A class to keep information about faces of a polyhedron

This module gives you a tool to work with the faces of a polyhedron and their relative position. First, you need to find the faces. To get the faces in a particular dimension, use the face() method:

sage: P = polytopes.cross_polytope(3)
sage: P.faces(3)
(A 3-dimensional face of a Polyhedron in ZZ^3 defined as the convex hull of 6 vertices,)
sage: [f.ambient_V_indices() for f in P.facets()]
[(3, 4, 5),
 (2, 4, 5),
 (1, 3, 5),
 (1, 2, 5),
 (0, 3, 4),
 (0, 2, 4),
 (0, 1, 3),
 (0, 1, 2)]
sage: [f.ambient_V_indices() for f in P.faces(1)]
[(4, 5),
 (3, 5),
 (2, 5),
 (1, 5),
 (3, 4),
 (2, 4),
 (0, 4),
 (1, 3),
 (0, 3),
 (1, 2),
 (0, 2),
 (0, 1)]

>>> from sage.all import *
>>> P = polytopes.cross_polytope(Integer(3))
>>> P.faces(Integer(3))
(A 3-dimensional face of a Polyhedron in ZZ^3 defined as the convex hull of 6 vertices,)
>>> [f.ambient_V_indices() for f in P.facets()]
[(3, 4, 5),
 (2, 4, 5),
 (1, 3, 5),
 (1, 2, 5),
 (0, 3, 4),
 (0, 2, 4),
 (0, 1, 3),
 (0, 1, 2)]
>>> [f.ambient_V_indices() for f in P.faces(Integer(1))]
[(4, 5),
 (3, 5),
 (2, 5),
 (1, 5),
 (3, 4),
 (2, 4),
 (0, 4),
 (1, 3),
 (0, 3),
 (1, 2),
 (0, 2),
 (0, 1)]

or face_lattice() to get the whole face lattice as a poset:

sage: P.face_lattice()                                                              # needs sage.combinat
Finite lattice containing 28 elements

>>> from sage.all import *
>>> P.face_lattice()                                                              # needs sage.combinat
Finite lattice containing 28 elements

The faces are printed in shorthand notation where each integer is the index of a vertex/ray/line in the same order as the containing Polyhedron’s Vrepresentation()

sage: face = P.faces(1)[8];  face
A 1-dimensional face of a Polyhedron in ZZ^3 defined as the convex hull of 2 vertices
sage: face.ambient_V_indices()
(0, 3)
sage: P.Vrepresentation(0)
A vertex at (-1, 0, 0)
sage: P.Vrepresentation(3)
A vertex at (0, 0, 1)
sage: face.vertices()
(A vertex at (-1, 0, 0), A vertex at (0, 0, 1))

>>> from sage.all import *
>>> face = P.faces(Integer(1))[Integer(8)];  face
A 1-dimensional face of a Polyhedron in ZZ^3 defined as the convex hull of 2 vertices
>>> face.ambient_V_indices()
(0, 3)
>>> P.Vrepresentation(Integer(0))
A vertex at (-1, 0, 0)
>>> P.Vrepresentation(Integer(3))
A vertex at (0, 0, 1)
>>> face.vertices()
(A vertex at (-1, 0, 0), A vertex at (0, 0, 1))

The face itself is not represented by Sage’s sage.geometry.polyhedron.constructor.Polyhedron() class, but by an auxiliary class to keep the information. You can get the face as a polyhedron with the PolyhedronFace.as_polyhedron() method:

sage: face.as_polyhedron()
A 1-dimensional polyhedron in ZZ^3 defined as the convex hull of 2 vertices
sage: _.equations()
(An equation (0, 1, 0) x + 0 == 0,
 An equation (1, 0, -1) x + 1 == 0)

>>> from sage.all import *
>>> face.as_polyhedron()
A 1-dimensional polyhedron in ZZ^3 defined as the convex hull of 2 vertices
>>> _.equations()
(An equation (0, 1, 0) x + 0 == 0,
 An equation (1, 0, -1) x + 1 == 0)
class sage.geometry.polyhedron.face.PolyhedronFace(polyhedron, V_indices, H_indices)[source]

Bases: ConvexSet_closed

A face of a polyhedron.

This class is for use in face_lattice().

INPUT:

No checking is performed whether the H/V-representation indices actually determine a face of the polyhedron. You should not manually create PolyhedronFace objects unless you know what you are doing.

