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

## Synopsis

• Usage:
b = commonFace(C1,C2)
b = commonFace(P1,P2)
b = commonFace(X,F)
b = commonFace(F,X)
b = commonFace L
• Inputs:
• C1,
• C2,
• P1,
• P2,
• F, an instance of the type Fan
• X, Cone or Fan
• L, a list
• Outputs:
• b, , 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 .