Parents for Polyhedra¶

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

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

sage: from sage.geometry.polyhedron.parent import Polyhedra
sage: Polyhedra(AA, 3)                                                          # needs sage.rings.number_field
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'>

Python

>>> from sage.all import *
>>> from sage.geometry.polyhedron.parent import Polyhedra
>>> Polyhedra(AA, Integer(3))                                                          # needs sage.rings.number_field
Polyhedra in AA^3
>>> Polyhedra(ZZ, Integer(3))
Polyhedra in ZZ^3
>>> type(_)
<class 'sage.geometry.polyhedron.parent.Polyhedra_ZZ_ppl_with_category'>
>>> Polyhedra(QQ, Integer(3), backend='cdd')
Polyhedra in QQ^3
>>> type(_)
<class 'sage.geometry.polyhedron.parent.Polyhedra_QQ_cdd_with_category'>

Sage Live

from sage.geometry.polyhedron.parent import Polyhedra
Polyhedra(AA, 3)                                                          # needs sage.rings.number_field
Polyhedra(ZZ, 3)
type(_)
Polyhedra(QQ, 3, backend='cdd')
type(_)

CDD does not support integer polytopes directly:

Sage

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

Python

>>> from sage.all import *
>>> Polyhedra(ZZ, Integer(3), backend='cdd')
Polyhedra in QQ^3

Sage Live

Polyhedra(ZZ, 3, backend='cdd')

Using a more general form of the constructor:

Sage

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

Python

>>> from sage.all import *
>>> V = VectorSpace(QQ, Integer(3))
>>> Polyhedra(V) is Polyhedra(QQ, Integer(3))
True
>>> Polyhedra(V, backend='field') is Polyhedra(QQ, Integer(3), 'field')
True
>>> Polyhedra(backend='field', ambient_space=V) is Polyhedra(QQ, Integer(3), 'field')
True

>>> M = FreeModule(ZZ, Integer(2))
>>> Polyhedra(M, backend='ppl') is Polyhedra(ZZ, Integer(2), 'ppl')
True

Sage Live

V = VectorSpace(QQ, 3)
Polyhedra(V) is Polyhedra(QQ, 3)
Polyhedra(V, backend='field') is Polyhedra(QQ, 3, 'field')
Polyhedra(backend='field', ambient_space=V) is Polyhedra(QQ, 3, 'field')
M = FreeModule(ZZ, 2)
Polyhedra(M, backend='ppl') is Polyhedra(ZZ, 2, 'ppl')

class sage.geometry.polyhedron.parent.Polyhedra_QQ_cdd(base_ring, ambient_dim, backend)[source]¶

Bases: Polyhedra_base

Element[source]¶: alias of Polyhedron_QQ_cdd

class sage.geometry.polyhedron.parent.Polyhedra_QQ_normaliz(base_ring, ambient_dim, backend)[source]¶

Bases: Polyhedra_base

Element[source]¶: alias of Polyhedron_QQ_normaliz

class sage.geometry.polyhedron.parent.Polyhedra_QQ_ppl(base_ring, ambient_dim, backend)[source]¶

Bases: Polyhedra_base

Element[source]¶: alias of Polyhedron_QQ_ppl

class sage.geometry.polyhedron.parent.Polyhedra_RDF_cdd(base_ring, ambient_dim, backend)[source]¶

Bases: Polyhedra_base

Element[source]¶: alias of Polyhedron_RDF_cdd

class sage.geometry.polyhedron.parent.Polyhedra_ZZ_normaliz(base_ring, ambient_dim, backend)[source]¶

Bases: Polyhedra_base

Element[source]¶: alias of Polyhedron_ZZ_normaliz

class sage.geometry.polyhedron.parent.Polyhedra_ZZ_ppl(base_ring, ambient_dim, backend)[source]¶

Bases: Polyhedra_base

Element[source]¶: alias of Polyhedron_ZZ_ppl

class sage.geometry.polyhedron.parent.Polyhedra_base(base_ring, ambient_dim, backend)[source]¶

Bases: UniqueRepresentation, 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

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

Python

>>> from sage.all import *
>>> from sage.geometry.polyhedron.parent import Polyhedra
>>> Polyhedra(ZZ, Integer(3))
Polyhedra in ZZ^3

Sage Live

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

Hrepresentation_space()[source]¶

Return the linear space containing the H-representation vectors.

OUTPUT: a free module over the base ring of dimension ambient_dim() + 1

EXAMPLES:

Sage

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

Python

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

Sage Live

from sage.geometry.polyhedron.parent import Polyhedra
Polyhedra(ZZ, 2).Hrepresentation_space()

Vrepresentation_space()[source]¶

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

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

Python

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

Sage Live

from sage.geometry.polyhedron.parent import Polyhedra
Polyhedra(QQ, 4).Vrepresentation_space()
Polyhedra(QQ, 4).ambient_space()

ambient_dim()[source]¶

Return the dimension of the ambient space.

EXAMPLES:

Sage

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

Python

>>> from sage.all import *
>>> from sage.geometry.polyhedron.parent import Polyhedra
>>> Polyhedra(QQ, Integer(3)).ambient_dim()
3

Sage Live

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

ambient_space()[source]¶

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

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

Python

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

Sage Live

from sage.geometry.polyhedron.parent import Polyhedra
Polyhedra(QQ, 4).Vrepresentation_space()
Polyhedra(QQ, 4).ambient_space()

an_element()[source]¶

Return a Polyhedron.

EXAMPLES:

Sage

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

Python

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

Sage Live

from sage.geometry.polyhedron.parent import Polyhedra
Polyhedra(QQ, 4).an_element()

backend()[source]¶

Return the backend.

