def test_associative_law(self):
A = Moeb.rnd(-50, +50, NUMBER_TESTS)
B = Moeb.rnd(-50, +50, NUMBER_TESTS)
C = Moeb.rnd(-50, +50, NUMBER_TESTS)
for i in range(0, NUMBER_TESTS):
a, b, c = A[i], B[i], C[i]
self.assertEqual(associative_property(a, b, c), moeb_id)
def moeb_to(z0, z1):
"""Returns the Moebius transformation mapping the complex number z0 to z1.
Parameters
----------
z0 : ComplexNumber
Complex number to be mapped to
z1 : ComplexNumber
Target complex number
Returns
-------
Moeb
Moebius transformation mapping the complex number z0 to z1
"""
re_z0 = z0.re
im_z0 = z0.im
re_z1 = z1.re
im_z1 = z1.im
a = math.sqrt(im_z1 / im_z0)
b = -re_z0 * math.sqrt(im_z1 / im_z0) + re_z1 * math.sqrt(im_z0 / im_z1)
c = .0
d = math.sqrt(im_z0 / im_z1)
return Moeb(a, b, c, d)
def parab(t):
"""Returns an element of the one parameter group of Moebius translations in x-direction.
Parameters
----------
t : Float
Flow parameter
Returns
-------
Moeb
The Moebius translation in y-direction by an offset of t
"""
return Moeb(1.0, t, .0, 1.0)
def loxod(t):
"""Returns an element of the one parameter group of Moebius loxodromic transformations (dilatation in y-direction).
Parameters
----------
t : Float
Flow parameter
Returns
-------
Moeb
The element of the one parameter group of Moebius loxodromic dilatation evaluated for time t
"""
return Moeb(math.exp(.5 * t), .0, .0, math.exp(-.5 * t))
def ellip_0(t):
"""Returns an element of the one parameter group of Moebius rotations (element of elliptic transformations)
around imaginary unit.
Parameters
----------
t : Float
Flow parameter
Returns
-------
Moeb
The element of the one parameter group of Moebius rotations around the imaginary unit for time t
"""
return Moeb(math.cos(t), math.sin(t), -math.sin(t), math.cos(t))
def vert_to_vert(v_0, v_1):
"""Returns the Moebius transformation mapping the geodesic, vertical line v_0 to the vertical line v_1.
Parameters
----------
v_0 : Line of LineType VERTICAL
Vertical line to be mapped to
v_1 : Line of LineType VERTICAL
Target Vertical line
Returns
-------
Moeb
Moebius transformation mapping the geodesic, vertical line v_0 to the vertical line v_1
"""
return Moeb(1.0, v_1.Absc - v_0.Absc, .0, 1.0)
def circ_to_vert(c, v):
"""Returns the Moebius transformation mapping the geodesic circle to the geodesic vertical line.
Parameters
----------
c : Line of LineType CIRCLE
Geodesic circle to be mapped to
v : Line of LineType VERTICAL
Target vertical line
Returns
-------
Moeb
Moebius transformation mapping the geodesic circle to the geodesic vertical line
"""
u = v.Absc
return Moeb(1.0, u, .0, 1.0) * circ_to_Yaxis(c)
def circ_to_Yaxis(circ):
"""Returns the Moebius transformation mapping the geodesic circle to the y-axis.
Parameters
----------
c : Line of LineType CIRCLE
Geodesic circle to be mapped to the y-axis
Returns
-------
Moeb
Moebius transformation mapping the geodesic circle to the y-axis
"""
c = circ.Center
r = circ.Radius
return Moeb(1 / (math.sqrt(2.0 * r)),
-c / (math.sqrt(2.0 * r)) + math.sqrt(r / 2.0),
-1 / (math.sqrt(2.0 * r)),
c / (math.sqrt(2.0 * r)) + math.sqrt(r / 2.0))
def test_inv_element(self):
M = Moeb.rnd(-50, +50, NUMBER_TESTS)
for m in M:
self.assertEqual(inv_element_r(m, m.inv()), moeb_id)
self.assertEqual(inv_element_r(m, m.inv()), moeb_id)
def test_neutral_element(self):
M = Moeb.rnd(-50, +50, NUMBER_TESTS)
for m in M:
self.assertEqual(neutral_element_r(m, moeb_id), moeb_id)
self.assertEqual(neutral_element_l(m, moeb_id), moeb_id)