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_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')