Parents for Polyhedra

sage.geometry.polyhedron.parent.Polyhedra(ambient_space_or_base_ring, ambient_dim, backend=None, ambient_space=None, base_ring=None)

Construct a suitable parent class for polyhedra

INPUT:

• base_ring – A ring. Currently there are backends for $\ZZ$, $\QQ$, and $\RDF$.

• ambient_dim – integer. The ambient space dimension.

• ambient_space – A free module.

• backend – string. The name of the backend for computations. There are

  several backends implemented:

  • backend="ppl" uses the Parma Polyhedra Library

  • backend="cdd" uses CDD

  • backend="normaliz" uses normaliz

  • backend="polymake" uses polymake

  • backend="field" a generic Sage implementation

OUTPUT:

A parent class for polyhedra over the given base ring if the backend supports it. If not, the parent base ring can be larger (for example, $\QQ$ instead of $\ZZ$). If there is no implementation at all, a ValueError is raised.

EXAMPLES:

sage: from sage.geometry.polyhedron.parent import Polyhedra
sage: Polyhedra(AA, 3)
Polyhedra in AA^3
sage: Polyhedra(ZZ, 3)
Polyhedra in ZZ^3
sage: type(_)
<class 'sage.geometry.polyhedron.parent.Polyhedra_ZZ_ppl_with_category'>
sage: Polyhedra(QQ, 3, backend='cdd')
Polyhedra in QQ^3
sage: type(_)
<class 'sage.geometry.polyhedron.parent.Polyhedra_QQ_cdd_with_category'>

CDD does not support integer polytopes directly:

sage: Polyhedra(ZZ, 3, backend='cdd')
Polyhedra in QQ^3

Using a more general form of the constructor:

sage: V = VectorSpace(QQ, 3)
sage: Polyhedra(V) is Polyhedra(QQ, 3)
True
sage: Polyhedra(V, backend='field') is Polyhedra(QQ, 3, 'field')
True
sage: Polyhedra(backend='field', ambient_space=V) is Polyhedra(QQ, 3, 'field')
True

sage: M = FreeModule(ZZ, 2)
sage: Polyhedra(M, backend='ppl') is Polyhedra(ZZ, 2, 'ppl')
True

class sage.geometry.polyhedron.parent.Polyhedra_QQ_cdd(base_ring, ambient_dim, backend)

Bases: sage.geometry.polyhedron.parent.Polyhedra_base

Element

alias of sage.geometry.polyhedron.backend_cdd.Polyhedron_QQ_cdd

class sage.geometry.polyhedron.parent.Polyhedra_QQ_normaliz(base_ring, ambient_dim, backend)

Bases: sage.geometry.polyhedron.parent.Polyhedra_base

Element

alias of sage.geometry.polyhedron.backend_normaliz.Polyhedron_QQ_normaliz

class sage.geometry.polyhedron.parent.Polyhedra_QQ_ppl(base_ring, ambient_dim, backend)

Bases: sage.geometry.polyhedron.parent.Polyhedra_base

Element

alias of sage.geometry.polyhedron.backend_ppl.Polyhedron_QQ_ppl

class sage.geometry.polyhedron.parent.Polyhedra_RDF_cdd(base_ring, ambient_dim, backend)

Bases: sage.geometry.polyhedron.parent.Polyhedra_base

Element

alias of sage.geometry.polyhedron.backend_cdd.Polyhedron_RDF_cdd

class sage.geometry.polyhedron.parent.Polyhedra_ZZ_normaliz(base_ring, ambient_dim, backend)

Bases: sage.geometry.polyhedron.parent.Polyhedra_base

Element

alias of sage.geometry.polyhedron.backend_normaliz.Polyhedron_ZZ_normaliz

class sage.geometry.polyhedron.parent.Polyhedra_ZZ_ppl(base_ring, ambient_dim, backend)

Bases: sage.geometry.polyhedron.parent.Polyhedra_base

Element

alias of sage.geometry.polyhedron.backend_ppl.Polyhedron_ZZ_ppl

class sage.geometry.polyhedron.parent.Polyhedra_base(base_ring, ambient_dim, backend)

Bases: sage.structure.unique_representation.UniqueRepresentation, sage.structure.parent.Parent

Polyhedra in a fixed ambient space.

