def _dist_geod_point(self, start, end, p):
r"""
Return the hyperbolic distance from a given hyperbolic geodesic
and a hyperbolic point.
INPUT:
- ``start`` -- the start ideal point coordinates of the geodesic
- ``end`` -- the end ideal point coordinates of the geodesic
- ``p`` -- the coordinates of the point
OUTPUT:
- the hyperbolic distance
EXAMPLES::
sage: UHP = HyperbolicPlane().UHP()
sage: UHP._dist_geod_point(2, infinity, I)
arccosh(1/10*sqrt(5)*((sqrt(5) - 1)^2 + 4) + 1)
If `p` is a boundary point, the distance is infinity::
sage: HyperbolicPlane().UHP()._dist_geod_point(2, infinity, 5)
+Infinity
"""
# Here is the trick for computing distance to a geodesic:
# find an isometry mapping the geodesic to the geodesic between
# 0 and infinity (so corresponding to the line imag(z) = 0.
# then any complex number is r exp(i*theta) in polar coordinates.
# the mutual perpendicular between this point and imag(z) = 0
# intersects imag(z) = 0 at ri. So we calculate the distance
# between r exp(i*theta) and ri after we transform the original
# point.
if start + end != infinity:
# Not a straight line:
# Map the endpoints to 0 and infinity and the midpoint to 1.
T = HyperbolicGeodesicUHP._crossratio_matrix(
start, (start + end) / 2, end)
else:
# Is a straight line:
# Map the endpoints to 0 and infinity and another endpoint to 1.
T = HyperbolicGeodesicUHP._crossratio_matrix(start, start + 1, end)
x = moebius_transform(T, p)
return self._dist_points(x, abs(x) * I)
def _dist_geod_point(self, start, end, p):
r"""
Return the hyperbolic distance from a given hyperbolic geodesic
and a hyperbolic point.
INPUT:
- ``start`` -- the start ideal point coordinates of the geodesic
- ``end`` -- the end ideal point coordinates of the geodesic
- ``p`` -- the coordinates of the point
OUTPUT:
- the hyperbolic distance
EXAMPLES::
sage: UHP = HyperbolicPlane().UHP()
sage: UHP._dist_geod_point(2, infinity, I)
arccosh(1/10*sqrt(5)*((sqrt(5) - 1)^2 + 4) + 1)
If `p` is a boundary point, the distance is infinity::
sage: HyperbolicPlane().UHP()._dist_geod_point(2, infinity, 5)
+Infinity
"""
# Here is the trick for computing distance to a geodesic:
# find an isometry mapping the geodesic to the geodesic between
# 0 and infinity (so corresponding to the line imag(z) = 0.
# then any complex number is r exp(i*theta) in polar coordinates.
# the mutual perpendicular between this point and imag(z) = 0
# intersects imag(z) = 0 at ri. So we calculate the distance
# between r exp(i*theta) and ri after we transform the original
# point.
if start + end != infinity:
# Not a straight line:
# Map the endpoints to 0 and infinity and the midpoint to 1.
T = HyperbolicGeodesicUHP._crossratio_matrix(start, (start + end)/2, end)
else:
# Is a straight line:
# Map the endpoints to 0 and infinity and another endpoint to 1.
T = HyperbolicGeodesicUHP._crossratio_matrix(start, start + 1, end)
x = moebius_transform(T, p)
return self._dist_points(x, abs(x)*I)
def _moebius_sending(z, w): #UHP
r"""
Given two lists ``z`` and ``w`` of three points each in
`\mathbb{CP}^1`, return the linear fractional transformation
taking the points in ``z`` to the points in ``w``.
EXAMPLES::
sage: from sage.geometry.hyperbolic_space.hyperbolic_model import HyperbolicModelUHP
sage: from sage.geometry.hyperbolic_space.hyperbolic_isometry import moebius_transform
sage: bool(abs(moebius_transform(HyperbolicModelUHP._moebius_sending([1,2,infinity],[3 - I, 5*I,-12]),1) - 3 + I) < 10^-4)
True
sage: bool(abs(moebius_transform(HyperbolicModelUHP._moebius_sending([1,2,infinity],[3 - I, 5*I,-12]),2) - 5*I) < 10^-4)
True
sage: bool(abs(moebius_transform(HyperbolicModelUHP._moebius_sending([1,2,infinity],[3 - I, 5*I,-12]),infinity) + 12) < 10^-4)
True
"""
if len(z) != 3 or len(w) != 3:
raise TypeError("moebius_sending requires each list to be three points long")
A = HyperbolicGeodesicUHP._crossratio_matrix(z[0],z[1],z[2])
B = HyperbolicGeodesicUHP._crossratio_matrix(w[0],w[1],w[2])
return B.inverse() * A
def _mobius_sending(z, w): #UHP
r"""
Given two lists ``z`` and ``w`` of three points each in
`\mathbb{CP}^1`, return the linear fractional transformation
taking the points in ``z`` to the points in ``w``.
EXAMPLES::
sage: from sage.geometry.hyperbolic_space.hyperbolic_model import HyperbolicModelUHP
sage: from sage.geometry.hyperbolic_space.hyperbolic_isometry import mobius_transform
sage: bool(abs(mobius_transform(HyperbolicModelUHP._mobius_sending([1,2,infinity],[3 - I, 5*I,-12]),1) - 3 + I) < 10^-4)
True
sage: bool(abs(mobius_transform(HyperbolicModelUHP._mobius_sending([1,2,infinity],[3 - I, 5*I,-12]),2) - 5*I) < 10^-4)
True
sage: bool(abs(mobius_transform(HyperbolicModelUHP._mobius_sending([1,2,infinity],[3 - I, 5*I,-12]),infinity) + 12) < 10^-4)
True
"""
if len(z) != 3 or len(w) != 3:
raise TypeError("mobius_sending requires each list to be three points long")
A = HyperbolicGeodesicUHP._crossratio_matrix(z[0],z[1],z[2])
B = HyperbolicGeodesicUHP._crossratio_matrix(w[0],w[1],w[2])
return B.inverse() * A