next | previous | forward | backward | up | top | index | toc | Macaulay2 website
OldPolyhedra :: halfspaces

halfspaces -- computes the defining half-spaces of a Cone or a Polyhedron

Synopsis

Description

halfspaces returns the defining affine half-spaces. For a polyhedron P the output is (M,v), where the source of M has the dimension of the ambient space of P and v is a one column matrix in the target space of M such that P = {p in H | M*p =< v} where H is the intersection of the defining affine hyperplanes.

For a cone C the output is the matrixM that is the same matrix as before but v is omitted since it is 0, so C = {c in H | M*c =< 0} and H is the intersection of the defining linear hyperplanes.

i1 : R = matrix {{1,1,2,2},{2,3,1,3},{3,2,3,1}};

              3        4
o1 : Matrix ZZ  <--- ZZ
i2 : V = matrix {{1,-1},{0,0},{0,0}};

              3        2
o2 : Matrix ZZ  <--- ZZ
i3 : C = posHull R

o3 = {ambient dimension => 3           }
      dimension of lineality space => 0
      dimension of the cone => 3
      number of facets => 4
      number of rays => 4

o3 : Cone
i4 : halfspaces C

o4 = | -2 1  1  |
     | 1  -1 1  |
     | 1  1  -1 |
     | 5  -1 -1 |

              4        3
o4 : Matrix ZZ  <--- ZZ

Now we take this cone over a line and get a polyhedron.

i5 : P = convexHull(V,R)

o5 = {ambient dimension => 3           }
      dimension of lineality space => 0
      dimension of polyhedron => 3
      number of facets => 6
      number of rays => 4
      number of vertices => 2

o5 : Polyhedron
i6 : halfspaces P

o6 = (| 0  1  -3 |, | 0 |)
      | 2  -1 -1 |  | 2 |
      | -1 1  -1 |  | 1 |
      | 0  -3 1  |  | 0 |
      | -1 -1 1  |  | 1 |
      | -5 1  1  |  | 5 |

o6 : Sequence

Ways to use halfspaces :

For the programmer

The object halfspaces is a method function.