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()
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:
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 ofself
.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 orNone
(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, theindex
-th element of the tuple is returned.EXAMPLES:
sage: square = polytopes.hypercube(2) sage: for face in square.face_lattice(): ....: 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 orNone
(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, theindex
-th element of the tuple is returned.EXAMPLES:
sage: square = polytopes.hypercube(2) sage: for fl in square.face_lattice(): ....: 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) 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() sage: sorted([ x.dim() for x in fl ]) [-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] sage: face A 1-dimensional face of a Polyhedron in ZZ^4 defined as the convex hull of 2 vertices sage: face.ambient_Hrepresentation() (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() 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] sage: face A 1-dimensional face of a Polyhedron in ZZ^4 defined as the convex hull of 2 vertices sage: face.ambient_Vrepresentation() (A vertex at (-1, 0, 0, 0), A vertex at (0, 0, -1, 0)) sage: face.n_ambient_Vrepresentation() 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 containingcombinatorial_face
combinatorial_face
– aCombinatorialFace
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