def _basis_vectors_sl2c(CF):
return [
matrix([[1, 0], [0, 1]], ring=CF),
matrix([[1, 0], [0, -1]], ring=CF),
matrix([[0, 1], [1, 0]], ring=CF),
matrix([[0, 1j], [-1j, 0]], ring=CF)
]
def update_view_state(self, boost_and_tet_num,
m = matrix([[1.0, 0.0, 0.0, 0.0],
[0.0, 1.0, 0.0, 0.0],
[0.0, 0.0, 1.0, 0.0],
[0.0, 0.0, 0.0, 1.0]])):
boost, tet_num = boost_and_tet_num
boost = matrix(boost, ring = self.RF)
m = matrix(m, ring = self.RF)
boost = O13_orthonormalize(boost * m)
entry_F = -1
for i in range(100):
pos = boost.transpose()[0]
tet = self.mcomplex.Tetrahedra[tet_num]
amount, F = max(
[ (R13_dot(pos, tet.R13_planes[F]), F)
for F in t3m.TwoSubsimplices ])
if F == entry_F:
break
if amount < 0.0000001:
break
boost = O13_orthonormalize(tet.O13_matrices[F] * boost)
tet_num = tet.Neighbor[F].Index
entry_F = tet.Gluing[F].image(F)
return boost, tet_num
def compute_so13_edge_involution(idealPoint0, idealPoint1):
ComplexField = idealPoint0.parent()
m1 = matrix([[idealPoint0, idealPoint1], [1, 1]], ring=ComplexField)
m2 = matrix([[-1, 0], [0, 1]], ring=ComplexField)
gl2c_matrix = m1 * m2 * _adjoint2(m1)
return GL2C_to_O13(gl2c_matrix)
def O13_orthonormalize(m):
try:
ring = m[0][0].parent()
except AttributeError:
ring = None
id_matrix = matrix([[1.0, 0.0, 0.0, 0.0], [0.0, 1.0, 0.0, 0.0],
[0.0, 0.0, 1.0, 0.0], [0.0, 0.0, 0.0, 1.0]],
ring=ring)
result = []
for row, id_row, sign in zip(m, id_matrix, _signature):
result.append(_orthonormalize_row_sane(row, id_row, result, sign))
return matrix(result, ring=ring)
def _check_matrix_o13(m):
s = matrix([[-1, 0,0,0],
[0, 1, 0, 0],
[0, 0, 1, 0],
[0, 0, 0, 1]])
_check_matrices_equal(s, m * s * m.transpose())
def _compute_cusp_triangle_vertex_positions(tet, V, i):
z = tet.ShapeParameters[t3m.E01]
CF = z.parent()
triangle = tet.horotriangles[V]
otherVerts = [ t3m.ZeroSubsimplices[(i + j) % 4] for j in range(1, 4) ]
vertex_positions = [ triangle.vertex_positions[V | otherVert ]
for otherVert in otherVerts ]
if tet.Class[V].is_complete:
m_translation, l_translation = tet.Class[V].Translations
a, c = m_translation.real(), m_translation.imag()
b, d = l_translation.real(), l_translation.imag()
log_z0 = CF(0)
# Inverting matrix here since SageMath screws up :(
translations_to_ml = matrix([[d,-b], [-c, a]]) / (a*d - b * c)
vertex_positions = [ translations_to_ml * complex_to_pair(z)
for z in vertex_positions ]
else:
log_z0 = triangle.lifted_vertex_positions[V | otherVerts[0]]
z0 = vertex_positions[0]
vertex_positions = [ complex_to_pair(z / z0)
for z in vertex_positions ]
return log_z0, vertex_positions
def initial_view_state(self):
boost = matrix([[1.0,0.0,0.0,0.0],
[0.0,1.0,0.0,0.0],
[0.0,0.0,1.0,0.0],
[0.0,0.0,0.0,1.0]])
tet_num = 0
return (boost, tet_num)
def _add_log_holonomies_to_cusp(self, cusp, shapes):
i = cusp.Index
if cusp.is_complete:
m_param, l_param = cusp.Translations
else:
m_param, l_param = [
sum(shape * expo
for shape, expo
in zip(shapes, self.peripheral_gluing_equations[2 * i + j]))
