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OldPolyhedra :: polarFace

polarFace -- computes the dual face of the polar polyhedron

Synopsis

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 :

For the programmer

The object polarFace is a method function.