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

## Synopsis

• Usage:
N = hyperplanes C
(N,w) = hyperplanes P
• Inputs:
• Outputs:
• N, , with entries over QQ
• w, , 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

## Ways to use hyperplanes :

• "hyperplanes(Cone)"
• "hyperplanes(Polyhedron)"

## For the programmer

The object hyperplanes is .