# 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()                                                              # 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);  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)#

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 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); 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(); 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():                                    # 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()#

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():                                      # 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()#

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)
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); 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
```
as_polyhedron(**kwds)#

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); 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); 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()                          # 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]
```
is_compact()#

Return whether `self` is compact.

OUTPUT: Boolean.

EXAMPLES:

```sage: half_plane = Polyhedron(ieqs=[(0,1,0)])
sage: line = half_plane.faces(1); 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); 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)
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(); 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
```
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(); 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
```
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)
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)
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().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().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)
sage: f2 = p.faces(1)
sage: f3 = p.faces(2)
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)
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(); 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)
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)
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()
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)
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()
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)
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
```