chap5.txt

  
  5 Cone
  
  
  5.1 Cone: Category and Representations
  
  5.1-1 IsCone
  
  IsCone( M )  Category
  Returns:  true or false
  
  The GAP category of a cone.
  
  Remember: Every cone is a convex object.
  
  
  5.2 Cone: Properties
  
  5.2-1 IsRay
  
  IsRay( cone )  property
  Returns:  true or false
  
  Checks if the cone cone is a ray, i.e. if it has only one ray generator.
  
  
  5.3 Cone: Attributes
  
  5.3-1 DualCone
  
  DualCone( cone )  attribute
  Returns:  a cone
  
  Returns the dual cone of the cone cone.
  
  5.3-2 HilbertBasis
  
  HilbertBasis( cone )  attribute
  Returns:  a list
  
  Returns a Hilbert Basis of the cone cone.
  
  5.3-3 RaysInFacets
  
  RaysInFacets( cone )  attribute
  Returns:  a list
  
  Returns an incidence matrix for the rays in the facets of the cone cone. The
  ith  entry of the result corresponds to the ith facet, the jth entry of this
  is 1 if the jth ray is in th ith facet, 0 otherwise.
  
  5.3-4 Facets
  
  Facets( cone )  attribute
  Returns:  a list
  
  Returns a list of the facets of the cone cone as homalg cones.
  
  5.3-5 GridGeneratedByCone
  
  GridGeneratedByCone( cone )  attribute
  Returns:  a homalg module
  
  Returns  the  grid  generated  by  the  lattice points of the cone cone as a
  homalg module.
  
  5.3-6 FactorGrid
  
  FactorGrid( cone )  attribute
  Returns:  a homalg module
  
  Returns  the  factor  of  the  containing grid of the cone cone and the grid
  generated by cone.
  
  5.3-7 GridGeneratedByOrthogonalCone
  
  GridGeneratedByOrthogonalCone( cone )  attribute
  Returns:  a homalg module
  
  Returns  the  grid generated by the lattice points of the orthogonal cone of
  the cone cone.
  
  5.3-8 DefiningInequalities
  
  DefiningInequalities( cone )  attribute
  Returns:  a list
  
  Returns a list of the defining inequalities of the cone cone.
  
  5.3-9 IsContainedInFan
  
  IsContainedInFan( cone )  attribute
  Returns:  a fan
  
  If  the  cone  cone is constructed as part of a fan, this method returns the
  fan.
  
  5.3-10 FactorGridMorphism
  
  FactorGridMorphism( cone )  attribute
  Returns:  a morphism
  
  Returns the morphism to the factor grid of the cone cone.
  
  
  5.4 Cone: Methods
  
  5.4-1 IntersectionOfCones
  
  IntersectionOfCones( cone1, cone2 )  operation
  Returns:  a cone
  
  If  the  cones  cone1  and  cone2  share  a  face,  the method returns their
  intersection,
  
  5.4-2 Contains
  
  Contains( cone1, cone2 )  operation
  Returns:  true or false
  
  Returns true if the cone cone1 contains the cone cone2, false otherwise.
  
  5.4-3 StarFan
  
  StarFan( cone )  operation
  Returns:  a fan
  
  Returns the star fan of the cone cone, as described in cox, 3.2.7
  
  5.4-4 StarFan
  
  StarFan( cone, fan )  operation
  Returns:  a fan
  
  Returns  the  star  fan  of the fan fan along the cone cone, as described in
  cox, 3.2.7
  
  5.4-5 StarSubdivisionOfIthMaximalCone
  
  StarSubdivisionOfIthMaximalCone( fan, numb )  operation
  Returns:  a fan
  
  Returns the star subdivision of the fan fan on the numbth maximal cone as in
  cox, 3.3.13.
  
  
  5.5 Cone: Constructors
  
  5.5-1 Cone
  
  Cone( cone )  operation
  Returns:  a cone
  
  Returns a cone generated by the rays in cone.
  
  
  5.6 Cone: Examples
  
  
  5.6-1 Cone example
  
    Example  
    gap> C := Cone([[1,2,3],[2,1,1],[1,0,0],[0,1,1]]);
    <A cone in |R^3>
    gap> Length( RayGenerators( C ) );
    3
    gap> IsSmooth( C );
    true
    gap> Length( HilbertBasis( C ) );
    3
    gap> IsSimplicial( C );
    true
    gap> DC := DualCone( C );
    <A cone in |R^3>
    gap> Length( HilbertBasis( DC ) );
    3