Relative Interiors of Polyhedra and Cones#
- class sage.geometry.relative_interior.RelativeInterior(polyhedron)#
Bases:
sage.geometry.convex_set.ConvexSet_relatively_open
The relative interior of a polyhedron or cone
This class should not be used directly. Use methods
relative_interior()
,interior()
,relative_interior()
,interior()
instead.EXAMPLES:
sage: segment = Polyhedron([[1, 2], [3, 4]]) sage: segment.relative_interior() Relative interior of a 1-dimensional polyhedron in ZZ^2 defined as the convex hull of 2 vertices sage: octant = Cone([(1,0,0), (0,1,0), (0,0,1)]) sage: octant.relative_interior() Relative interior of 3-d cone in 3-d lattice N
- ambient()#
Return the ambient convex set or space.
EXAMPLES:
sage: segment = Polyhedron([[1, 2], [3, 4]]) sage: ri_segment = segment.relative_interior(); ri_segment Relative interior of a 1-dimensional polyhedron in ZZ^2 defined as the convex hull of 2 vertices sage: ri_segment.ambient() Vector space of dimension 2 over Rational Field
- ambient_dim()#
Return the dimension of the ambient space.
EXAMPLES:
sage: segment = Polyhedron([[1, 2], [3, 4]]) sage: segment.ambient_dim() 2 sage: ri_segment = segment.relative_interior(); ri_segment Relative interior of a 1-dimensional polyhedron in ZZ^2 defined as the convex hull of 2 vertices sage: ri_segment.ambient_dim() 2
- ambient_vector_space(base_field=None)#
Return the ambient vector space.
EXAMPLES:
sage: segment = Polyhedron([[1, 2], [3, 4]]) sage: ri_segment = segment.relative_interior(); ri_segment Relative interior of a 1-dimensional polyhedron in ZZ^2 defined as the convex hull of 2 vertices sage: ri_segment.ambient_vector_space() Vector space of dimension 2 over Rational Field
- an_affine_basis()#
Return points that form an affine basis for the affine hull.
The points are guaranteed to lie in the topological closure of
self
.EXAMPLES:
sage: segment = Polyhedron([[1, 0], [0, 1]]) sage: segment.relative_interior().an_affine_basis() [A vertex at (1, 0), A vertex at (0, 1)]
- closure()#
Return the topological closure of
self
.EXAMPLES:
sage: segment = Polyhedron([[1, 2], [3, 4]]) sage: ri_segment = segment.relative_interior(); ri_segment Relative interior of a 1-dimensional polyhedron in ZZ^2 defined as the convex hull of 2 vertices sage: ri_segment.closure() is segment True
- dilation(scalar)#
Return the dilated (uniformly stretched) set.
INPUT:
scalar
– A scalar
EXAMPLES:
sage: segment = Polyhedron([[1, 2], [3, 4]]) sage: ri_segment = segment.relative_interior(); ri_segment Relative interior of a 1-dimensional polyhedron in ZZ^2 defined as the convex hull of 2 vertices sage: A = ri_segment.dilation(2); A Relative interior of a 1-dimensional polyhedron in ZZ^2 defined as the convex hull of 2 vertices sage: A.closure().vertices() (A vertex at (2, 4), A vertex at (6, 8)) sage: B = ri_segment.dilation(-1/3); B Relative interior of a 1-dimensional polyhedron in QQ^2 defined as the convex hull of 2 vertices sage: B.closure().vertices() (A vertex at (-1, -4/3), A vertex at (-1/3, -2/3)) sage: C = ri_segment.dilation(0); C A 0-dimensional polyhedron in ZZ^2 defined as the convex hull of 1 vertex sage: C.vertices() (A vertex at (0, 0),)
- dim()#
Return the dimension of
self
.EXAMPLES:
sage: segment = Polyhedron([[1, 2], [3, 4]]) sage: segment.dim() 1 sage: ri_segment = segment.relative_interior(); ri_segment Relative interior of a 1-dimensional polyhedron in ZZ^2 defined as the convex hull of 2 vertices sage: ri_segment.dim() 1
- interior()#
Return the interior of
self
.EXAMPLES:
sage: segment = Polyhedron([[1, 2], [3, 4]]) sage: ri_segment = segment.relative_interior(); ri_segment Relative interior of a 1-dimensional polyhedron in ZZ^2 defined as the convex hull of 2 vertices sage: ri_segment.interior() The empty polyhedron in ZZ^2 sage: octant = Cone([(1,0,0), (0,1,0), (0,0,1)]) sage: ri_octant = octant.relative_interior(); ri_octant Relative interior of 3-d cone in 3-d lattice N sage: ri_octant.interior() is ri_octant True
- is_closed()#
Return whether
self
is closed.OUTPUT:
Boolean.
