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# 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 /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/OldPolyhedra.m2:5496:0.