chap4.txt

  
  4 Fan
  
  
  4.1 Fan: Category and Representations
  
  4.1-1 IsFan
  
  IsFan( M )  Category
  Returns:  true or false
  
  The GAP category of a fan. Every fan is a convex object.
  
  Remember: Every fan is a convex object.
  
  
  4.2 Fan: Properties
  
  4.2-1 IsComplete
  
  IsComplete( fan )  property
  Returns:  true or false
  
  Checks if the fan fan is complete, i. e. if it's support is the whole space.
  
  4.2-2 IsPointed
  
  IsPointed( fan )  property
  Returns:  true or false
  
  Checks if the fan fan is pointed, which means that every cone it contains is
  strictly convex.
  
  4.2-3 IsSmooth
  
  IsSmooth( fan )  property
  Returns:  true or false
  
  Checks if the fan fan is smooth, i. e. if every cone in the fan is smooth.
  
  4.2-4 IsRegularFan
  
  IsRegularFan( fan )  property
  Returns:  true or false
  
  Checks  if  the  fan  fan  is  regular,  i.  e. if it is the normal fan of a
  polytope.
  
  4.2-5 IsSimplicial
  
  IsSimplicial( fan )  property
  Returns:  true or false
  
  Checks  if  the  fan  fan  is  simplicial, i. e. if every cone in the fan is
  simplicial.
  
  4.2-6 HasConvexSupport
  
  HasConvexSupport( fan )  property
  Returns:  true or false
  
  Checks  if  the  fan  fan  is  simplicial, i. e. if every cone in the fan is
  simplicial.
  
  
  4.3 Fan: Attributes
  
  4.3-1 Rays
  
  Rays( fan )  attribute
  Returns:  a list
  
  Returns the rays of the fan fan as a list of cones.
  
  4.3-2 RayGenerators
  
  RayGenerators( fan )  attribute
  Returns:  a list
  
  Returns the generators rays of the fan fan as a list of of list of integers.
  
  4.3-3 RaysInMaximalCones
  
  RaysInMaximalCones( fan )  attribute
  Returns:  a list
  
  Returns  a  list  of  lists,  which  represent  an  incidence matrix for the
  correspondence  of  the  rays  and the maximal cones of the fan fan. The ith
  list  in  the result represents the ith maximal cone of fan. In such a list,
  the jth entry is 1 if the jth ray is in the cone, 0 otherwise.
  
  4.3-4 MaximalCones
  
  MaximalCones( fan )  attribute
  Returns:  a list
  
  Returns the maximal cones of the fan fan as a list of cones.
  
  
  4.4 Fan: Methods
  
  4.4-1 *
  
  *( fan1, fan2 )  operation
  Returns:  a fan
  
  Returns the product of the fans fan1 and fan2.
  
  
  4.5 Fan: Constructors
  
  4.5-1 Fan
  
  Fan( fan )  operation
  Returns:  a fan
  
  Copy constructor for fans. For completeness reasons.
  
  4.5-2 Fan
  
  Fan( rays, cones )  operation
  Returns:  a fan
  
  Constructs  the  fan  out  of  the given rays and a list of cones given by a
  lists of numbers of rays.
  
  
  4.6 Fan: Examples
  
  
  4.6-1 Fan example
  
    Example  
    gap> F := Fan( [[-1,5],[0,1],[1,0],[0,-1]],[[1,2],[2,3],[3,4],[4,1]] );
    <A fan in |R^2>
    gap> RayGenerators( F );
    [ [ -1, 5 ], [ 0, 1 ], [ 1, 0 ], [ 0, -1 ] ]
    gap> RaysInMaximalCones( F );
    [ [ 1, 1, 0, 0 ], [ 0, 1, 1, 0 ], [ 0, 0, 1, 1 ], [ 1, 0, 0, 1 ] ]
    gap> IsRegularFan( F );
    true
    gap> IsComplete( F );
    true
    gap> IsSmooth( F );
    true
    gap> F1 := MaximalCones( F )[ 1 ];
    <A cone in |R^2>
    gap> DualCone( F1 );
    <A cone in |R^2>
    gap> RayGenerators( F1 );
    [ [ -1, 5 ], [ 0, 1 ] ]
    gap> F2 := StarSubdivisionOfIthMaximalCone( F, 1 );
    <A fan in |R^2>
    gap> IsSmooth( F2 );
    true
    gap> RayGenerators( F2 );
    [ [ -1, 5 ], [ -1, 6 ], [ 0, -1 ], [ 0, 1 ], [ 1, 0 ] ]