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 _find_isomorphism_degenerate(self, polytope):
"""
Helper to pick an isomorphism of degenerate polygons
INPUT:
- ``polytope`` -- a :class:`LatticePolytope_PPL_class`. The
polytope to compare with.
EXAMPLES::
sage: from sage.geometry.polyhedron.ppl_lattice_polytope import LatticePolytope_PPL, C_Polyhedron
sage: L1 = LatticePolytope_PPL(C_Polyhedron(2, 'empty'))
sage: L2 = LatticePolytope_PPL(C_Polyhedron(3, 'empty'))
sage: iso = L1.find_isomorphism(L2) # indirect doctest
sage: iso(L1) == L2
True
sage: iso = L1._find_isomorphism_degenerate(L2)
sage: iso(L1) == L2
True
sage: L1 = LatticePolytope_PPL((-1,4))
sage: L2 = LatticePolytope_PPL((2,1,5))
sage: iso = L1.find_isomorphism(L2)
sage: iso(L1) == L2
True
sage: L1 = LatticePolytope_PPL((-1,), (3,))
sage: L2 = LatticePolytope_PPL((2,1,5), (2,-3,5))
sage: iso = L1.find_isomorphism(L2)
sage: iso(L1) == L2
True
sage: L1 = LatticePolytope_PPL((-1,-1), (3,-1))
sage: L2 = LatticePolytope_PPL((2,1,5), (2,-3,5))
sage: iso = L1.find_isomorphism(L2)
sage: iso(L1) == L2
True
sage: L1 = LatticePolytope_PPL((-1,2), (3,1))
sage: L2 = LatticePolytope_PPL((1,2,3),(1,2,4))
sage: iso = L1.find_isomorphism(L2)
sage: iso(L1) == L2
True
sage: L1 = LatticePolytope_PPL((-1,2), (3,2))
sage: L2 = LatticePolytope_PPL((1,2,3),(1,2,4))
sage: L1.find_isomorphism(L2)
Traceback (most recent call last):
...
LatticePolytopesNotIsomorphicError: different number of integral points
sage: L1 = LatticePolytope_PPL((-1,2), (3,1))
sage: L2 = LatticePolytope_PPL((1,2,3),(1,2,5))
sage: L1.find_isomorphism(L2)
Traceback (most recent call last):
...
LatticePolytopesNotIsomorphicError: different number of integral points
"""
from sage.geometry.polyhedron.lattice_euclidean_group_element import \
LatticePolytopesNotIsomorphicError
polytope_vertices = polytope.vertices()
self_vertices = self.ordered_vertices()
# handle degenerate cases
if self.n_vertices() == 0:
A = zero_matrix(ZZ, polytope.space_dimension(),
self.space_dimension())
b = zero_vector(ZZ, polytope.space_dimension())
return LatticeEuclideanGroupElement(A, b)
if self.n_vertices() == 1:
A = zero_matrix(ZZ, polytope.space_dimension(),
self.space_dimension())
b = polytope_vertices[0]
return LatticeEuclideanGroupElement(A, b)
if self.n_vertices() == 2:
self_origin = self_vertices[0]
self_ray = self_vertices[1] - self_origin
polytope_origin = polytope_vertices[0]
polytope_ray = polytope_vertices[1] - polytope_origin
Ds, Us, Vs = self_ray.column().smith_form()
Dp, Up, Vp = polytope_ray.column().smith_form()
assert Vs.nrows() == Vs.ncols() == Vp.nrows() == Vp.ncols() == 1
assert abs(Vs[0, 0]) == abs(Vp[0, 0]) == 1
A = zero_matrix(ZZ, Dp.nrows(), Ds.nrows())
A[0, 0] = 1
A = Up.inverse() * A * Us * (Vs[0, 0] * Vp[0, 0])
b = polytope_origin - A * self_origin
try:
A = matrix(ZZ, A)
b = vector(ZZ, b)
except TypeError:
raise LatticePolytopesNotIsomorphicError('different lattice')
hom = LatticeEuclideanGroupElement(A, b)
if hom(self) == polytope:
return hom
raise LatticePolytopesNotIsomorphicError('different polygons')
def _find_cyclic_isomorphism_matching_edge(self, polytope, polytope_origin,
p_ray_left, p_ray_right):
"""
Helper to find an isomorphism of polygons
INPUT:
- ``polytope`` -- the lattice polytope to compare to.
- ``polytope_origin`` -- `\ZZ`-vector. a vertex of ``polytope``
- ``p_ray_left`` - vector. the vector from ``polytope_origin``
to one of its neighboring vertices.
- ``p_ray_right`` - vector. the vector from
``polytope_origin`` to the other neighboring vertices.
OUTPUT:
The element of the lattice Euclidean group that maps ``self``
to ``polytope`` with given origin and left/right neighboring
vertex. 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: v0, v1, v2 = L2.vertices()
sage: L1._find_cyclic_isomorphism_matching_edge(L2, v0, v1-v0, v2-v0)
The map A*x+b with A=
[ 0 1]
[-1 -1]
[ 1 3]
b =
(0, 1, 0)
"""
from sage.geometry.polyhedron.lattice_euclidean_group_element import \
LatticePolytopesNotIsomorphicError
polytope_matrix = block_matrix(
1, 2,
[p_ray_left.column(), p_ray_right.column()])
self_vertices = self.ordered_vertices()
for i in range(len(self_vertices)):
# three consecutive vertices
v_left = self_vertices[(i + 0) % len(self_vertices)]
v_origin = self_vertices[(i + 1) % len(self_vertices)]
v_right = self_vertices[(i + 2) % len(self_vertices)]
r_left = v_left - v_origin
r_right = v_right - v_origin
self_matrix = block_matrix(
1, 2, [r_left.column(), r_right.column()])
A = self_matrix.solve_left(polytope_matrix)
b = polytope_origin - A * v_origin
try:
A = matrix(ZZ, A)
b = vector(ZZ, b)
except TypeError:
continue
if A.elementary_divisors()[0:2] != [1, 1]:
continue
hom = LatticeEuclideanGroupElement(A, b)
if hom(self) == polytope:
return hom
raise LatticePolytopesNotIsomorphicError('different polygons')