EXAMPLES:
sage: segment = Polyhedron([[1, 2], [3, 4]]) sage: ri_segment = segment.relative_interior(); ri_segment Relative interior of a 1-dimensional polyhedron in ZZ^2 defined as the convex hull of 2 vertices sage: ri_segment.is_closed() False
- is_universe()#
Return whether
self
is the whole ambient spaceOUTPUT:
Boolean.
EXAMPLES:
sage: segment = Polyhedron([[1, 2], [3, 4]]) sage: ri_segment = segment.relative_interior(); ri_segment Relative interior of a 1-dimensional polyhedron in ZZ^2 defined as the convex hull of 2 vertices sage: ri_segment.is_universe() False
- linear_transformation(linear_transf, **kwds)#
Return the linear transformation of
self
.By [Roc1970], Theorem 6.6, the linear transformation of a relative interior is the relative interior of the linear transformation.
INPUT:
linear_transf
– a matrix**kwds
– passed to thelinear_transformation()
method of the closure ofself
.
EXAMPLES:
sage: segment = Polyhedron([[1, 2], [3, 4]]) sage: ri_segment = segment.relative_interior(); ri_segment Relative interior of a 1-dimensional polyhedron in ZZ^2 defined as the convex hull of 2 vertices sage: T = matrix([[1, 1]]) sage: A = ri_segment.linear_transformation(T); A Relative interior of a 1-dimensional polyhedron in ZZ^1 defined as the convex hull of 2 vertices sage: A.closure().vertices() (A vertex at (3), A vertex at (7))
- relative_interior()#
Return the relative interior of
self
.As
self
is already relatively open, this method just returnsself
.EXAMPLES:
sage: segment = Polyhedron([[1, 2], [3, 4]]) sage: ri_segment = segment.relative_interior(); ri_segment Relative interior of a 1-dimensional polyhedron in ZZ^2 defined as the convex hull of 2 vertices sage: ri_segment.relative_interior() is ri_segment True
- representative_point()#
Return a “generic” point of
self
.OUTPUT:
A point in
self
(thus, in the relative interior ofself
) as a coordinate vector.EXAMPLES:
sage: C = Cone([[1, 2, 0], [2, 1, 0]]) sage: C.relative_interior().representative_point() (1, 1, 0)
- translation(displacement)#
Return the translation of
self
by adisplacement
vector.INPUT:
displacement
– a displacement vector or a list/tuple of coordinates that determines a displacement vector
EXAMPLES:
sage: segment = Polyhedron([[1, 2], [3, 4]]) sage: ri_segment = segment.relative_interior(); ri_segment Relative interior of a 1-dimensional polyhedron in ZZ^2 defined as the convex hull of 2 vertices sage: t = vector([100, 100]) sage: ri_segment.translation(t) Relative interior of a 1-dimensional polyhedron in ZZ^2 defined as the convex hull of 2 vertices sage: ri_segment.closure().vertices() (A vertex at (1, 2), A vertex at (3, 4))