INPUT:

• base_ring – either ZZ, QQ, or RDF. The base ring of the ambient module/vector space.

• ambient_dim – integer. The ambient space dimension.

• backend – string. The name of the backend for computations. There are

  several backends implemented:

  • backend="ppl" uses the Parma Polyhedra Library

  • backend="cdd" uses CDD

  • backend="normaliz" uses normaliz

  • backend="polymake" uses polymake

  • backend="field" a generic Sage implementation

EXAMPLES:

sage: from sage.geometry.polyhedron.parent import Polyhedra
sage: Polyhedra(ZZ, 3)
Polyhedra in ZZ^3

Hrepresentation_space()

Return the linear space containing the H-representation vectors.

OUTPUT:

A free module over the base ring of dimension ambient_dim() + 1.

EXAMPLES:

sage: from sage.geometry.polyhedron.parent import Polyhedra
sage: Polyhedra(ZZ, 2).Hrepresentation_space()
Ambient free module of rank 3 over the principal ideal domain Integer Ring

Vrepresentation_space()

Return the ambient vector space.

This is the vector space or module containing the Vrepresentation vectors.

OUTPUT:

A free module over the base ring of dimension ambient_dim().

EXAMPLES:

sage: from sage.geometry.polyhedron.parent import Polyhedra
sage: Polyhedra(QQ, 4).Vrepresentation_space()
Vector space of dimension 4 over Rational Field
sage: Polyhedra(QQ, 4).ambient_space()
Vector space of dimension 4 over Rational Field

ambient_dim()

Return the dimension of the ambient space.

EXAMPLES:

sage: from sage.geometry.polyhedron.parent import Polyhedra
sage: Polyhedra(QQ, 3).ambient_dim()
3

ambient_space()

Return the ambient vector space.

This is the vector space or module containing the Vrepresentation vectors.

OUTPUT:

A free module over the base ring of dimension ambient_dim().

EXAMPLES:

sage: from sage.geometry.polyhedron.parent import Polyhedra
sage: Polyhedra(QQ, 4).Vrepresentation_space()
Vector space of dimension 4 over Rational Field
sage: Polyhedra(QQ, 4).ambient_space()
Vector space of dimension 4 over Rational Field

an_element()

Return a Polyhedron.

EXAMPLES:

sage: from sage.geometry.polyhedron.parent import Polyhedra
sage: Polyhedra(QQ, 4).an_element()
A 4-dimensional polyhedron in QQ^4 defined as the convex hull of 5 vertices

backend()

Return the backend.

EXAMPLES:

sage: from sage.geometry.polyhedron.parent import Polyhedra
sage: Polyhedra(QQ, 3).backend()
'ppl'

base_extend(base_ring, backend=None, ambient_dim=None)

Return the base extended parent.

INPUT:

• base_ring, backend – see Polyhedron().

• ambient_dim – if not None change ambient dimension accordingly.

EXAMPLES:

sage: from sage.geometry.polyhedron.parent import Polyhedra
sage: Polyhedra(ZZ,3).base_extend(QQ)
Polyhedra in QQ^3
sage: Polyhedra(ZZ,3).an_element().base_extend(QQ)
A 3-dimensional polyhedron in QQ^3 defined as the convex hull of 4 vertices
sage: Polyhedra(QQ, 2).base_extend(ZZ)
Polyhedra in QQ^2

change_ring(base_ring, backend=None, ambient_dim=None)

Return the parent with the new base ring.

INPUT:

• base_ring, backend – see Polyhedron().

• ambient_dim – if not None change ambient dimension accordingly.

EXAMPLES:

sage: from sage.geometry.polyhedron.parent import Polyhedra
sage: Polyhedra(ZZ,3).change_ring(QQ)
Polyhedra in QQ^3
sage: Polyhedra(ZZ,3).an_element().change_ring(QQ)
A 3-dimensional polyhedron in QQ^3 defined as the convex hull of 4 vertices

sage: Polyhedra(RDF, 3).change_ring(QQ).backend()
'cdd'
sage: Polyhedra(QQ, 3).change_ring(ZZ, ambient_dim=4)
Polyhedra in ZZ^4
sage: Polyhedra(QQ, 3, backend='cdd').change_ring(QQ, ambient_dim=4).backend()
'cdd'

empty()

Return the empty polyhedron.

