def random_isometry(self, preserve_orientation=True, **kwargs):
r"""
Return a random isometry in the Upper Half Plane model.
INPUT:
- ``preserve_orientation`` -- if ``True`` return an
orientation-preserving isometry
OUTPUT:
- a hyperbolic isometry
EXAMPLES::
sage: A = HyperbolicPlane().UHP().random_isometry()
sage: B = HyperbolicPlane().UHP().random_isometry(preserve_orientation=False)
sage: B.preserves_orientation()
False
"""
[a,b,c,d] = [RR.random_element() for k in range(4)]
while abs(a*d - b*c) < EPSILON:
[a,b,c,d] = [RR.random_element() for k in range(4)]
M = matrix(RDF, 2,[a,b,c,d])
M = M / (M.det()).abs().sqrt()
if M.det() > 0:
if not preserve_orientation:
M = M * matrix(2,[0,1,1,0])
elif preserve_orientation:
M = M * matrix(2,[0,1,1,0])
return self._Isometry(self, M, check=False)
def random_isometry(self, preserve_orientation=True, **kwargs):
r"""
Return a random isometry in the Upper Half Plane model.
INPUT:
- ``preserve_orientation`` -- if ``True`` return an
orientation-preserving isometry
OUTPUT:
- a hyperbolic isometry
EXAMPLES::
sage: A = HyperbolicPlane().UHP().random_isometry()
sage: B = HyperbolicPlane().UHP().random_isometry(preserve_orientation=False)
sage: B.preserves_orientation()
False
"""
[a, b, c, d] = [RR.random_element() for k in range(4)]
while abs(a * d - b * c) < EPSILON:
[a, b, c, d] = [RR.random_element() for k in range(4)]
M = matrix(RDF, 2, [a, b, c, d])
M = M / (M.det()).abs().sqrt()
if M.det() > 0:
if not preserve_orientation:
M = M * matrix(2, [0, 1, 1, 0])
elif preserve_orientation:
M = M * matrix(2, [0, 1, 1, 0])
return self._Isometry(self, M, check=False)
def random_point(self, **kwargs):
r"""
Return a random point in the upper half plane. The points are
uniformly distributed over the rectangle `[-10, 10] \times [0, 10i]`.
EXAMPLES::
sage: p = HyperbolicPlane().UHP().random_point().coordinates()
sage: bool((p.imag()) > 0)
True
"""
# TODO: use **kwargs to allow these to be set
real_min = -10
real_max = 10
imag_min = 0
imag_max = 10
p = RR.random_element(min=real_min, max=real_max) \
+ I * RR.random_element(min=imag_min, max=imag_max)
return self.get_point(p)
def random_point(self, **kwargs):
r"""
Return a random point in the upper half plane. The points are
uniformly distributed over the rectangle `[-10, 10] \times [0, 10i]`.
EXAMPLES::
sage: p = HyperbolicPlane().UHP().random_point().coordinates()
sage: bool((p.imag()) > 0)
True
"""
# TODO: use **kwargs to allow these to be set
real_min = -10
real_max = 10
imag_min = 0
imag_max = 10
p = RR.random_element(min=real_min, max=real_max) \
+ I * RR.random_element(min=imag_min, max=imag_max)
return self.get_point(p)