Double Description for Arbitrary Polyhedra¶
This module is part of the python backend for polyhedra. It uses the
double description method for cones
double_description
to find minimal
H/Vrepresentations of polyhedra. The latter works with cones
only. This is sufficient to treat general polyhedra by the following
construction: Any polyhedron can be embedded in one dimension higher
in the hyperplane \((1,*,\dots,*)\). The cone over the embedded
polyhedron will be called the homogenized cone in the
following. Conversely, intersecting the homogenized cone with the
hyperplane \(x_0=1\) gives you back the original polyhedron.
While slower than specialized C/C++ implementations, the implementation is general and works with any field in Sage that allows you to define polyhedra.
Note
If you just want polyhedra over arbitrary fields then you should
just use the
Polyhedron()
constructor.
EXAMPLES:
sage: from sage.geometry.polyhedron.double_description_inhomogeneous \
....: import Hrep2Vrep, Vrep2Hrep
sage: Hrep2Vrep(QQ, 2, [(1,2,3), (2,4,3)], [])
[1/21/2 1/2]
[ 0 2/3 1/3]
Note that the columns of the printed matrix are the vertices, rays, and lines of the minimal Vrepresentation. Dually, the rows of the following are the inequalities and equations:
sage: Vrep2Hrep(QQ, 2, [(1/2,0)], [(1/2,2/3), (1/2,1/3)], [])
[1 2 3]
[2 4 3]
[]

class
sage.geometry.polyhedron.double_description_inhomogeneous.
Hrep2Vrep
(base_ring, dim, inequalities, equations)¶ Bases:
sage.geometry.polyhedron.double_description_inhomogeneous.PivotedInequalities
Convert Hrepresentation to a minimal Vrepresentation.
INPUT:
base_ring
– a field.dim
– integer. The ambient space dimension.inequalities
– list of inequalities. Each inequalitiy is given as constant term,dim
coefficients.equations
– list of equations. Same notation as for inequalities.
EXAMPLES:
sage: from sage.geometry.polyhedron.double_description_inhomogeneous import Hrep2Vrep sage: Hrep2Vrep(QQ, 2, [(1,2,3), (2,4,3)], []) [1/21/2 1/2] [ 0 2/3 1/3] sage: Hrep2Vrep(QQ, 2, [(1,2,3), (2,2,3)], []) [ 1 1/2 1] [ 0 02/3] sage: Hrep2Vrep(QQ, 2, [(1,2,3), (2,2,3)], []) [1/2 1/2 1] [ 0 02/3] sage: Hrep2Vrep(QQ, 2, [(8,7,2), (1,4,3), (4,3,1)], []) [ 1 0 2] [ 1 4 3] sage: Hrep2Vrep(QQ, 2, [(1,2,3), (2,4,3), (5,1,2)], []) [19/5 1/2 2/33 1/11] [ 22/5 01/33 2/33] sage: Hrep2Vrep(QQ, 2, [(0,2,3), (0,4,3), (0,1,2)], []) [ 0 1/2 1/3] [ 01/3 1/6] sage: Hrep2Vrep(QQ, 2, [], [(1,2,3), (7,8,9)]) [2] [ 1] sage: Hrep2Vrep(QQ, 2, [(1,0,0)], []) # universe [01 0] [00 1] sage: Hrep2Vrep(QQ, 2, [(1,0,0)], []) # empty [] sage: Hrep2Vrep(QQ, 2, [], []) # empty []

verify
(inequalities, equations)¶ Compare result to PPL if the base ring is QQ.
This method is for debugging purposes and compares the computation with another backend if available.
INPUT:
inequalities
,equations
– seeHrep2Vrep
.
EXAMPLES:
sage: from sage.geometry.polyhedron.double_description_inhomogeneous import Hrep2Vrep sage: H = Hrep2Vrep(QQ, 1, [(1,2)], []) sage: H.verify([(1,2)], [])

class
sage.geometry.polyhedron.double_description_inhomogeneous.
PivotedInequalities
(base_ring, dim)¶ Bases:
sage.structure.sage_object.SageObject
Base class for inequalities that may contain linear subspaces
INPUT:
base_ring
– a field.dim
– integer. The ambient space dimension.
EXAMPLES:
sage: from sage.geometry.polyhedron.double_description_inhomogeneous ....: import PivotedInequalities sage: piv = PivotedInequalities(QQ, 2) sage: piv._pivot_inequalities(matrix([(1,1,3), (5,5,7)])) [1 3] [5 7] sage: piv._pivots (0, 2) sage: piv._linear_subspace Free module of degree 3 and rank 1 over Integer Ring Echelon basis matrix: [ 1 1 0]

class
sage.geometry.polyhedron.double_description_inhomogeneous.
Vrep2Hrep
(base_ring, dim, vertices, rays, lines)¶ Bases:
sage.geometry.polyhedron.double_description_inhomogeneous.PivotedInequalities
Convert Vrepresentation to a minimal Hrepresentation.
INPUT:
base_ring
– a field.dim
– integer. The ambient space dimension.vertices
– list of vertices. Each vertex is given as list ofdim
coordinates.rays
– list of rays. Each ray is given as list ofdim
coordinates, not all zero.lines
– list of line generators. Each line is given as list ofdim
coordinates, not all zero.
EXAMPLES:
sage: from sage.geometry.polyhedron.double_description_inhomogeneous import Vrep2Hrep sage: Vrep2Hrep(QQ, 2, [(1/2,0)], [(1/2,2/3), (1/2,1/3)], []) [1 2 3] [2 4 3] [] sage: Vrep2Hrep(QQ, 2, [(1,0), (1/2,0)], [], [(1,2/3)]) [ 1/3 2/3 1] [ 2/3 2/3 1] [] sage: Vrep2Hrep(QQ, 2, [(1/2,0)], [(1/2,0)], [(1,2/3)]) [1 2 3] [] sage: Vrep2Hrep(QQ, 2, [(1,1), (0,4), (2,3)], [], []) [ 8/13 7/13 2/13] [ 1/13 4/13 3/13] [ 4/13 3/13 1/13] [] sage: Vrep2Hrep(QQ, 2, [(19/5,22/5), (1/2,0)], [(2/33,1/33), (1/11,2/33)], []) [10/11 2/11 4/11] [ 66/5 132/5 99/5] [ 2/11 4/11 6/11] [] sage: Vrep2Hrep(QQ, 2, [(0,0)], [(1/2,1/3), (1/3,1/6)], []) [ 0 6 12] [ 0 12 18] [] sage: Vrep2Hrep(QQ, 2, [(1/2,0)], [], [(1,2/3)]) [] [1 2 3] sage: Vrep2Hrep(QQ, 2, [(1/2,0)], [], [(1,2/3), (1,0)]) []

verify
(vertices, rays, lines)¶ Compare result to PPL if the base ring is QQ.
This method is for debugging purposes and compares the computation with another backend if available.
INPUT:
vertices
,rays
,lines
– seeVrep2Hrep
.
EXAMPLES:
sage: from sage.geometry.polyhedron.double_description_inhomogeneous import Vrep2Hrep sage: vertices = [(1/2,0)] sage: rays = [(1/2,2/3), (1/2,1/3)] sage: lines = [] sage: V2H = Vrep2Hrep(QQ, 2, vertices, rays, lines) sage: V2H.verify(vertices, rays, lines)