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)]

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

sage: P.face_lattice()                                                      # optional - 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))

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)
class sage.geometry.polyhedron.face.PolyhedronFace(polyhedron, V_indices, H_indices)

Bases: sage.geometry.convex_set.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))
affine_tangent_cone()

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))
ambient()

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]
sage: 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
ambient_H_indices()

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)]
ambient_Hrepresentation(index=None)

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

INPUT:

  • index – optional. Either an 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():                            # optional - 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()

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)]
ambient_Vrepresentation(index=None)

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

INPUT:

  • index – optional. Either an 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():                              # optional - 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()

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
ambient_vector_space(base_field=None)

Return the ambient vector space.

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

INPUT:

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

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)                               # optional - sage.rings.number_field
Vector space of dimension 2 over Algebraic Real Field
as_polyhedron()

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]
sage: 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
contains(point)

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

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

sage: [5, 7] in line
False
dim()

Return the dimension of the face.

OUTPUT:

Integer.

EXAMPLES:

sage: fl = polytopes.dodecahedron().face_lattice()  # optional - sage.combinat  # optional - sage.rings.number_field
sage: sorted([ x.dim() for x in fl ])               # optional - sage.combinat  # optional - 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()

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
is_relatively_open()

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
line_generator()

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)
lines()

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),)
n_ambient_Hrepresentation()

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]                                    # optional - sage.combinat
sage: face                                                          # optional - sage.combinat
A 1-dimensional face of a Polyhedron in ZZ^4 defined as the convex hull of 2 vertices
sage: face.ambient_Hrepresentation()                                # optional - 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()                              # optional - sage.combinat
4
n_ambient_Vrepresentation()

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]                                    # optional - sage.combinat
sage: face                                                          # optional - sage.combinat
A 1-dimensional face of a Polyhedron in ZZ^4 defined as the convex hull of 2 vertices
sage: face.ambient_Vrepresentation()                                # optional - sage.combinat
(A vertex at (-1, 0, 0, 0), A vertex at (0, 0, -1, 0))
sage: face.n_ambient_Vrepresentation()                              # optional - sage.combinat
2
n_lines()

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
n_rays()

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
n_vertices()

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
normal_cone(direction='outer')

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]]

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

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
polyhedron()

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]
sage: 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
ray_generator()

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)
rays()

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))
stacking_locus()

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))
vertex_generator()

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'>
vertices()

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))
sage.geometry.polyhedron.face.combinatorial_face_to_polyhedral_face(polyhedron, combinatorial_face)

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