polarFace -- computes the dual face of the polar polyhedron

Usage:

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

Description

Given a polyhedron f which is a face of a polyhedron P 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 = P

o1 : Polyhedron

i2 : f = first faces(1,P)

o2 = ({0, 2, 4, 6}, {})

o2 : Sequence

i3 : f = convexHull (vertices P)_(f#0)

o3 = f

o3 : Polyhedron

i4 : fv = polarFace(f, P)

o4 = fv

o4 : Polyhedron

i5 : vertices fv

o5 = | 1 |
     | 0 |
     | 0 |

              3       1
o5 : Matrix QQ  <-- QQ

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.

i6 : P = hypercube 3

o6 = P

o6 : Polyhedron

i7 : polarFace(P, P)

o7 = Polyhedron{...1...}

o7 : Polyhedron

Ways to use polarFace:

polarFace(Polyhedron,Polyhedron)

For the programmer

The object polarFace is a method function.

The source of this document is in Polyhedra/documentation/old_documentation.m2:2798:0.