OUTPUT: a PolyhedronFace

EXAMPLES:

sage: octahedron = polytopes.cross_polytope(3)
sage: inequality = octahedron.Hrepresentation(2)
sage: face_h = tuple([ inequality ])
sage: face_v = tuple( inequality.incident() )
sage: face_h_indices = [ h.index() for h in face_h ]
sage: face_v_indices = [ v.index() for v in face_v ]
sage: from sage.geometry.polyhedron.face import PolyhedronFace
sage: face = PolyhedronFace(octahedron, face_v_indices, face_h_indices)
sage: face
A 2-dimensional face of a Polyhedron in ZZ^3 defined as the convex hull of 3 vertices
sage: face.dim()
2
sage: face.ambient_V_indices()
(0, 1, 2)
sage: face.ambient_Hrepresentation()
(An inequality (1, 1, 1) x + 1 >= 0,)
sage: face.ambient_Vrepresentation()
(A vertex at (-1, 0, 0), A vertex at (0, -1, 0), A vertex at (0, 0, -1))

>>> from sage.all import *
>>> octahedron = polytopes.cross_polytope(Integer(3))
>>> inequality = octahedron.Hrepresentation(Integer(2))
>>> face_h = tuple([ inequality ])
>>> face_v = tuple( inequality.incident() )
>>> face_h_indices = [ h.index() for h in face_h ]
>>> face_v_indices = [ v.index() for v in face_v ]
>>> from sage.geometry.polyhedron.face import PolyhedronFace
>>> face = PolyhedronFace(octahedron, face_v_indices, face_h_indices)
>>> face
A 2-dimensional face of a Polyhedron in ZZ^3 defined as the convex hull of 3 vertices
>>> face.dim()
2
>>> face.ambient_V_indices()
(0, 1, 2)
>>> face.ambient_Hrepresentation()
(An inequality (1, 1, 1) x + 1 >= 0,)
>>> face.ambient_Vrepresentation()
(A vertex at (-1, 0, 0), A vertex at (0, -1, 0), A vertex at (0, 0, -1))
affine_tangent_cone()[source]

Return the affine tangent cone of self as a polyhedron.

It is equal to the sum of self and the cone of feasible directions at any point of the relative interior of self.

OUTPUT: a polyhedron

EXAMPLES:

sage: half_plane_in_space = Polyhedron(ieqs=[(0,1,0,0)], eqns=[(0,0,0,1)])
sage: line = half_plane_in_space.faces(1)[0]; line
A 1-dimensional face of a
 Polyhedron in QQ^3 defined as the convex hull of 1 vertex and 1 line
sage: T_line = line.affine_tangent_cone()
sage: T_line == half_plane_in_space
True

sage: c = polytopes.cube()
sage: edge = min(c.faces(1))
sage: edge.vertices()
(A vertex at (1, -1, -1), A vertex at (1, 1, -1))
sage: T_edge = edge.affine_tangent_cone()
sage: T_edge.Vrepresentation()
(A line in the direction (0, 1, 0),
 A ray in the direction (0, 0, 1),
 A vertex at (1, 0, -1),
 A ray in the direction (-1, 0, 0))

>>> from sage.all import *
>>> half_plane_in_space = Polyhedron(ieqs=[(Integer(0),Integer(1),Integer(0),Integer(0))], eqns=[(Integer(0),Integer(0),Integer(0),Integer(1))])
>>> line = half_plane_in_space.faces(Integer(1))[Integer(0)]; line
A 1-dimensional face of a
 Polyhedron in QQ^3 defined as the convex hull of 1 vertex and 1 line
>>> T_line = line.affine_tangent_cone()
>>> T_line == half_plane_in_space
True

>>> c = polytopes.cube()
>>> edge = min(c.faces(Integer(1)))
>>> edge.vertices()
(A vertex at (1, -1, -1), A vertex at (1, 1, -1))
>>> T_edge = edge.affine_tangent_cone()
>>> T_edge.Vrepresentation()
(A line in the direction (0, 1, 0),
 A ray in the direction (0, 0, 1),
 A vertex at (1, 0, -1),
 A ray in the direction (-1, 0, 0))
ambient()[source]

Return the containing polyhedron.

EXAMPLES:

sage: P = polytopes.cross_polytope(3); P
A 3-dimensional polyhedron in ZZ^3 defined as the convex hull of 6 vertices
sage: face = P.facets()[3]; face
A 2-dimensional face of a Polyhedron in ZZ^3 defined as the convex hull of 3 vertices
sage: face.polyhedron()
A 3-dimensional polyhedron in ZZ^3 defined as the convex hull of 6 vertices

>>> from sage.all import *
>>> P = polytopes.cross_polytope(Integer(3)); P
A 3-dimensional polyhedron in ZZ^3 defined as the convex hull of 6 vertices
>>> face = P.facets()[Integer(3)]; face
A 2-dimensional face of a Polyhedron in ZZ^3 defined as the convex hull of 3 vertices
>>> face.polyhedron()
A 3-dimensional polyhedron in ZZ^3 defined as the convex hull of 6 vertices
ambient_H_indices()[source]

Return the indices of the H-representation objects of the ambient polyhedron that make up the H-representation of self.

See also ambient_Hrepresentation().

