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OldPolyhedra :: affinePreimage(Matrix,Cone,Matrix)

affinePreimage(Matrix,Cone,Matrix) -- computes the affine preimage of a cone

Synopsis

Description

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

For example, consider the following three dimensional cone:

i1 : C = posHull matrix {{1,2,3},{3,1,2},{2,3,1}}

o1 = {ambient dimension => 3           }
      dimension of lineality space => 0
      dimension of the cone => 3
      number of facets => 3
      number of rays => 3

o1 : Cone

We can look at its preimage under the following map:

i2 : A = matrix  {{-5,7,1},{1,-5,7},{7,1,-5}}

o2 = | -5 7  1  |
     | 1  -5 7  |
     | 7  1  -5 |

              3        3
o2 : Matrix ZZ  <--- ZZ
i3 : C1 = affinePreimage(A,C)

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

o3 : Cone
i4 : rays C1

o4 = | 13 13 10 |
     | 13 10 13 |
     | 10 13 13 |

              3        3
o4 : Matrix ZZ  <--- ZZ