hyperplanes -- computes the defining hyperplanes of a Cone or a Polyhedron

Synopsis

• Usage:

  N = hyperplanes C

  (N,w) = hyperplanes P

• Inputs:
  • C, a convex rational cone
  • P, a convex polyhedron
• Outputs:
  • N, a matrix, with entries over QQ
  • w, a matrix, with entries over QQ and only one column

Description

hyperplanes returns the defining affine hyperplanes for a polyhedron P. The output is (N,w), where the source of N has the dimension of the ambient space of P and w is a one column matrix in the target space of N such that P = {p in H | N*p = w} where H is the intersection of the defining affine half-spaces.

For a cone C the output is the matrix N, that is the same matrix as before but w is omitted since it is 0, so C = {c in H | N*c = 0} and H is the intersection of the defining linear half-spaces.

i1 : P = stdSimplex 2

o1 = {ambient dimension => 3           }
      dimension of lineality space => 0
      dimension of polyhedron => 2
      number of facets => 3
      number of rays => 0
      number of vertices => 3

o1 : Polyhedron

i2 : hyperplanes P

o2 = (| 1 1 1 |, | 1 |)

o2 : Sequence

i3 : C = posHull matrix {{1,2,4},{2,3,5},{3,4,6}}

o3 = {ambient dimension => 3           }
      dimension of lineality space => 0
      dimension of the cone => 2
      number of facets => 2
      number of rays => 2

o3 : Cone

i4 : hyperplanes C

o4 = | 1 -2 1 |

              1        3
o4 : Matrix ZZ  <--- ZZ