def fixed_point_set(self): #UHP
r"""
Return the a list or geodesic containing the fixed point set of
orientation-preserving isometries.
OUTPUT:
- a list of hyperbolic points or a hyperbolic geodesic
EXAMPLES::
sage: UHP = HyperbolicPlane().UHP()
sage: H = UHP.get_isometry(matrix(2, [-2/3,-1/3,-1/3,-2/3]))
sage: g = H.fixed_point_set(); g
Geodesic in UHP from -1 to 1
sage: H(g.start()) == g.start()
True
sage: H(g.end()) == g.end()
True
sage: A = UHP.get_isometry(matrix(2,[0,1,1,0]))
sage: A.preserves_orientation()
False
sage: A.fixed_point_set()
Geodesic in UHP from 1 to -1
::
sage: B = UHP.get_isometry(identity_matrix(2))
sage: B.fixed_point_set()
Traceback (most recent call last):
...
ValueError: the identity transformation fixes the entire hyperbolic plane
"""
d = sqrt(self._matrix.det() ** 2)
M = self._matrix / sqrt(d)
tau = M.trace() ** 2
M_cls = self.classification()
if M_cls == 'identity':
raise ValueError("the identity transformation fixes the entire "
"hyperbolic plane")
pt = self.domain().get_point
if M_cls == 'parabolic':
if abs(M[1, 0]) < EPSILON:
return [pt(infinity)]
else:
# boundary point
return [pt((M[0,0] - M[1,1]) / (2*M[1,0]))]
elif M_cls == 'elliptic':
d = sqrt(tau - 4)
return [pt((M[0,0] - M[1,1] + sign(M[1,0])*d) / (2*M[1,0]))]
elif M_cls == 'hyperbolic':
if M[1,0] != 0: #if the isometry doesn't fix infinity
d = sqrt(tau - 4)
p_1 = (M[0,0] - M[1,1]+d) / (2*M[1,0])
p_2 = (M[0,0] - M[1,1]-d) / (2*M[1,0])
return self.domain().get_geodesic(pt(p_1), pt(p_2))
#else, it fixes infinity.
p_1 = M[0,1] / (M[1,1] - M[0,0])
p_2 = infinity
return self.domain().get_geodesic(pt(p_1), pt(p_2))
try:
p, q = [M.eigenvectors_right()[k][1][0] for k in range(2)]
except IndexError:
M = M.change_ring(RDF)
p, q = [M.eigenvectors_right()[k][1][0] for k in range(2)]
pts = []
if p[1] == 0:
pts.append(infinity)
else:
p = p[0] / p[1]
if imag(p) >= 0:
pts.append(p)
if q[1] == 0:
pts.append(infinity)
else:
q = q[0] / q[1]
if imag(q) >= 0:
pts.append(q)
pts = [pt(k) for k in pts]
if len(pts) == 2:
return self.domain().get_geodesic(*pts)
return pts
def fixed_point_set(self): #UHP
r"""
Return the a list or geodesic containing the fixed point set of
orientation-preserving isometries.
OUTPUT:
- a list of hyperbolic points or a hyperbolic geodesic
EXAMPLES::
sage: UHP = HyperbolicPlane().UHP()
sage: H = UHP.get_isometry(matrix(2, [-2/3,-1/3,-1/3,-2/3]))
sage: g = H.fixed_point_set(); g
Geodesic in UHP from -1 to 1
sage: H(g.start()) == g.start()
True
sage: H(g.end()) == g.end()
True
sage: A = UHP.get_isometry(matrix(2,[0,1,1,0]))
sage: A.preserves_orientation()
False
sage: A.fixed_point_set()
Geodesic in UHP from 1 to -1
::
sage: B = UHP.get_isometry(identity_matrix(2))
sage: B.fixed_point_set()
Traceback (most recent call last):
...
ValueError: the identity transformation fixes the entire hyperbolic plane
"""
d = sqrt(self._matrix.det()**2)
M = self._matrix / sqrt(d)
tau = M.trace()**2
M_cls = self.classification()
if M_cls == 'identity':
raise ValueError("the identity transformation fixes the entire "
"hyperbolic plane")
pt = self.domain().get_point
if M_cls == 'parabolic':
if abs(M[1, 0]) < EPSILON:
return [pt(infinity)]
else:
# boundary point
return [pt((M[0, 0] - M[1, 1]) / (2 * M[1, 0]))]
elif M_cls == 'elliptic':
d = sqrt(tau - 4)
return [
pt((M[0, 0] - M[1, 1] + sign(M[1, 0]) * d) / (2 * M[1, 0]))
]
elif M_cls == 'hyperbolic':
if M[1, 0] != 0: #if the isometry doesn't fix infinity
d = sqrt(tau - 4)
p_1 = (M[0, 0] - M[1, 1] + d) / (2 * M[1, 0])
p_2 = (M[0, 0] - M[1, 1] - d) / (2 * M[1, 0])
return self.domain().get_geodesic(pt(p_1), pt(p_2))
#else, it fixes infinity.
p_1 = M[0, 1] / (M[1, 1] - M[0, 0])
p_2 = infinity
return self.domain().get_geodesic(pt(p_1), pt(p_2))
try:
p, q = [M.eigenvectors_right()[k][1][0] for k in range(2)]
except IndexError:
M = M.change_ring(RDF)
p, q = [M.eigenvectors_right()[k][1][0] for k in range(2)]
pts = []
if p[1] == 0:
pts.append(infinity)
else:
p = p[0] / p[1]
if imag(p) >= 0:
pts.append(p)
if q[1] == 0:
pts.append(infinity)
else:
q = q[0] / q[1]
if imag(q) >= 0:
pts.append(q)
pts = [pt(k) for k in pts]
if len(pts) == 2:
return self.domain().get_geodesic(*pts)
return pts