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Polyhedra :: affinePreimage(Matrix,Polyhedron,Matrix)

affinePreimage(Matrix,Polyhedron,Matrix) -- computes the affine preimage of a polyhedron

Synopsis

Description

A must be a matrix from some source space to the ambient space of the polyhedron P and v must be a vector in that ambient space, i.e. the number of rows of A must equal the ambient dimension of P and the number of rows of v. affinePreimage then computes the polyhedron {q | (A*q)+v in P} where v is set to 0 if omitted and A is the identity if omitted.

For example, consider the following two dimensional polytope

i1 : P = convexHull matrix {{-2,0,2,4},{-8,-2,2,8}}

o1 = P

o1 : Polyhedron

and its affine preimage under the following map:

i2 : A = matrix {{-5,2},{3,-1}}

o2 = | -5 2  |
     | 3  -1 |

              2        2
o2 : Matrix ZZ  <--- ZZ
i3 : v = matrix {{5},{-3}}

o3 = | 5  |
     | -3 |

              2        1
o3 : Matrix ZZ  <--- ZZ
i4 : Q = affinePreimage(A,P,v)

o4 = Q

o4 : Polyhedron
i5 : vertices Q

o5 = | -17 -3  7  21 |
     | -46 -10 16 52 |

              2        4
o5 : Matrix QQ  <--- QQ