def isometry_in_model(self, A):
r"""
Check if the given matrix ``A`` is in the group `U(1,1)`.
EXAMPLES::
sage: z = [CC.random_element() for k in range(2)]; z.sort(key=abs)
sage: A = matrix(2,[z[1], z[0],z[0].conjugate(),z[1].conjugate()])
sage: HyperbolicPlane().PD().isometry_in_model(A)
True
"""
if isinstance(A, HyperbolicIsometry):
return True
# alpha = A[0][0]
# beta = A[0][1]
# Orientation preserving and reversing
return (HyperbolicIsometryPD._orientation_preserving(A)
or HyperbolicIsometryPD._orientation_preserving(I * A))
def isometry_in_model(self, A):
r"""
Check if the given matrix ``A`` is in the group `U(1,1)`.
EXAMPLES::
sage: z = [CC.random_element() for k in range(2)]; z.sort(key=abs)
sage: A = matrix(2,[z[1], z[0],z[0].conjugate(),z[1].conjugate()])
sage: HyperbolicPlane().PD().isometry_in_model(A)
True
"""
if isinstance(A, HyperbolicIsometry):
return True
# alpha = A[0][0]
# beta = A[0][1]
# Orientation preserving and reversing
return (HyperbolicIsometryPD._orientation_preserving(A) or
HyperbolicIsometryPD._orientation_preserving(I * A))
def image_isometry_matrix(self, x):
"""
Return the image of the matrix of the hyperbolic isometry ``x``
under ``self``.
EXAMPLES:
We check that orientation-reversing isometries behave as they
should::
sage: PD = HyperbolicPlane().PD()
sage: UHP = HyperbolicPlane().UHP()
sage: phi = UHP.coerce_map_from(PD)
sage: phi.image_isometry_matrix(matrix([[0,I],[I,0]]))
[-1 0]
[ 0 -1]
"""
from sage.geometry.hyperbolic_space.hyperbolic_isometry import HyperbolicIsometryPD
if not HyperbolicIsometryPD._orientation_preserving(x):
return matrix([[1,I],[I,1]]) * x * matrix([[1,-I],[-I,1]]).conjugate() / Integer(2)
return matrix([[1,I],[I,1]]) * x * matrix([[1,-I],[-I,1]]) / Integer(2)
def image_isometry_matrix(self, x):
"""
Return the image of the matrix of the hyperbolic isometry ``x``
under ``self``.
EXAMPLES:
We check that orientation-reversing isometries behave as they
should::
sage: PD = HyperbolicPlane().PD()
sage: UHP = HyperbolicPlane().UHP()
sage: phi = UHP.coerce_map_from(PD)
sage: phi.image_isometry_matrix(matrix([[0,I],[I,0]]))
[-1 0]
[ 0 -1]
"""
from sage.geometry.hyperbolic_space.hyperbolic_isometry import HyperbolicIsometryPD
if not HyperbolicIsometryPD._orientation_preserving(x):
return matrix([[1,I],[I,1]]) * x * matrix([[1,-I],[-I,1]]).conjugate() / Integer(2)
return matrix([[1,I],[I,1]]) * x * matrix([[1,-I],[-I,1]]) / Integer(2)