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Polyhedra :: polarFace

polarFace -- computes the dual face of the polar polyhedron

Synopsis

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 :