EXAMPLES:

Sage

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

Python

>>> from sage.all import *
>>> from sage.geometry.polyhedron.parent import Polyhedra
>>> Polyhedra(QQ, Integer(3)).backend()
'ppl'

Sage Live

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

base_extend(base_ring, backend=None, ambient_dim=None)[source]¶

Return the base extended parent.

INPUT:

base_ring, backend – see Polyhedron()
ambient_dim – if not None change ambient dimension accordingly

EXAMPLES:

Sage

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

Python

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

Sage Live

from sage.geometry.polyhedron.parent import Polyhedra
Polyhedra(ZZ, 3).base_extend(QQ)
Polyhedra(ZZ, 3).an_element().base_extend(QQ)
Polyhedra(QQ, 2).base_extend(ZZ)

change_ring(base_ring, backend=None, ambient_dim=None)[source]¶

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

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'

Python

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

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

Sage Live

from sage.geometry.polyhedron.parent import Polyhedra
Polyhedra(ZZ, 3).change_ring(QQ)
Polyhedra(ZZ, 3).an_element().change_ring(QQ)
Polyhedra(RDF, 3).change_ring(QQ).backend()
Polyhedra(QQ, 3).change_ring(ZZ, ambient_dim=4)
Polyhedra(QQ, 3, backend='cdd').change_ring(QQ, ambient_dim=4).backend()

empty()[source]¶

Return the empty polyhedron.

EXAMPLES:

Sage

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

Python

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

Sage Live

from sage.geometry.polyhedron.parent import Polyhedra
P = Polyhedra(QQ, 4)
P.empty()
P.empty().is_empty()

list()[source]¶

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

EXAMPLES:

Sage

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

sage: # needs sage.rings.number_field
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

Python

>>> from sage.all import *
>>> from sage.geometry.polyhedron.parent import Polyhedra
>>> P = Polyhedra(QQ, Integer(3))
>>> P.cardinality()
+Infinity

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

Sage Live

from sage.geometry.polyhedron.parent import Polyhedra
P = Polyhedra(QQ, 3)
P.cardinality()
# needs sage.rings.number_field
P = Polyhedra(AA, 0)
P.category()
P.list()
P.cardinality()

recycle(polyhedron)[source]¶

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

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

Python

>>> from sage.all import *
>>> p = Polyhedron([(Integer(0),Integer(0)),(Integer(1),Integer(0)),(Integer(0),Integer(1))])
>>> p.parent().recycle(p)

Sage Live

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

some_elements()[source]¶

Return a list of some elements of the semigroup.

EXAMPLES:

Sage

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]

Python

>>> from sage.all import *
>>> from sage.geometry.polyhedron.parent import Polyhedra
>>> Polyhedra(QQ, Integer(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]
>>> Polyhedra(ZZ, Integer(0)).some_elements()
[The empty polyhedron in ZZ^0,
 A 0-dimensional polyhedron in ZZ^0 defined as the convex hull of 1 vertex]

Sage Live

from sage.geometry.polyhedron.parent import Polyhedra
Polyhedra(QQ, 4).some_elements()
Polyhedra(ZZ, 0).some_elements()

universe()[source]¶

Return the entire ambient space as polyhedron.

EXAMPLES:

Sage

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

Python

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

Sage Live

from sage.geometry.polyhedron.parent import Polyhedra
P = Polyhedra(QQ, 4)
P.universe()
P.universe().is_universe()

zero()[source]¶

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

EXAMPLES:

Sage

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

Python

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

Sage Live

from sage.geometry.polyhedron.parent import Polyhedra
p = Polyhedra(QQ, 4).zero();  p
p + p == p

class sage.geometry.polyhedron.parent.Polyhedra_field(base_ring, ambient_dim, backend)[source]¶

Bases: Polyhedra_base

Element[source]¶: alias of Polyhedron_field

class sage.geometry.polyhedron.parent.Polyhedra_normaliz(base_ring, ambient_dim, backend)[source]¶

Bases: Polyhedra_base

Element[source]¶: alias of Polyhedron_normaliz

class sage.geometry.polyhedron.parent.Polyhedra_number_field(base_ring, ambient_dim, backend)[source]¶

Bases: Polyhedra_base

Element[source]¶: alias of Polyhedron_number_field

class sage.geometry.polyhedron.parent.Polyhedra_polymake(base_ring, ambient_dim, backend)[source]¶

Bases: Polyhedra_base

Element[source]¶: alias of Polyhedron_polymake

sage.geometry.polyhedron.parent.does_backend_handle_base_ring(backend)[source]¶

Return true, if backend can handle base_ring.

EXAMPLES:

Sage

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')                         # needs sage.rings.number_field sage.symbolic
False
sage: does_backend_handle_base_ring(QQ[sqrt(5)], 'field')                       # needs sage.rings.number_field sage.symbolic
True

Python

>>> from sage.all import *
>>> from sage.geometry.polyhedron.parent import does_backend_handle_base_ring
>>> does_backend_handle_base_ring(QQ, 'ppl')
True
>>> does_backend_handle_base_ring(QQ[sqrt(Integer(5))], 'ppl')                         # needs sage.rings.number_field sage.symbolic
False
>>> does_backend_handle_base_ring(QQ[sqrt(Integer(5))], 'field')                       # needs sage.rings.number_field sage.symbolic
True

Sage Live

from sage.geometry.polyhedron.parent import does_backend_handle_base_ring
does_backend_handle_base_ring(QQ, 'ppl')
does_backend_handle_base_ring(QQ[sqrt(5)], 'ppl')                         # needs sage.rings.number_field sage.symbolic
does_backend_handle_base_ring(QQ[sqrt(5)], 'field')                       # needs sage.rings.number_field sage.symbolic