next | previous | forward | backward | up | top | index | toc | Macaulay2 website
OldPolyhedra :: affineImage(Matrix,Polyhedron,Matrix)

affineImage(Matrix,Polyhedron,Matrix) -- computes the affine image of a polyhedron

Synopsis

Description

A must be a matrix from the ambient space of the polyhedron P to some other target space and v must be a vector in that target space, i.e. the number of columns of A must equal the ambient dimension of P and A and v must have the same number of rows. Then affineImage computes the polyhedron {(A*p)+v | p 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 = {ambient dimension => 2           }
      dimension of lineality space => 0
      dimension of polyhedron => 2
      number of facets => 4
      number of rays => 0
      number of vertices => 4

o1 : Polyhedron

This polytope is the affine image of the square:

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 = affineImage(A,P,v)

o4 = {ambient dimension => 2           }
      dimension of lineality space => 0
      dimension of polyhedron => 2
      number of facets => 4
      number of rays => 0
      number of vertices => 4

o4 : Polyhedron
i5 : vertices Q

o5 = | -1 1  -1 1 |
     | -1 -1 1  1 |

              2        4
o5 : Matrix QQ  <--- QQ