commonFace -- checks if the intersection is a face of both Cones or Polyhedra, or of cones with fans

Usage:

b = commonFace(C1,C2)

b = commonFace(P1,P2)

b = commonFace(X,F)

b = commonFace(F,X)

b = commonFace L
Inputs:
- C1, a convex rational cone
- C2, a convex rational cone
- P1, a convex polyhedron
- P2, a convex polyhedron
- F, an instance of the type Fan
- X, Cone or Fan
- L, a list
Outputs:
- b, a Boolean value, true if the intersection is a face both, and false otherwise.

Description

commonFace checks if the intersection of C1 and C2 or the intersection of P1 and P2 is a face of both. If it is applied to a pair of a cone C and a fan F then it checks if the intersection of C with every generating cone of F is a face of each. For two fans it checks this condition for every pair of generating cones. If applied to a list then the list must contain Fans and Cones and it checks pairwise for a common face.

For example, consider the following three cones:

i1 : C1 = posHull matrix {{1,0},{0,1}};

i2 : C2 = posHull matrix {{1,-1},{0,-1}};

i3 : C3 = posHull matrix {{1,-1},{2,-1}};

for each pair of two of them we can check if their intersection is a common face:

i4 : commonFace(C1,C2)

o4 = true

i5 : commonFace(C2,C3)

o5 = true

i6 : commonFace(C3,C1)

o6 = false

Ways to use commonFace:

commonFace(Cone,Cone)
commonFace(Cone,Fan)
commonFace(Fan,Cone)
commonFace(Fan,Fan)
commonFace(List)
commonFace(Polyhedron,Polyhedron)

For the programmer

The object commonFace is a method function.

The source of this document is in OldPolyhedra.m2:5496:0.