OUTPUT: tuple of indices

EXAMPLES:

sage: Q = polytopes.cross_polytope(3)
sage: F = Q.faces(1)
sage: [f.ambient_H_indices() for f in F]
[(4, 5),
 (5, 6),
 (4, 7),
 (6, 7),
 (0, 5),
 (3, 4),
 (0, 3),
 (1, 6),
 (0, 1),
 (2, 7),
 (2, 3),
 (1, 2)]

>>> from sage.all import *
>>> Q = polytopes.cross_polytope(Integer(3))
>>> F = Q.faces(Integer(1))
>>> [f.ambient_H_indices() for f in F]
[(4, 5),
 (5, 6),
 (4, 7),
 (6, 7),
 (0, 5),
 (3, 4),
 (0, 3),
 (1, 6),
 (0, 1),
 (2, 7),
 (2, 3),
 (1, 2)]
ambient_Hrepresentation(index=None)[source]

Return the H-representation objects of the ambient polytope defining the face.

INPUT:

  • index – integer or None (default)

OUTPUT:

If the optional argument is not present, a tuple of H-representation objects. Each entry is either an inequality or an equation.

If the optional integer index is specified, the index-th element of the tuple is returned.

EXAMPLES:

sage: square = polytopes.hypercube(2)
sage: for face in square.face_lattice():                                    # needs sage.combinat
....:     print(face.ambient_Hrepresentation())
(An inequality (-1, 0) x + 1 >= 0, An inequality (0, -1) x + 1 >= 0,
 An inequality (1, 0) x + 1 >= 0, An inequality (0, 1) x + 1 >= 0)
(An inequality (-1, 0) x + 1 >= 0, An inequality (0, 1) x + 1 >= 0)
(An inequality (-1, 0) x + 1 >= 0, An inequality (0, -1) x + 1 >= 0)
(An inequality (-1, 0) x + 1 >= 0,)
(An inequality (0, -1) x + 1 >= 0, An inequality (1, 0) x + 1 >= 0)
(An inequality (0, -1) x + 1 >= 0,)
(An inequality (1, 0) x + 1 >= 0, An inequality (0, 1) x + 1 >= 0)
(An inequality (0, 1) x + 1 >= 0,)
(An inequality (1, 0) x + 1 >= 0,)
()

>>> from sage.all import *
>>> square = polytopes.hypercube(Integer(2))
>>> for face in square.face_lattice():                                    # needs sage.combinat
...     print(face.ambient_Hrepresentation())
(An inequality (-1, 0) x + 1 >= 0, An inequality (0, -1) x + 1 >= 0,
 An inequality (1, 0) x + 1 >= 0, An inequality (0, 1) x + 1 >= 0)
(An inequality (-1, 0) x + 1 >= 0, An inequality (0, 1) x + 1 >= 0)
(An inequality (-1, 0) x + 1 >= 0, An inequality (0, -1) x + 1 >= 0)
(An inequality (-1, 0) x + 1 >= 0,)
(An inequality (0, -1) x + 1 >= 0, An inequality (1, 0) x + 1 >= 0)
(An inequality (0, -1) x + 1 >= 0,)
(An inequality (1, 0) x + 1 >= 0, An inequality (0, 1) x + 1 >= 0)
(An inequality (0, 1) x + 1 >= 0,)
(An inequality (1, 0) x + 1 >= 0,)
()
ambient_V_indices()[source]

Return the indices of the V-representation objects of the ambient polyhedron that make up the V-representation of self.

See also ambient_Vrepresentation().

OUTPUT: tuple of indices

EXAMPLES:

sage: P = polytopes.cube()
sage: F = P.faces(2)
sage: [f.ambient_V_indices() for f in F]
[(0, 3, 4, 5),
 (0, 1, 5, 6),
 (4, 5, 6, 7),
 (2, 3, 4, 7),
 (1, 2, 6, 7),
 (0, 1, 2, 3)]

>>> from sage.all import *
>>> P = polytopes.cube()
>>> F = P.faces(Integer(2))
>>> [f.ambient_V_indices() for f in F]
[(0, 3, 4, 5),
 (0, 1, 5, 6),
 (4, 5, 6, 7),
 (2, 3, 4, 7),
 (1, 2, 6, 7),
 (0, 1, 2, 3)]
ambient_Vrepresentation(index=None)[source]

Return the V-representation objects of the ambient polytope defining the face.

INPUT:

  • index – integer or None (default)

OUTPUT:

If the optional argument is not present, a tuple of V-representation objects. Each entry is either a vertex, a ray, or a line.

If the optional integer index is specified, the index-th element of the tuple is returned.

