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relative_interior.py
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relative_interior.py
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r"""
Relative Interiors of Polyhedra and Cones
"""
# ****************************************************************************
# Copyright (C) 2021 Matthias Koeppe
#
# This program is free software: you can redistribute it and/or modify
# it under the terms of the GNU General Public License as published by
# the Free Software Foundation, either version 2 of the License, or
# (at your option) any later version.
# https://www.gnu.org/licenses/
# ****************************************************************************
from sage.geometry.convex_set import ConvexSet_relatively_open
class RelativeInterior(ConvexSet_relatively_open):
r"""
The relative interior of a polyhedron or cone
This class should not be used directly. Use methods
:meth:`~sage.geometry.polyhedron.Polyhedron_base.relative_interior`,
:meth:`~sage.geometry.polyhedron.Polyhedron_base.interior`,
:meth:`~sage.geometry.cone.ConvexRationalPolyhedralCone.relative_interior`,
:meth:`~sage.geometry.cone.ConvexRationalPolyhedralCone.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
"""
def __init__(self, polyhedron):
r"""
Initialize ``self``.
INPUT:
- ``polyhedron`` - an instance of :class:`Polyhedron_base` or
:class:`ConvexRationalPolyhedralCone`.
TESTS::
sage: P = Polyhedron([[1, 2], [3, 4]])
sage: from sage.geometry.relative_interior import RelativeInterior
sage: TestSuite(RelativeInterior(P)).run()
"""
self._polyhedron = polyhedron
def __contains__(self, point):
r"""
Return whether ``self`` contains ``point``.
EXAMPLES::
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: (1, 1, 1) in ri_octant
True
sage: (1, 0, 0) in ri_octant
False
"""
return self._polyhedron.relative_interior_contains(point)
def ambient(self):
r"""
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
"""
return self._polyhedron.ambient()
def ambient_vector_space(self, base_field=None):
r"""
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
"""
return self._polyhedron.ambient_vector_space(base_field=base_field)
def ambient_dim(self):
r"""
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
"""
return self._polyhedron.ambient_dim()
def dim(self):
r"""
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
"""
return self._polyhedron.dim()
def interior(self):
r"""
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
"""
return self._polyhedron.interior()
def relative_interior(self):
r"""
Return the relative interior of ``self``.
As ``self`` is already relatively open, this method just returns ``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.relative_interior() is ri_segment
True
"""
return self
def closure(self):
r"""
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
"""
return self._polyhedron
def is_universe(self):
r"""
Return whether ``self`` is the whole ambient space
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_universe()
False
"""
# Relies on ``self`` not set up for polyhedra that are already
# relatively open themselves.
assert not self._polyhedron.is_universe()
return False
def is_closed(self):
r"""
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
"""
# Relies on ``self`` not set up for polyhedra that are already
# relatively open themselves.
assert not self._polyhedron.is_relatively_open()
return False
def _repr_(self):
r"""
Return a description of ``self``.
EXAMPLES::
sage: P = Polyhedron(vertices = [[1,2,3,4],[2,1,3,4],[4,3,2,1]])
sage: P.relative_interior()._repr_()
'Relative interior of a 2-dimensional polyhedron in ZZ^4 defined as the convex hull of 3 vertices'
sage: P.rename('A')
sage: P.relative_interior()._repr_()
'Relative interior of A'
"""
repr_P = repr(self._polyhedron)
if repr_P.startswith('A '):
repr_P = 'a ' + repr_P[2:]
return 'Relative interior of ' + repr_P
def __eq__(self, other):
r"""
Compare ``self`` and ``other``.
INPUT:
- ``other`` -- any object
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: segment2 = Polyhedron([[1, 2], [3, 4]], base_ring=AA)
sage: ri_segment2 = segment2.relative_interior(); ri_segment2
Relative interior of
a 1-dimensional polyhedron in AA^2 defined as the convex hull of 2 vertices
sage: ri_segment == ri_segment2
True
TESTS::
sage: empty = Polyhedron(ambient_dim=2)
sage: ri_segment == empty
False
"""
if type(self) != type(other):
return False
return self._polyhedron == other._polyhedron
def __ne__(self, other):
r"""
Compare ``self`` and ``other``.
INPUT:
- ``other`` -- any object
TESTS::
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: segment2 = Polyhedron([[1, 2], [3, 4]], base_ring=AA)
sage: ri_segment2 = segment2.relative_interior(); ri_segment2
Relative interior of
a 1-dimensional polyhedron in AA^2 defined as the convex hull of 2 vertices
sage: ri_segment != ri_segment2
False
"""
return not (self == other)