EXAMPLES:

sage: from sage.geometry.polyhedron.parent import Polyhedra
sage: P = Polyhedra(QQ, 4)
sage: P.empty()
The empty polyhedron in QQ^4
sage: P.empty().is_empty()
True

list()

Return the two polyhedra in ambient dimension 0, raise an error otherwise

EXAMPLES:

sage: from sage.geometry.polyhedron.parent import Polyhedra
sage: P = Polyhedra(QQ, 3)
sage: P.cardinality()
+Infinity

sage: P = Polyhedra(AA, 0)
sage: P.category()
Category of finite enumerated polyhedral sets over Algebraic Real Field
sage: P.list()
[The empty polyhedron in AA^0,
 A 0-dimensional polyhedron in AA^0 defined as the convex hull of 1 vertex]
sage: P.cardinality()
2

recycle(polyhedron)

Recycle the H/V-representation objects of a polyhedron.

This speeds up creation of new polyhedra by reusing objects. After recycling a polyhedron object, it is not in a consistent state any more and neither the polyhedron nor its H/V-representation objects may be used any more.

INPUT:

• polyhedron – a polyhedron whose parent is self.

EXAMPLES:

sage: p = Polyhedron([(0,0),(1,0),(0,1)])
sage: p.parent().recycle(p)

some_elements()

Return a list of some elements of the semigroup.

EXAMPLES:

sage: from sage.geometry.polyhedron.parent import Polyhedra
sage: Polyhedra(QQ, 4).some_elements()
[A 3-dimensional polyhedron in QQ^4 defined as the convex hull of 4 vertices,
 A 4-dimensional polyhedron in QQ^4 defined as the convex hull of 1 vertex and 4 rays,
 A 2-dimensional polyhedron in QQ^4 defined as the convex hull of 2 vertices and 1 ray,
 The empty polyhedron in QQ^4]
sage: Polyhedra(ZZ,0).some_elements()
[The empty polyhedron in ZZ^0,
 A 0-dimensional polyhedron in ZZ^0 defined as the convex hull of 1 vertex]

universe()

Return the entire ambient space as polyhedron.

EXAMPLES:

sage: from sage.geometry.polyhedron.parent import Polyhedra
sage: P = Polyhedra(QQ, 4)
sage: P.universe()
A 4-dimensional polyhedron in QQ^4 defined as the convex hull of 1 vertex and 4 lines
sage: P.universe().is_universe()
True

zero()

Return the polyhedron consisting of the origin, which is the neutral element for Minkowski addition.

EXAMPLES:

sage: from sage.geometry.polyhedron.parent import Polyhedra
sage: p = Polyhedra(QQ, 4).zero();  p
A 0-dimensional polyhedron in QQ^4 defined as the convex hull of 1 vertex
sage: p+p == p
True

class sage.geometry.polyhedron.parent.Polyhedra_field(base_ring, ambient_dim, backend)

Bases: sage.geometry.polyhedron.parent.Polyhedra_base

Element

alias of sage.geometry.polyhedron.backend_field.Polyhedron_field

class sage.geometry.polyhedron.parent.Polyhedra_normaliz(base_ring, ambient_dim, backend)

Bases: sage.geometry.polyhedron.parent.Polyhedra_base

Element

alias of sage.geometry.polyhedron.backend_normaliz.Polyhedron_normaliz

class sage.geometry.polyhedron.parent.Polyhedra_polymake(base_ring, ambient_dim, backend)

Bases: sage.geometry.polyhedron.parent.Polyhedra_base

Element

alias of sage.geometry.polyhedron.backend_polymake.Polyhedron_polymake

sage.geometry.polyhedron.parent.does_backend_handle_base_ring(base_ring, backend)

Return true, if backend can handle base_ring.

EXAMPLES:

sage: from sage.geometry.polyhedron.parent import does_backend_handle_base_ring
sage: does_backend_handle_base_ring(QQ, 'ppl')
True
sage: does_backend_handle_base_ring(QQ[sqrt(5)], 'ppl')
False
sage: does_backend_handle_base_ring(QQ[sqrt(5)], 'field')
True