EXAMPLES:

sage: square = polytopes.hypercube(2)
sage: for fl in square.face_lattice():                                      # needs sage.combinat
....:     print(fl.ambient_Vrepresentation())
()
(A vertex at (1, -1),)
(A vertex at (1, 1),)
(A vertex at (1, -1), A vertex at (1, 1))
(A vertex at (-1, 1),)
(A vertex at (1, 1), A vertex at (-1, 1))
(A vertex at (-1, -1),)
(A vertex at (1, -1), A vertex at (-1, -1))
(A vertex at (-1, 1), A vertex at (-1, -1))
(A vertex at (1, -1), A vertex at (1, 1),
 A vertex at (-1, 1), A vertex at (-1, -1))

>>> from sage.all import *
>>> square = polytopes.hypercube(Integer(2))
>>> for fl in square.face_lattice():                                      # needs sage.combinat
...     print(fl.ambient_Vrepresentation())
()
(A vertex at (1, -1),)
(A vertex at (1, 1),)
(A vertex at (1, -1), A vertex at (1, 1))
(A vertex at (-1, 1),)
(A vertex at (1, 1), A vertex at (-1, 1))
(A vertex at (-1, -1),)
(A vertex at (1, -1), A vertex at (-1, -1))
(A vertex at (-1, 1), A vertex at (-1, -1))
(A vertex at (1, -1), A vertex at (1, 1),
 A vertex at (-1, 1), A vertex at (-1, -1))
ambient_dim()[source]

Return the dimension of the containing polyhedron.

EXAMPLES:

sage: P = Polyhedron(vertices = [[1,0,0,0],[0,1,0,0]])
sage: face = P.faces(1)[0]
sage: face.ambient_dim()
4

>>> from sage.all import *
>>> P = Polyhedron(vertices = [[Integer(1),Integer(0),Integer(0),Integer(0)],[Integer(0),Integer(1),Integer(0),Integer(0)]])
>>> face = P.faces(Integer(1))[Integer(0)]
>>> face.ambient_dim()
4
ambient_vector_space(base_field=None)[source]

Return the ambient vector space.

It is the ambient free module of the containing polyhedron tensored with a field.

INPUT:

  • base_field – a field (default: the fraction field of the base ring)

EXAMPLES:

sage: half_plane = Polyhedron(ieqs=[(0,1,0)])
sage: line = half_plane.faces(1)[0]; line
A 1-dimensional face of a
 Polyhedron in QQ^2 defined as the convex hull of 1 vertex and 1 line
sage: line.ambient_vector_space()
Vector space of dimension 2 over Rational Field
sage: line.ambient_vector_space(AA)                                         # needs sage.rings.number_field
Vector space of dimension 2 over Algebraic Real Field

>>> from sage.all import *
>>> half_plane = Polyhedron(ieqs=[(Integer(0),Integer(1),Integer(0))])
>>> line = half_plane.faces(Integer(1))[Integer(0)]; line
A 1-dimensional face of a
 Polyhedron in QQ^2 defined as the convex hull of 1 vertex and 1 line
>>> line.ambient_vector_space()
Vector space of dimension 2 over Rational Field
>>> line.ambient_vector_space(AA)                                         # needs sage.rings.number_field
Vector space of dimension 2 over Algebraic Real Field
as_polyhedron(**kwds)[source]

Return the face as an independent polyhedron.

OUTPUT: a polyhedron

EXAMPLES:

sage: P = polytopes.cross_polytope(3);  P
A 3-dimensional polyhedron in ZZ^3 defined as the convex hull of 6 vertices
sage: face = P.faces(2)[3]; face
A 2-dimensional face of a
 Polyhedron in ZZ^3 defined as the convex hull of 3 vertices
sage: face.as_polyhedron()
A 2-dimensional polyhedron in ZZ^3 defined as the convex hull of 3 vertices

sage: P.intersection(face.as_polyhedron()) == face.as_polyhedron()
True

>>> from sage.all import *
>>> P = polytopes.cross_polytope(Integer(3));  P
A 3-dimensional polyhedron in ZZ^3 defined as the convex hull of 6 vertices
>>> face = P.faces(Integer(2))[Integer(3)]; face
A 2-dimensional face of a
 Polyhedron in ZZ^3 defined as the convex hull of 3 vertices
>>> face.as_polyhedron()
A 2-dimensional polyhedron in ZZ^3 defined as the convex hull of 3 vertices

>>> P.intersection(face.as_polyhedron()) == face.as_polyhedron()
True
contains(point)[source]

Test whether the polyhedron contains the given point.

INPUT:

  • point – a point or its coordinates

EXAMPLES:

sage: half_plane = Polyhedron(ieqs=[(0,1,0)])
sage: line = half_plane.faces(1)[0]; line
A 1-dimensional face of a
 Polyhedron in QQ^2 defined as the convex hull of 1 vertex and 1 line
sage: line.contains([0, 1])
True

>>> from sage.all import *
>>> half_plane = Polyhedron(ieqs=[(Integer(0),Integer(1),Integer(0))])
>>> line = half_plane.faces(Integer(1))[Integer(0)]; line
A 1-dimensional face of a
 Polyhedron in QQ^2 defined as the convex hull of 1 vertex and 1 line
>>> line.contains([Integer(0), Integer(1)])
True

As a shorthand, one may use the usual in operator:

sage: [5, 7] in line
False

>>> from sage.all import *
>>> [Integer(5), Integer(7)] in line
False
dim()[source]

Return the dimension of the face.

OUTPUT: integer

EXAMPLES:

sage: fl = polytopes.dodecahedron().face_lattice()                          # needs sage.combinat sage.rings.number_field
sage: sorted(x.dim() for x in fl)                                           # needs sage.combinat sage.rings.number_field
[-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
  1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
  1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3]

>>> from sage.all import *
>>> fl = polytopes.dodecahedron().face_lattice()                          # needs sage.combinat sage.rings.number_field
>>> sorted(x.dim() for x in fl)                                           # needs sage.combinat sage.rings.number_field
[-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
  1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
  1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3]
is_compact()[source]

Return whether self is compact.

OUTPUT: boolean

EXAMPLES:

sage: half_plane = Polyhedron(ieqs=[(0,1,0)])
sage: line = half_plane.faces(1)[0]; line
A 1-dimensional face of a
 Polyhedron in QQ^2 defined as the convex hull of 1 vertex and 1 line
sage: line.is_compact()
False

>>> from sage.all import *
>>> half_plane = Polyhedron(ieqs=[(Integer(0),Integer(1),Integer(0))])
>>> line = half_plane.faces(Integer(1))[Integer(0)]; line
A 1-dimensional face of a
 Polyhedron in QQ^2 defined as the convex hull of 1 vertex and 1 line
>>> line.is_compact()
False
is_relatively_open()[source]

Return whether self is relatively open.

OUTPUT: boolean

EXAMPLES:

sage: half_plane = Polyhedron(ieqs=[(0,1,0)])
sage: line = half_plane.faces(1)[0]; line
A 1-dimensional face of a
 Polyhedron in QQ^2 defined as the convex hull of 1 vertex and 1 line
sage: line.is_relatively_open()
True

>>> from sage.all import *
>>> half_plane = Polyhedron(ieqs=[(Integer(0),Integer(1),Integer(0))])
>>> line = half_plane.faces(Integer(1))[Integer(0)]; line
A 1-dimensional face of a
 Polyhedron in QQ^2 defined as the convex hull of 1 vertex and 1 line
>>> line.is_relatively_open()
True
line_generator()[source]

Return a generator for the lines of the face.

EXAMPLES:

sage: pr = Polyhedron(rays = [[1,0],[-1,0],[0,1]], vertices = [[-1,-1]])
sage: face = pr.faces(1)[0]
sage: next(face.line_generator())
A line in the direction (1, 0)

>>> from sage.all import *
>>> pr = Polyhedron(rays = [[Integer(1),Integer(0)],[-Integer(1),Integer(0)],[Integer(0),Integer(1)]], vertices = [[-Integer(1),-Integer(1)]])
>>> face = pr.faces(Integer(1))[Integer(0)]
>>> next(face.line_generator())
A line in the direction (1, 0)
lines()[source]

Return all lines of the face.

OUTPUT: a tuple of lines

EXAMPLES:

sage: p = Polyhedron(rays = [[1,0],[-1,0],[0,1],[1,1]], vertices = [[-2,-2],[2,3]])
sage: p.lines()
(A line in the direction (1, 0),)

>>> from sage.all import *
>>> p = Polyhedron(rays = [[Integer(1),Integer(0)],[-Integer(1),Integer(0)],[Integer(0),Integer(1)],[Integer(1),Integer(1)]], vertices = [[-Integer(2),-Integer(2)],[Integer(2),Integer(3)]])
>>> p.lines()
(A line in the direction (1, 0),)
n_ambient_Hrepresentation()[source]

Return the number of objects that make up the ambient H-representation of the polyhedron.

See also ambient_Hrepresentation().

OUTPUT: integer

EXAMPLES:

sage: p = polytopes.cross_polytope(4)
sage: face = p.face_lattice()[5]; face                                      # needs sage.combinat
A 1-dimensional face of a Polyhedron in ZZ^4 defined as the convex hull of 2 vertices
sage: face.ambient_Hrepresentation()                                        # needs sage.combinat
(An inequality (1, -1, 1, -1) x + 1 >= 0,
 An inequality (1, 1, 1, 1) x + 1 >= 0,
 An inequality (1, 1, 1, -1) x + 1 >= 0,
 An inequality (1, -1, 1, 1) x + 1 >= 0)
sage: face.n_ambient_Hrepresentation()                                      # needs sage.combinat
4

>>> from sage.all import *
>>> p = polytopes.cross_polytope(Integer(4))
>>> face = p.face_lattice()[Integer(5)]; face                                      # needs sage.combinat
A 1-dimensional face of a Polyhedron in ZZ^4 defined as the convex hull of 2 vertices
>>> face.ambient_Hrepresentation()                                        # needs sage.combinat
(An inequality (1, -1, 1, -1) x + 1 >= 0,
 An inequality (1, 1, 1, 1) x + 1 >= 0,
 An inequality (1, 1, 1, -1) x + 1 >= 0,
 An inequality (1, -1, 1, 1) x + 1 >= 0)
>>> face.n_ambient_Hrepresentation()                                      # needs sage.combinat
4
n_ambient_Vrepresentation()[source]

Return the number of objects that make up the ambient V-representation of the polyhedron.

See also ambient_Vrepresentation().

OUTPUT: integer

EXAMPLES:

sage: p = polytopes.cross_polytope(4)
sage: face = p.face_lattice()[5]; face                                      # needs sage.combinat
A 1-dimensional face of a Polyhedron in ZZ^4 defined as the convex hull of 2 vertices
sage: face.ambient_Vrepresentation()                                        # needs sage.combinat
(A vertex at (-1, 0, 0, 0), A vertex at (0, 0, -1, 0))
sage: face.n_ambient_Vrepresentation()                                      # needs sage.combinat
2

>>> from sage.all import *
>>> p = polytopes.cross_polytope(Integer(4))
>>> face = p.face_lattice()[Integer(5)]; face                                      # needs sage.combinat
A 1-dimensional face of a Polyhedron in ZZ^4 defined as the convex hull of 2 vertices
>>> face.ambient_Vrepresentation()                                        # needs sage.combinat
(A vertex at (-1, 0, 0, 0), A vertex at (0, 0, -1, 0))
>>> face.n_ambient_Vrepresentation()                                      # needs sage.combinat
2
n_lines()[source]

Return the number of lines of the face.

OUTPUT: integer

EXAMPLES:

sage: p = Polyhedron(rays = [[1,0],[-1,0],[0,1],[1,1]], vertices = [[-2,-2],[2,3]])
sage: p.n_lines()
1

>>> from sage.all import *
>>> p = Polyhedron(rays = [[Integer(1),Integer(0)],[-Integer(1),Integer(0)],[Integer(0),Integer(1)],[Integer(1),Integer(1)]], vertices = [[-Integer(2),-Integer(2)],[Integer(2),Integer(3)]])
>>> p.n_lines()
1
n_rays()[source]

Return the number of rays of the face.

OUTPUT: integer

EXAMPLES:

sage: p = Polyhedron(ieqs = [[0,0,0,1],[0,0,1,0],[1,1,0,0]])
sage: face = p.faces(2)[0]
sage: face.n_rays()
2

>>> from sage.all import *
>>> p = Polyhedron(ieqs = [[Integer(0),Integer(0),Integer(0),Integer(1)],[Integer(0),Integer(0),Integer(1),Integer(0)],[Integer(1),Integer(1),Integer(0),Integer(0)]])
>>> face = p.faces(Integer(2))[Integer(0)]
>>> face.n_rays()
2
n_vertices()[source]

Return the number of vertices of the face.

OUTPUT: integer

EXAMPLES:

sage: Q = polytopes.cross_polytope(3)
sage: face = Q.faces(2)[0]
sage: face.n_vertices()
3

>>> from sage.all import *
>>> Q = polytopes.cross_polytope(Integer(3))
>>> face = Q.faces(Integer(2))[Integer(0)]
>>> face.n_vertices()
3
normal_cone(direction='outer')[source]

Return the polyhedral cone consisting of normal vectors to hyperplanes supporting self.

INPUT:

  • direction – string (default: 'outer'); the direction in which to consider the normals. The other allowed option is 'inner'.

OUTPUT: a polyhedron

EXAMPLES:

sage: p = Polyhedron(vertices=[[1,2], [2,1], [-2,2], [-2,-2], [2,-2]])
sage: for v in p.face_generator(0):
....:     vect = v.vertices()[0].vector()
....:     nc = v.normal_cone().rays_list()
....:     print("{} has outer normal cone spanned by {}".format(vect,nc))
....:
(2, 1) has outer normal cone spanned by [[1, 0], [1, 1]]
(1, 2) has outer normal cone spanned by [[0, 1], [1, 1]]
(2, -2) has outer normal cone spanned by [[0, -1], [1, 0]]
(-2, -2) has outer normal cone spanned by [[-1, 0], [0, -1]]
(-2, 2) has outer normal cone spanned by [[-1, 0], [0, 1]]

sage: for v in p.face_generator(0):
....:     vect = v.vertices()[0].vector()
....:     nc = v.normal_cone(direction='inner').rays_list()
....:     print("{} has inner normal cone spanned by {}".format(vect,nc))
....:
(2, 1) has inner normal cone spanned by [[-1, -1], [-1, 0]]
(1, 2) has inner normal cone spanned by [[-1, -1], [0, -1]]
(2, -2) has inner normal cone spanned by [[-1, 0], [0, 1]]
(-2, -2) has inner normal cone spanned by [[0, 1], [1, 0]]
(-2, 2) has inner normal cone spanned by [[0, -1], [1, 0]]

>>> from sage.all import *
>>> p = Polyhedron(vertices=[[Integer(1),Integer(2)], [Integer(2),Integer(1)], [-Integer(2),Integer(2)], [-Integer(2),-Integer(2)], [Integer(2),-Integer(2)]])
>>> for v in p.face_generator(Integer(0)):
...     vect = v.vertices()[Integer(0)].vector()
...     nc = v.normal_cone().rays_list()
...     print("{} has outer normal cone spanned by {}".format(vect,nc))
....:
(2, 1) has outer normal cone spanned by [[1, 0], [1, 1]]
(1, 2) has outer normal cone spanned by [[0, 1], [1, 1]]
(2, -2) has outer normal cone spanned by [[0, -1], [1, 0]]
(-2, -2) has outer normal cone spanned by [[-1, 0], [0, -1]]
(-2, 2) has outer normal cone spanned by [[-1, 0], [0, 1]]

>>> for v in p.face_generator(Integer(0)):
...     vect = v.vertices()[Integer(0)].vector()
...     nc = v.normal_cone(direction='inner').rays_list()
...     print("{} has inner normal cone spanned by {}".format(vect,nc))
....:
(2, 1) has inner normal cone spanned by [[-1, -1], [-1, 0]]
(1, 2) has inner normal cone spanned by [[-1, -1], [0, -1]]
(2, -2) has inner normal cone spanned by [[-1, 0], [0, 1]]
(-2, -2) has inner normal cone spanned by [[0, 1], [1, 0]]
(-2, 2) has inner normal cone spanned by [[0, -1], [1, 0]]

The function works for polytopes that are not full-dimensional:

sage: p = polytopes.permutahedron(3)
sage: f1 = p.faces(0)[0]
sage: f2 = p.faces(1)[0]
sage: f3 = p.faces(2)[0]
sage: f1.normal_cone()
A 3-dimensional polyhedron in ZZ^3 defined as
 the convex hull of 1 vertex, 2 rays, 1 line
sage: f2.normal_cone()
A 2-dimensional polyhedron in ZZ^3 defined as
 the convex hull of 1 vertex, 1 ray, 1 line
sage: f3.normal_cone()
A 1-dimensional polyhedron in ZZ^3 defined as
 the convex hull of 1 vertex and 1 line

>>> from sage.all import *
>>> p = polytopes.permutahedron(Integer(3))
>>> f1 = p.faces(Integer(0))[Integer(0)]
>>> f2 = p.faces(Integer(1))[Integer(0)]
>>> f3 = p.faces(Integer(2))[Integer(0)]
>>> f1.normal_cone()
A 3-dimensional polyhedron in ZZ^3 defined as
 the convex hull of 1 vertex, 2 rays, 1 line
>>> f2.normal_cone()
A 2-dimensional polyhedron in ZZ^3 defined as
 the convex hull of 1 vertex, 1 ray, 1 line
>>> f3.normal_cone()
A 1-dimensional polyhedron in ZZ^3 defined as
 the convex hull of 1 vertex and 1 line

Normal cones are only defined for non-empty faces:

sage: f0 = p.faces(-1)[0]
sage: f0.normal_cone()
Traceback (most recent call last):
...
ValueError: the empty face does not have a normal cone

>>> from sage.all import *
>>> f0 = p.faces(-Integer(1))[Integer(0)]
>>> f0.normal_cone()
Traceback (most recent call last):
...
ValueError: the empty face does not have a normal cone
polyhedron()[source]

Return the containing polyhedron.

EXAMPLES:

sage: P = polytopes.cross_polytope(3); P
A 3-dimensional polyhedron in ZZ^3 defined as the convex hull of 6 vertices
sage: face = P.facets()[3]; face
A 2-dimensional face of a Polyhedron in ZZ^3 defined as the convex hull of 3 vertices
sage: face.polyhedron()
A 3-dimensional polyhedron in ZZ^3 defined as the convex hull of 6 vertices

>>> from sage.all import *
>>> P = polytopes.cross_polytope(Integer(3)); P
A 3-dimensional polyhedron in ZZ^3 defined as the convex hull of 6 vertices
>>> face = P.facets()[Integer(3)]; face
A 2-dimensional face of a Polyhedron in ZZ^3 defined as the convex hull of 3 vertices
>>> face.polyhedron()
A 3-dimensional polyhedron in ZZ^3 defined as the convex hull of 6 vertices
ray_generator()[source]

Return a generator for the rays of the face.

EXAMPLES:

sage: pi = Polyhedron(ieqs = [[1,1,0],[1,0,1]])
sage: face = pi.faces(1)[1]
sage: next(face.ray_generator())
A ray in the direction (1, 0)

>>> from sage.all import *
>>> pi = Polyhedron(ieqs = [[Integer(1),Integer(1),Integer(0)],[Integer(1),Integer(0),Integer(1)]])
>>> face = pi.faces(Integer(1))[Integer(1)]
>>> next(face.ray_generator())
A ray in the direction (1, 0)
rays()[source]

Return the rays of the face.

