affineImage(Matrix,Cone,Matrix) -- computes the affine image of a cone

Synopsis

Function: affineImage
Usage:

C1 = affineImage(A,C,b)

C1 = affineImage(A,C)

C1 = affineImage(C,b)
Inputs:
- A, a matrix, with entries in ZZ or QQ
- C, a convex rational cone
- b, a matrix, with entries in ZZ or QQ and only one column representing a vector
Outputs:
- C1, a convex polyhedron, of class Cone or Polyhedron

Description

A must be a matrix from the ambient space of the cone C to some other target space and b must be a vector in that target space, i.e. the number of columns of A must equal the ambient dimension of C and A and b must have the same number of rows. Then affineImage computes the polyhedron {(A*c)+b | c in C} and the cone {A*c | c 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 = coneFromVData matrix {{1,2,3},{3,1,2},{2,3,1}}

o1 = C

o1 : Cone

This Cone can be mapped to the positive orthant:

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 = affineImage(A,C)

o3 = C1

o3 : Cone

i4 : rays C1

o4 = | 1 0 0 |
     | 0 1 0 |
     | 0 0 1 |

              3       3
o4 : Matrix ZZ  <-- ZZ

Ways to use this method: