def find_isomorphism(self, polytope):
"""
Return a lattice isomorphism with ``polytope``.
INPUT:
- ``polytope`` -- a polytope, potentially higher-dimensional.
OUTPUT:
A
:class:`~sage.geometry.polyhedron.lattice_euclidean_group_element.LatticeEuclideanGroupElement`. It
is not necessarily invertible if the affine dimension of
``self`` or ``polytope`` is not two. A
:class:`~sage.geometry.polyhedron.lattice_euclidean_group_element.LatticePolytopesNotIsomorphicError`
is raised if no such isomorphism exists.
EXAMPLES::
sage: from sage.geometry.polyhedron.ppl_lattice_polytope import LatticePolytope_PPL
sage: L1 = LatticePolytope_PPL((1,0),(0,1),(0,0))
sage: L2 = LatticePolytope_PPL((1,0,3),(0,1,0),(0,0,1))
sage: iso = L1.find_isomorphism(L2)
sage: iso(L1) == L2
True
sage: L1 = LatticePolytope_PPL((0, 1), (3, 0), (0, 3), (1, 0))
sage: L2 = LatticePolytope_PPL((0,0,2,1),(0,1,2,0),(2,0,0,3),(2,3,0,0))
sage: iso = L1.find_isomorphism(L2)
sage: iso(L1) == L2
True
The following polygons are isomorphic over `\QQ`, but not as
lattice polytopes::
sage: L1 = LatticePolytope_PPL((1,0),(0,1),(-1,-1))
sage: L2 = LatticePolytope_PPL((0, 0), (0, 1), (1, 0))
sage: L1.find_isomorphism(L2)
Traceback (most recent call last):
...
LatticePolytopesNotIsomorphicError: different number of integral points
sage: L2.find_isomorphism(L1)
Traceback (most recent call last):
...
LatticePolytopesNotIsomorphicError: different number of integral points
"""
from sage.geometry.polyhedron.lattice_euclidean_group_element import \
LatticePolytopesNotIsomorphicError
if polytope.affine_dimension() != self.affine_dimension():
raise LatticePolytopesNotIsomorphicError('different dimension')
polytope_vertices = polytope.vertices()
if len(polytope_vertices) != self.n_vertices():
raise LatticePolytopesNotIsomorphicError(
'different number of vertices')
self_vertices = self.ordered_vertices()
if len(polytope.integral_points()) != len(self.integral_points()):
raise LatticePolytopesNotIsomorphicError(
'different number of integral points')
if len(self_vertices) < 3:
return self._find_isomorphism_degenerate(polytope)
polytope_origin = polytope_vertices[0]
origin_P = C_Polyhedron(
next(Generator_System_iterator(polytope.minimized_generators())))
neighbors = []
for c in polytope.minimized_constraints():
if not c.is_inequality():
continue
if origin_P.relation_with(c).implies(
Poly_Con_Relation.saturates()):
for i, g in enumerate(polytope.minimized_generators()):
if i == 0:
continue
g = C_Polyhedron(g)
if g.relation_with(c).implies(
Poly_Con_Relation.saturates()):
neighbors.append(polytope_vertices[i])
break
p_ray_left = neighbors[0] - polytope_origin
p_ray_right = neighbors[1] - polytope_origin
try:
return self._find_cyclic_isomorphism_matching_edge(
polytope, polytope_origin, p_ray_left, p_ray_right)
except LatticePolytopesNotIsomorphicError:
pass
try:
return self._find_cyclic_isomorphism_matching_edge(
polytope, polytope_origin, p_ray_right, p_ray_left)
except LatticePolytopesNotIsomorphicError:
pass
raise LatticePolytopesNotIsomorphicError('different polygons')
def LatticePolytope_PPL(*args):
"""
Construct a new instance of the PPL-based lattice polytope class.
EXAMPLES::
sage: from sage.geometry.polyhedron.ppl_lattice_polytope import LatticePolytope_PPL
sage: LatticePolytope_PPL((0,0),(1,0),(0,1))
A 2-dimensional lattice polytope in ZZ^2 with 3 vertices
sage: from sage.libs.ppl import point, Generator_System, C_Polyhedron, Linear_Expression, Variable
sage: p = point(Linear_Expression([2,3],0)); p
point(2/1, 3/1)
sage: LatticePolytope_PPL(p)
A 0-dimensional lattice polytope in ZZ^2 with 1 vertex
sage: P = C_Polyhedron(Generator_System(p)); P
A 0-dimensional polyhedron in QQ^2 defined as the convex hull of 1 point
sage: LatticePolytope_PPL(P)
A 0-dimensional lattice polytope in ZZ^2 with 1 vertex
A ``TypeError`` is raised if the arguments do not specify a lattice polytope::
sage: from sage.geometry.polyhedron.ppl_lattice_polytope import LatticePolytope_PPL
sage: LatticePolytope_PPL((0,0),(1/2,1))
Traceback (most recent call last):
...
TypeError: no conversion of this rational to integer
sage: from sage.libs.ppl import point, Generator_System, C_Polyhedron, Linear_Expression, Variable
sage: p = point(Linear_Expression([2,3],0), 5); p
point(2/5, 3/5)
sage: LatticePolytope_PPL(p)
Traceback (most recent call last):
...
TypeError: generator is not a lattice polytope generator
sage: P = C_Polyhedron(Generator_System(p)); P
A 0-dimensional polyhedron in QQ^2 defined as the convex hull of 1 point
sage: LatticePolytope_PPL(P)
Traceback (most recent call last):
...
TypeError: polyhedron has non-integral generators
"""
polytope_class = LatticePolytope_PPL_class
if len(args)==1 and isinstance(args[0], C_Polyhedron):
polyhedron = args[0]
polytope_class = _class_for_LatticePolytope(polyhedron.space_dimension())
if not all(p.is_point() and p.divisor().is_one() for p in polyhedron.generators()):
raise TypeError('polyhedron has non-integral generators')
return polytope_class(polyhedron)
if len(args)==1 \
and isinstance(args[0], (list, tuple)) \
and isinstance(args[0][0], (list,tuple)):
vertices = args[0]
else:
vertices = args
gs = Generator_System()
for v in vertices:
if isinstance(v, Generator):
if (not v.is_point()) or (not v.divisor().is_one()):
raise TypeError('generator is not a lattice polytope generator')
gs.insert(v)
else:
gs.insert(point(Linear_Expression(v, 0)))
if not gs.empty():
dim = Generator_System_iterator(gs).next().space_dimension()
polytope_class = _class_for_LatticePolytope(dim)
return polytope_class(gs)
