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# polarFace -- computes the dual face of the polar polyhedron

- Usage:
    `fv = polarFace f`
- Inputs:
  - f, a convex polyhedron
- Outputs:
  - fv, a convex polyhedron

## Description

When computing a face f of a polyhedron P with the function faces, it is stored in the cache that f is a face of P. Then the function polarFace computes the polar P' of P and the corresponding face of P' on which all points of f attain their minimum. Note that this function only works correctly for polyhedra with the origin in its relative interior.

    i1 : P = hypercube 3

    o1 = {ambient dimension => 3           }
          dimension of lineality space => 0
          dimension of polyhedron => 3
          number of facets => 6
          number of rays => 0
          number of vertices => 8

    o1 : Polyhedron

    i2 : f = first faces(1,P)

    o2 = {ambient dimension => 3           }
          dimension of lineality space => 0
          dimension of polyhedron => 2
          number of facets => 4
          number of rays => 0
          number of vertices => 4

    o2 : Polyhedron

    i3 : fv = polarFace f

    o3 = {ambient dimension => 3           }
          dimension of lineality space => 0
          dimension of polyhedron => -1
          number of facets => 0
          number of rays => 0
          number of vertices => 0

    o3 : Polyhedron

    i4 : vertices fv

    o4 = 0

                  3
    o4 : Matrix QQ  <-- 0

If f is not a face of another polytope, then it considers f as a face of itself. Thus, it computes the polar of f, and returns the empty polyhedron as a face of the polar of f.

    i5 : P = hypercube 3

    o5 = {ambient dimension => 3           }
          dimension of lineality space => 0
          dimension of polyhedron => 3
          number of facets => 6
          number of rays => 0
          number of vertices => 8

    o5 : Polyhedron

    i6 : polarFace P

    o6 = {ambient dimension => 3           }
          dimension of lineality space => 0
          dimension of polyhedron => -1
          number of facets => 0
          number of rays => 0
          number of vertices => 0

    o6 : Polyhedron

## Ways to use polarFace:

- `polarFace(Polyhedron)`

## For the programmer

The object polarFace is a method function.

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/OldPolyhedra.m2:7326:0.