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<p><a id="X7DC480E57D26429A" name="X7DC480E57D26429A"></a></p>
<div class="ChapSects"><a href="chap4_mj.html#X7DC480E57D26429A">4 <span class="Heading">Attributes and properties</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4_mj.html#X7A2BE11B87B4A521">4.1 <span class="Heading">Attributes and properties of polyhedron</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X8385F69A87131EAD">4.1-1 Cdd_Canonicalize</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X85E888C97D4E105B">4.1-2 Cdd_V_Rep</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X8414F54E80B312C5">4.1-3 Cdd_H_Rep</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X7CF2AEA1810DCDB9">4.1-4 Cdd_AmbientSpaceDimension</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X81C260A1830AB7D4">4.1-5 Cdd_Dimension</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X82FBBDE082F8D36E">4.1-6 Cdd_GeneratingVertices</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X7E50781780D8F8EB">4.1-7 Cdd_GeneratingRays</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X7A31F87C7903D209">4.1-8 Cdd_Equalities</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X86B3B0468420F573">4.1-9 Cdd_Inequalities</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X869C857285945057">4.1-10 Cdd_InteriorPoint</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X8040DAA2872555F5">4.1-11 Cdd_Faces</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X84B3FAAE7ECF5ACF">4.1-12 Cdd_FacesWithFixedDimension</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X872EC7F186282090">4.1-13 Cdd_FacesWithInteriorPoints</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X7DE87B0984B3F177">4.1-14 Cdd_FacesWithFixedDimensionAndInteriorPoints</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X7B448A2578CB679F">4.1-15 Cdd_Facets</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X7F1FB2B17B561A45">4.1-16 Cdd_Lines</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X7FABEAFE790C7ABD">4.1-17 Cdd_Vertices</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X7D4B6C2F7E71756C">4.1-18 Cdd_IsEmpty</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X79FC1D7D7C266683">4.1-19 Cdd_IsCone</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X80784F1B7FE8419D">4.1-20 Cdd_IsPointed</a></span>
</div></div>
</div>

<h3>4 <span class="Heading">Attributes and properties</span></h3>

<p><a id="X7A2BE11B87B4A521" name="X7A2BE11B87B4A521"></a></p>

<h4>4.1 <span class="Heading">Attributes and properties of polyhedron</span></h4>

<p><a id="X8385F69A87131EAD" name="X8385F69A87131EAD"></a></p>

<h5>4.1-1 Cdd_Canonicalize</h5>

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

<p>The function takes a polyhedron and reduces its defining inequalities (generators set) by deleting all redundant inequalities (generators).</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">A:= Cdd_PolyhedronByInequalities( [ [ 0, 2, 6 ], [ 0, 1, 3 ], [1, 4, 10 ] ] );</span>
<Polyhedron given by its H-representation>
<span class="GAPprompt">gap></span> <span class="GAPinput">B:= Cdd_Canonicalize( A );</span>
<Polyhedron given by its H-representation>
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( B );</span>
H-representation
begin
   2 X 3  rational

   0   1   3
   1   4  10
end
</pre></div>

<p><a id="X85E888C97D4E105B" name="X85E888C97D4E105B"></a></p>

<h5>4.1-2 Cdd_V_Rep</h5>

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

<p>The function takes a polyhedron and returns its reduced <span class="SimpleMath">\(V\)</span>-representation.</p>

<p><a id="X8414F54E80B312C5" name="X8414F54E80B312C5"></a></p>

<h5>4.1-3 Cdd_H_Rep</h5>

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

<p>The function takes a polyhedron and returns its reduced <span class="SimpleMath">\(H\)</span>-representation.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">A:= Cdd_PolyhedronByInequalities( [ [ 0, 1, 1 ], [ 0, 5, 5 ] ] );</span>
<Polyhedron given by its H-representation>
<span class="GAPprompt">gap></span> <span class="GAPinput">B:= Cdd_V_Rep( A );</span>
<Polyhedron given by its V-representation>
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( B );</span>
V-representation
linearity 1, [ 2 ]
begin
   2 X 3  rational

   0   1   0
   0  -1   1
end
<span class="GAPprompt">gap></span> <span class="GAPinput">C:= Cdd_H_Rep( B );</span>
<Polyhedron given by its H-representation>
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( C );</span>
H-representation
begin
   1 X 3  rational

   0  1  1
end
<span class="GAPprompt">gap></span> <span class="GAPinput">D:= Cdd_PolyhedronByInequalities( [ [ 0, 1, 1, 34, 22, 43 ],</span>
<span class="GAPprompt">></span> <span class="GAPinput">[ 11, 2, 2, 54, 53, 221 ], [33, 23, 45, 2, 40, 11 ] ] );</span>
<Polyhedron given by its H-representation>
<span class="GAPprompt">gap></span> <span class="GAPinput">Cdd_V_Rep( D );</span>
<Polyhedron given by its V-representation>
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( last );</span>
V-representation
linearity 2, [ 5, 6 ]
begin
   6 X 6  rational