for j in range(2) ]
a, c = m_param.real(), m_param.imag()
b, d = l_param.real(), l_param.imag()
det = a*d - b * c
cusp.mat_log = matrix([[d,-b], [-c, a]]) / det
if cusp.is_complete:
cusp.margulisTubeRadiusParam = 0.0
else:
slope = 2 * self.areas[i] / abs(det)
x = (slope ** 2 / (slope ** 2 + 1)).sqrt()
y = (1 / (slope ** 2 + 1)).sqrt()
rSqr = 1 + (x ** 2 + (1 - y) ** 2) / (2 * y)
cusp.margulisTubeRadiusParam = 0.25 * (1.0 + rSqr)
def update_view_state(self,
boost_tet_num_and_weight,
m=matrix([[1.0, 0.0, 0.0, 0.0], [0.0, 1.0, 0.0, 0.0],
[0.0, 0.0, 1.0, 0.0], [0.0, 0.0, 0.0,
1.0]])):
boost, tet_num, weight = boost_tet_num_and_weight
boost = boost * m
return boost, tet_num, weight
def O13_z_rotation(angle):
c = cos(angle)
s = sin(angle)
return matrix(
[[ 1.0, 0.0, 0.0, 0.0],
[ 0.0, c, s, 0.0],
[ 0.0, -s, c, 0.0],
[ 0.0, 0.0, 0.0, 1.0]])
def update_view_state(self, boost_and_tet_num,
m = matrix([[1.0, 0.0, 0.0, 0.0],
[0.0, 1.0, 0.0, 0.0],
[0.0, 0.0, 1.0, 0.0],
[0.0, 0.0, 0.0, 1.0]])):
boost, tet_num = boost_and_tet_num
boost = boost * m
return boost, tet_num
def recompute_raytracing_data_and_redraw(self):
self._initialize_raytracing_data()
self.view_state = self.raytracing_data.update_view_state(
self.view_state,
matrix([[1,0,0,0],
[0,1,0,0],
[0,0,1,0],
[0,0,0,1]]))
self.redraw_if_initialized()
def O13_orthonormalize(m):
result = []
for i in range(0, 4):
v = m[(i + 1) % 4]
for j in range(i):
s = R13_dot(v, result[j])
v = [x - s * y for x, y in zip(v, result[j])]
result.append(R13_normalise(v))
return matrix([result[3]] + result[:3])
def unit_time_vector_to_O13_hyperbolic_translation(v):
def diag(i, j):
if i == j:
if i == 0:
return -1
else:
return 1
return 0
v1 = [1 + v[0]] + v[1:]
return matrix([[x * y / (1 + v[0]) + diag(i, j) for i, x in enumerate(v1)]
for j, y in enumerate(v1)])
def _add_log_holonomies(self):
shapes = [
tet.ShapeParameters[e].log() for tet in self.mcomplex.Tetrahedra
for e in [t3m.E01, t3m.E02, t3m.E03]
]
for cusp, cusp_info in zip(self.mcomplex.Vertices,
self.snappy_manifold.cusp_info()):
if cusp_info['complete?']:
cusp.mat_log = matrix([[1, 0], [0, 1]])
cusp.margulisTubeRadiusParam = 0.0
else:
self._add_log_holonomies_to_cusp(cusp, shapes)
def __init__(self, manifold, master, *args, **kwargs):
self.ui_uniform_dict = {
'maxSteps' : ('int', 20),
'maxDist' : ('float', 17),
'subpixelCount': ('int', 1),
'fov': ('float', 90),
'edgeThickness' : ('float', 0.0000001),
'lightBias' : ('float', 2.0),
'lightFalloff' : ('float', 1.65),
'brightness' : ('float', 1.9),
'fudge' : ('float', 1.0)
}
self.ui_parameter_dict = {
'insphere_scale' : ('float', 0.05),
'cuspAreas' : ('float[]', manifold.num_cusps() * [ 1.0 ]),
'edgeTubeRadius' : ('float', 0.08),
}
self.manifold = manifold
self._initialize_raytracing_data()
self.num_tets = len(self.raytracing_data.mcomplex.Tetrahedra)
shader_source = shaders.get_ideal_triangulation_shader_source(
self.raytracing_data.get_compile_time_constants())