OUTPUT: a tuple of rays

EXAMPLES:

sage: p = Polyhedron(ieqs = [[0,0,0,1],[0,0,1,0],[1,1,0,0]])
sage: face = p.faces(2)[2]
sage: face.rays()
(A ray in the direction (1, 0, 0), A ray in the direction (0, 1, 0))

>>> from sage.all import *
>>> p = Polyhedron(ieqs = [[Integer(0),Integer(0),Integer(0),Integer(1)],[Integer(0),Integer(0),Integer(1),Integer(0)],[Integer(1),Integer(1),Integer(0),Integer(0)]])
>>> face = p.faces(Integer(2))[Integer(2)]
>>> face.rays()
(A ray in the direction (1, 0, 0), A ray in the direction (0, 1, 0))
stacking_locus()[source]

Return the polyhedron containing the points that sees every facet containing self.

OUTPUT: a polyhedron

EXAMPLES:

sage: cp = polytopes.cross_polytope(4)
sage: facet = cp.facets()[0]
sage: facet.stacking_locus().vertices()
(A vertex at (1/2, 1/2, 1/2, 1/2),
 A vertex at (1, 0, 0, 0),
 A vertex at (0, 0, 0, 1),
 A vertex at (0, 0, 1, 0),
 A vertex at (0, 1, 0, 0))
sage: face = cp.faces(2)[0]
sage: face.stacking_locus().vertices()
(A vertex at (0, 1, 0, 0),
 A vertex at (0, 0, 1, 0),
 A vertex at (1, 0, 0, 0),
 A vertex at (1, 1, 1, 0),
 A vertex at (1/2, 1/2, 1/2, 1/2),
 A vertex at (1/2, 1/2, 1/2, -1/2))

>>> from sage.all import *
>>> cp = polytopes.cross_polytope(Integer(4))
>>> facet = cp.facets()[Integer(0)]
>>> facet.stacking_locus().vertices()
(A vertex at (1/2, 1/2, 1/2, 1/2),
 A vertex at (1, 0, 0, 0),
 A vertex at (0, 0, 0, 1),
 A vertex at (0, 0, 1, 0),
 A vertex at (0, 1, 0, 0))
>>> face = cp.faces(Integer(2))[Integer(0)]
>>> face.stacking_locus().vertices()
(A vertex at (0, 1, 0, 0),
 A vertex at (0, 0, 1, 0),
 A vertex at (1, 0, 0, 0),
 A vertex at (1, 1, 1, 0),
 A vertex at (1/2, 1/2, 1/2, 1/2),
 A vertex at (1/2, 1/2, 1/2, -1/2))
vertex_generator()[source]

Return a generator for the vertices of the face.

EXAMPLES:

sage: triangle = Polyhedron(vertices=[[1,0],[0,1],[1,1]])
sage: face = triangle.facets()[0]
sage: for v in face.vertex_generator(): print(v)
A vertex at (1, 0)
A vertex at (1, 1)
sage: type(face.vertex_generator())
<... 'generator'>

>>> from sage.all import *
>>> triangle = Polyhedron(vertices=[[Integer(1),Integer(0)],[Integer(0),Integer(1)],[Integer(1),Integer(1)]])
>>> face = triangle.facets()[Integer(0)]
>>> for v in face.vertex_generator(): print(v)
A vertex at (1, 0)
A vertex at (1, 1)
>>> type(face.vertex_generator())
<... 'generator'>
vertices()[source]

Return all vertices of the face.

OUTPUT: a tuple of vertices

EXAMPLES:

sage: triangle = Polyhedron(vertices=[[1,0],[0,1],[1,1]])
sage: face = triangle.faces(1)[2]
sage: face.vertices()
(A vertex at (0, 1), A vertex at (1, 0))

>>> from sage.all import *
>>> triangle = Polyhedron(vertices=[[Integer(1),Integer(0)],[Integer(0),Integer(1)],[Integer(1),Integer(1)]])
>>> face = triangle.faces(Integer(1))[Integer(2)]
>>> face.vertices()
(A vertex at (0, 1), A vertex at (1, 0))
sage.geometry.polyhedron.face.combinatorial_face_to_polyhedral_face(polyhedron, combinatorial_face)[source]

Convert a combinatorial face to a face of a polyhedron.

INPUT:

  • polyhedron – a polyhedron containing combinatorial_face

  • combinatorial_face – a CombinatorialFace

OUTPUT: a PolyhedronFace

EXAMPLES:

sage: from sage.geometry.polyhedron.face import combinatorial_face_to_polyhedral_face
sage: P = polytopes.simplex()
sage: C = P.combinatorial_polyhedron()
sage: it = C.face_iter()
sage: comb_face = next(it)
sage: combinatorial_face_to_polyhedral_face(P, comb_face)
A 2-dimensional face of a Polyhedron in ZZ^4 defined as the convex hull of 3 vertices

>>> from sage.all import *
>>> from sage.geometry.polyhedron.face import combinatorial_face_to_polyhedral_face
>>> P = polytopes.simplex()
>>> C = P.combinatorial_polyhedron()
>>> it = C.face_iter()
>>> comb_face = next(it)
>>> combinatorial_face_to_polyhedral_face(P, comb_face)
A 2-dimensional face of a Polyhedron in ZZ^4 defined as the convex hull of 3 vertices