[ 0, 0, 1, 1, 0, 1 ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">H:= Polytope( A );</span>
<A polytope in |R^6>
<span class="GAPprompt">gap></span> <span class="GAPinput">IsVeryAmple( H );   </span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">IsNormalPolytope( H );</span>
false
<span class="GAPprompt">gap></span> <span class="GAPinput">H;</span>
<A very ample polytope in |R^6 with 10 vertices>
<span class="GAPprompt">gap></span> <span class="GAPinput">l:= [ [ 0, 0, 1 ], [ 0, 0, 0 ], [ 1, 0, 0 ], [ 1, 0, 1 ], [ 0, 1, 0 ], </span>
<span class="GAPprompt">></span> <span class="GAPinput">[ 0, 1, 1 ], [ 1, 1, 4 ], [ 1, 1, 5 ] ];;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">P:= Polytope( l );</span>
<A polytope in |R^3>
<span class="GAPprompt">gap></span> <span class="GAPinput">IsNormalPolytope( P );</span>
false
<span class="GAPprompt">gap></span> <span class="GAPinput">lattic_points:= LatticePoints( P );</span>
[ [ 0, 0, 0 ], [ 0, 0, 1 ], [ 0, 1, 0 ], [ 0, 1, 1 ], [ 1, 0, 0 ], [ 1, 0, 1 ], 
[ 1, 1, 4 ], [ 1, 1, 5 ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">u:= Cartesian( lattic_points, lattic_points );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">k:= Set( List( u, u-> u[1]+u[2] ) );</span>
[ [ 0, 0, 0 ], [ 0, 0, 1 ], [ 0, 0, 2 ], [ 0, 1, 0 ], [ 0, 1, 1 ], [ 0, 1, 2 ],
[ 0, 2, 0 ], [ 0, 2, 1 ], [ 0, 2, 2 ], [ 1, 0, 0 ], [ 1, 0, 1 ], [ 1, 0, 2 ], 
[ 1, 1, 0 ], [ 1, 1, 1 ], [ 1, 1, 2 ], [ 1, 1, 4 ], [ 1, 1, 5 ], [ 1, 1, 6 ], 
[ 1, 2, 4 ], [ 1, 2, 5 ], [ 1, 2, 6 ], [ 2, 0, 0 ], [ 2, 0, 1 ], [ 2, 0, 2 ], 
[ 2, 1, 4 ], [ 2, 1, 5 ], [ 2, 1, 6 ], [ 2, 2, 8 ], [ 2, 2, 9 ], [ 2, 2, 10 ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">Q:= 2*P;</span>
<A polytope in |R^3 with 8 vertices>
<span class="GAPprompt">gap></span> <span class="GAPinput">LatticePoints( Q );</span>
[ [ 0, 0, 0 ], [ 0, 0, 1 ], [ 0, 0, 2 ], [ 0, 1, 0 ], [ 0, 1, 1 ], [ 0, 1, 2 ],
[ 0, 2, 0 ], [ 0, 2, 1 ], [ 0, 2, 2 ], [ 1, 0, 0 ], 
  [ 1, 0, 1 ], [ 1, 0, 2 ], [ 1, 1, 0 ], [ 1, 1, 1 ], [ 1, 1, 2 ], [ 1, 1, 3 ], 
[ 1, 1, 4 ], [ 1, 1, 5 ], [ 1, 1, 6 ], [ 1, 2, 4 ], [ 1, 2, 5 ], [ 1, 2, 6 ], 
[ 2, 0, 0 ], [ 2, 0, 1 ], [ 2, 0, 2 ], [ 2, 1, 4 ], 
  [ 2, 1, 5 ], [ 2, 1, 6 ], [ 2, 2, 8 ], [ 2, 2, 9 ], [ 2, 2, 10 ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">P:= Polytope( [ [ 1, 1 ], [ 1, -1 ], [ -1, 1 ], [ -1, -1 ] ] );</span>
<A polytope in |R^2>
<span class="GAPprompt">gap></span> <span class="GAPinput">Q:= PolarPolytope( P );</span>
<A polytope in |R^2>
<span class="GAPprompt">gap></span> <span class="GAPinput">Vertices( Q );</span>
[ [ -1, 0 ], [ 0, -1 ], [ 0, 1 ], [ 1, 0 ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">T := PolarPolytope( Q );</span>
<A polytope in |R^2>
<span class="GAPprompt">gap></span> <span class="GAPinput">Vertices( T );</span>
[ [ -1, -1 ], [ -1, 1 ], [ 1, -1 ], [ 1, 1 ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">P:= Polytope( [ [ 0, 0 ], [ 1, -1], [ -1, 1 ], [ -1, -1 ] ] );</span>
<A polytope in |R^2>
<span class="GAPprompt">gap></span> <span class="GAPinput"># PolarPolytope( P );;</span>
</pre></div>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">P := PolytopeByInequalities( [ [ 0, 0, 1 ], [ 1, -1, -1 ], [ 1, 1, -1 ] ] );</span>
<A polytope in |R^2>
<span class="GAPprompt">gap></span> <span class="GAPinput">Vertices( P );</span>
[ [ -1, 0 ], [ 0, 1 ], [ 1, 0 ] ]
</pre></div>


<div class="chlinkprevnextbot"> <a href="chap0.html">[Top of Book]</a>   <a href="chap0.html#contents">[Contents]</a>    <a href="chap6.html">[Previous Chapter]</a>    <a href="chapInd.html">[Next Chapter]</a>   </div>


<div class="chlinkbot"><span class="chlink1">Goto Chapter: </span><a href="chap0.html">Top</a>  <a href="chap1.html">1</a>  <a href="chap2.html">2</a>  <a href="chap3.html">3</a>  <a href="chap4.html">4</a>  <a href="chap5.html">5</a>  <a href="chap6.html">6</a>  <a href="chap7.html">7</a>  <a href="chapInd.html">Ind</a>  </div>

<hr />
<p class="foot">generated by <a href="https://www.math.rwth-aachen.de/~Frank.Luebeck/GAPDoc">GAPDoc2HTML</a></p>
</body>
</html>

97%


¤ Dauer der Verarbeitung: 0.15 Sekunden  (vorverarbeitet)  ¤

*© Formatika GbR, Deutschland




Wurzel

Suchen

Beweissystem der NASA

Beweissystem Isabelle

NIST Cobol Testsuite

Cephes Mathematical Library

Wiener Entwicklungsmethode

Haftungshinweis

Die Informationen auf dieser Webseite wurden nach bestem Wissen sorgfältig zusammengestellt. Es wird jedoch weder Vollständigkeit, noch Richtigkeit, noch Qualität der bereit gestellten Informationen zugesichert.

Bemerkung:

Die farbliche Syntaxdarstellung ist noch experimentell.