user_guide:tutorials:apps_polytope

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user_guide:tutorials:apps_polytope [2019/01/25 09:35] – ↷ Links adapted because of a move operation oroehriguser_guide:tutorials:apps_polytope [2019/01/25 13:40] – ↷ Links adapted because of a move operation oroehrig
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 A //polytope// is the convex hull of finitely many points in some Euclidean space. Equivalently, a polytope is the bounded intersection of finitely many affine halfspaces. ''polymake'' can deal with polytopes in both representations and provides numerous tools for analysis.\\ A //polytope// is the convex hull of finitely many points in some Euclidean space. Equivalently, a polytope is the bounded intersection of finitely many affine halfspaces. ''polymake'' can deal with polytopes in both representations and provides numerous tools for analysis.\\
  
-This tutorial first shows basic ways of defining a polytope from scratch. For larger input (e.g. from a file generated by some other program) have a look at  [[..:data|how to load data]] in ''polymake''.\\+This tutorial first shows basic ways of defining a polytope from scratch. For larger input (e.g. from a file generated by some other program) have a look at  [[..:howto:data|how to load data]] in ''polymake''.\\
  
 The second part demonstrates some of the tool ''polymake'' provides for handling polytopes by examining a small example. For a complete list of properties of polytopes and functions that ''polymake'' provides, see the [[reldocs>3.0/polytope.html|polytope documentation]]. The second part demonstrates some of the tool ''polymake'' provides for handling polytopes by examining a small example. For a complete list of properties of polytopes and functions that ''polymake'' provides, see the [[reldocs>3.0/polytope.html|polytope documentation]].
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 ==== V-Description ==== ==== V-Description ====
-To define a polytope as the convex hull of finitely many points, you can pass a matrix of coordinates to the constructor. Since ''polymake'' uses [[user_guide:coordinates|homogeneous coordinates]], you need to set the additional coordinate x<sub>0</sub> to 1.+To define a polytope as the convex hull of finitely many points, you can pass a matrix of coordinates to the constructor. Since ''polymake'' uses [[user_guide:tutorials:coordinates|homogeneous coordinates]], you need to set the additional coordinate x<sub>0</sub> to 1.
 <code> <code>
 polytope > $p = new Polytope(POINTS=>[[1,-1,-1],[1,1,-1],[1,-1,1],[1,1,1],[1,0,0]]); polytope > $p = new Polytope(POINTS=>[[1,-1,-1],[1,1,-1],[1,-1,1],[1,1,1],[1,0,0]]);
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 polytope > $p2->VISUAL; polytope > $p2->VISUAL;
 </code> </code>
-See [[..:visual_tutorial#application polytope|here]] for details on visualizing polytopes.+See [[visual_tutorial#application polytope|here]] for details on visualizing polytopes.
  
  If you are sure that all the points really are //extreme points// (vertices) and your description of the lineality space is complete, you can define the polytope via the properties ''VERTICES'' and ''LINEALITY_SPACE'' instead of ''POINTS'' and ''INPUT_LINEALITY''. This way, you can avoid unnecessary redundancy checks.\\  If you are sure that all the points really are //extreme points// (vertices) and your description of the lineality space is complete, you can define the polytope via the properties ''VERTICES'' and ''LINEALITY_SPACE'' instead of ''POINTS'' and ''INPUT_LINEALITY''. This way, you can avoid unnecessary redundancy checks.\\
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 It is also possible to define a polytope as an intersection of finitely many halfspaces, i.e., a matrix of inequalities.\\ It is also possible to define a polytope as an intersection of finitely many halfspaces, i.e., a matrix of inequalities.\\
  
-An inequality a<sub>0</sub> + a<sub>1</sub> x<sub>1</sub> + ... + a<sub>d</sub> x<sub>d</sub> >= 0 is encoded as a row vector (a<sub>0</sub>,a<sub>1</sub>,...,a<sub>d</sub>), see also [[user_guide:coordinates|Coordinates for Polyhedra]]. Here is an example:+An inequality a<sub>0</sub> + a<sub>1</sub> x<sub>1</sub> + ... + a<sub>d</sub> x<sub>d</sub> >= 0 is encoded as a row vector (a<sub>0</sub>,a<sub>1</sub>,...,a<sub>d</sub>), see also [[user_guide:tutorials:coordinates|Coordinates for Polyhedra]]. Here is an example:
 <code> <code>
 polytope > $p4 = new Polytope(INEQUALITIES=>[[1,1,0],[1,0,1],[1,-1,0],[1,0,-1],[17,1,1]]); polytope > $p4 = new Polytope(INEQUALITIES=>[[1,1,0],[1,0,1],[1,-1,0],[1,0,-1],[17,1,1]]);
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 </code> </code>
  
-See the [[..:apps_graph|tutorial on graphs]] for more on that subject.+See the [[apps_graph|tutorial on graphs]] for more on that subject.
  
  • user_guide/tutorials/apps_polytope.txt
  • Last modified: 2019/02/04 22:55
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