   1  -743/14   369/14    11/14        0        0
   0    -1213      619       22        0        0
   0       -1        1        0        0        0
   0      764     -390      -11        0        0
   0   -13526     6772       99      154        0
   0  -116608    59496     1485        0      154
end
</pre></div>

<p><a id="X7CF2AEA1810DCDB9" name="X7CF2AEA1810DCDB9"></a></p>

<h5>4.1-4 Cdd_AmbientSpaceDimension</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Cdd_AmbientSpaceDimension</code>( <var class="Arg">P</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: The dimension of the ambient space of the polyhedron(i.e., the space that contains <span class="SimpleMath">\(P\)</span>).</p>

<p><a id="X81C260A1830AB7D4" name="X81C260A1830AB7D4"></a></p>

<h5>4.1-5 Cdd_Dimension</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Cdd_Dimension</code>( <var class="Arg">P</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: The dimension of the polyhedron, where the dimension, <span class="SimpleMath">\(\mathrm{dim}(P)\)</span>, of a polyhedron <span class="SimpleMath">\(P\)</span> is the maximum number of affinely independent points in <span class="SimpleMath">\(P\)</span> minus 1.</p>

<p><a id="X82FBBDE082F8D36E" name="X82FBBDE082F8D36E"></a></p>

<h5>4.1-6 Cdd_GeneratingVertices</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Cdd_GeneratingVertices</code>( <var class="Arg">P</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: The reduced generating vertices of the polyhedron</p>

<p><a id="X7E50781780D8F8EB" name="X7E50781780D8F8EB"></a></p>

<h5>4.1-7 Cdd_GeneratingRays</h5>

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

<p>The output is the reduced generating rays of the polyhedron</p>

<p><a id="X7A31F87C7903D209" name="X7A31F87C7903D209"></a></p>

<h5>4.1-8 Cdd_Equalities</h5>

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

<p>The output is the reduced equalities of the polyhedron.</p>

<p><a id="X86B3B0468420F573" name="X86B3B0468420F573"></a></p>

<h5>4.1-9 Cdd_Inequalities</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Cdd_Inequalities</code>( <var class="Arg">P</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>The output is the reduced inequalities of the polyhedron.</p>

<p><a id="X869C857285945057" name="X869C857285945057"></a></p>

<h5>4.1-10 Cdd_InteriorPoint</h5>

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

<p>The output is an interior point in the polyhedron</p>

<p><a id="X8040DAA2872555F5" name="X8040DAA2872555F5"></a></p>

<h5>4.1-11 Cdd_Faces</h5>

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

<p>This function takes a <span class="SimpleMath">\(H\)</span>-represented polyhedron <em>P</em> and returns a list. Every entry in this list is a again a list, contains the dimension and linearity of the face defined as a polyhedron over the same system of inequalities.</p>

<p><a id="X84B3FAAE7ECF5ACF" name="X84B3FAAE7ECF5ACF"></a></p>

<h5>4.1-12 Cdd_FacesWithFixedDimension</h5>

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

<p>This function takes a <span class="SimpleMath">\(H\)</span>-represented polyhedron <em>P</em> and a positive integer <em>d</em>. The output is a list. Every entry in this list is the linearity of an <em>d</em>- dimensional face of <em>P</em> defined as a polyhedron over the same system of inequalities.</p>

<p><a id="X872EC7F186282090" name="X872EC7F186282090"></a></p>

<h5>4.1-13 Cdd_FacesWithInteriorPoints</h5>

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

<p>This function takes a <span class="SimpleMath">\(H\)</span>-represented polyhedron <em>P</em> and returns a list. Every entry in this list is a again a list, contains the dimension, linearity of the face defined as a polyhedron over the same system of inequalities and an interior point in the face.</p>

<p><a id="X7DE87B0984B3F177" name="X7DE87B0984B3F177"></a></p>

<h5>4.1-14 Cdd_FacesWithFixedDimensionAndInteriorPoints</h5>

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

<p>This function takes a <span class="SimpleMath">\(H\)</span>-represented polyhedron <em>P</em> and a positive integer <em>d</em>. The output is a list. Every entry in this list is a again a list, contains the linearity of the face defined as a polyhedron over the same system of inequalities and an interior point in this face.</p>

<p><a id="X7B448A2578CB679F" name="X7B448A2578CB679F"></a></p>

<h5>4.1-15 Cdd_Facets</h5>

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

<p>This function takes a <span class="SimpleMath">\(H\)</span>-represented polyhedron <em>P</em> and returns a list. Every entry in this is the linearity of a facet defined as a polyhedron over the same system of inequalities.</p>

