user_guide:tutorials:polytope_semantics

Differences

This shows you the differences between two versions of the page.

Link to this comparison view

Both sides previous revision Previous revision
Next revision		 Previous revision
Next revisionBoth sides next revision
tutorial:polytope_semantics [2011/11/21 15:36] – [Special Cases] joswigtutorial:polytope_semantics [2014/01/03 15:45] – external edit 127.0.0.1
Line 25: Line 25:
 ==== Without coordinates: Combinatorics ==== ==== Without coordinates: Combinatorics ====
  
-''VERTICES_IN_FACETS'' always describes the combinatorics of a bounded polytope: this is any polytope which is projectively equivalent to the polyhedron defined by ''VERTICES'' or ''POINTS'' or dually) modulo its ''LINEALITY_SPACE''.+''VERTICES_IN_FACETS'' always describes the combinatorics of a bounded polytope: this is any polytope which is projectively equivalent to the polyhedron defined by ''VERTICES'' or ''POINTS'' or dually modulo its ''LINEALITY_SPACE''.
  
 Each property must clearly specify if it depends on the geometry or only on the combinatorics. Each property must clearly specify if it depends on the geometry or only on the combinatorics.
  
 ==== Special Cases ==== ==== Special Cases ====
 +
 +Most of what comes below is a consequence of the design decisions explained above.
  
 === Empty polytopes === === Empty polytopes ===
Line 42: Line 44:
 === Zero-dimensional polytopes === === Zero-dimensional polytopes ===
  
-A zero-dimensional polytope is a single point.  In our model it has one vertex and one facet.+A zero-dimensional polytope is a single point.  In our model it has one vertex and one facet (the far hyperplane).
 ''VERTICES_IN_FACETS'' is a 1-by-1 matrix with a zero entry.  This means that the single vertex does //not// lie on the single facet. ''VERTICES_IN_FACETS'' is a 1-by-1 matrix with a zero entry.  This means that the single vertex does //not// lie on the single facet.
  
 Such a polytope is both simple and simplicial, i.e. it is a simplex.  Such a polytope is both simple and simplicial, i.e. it is a simplex. 
 +
 +=== Zero-dimensional fans ===
 +
 +A zero-dimensional fan can e.g. be defined via <code>$f=new PolyhedralFan(RAYS=>[[ ]], MAXIMAL_CONES=>[[]]);</code>
 +==== Summing Up ====
 +For instance we have four possibilities which can occur for ''VERTICES''. The property
 +  * does not exist (it is not listed in ''list_properties''): This basically means that the property is not derived/calculated, yet.
 +  * exists and is set to ''undef'': Polymake is not able to derive this property with the given properties. The polytope may be empty or not.
 +  * exists and is empty: So the polytope is empty.
 +  * exists and is neither set to ''undef'' nor is empty: Our polytope is not empty and the property returns what you expect.
  
  • user_guide/tutorials/polytope_semantics.txt
  • Last modified: 2019/02/04 22:55
  • by 127.0.0.1