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OldPolyhedra :: skeleton

skeleton -- computes the k-skeleton of a Fan or PolyhedralComplex

Synopsis

Description

For a Fan F and an integer k between 0 and the dimension of F, skeleton computes the k-skeleton of the Fan F, i.e. the Fan F1 generated by all cones of dimension k in F.

For example, we can look at the 2-skeleton of the fan of projective 3-space: 

i1 : P = convexHull matrix{{1,0,0,0},{0,1,0,0},{0,0,1,0}}

o1 = {ambient dimension => 3           }
      dimension of lineality space => 0
      dimension of polyhedron => 3
      number of facets => 4
      number of rays => 0
      number of vertices => 4

o1 : Polyhedron
 
i2 : F = normalFan P

o2 = {ambient dimension => 3         }
      number of generating cones => 4
      number of rays => 4
      top dimension of the cones => 3

o2 : Fan
 
i3 : F1 = skeleton(2,F)

o3 = {ambient dimension => 3         }
      number of generating cones => 6
      number of rays => 4
      top dimension of the cones => 2

o3 : Fan
 
i4 : apply(maxCones F1,rays)

o4 = {| 1 -1 |, | 0 -1 |, | 1 0 |, | 0 0 |, | 1 0 |, | -1 0 |}
      | 0 -1 |  | 1 -1 |  | 0 0 |  | 1 0 |  | 0 1 |  | -1 0 |
      | 0 -1 |  | 0 -1 |  | 0 1 |  | 0 1 |  | 0 0 |  | -1 1 |

o4 : List

For a PolyhedralComplex PC and an integer k between 0 and the dimension of PC, skeleton computes the k-skeleton of the PolyhedralComplex PC, i.e. the PolyhedralComplex PC1 generated by all polyhedra of dimension k in PC. 

i5 : PC = polyhedralComplex hypercube 3

o5 = {ambient dimension => 3             }
      number of generating polyhedra => 1
      top dimension of the polyhedra => 3

o5 : PolyhedralComplex
 
i6 : PC1 = skeleton(2,PC)

o6 = {ambient dimension => 3             }
      number of generating polyhedra => 6
      top dimension of the polyhedra => 2

o6 : PolyhedralComplex
 
i7 : apply(maxPolyhedra PC1,vertices)

o7 = {| -1 1  -1 1  |, | -1 -1 -1 -1 |, | -1 1  -1 1 |, | 1  1  1  1 |, | -1 1  -1 1  |, | -1 1  -1 1 |}
      | -1 -1 1  1  |  | -1 1  -1 1  |  | 1  1  1  1 |  | -1 1  -1 1 |  | -1 -1 -1 -1 |  | -1 -1 1  1 |
      | -1 -1 -1 -1 |  | -1 -1 1  1  |  | -1 -1 1  1 |  | -1 -1 1  1 |  | -1 -1 1  1  |  | 1  1  1  1 |

o7 : List

Ways to use skeleton :

For the programmer

The object skeleton is a method function.