<p><a id="X7F1FB2B17B561A45" name="X7F1FB2B17B561A45"></a></p>

<h5>4.1-16 Cdd_Lines</h5>

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

<p>This function takes a <span class="SimpleMath">\(H\)</span>-represented polyhedron <em>P</em> and returns a list. Every entry in this is the linearity of a ray (<span class="SimpleMath">\(1\)</span>-dimensional face) defined as a polyhedron over the same system of inequalities.</p>

<p><a id="X7FABEAFE790C7ABD" name="X7FABEAFE790C7ABD"></a></p>

<h5>4.1-17 Cdd_Vertices</h5>

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

<p>This function takes a <span class="SimpleMath">\(H\)</span>-represented polyhedron <em>P</em> and returns a list. Every entry in this list is the linearity of a vertex defined as a polyhedron over the same system of inequalities.</p>

<p><a id="X7D4B6C2F7E71756C" name="X7D4B6C2F7E71756C"></a></p>

<h5>4.1-18 Cdd_IsEmpty</h5>

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

<p>The output is <code class="code">true</code> if the polyhedron is empty and <code class="code">false</code> otherwise</p>

<p><a id="X79FC1D7D7C266683" name="X79FC1D7D7C266683"></a></p>

<h5>4.1-19 Cdd_IsCone</h5>

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

<p>The output is <code class="code">true</code> if the polyhedron is cone and <code class="code">false</code> otherwise</p>

<p><a id="X80784F1B7FE8419D" name="X80784F1B7FE8419D"></a></p>

<h5>4.1-20 Cdd_IsPointed</h5>

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

<p>The output is <code class="code">true</code> if the polyhedron is pointed and <code class="code">false</code> otherwise</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">poly:= Cdd_PolyhedronByInequalities( [ [ 1, 3, 4, 5, 7 ], [ 1, 3, 5, 12, 34 ],</span>
<span class="GAPprompt">></span> <span class="GAPinput">[ 9, 3, 0, 2, 13 ]  ], [ 1 ] );</span>
<Polyhedron given by its H-representation>
<span class="GAPprompt">gap></span> <span class="GAPinput">Cdd_InteriorPoint( poly );</span>
[ -194/75, 46/25, -3/25, 0 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">Cdd_FacesWithInteriorPoints( poly );</span>
[ [ 3, [ 1 ], [ -194/75, 46/25, -3/25, 0 ] ], [ 2, [ 1, 2 ],
[ -62/25, 49/25, -7/25, 0 ] ], [ 1, [ 1, 2, 3 ],
[ -209/75, 56/25, -8/25, 0 ] ], [ 2, [ 1, 3 ], [ -217/75, 53/25, -4/25, 0 ] ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">Cdd_Dimension( poly );</span>
3
<span class="GAPprompt">gap></span> <span class="GAPinput">Cdd_IsPointed( poly );</span>
false
<span class="GAPprompt">gap></span> <span class="GAPinput">Cdd_IsEmpty( poly );</span>
false
<span class="GAPprompt">gap></span> <span class="GAPinput">Cdd_Faces( poly );</span>
[ [ 3, [ 1 ] ], [ 2, [ 1, 2 ] ], [ 1, [ 1, 2, 3 ] ], [ 2, [ 1, 3  ] ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">poly1 := Cdd_ExtendLinearity( poly, [ 1, 2, 3 ] );</span>
<Polyhedron given by its H-representation>
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( poly1 );</span>
H-representation 
linearity 3, [ 1, 2, 3 ]
begin
   3 X 5  rational

   1   3   4   5   7 
   1   3   5  12  34 
   9   3   0   2  13 
end
<span class="GAPprompt">gap></span> <span class="GAPinput">Cdd_Dimension( poly1 );</span>
1
<span class="GAPprompt">gap></span> <span class="GAPinput">Cdd_Facets( poly );</span>
[ [ 1, 2 ], [ 1, 3 ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">Cdd_GeneratingVertices( poly );</span>
[ [ -209/75, 56/25, -8/25, 0 ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">Cdd_GeneratingRays( poly );</span>
[ [ -97, 369, -342, 75 ], [ -8, -9, 12, 0 ],
[ 23, -21, 3, 0 ], [ 97, -369, 342, -75 ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">Cdd_Inequalities( poly );</span>
[ [ 1, 3, 5, 12, 34 ], [ 9, 3, 0, 2, 13 ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">Cdd_Equalities( poly );</span>
[ [ 1, 3, 4, 5, 7 ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">P := Cdd_FourierProjection( poly, 2);</span>
<Polyhedron given by its H-representation>
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( P );</span>
H-representation 
linearity 1, [ 3 ]
begin 
   3 X 5  rational

    9    3    0    2   13 
   -1   -3    0   23  101 
    0    0    1    0    0 
end
</pre></div>


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