SimpleImageShaderWidget.__init__(
self, master,
shader_source, *args, **kwargs)
boost = matrix([[1.0,0.0,0.0,0.0],
[0.0,1.0,0.0,0.0],
[0.0,0.0,1.0,0.0],
[0.0,0.0,0.0,1.0]])
tet_num = self.raytracing_data.get_initial_tet_num()
self.view_state = (boost, tet_num)
self.view = 0
self.perspectiveType = 0
HyperboloidNavigation.__init__(self)
def edge_involution(u, v):
b0 = vector(R13_normalise(u + v))
b1 = u - v
b1 = vector(R13_normalise(b1 + b0 * R13_dot(b0, b1)))
candidates = [ [ 0, 1, 0, 0], [ 0, 0, 1, 0], [ 0, 0, 0, 1] ]
penalties_and_candidates = [ (abs(R13_dot(b1, c)), vector(c)) for c in candidates ]
penalties_and_candidates.sort()
b2 = penalties_and_candidates[0][1]
b3 = penalties_and_candidates[1][1]
b2 = vector(R13_normalise(b2 + b0 * R13_dot(b0, b2) - b1 * R13_dot(b1, b2)))
b3 = vector(R13_normalise(b3 + b0 * R13_dot(b0, b3) - b1 * R13_dot(b1, b3) - b2 * R13_dot(b2, b3)))
bs = [ b0, b1, b2, b3 ]
m = [ matrix([[x * y for x in _change_first_sign(b)]
for y in b]) for b in bs ]
return - m[0] + m[1] - m[2] - m[3]
else:
return 1
return 0
v1 = [1 + v[0]] + v[1:]
return matrix(
[[ x * y / (1 + v[0]) + diag(i, j)
for i, x in enumerate(v1) ]
for j, y in enumerate(v1) ])
def unit_3_vector_and_distance_to_O13_hyperbolic_translation(v, d):
return unit_time_vector_to_O13_hyperbolic_translation(
[ cosh(d)] + [ sinh(d) * x for x in v])
_basis_vectors_sl2c = [ matrix([[ 1 , 0 ],
[ 0, 1 ]]),
matrix([[ 1 , 0 ],
[ 0 ,-1 ]]),
matrix([[ 0 , 1 ],
[ 1 , 0 ]]),
matrix([[ 0 , 1j],
[-1j, 0 ]]) ]
def _adjoint(m):
return matrix([[ m[0][0].conjugate(), m[1][0].conjugate()],
[ m[0][1].conjugate(), m[1][1].conjugate()]])
def _o13_matrix_column(A, m):
fAmj = A * m * _adjoint(A)
return [ (fAmj[0][0].real() + fAmj[1][1].real()) / 2,
def _adjoint2(m):
return matrix([[m[1, 1], -m[0, 1]], [-m[1, 0], m[0, 0]]])
def PSL2C_to_O13(A):
return matrix([
_o13_matrix_column(A, m) for m in _basis_vectors_sl2c(A[0, 0].parent())
]).transpose()
def _matrix_taking_0_1_inf_to_given_points(z0, z1, zinf):
l = z1 - z0
m = zinf - z1
return matrix([[ l * zinf, m * z0 ],
[ l, m ]])
return 1
return 0
v1 = [1 + v[0]] + v[1:]
return matrix([[x * y / (1 + v[0]) + diag(i, j) for i, x in enumerate(v1)]
for j, y in enumerate(v1)])
def unit_3_vector_and_distance_to_O13_hyperbolic_translation(v, d):
return unit_time_vector_to_O13_hyperbolic_translation(
[cosh(d)] + [sinh(d) * x for x in v])
_basis_vectors_sl2c = [
matrix([[1, 0], [0, 1]]),
matrix([[1, 0], [0, -1]]),
matrix([[0, 1], [1, 0]]),
matrix([[0, 1j], [-1j, 0]])
]
def _adjoint(m):
return matrix([[m[0][0].conjugate(), m[1][0].conjugate()],
[m[0][1].conjugate(), m[1][1].conjugate()]])
def _o13_matrix_column(A, m):
fAmj = A * m * _adjoint(A)
return [(fAmj[0][0].real() + fAmj[1][1].real()) / 2,
def _adjoint(m):
return matrix([[ m[1,1], -m[0,1]],
[-m[1,0], m[0,0]]], ring = m[0,0].parent())
def _adjoint(m):
return matrix([[m[0][0].conjugate(), m[1][0].conjugate()],
[m[0][1].conjugate(), m[1][1].conjugate()]])
