def _compute_margulis_tube_ends(tet, vertex):
trig = tet.horotriangles[vertex]
CF = tet.ShapeParameters[t3m.E01].parent()
if tet.Class[vertex].is_complete:
return [(0.0, 0.0, 0.0, 0.0), (0.0, 0.0, 0.0, 0.0)]
tet_vertices = [
tet.complex_vertices[v] for v in t3m.ZeroSubsimplices if v != vertex
]
cusp_vertices = [
trig.vertex_positions[vertex | v] for v in t3m.ZeroSubsimplices
if v != vertex
]
std_to_tet = _matrix_taking_0_1_inf_to_given_points(*tet_vertices)
cusp_to_std = _adjoint(
_matrix_taking_0_1_inf_to_given_points(*cusp_vertices))
cusp_to_tet = std_to_tet * cusp_to_std
cusp_to_tet_O13 = GL2C_to_O13(cusp_to_tet)
return [cusp_to_tet_O13 * vector([1.0, x, 0.0, 0.0]) for x in [-1.0, 1.0]]
def _compute_margulis_tube_ends(tet, vertex):
if tet.Class[vertex].is_complete:
return [(0.0, 0.0, 0.0, 0.0), (0.0, 0.0, 0.0, 0.0)]
return [ tet.cusp_to_tet_matrices[vertex] * vector([1.0, x, 0.0, 0.0])
for x in [-1.0, 1.0] ]
def _compute_edge_ends(self):
cs = [vector(self.RF, [1, 1, 0, 0]), vector(self.RF, [1, -1, 0, 0])]
def _compute_edge_ends(tet, perm):
m = tet.permutahedron_matrices[perm]
return [GL2C_to_O13(_adjoint(m)) * c for c in cs]
for tet in self.mcomplex.Tetrahedra:
tet.R13_edge_ends = {
t3m.E01: _compute_edge_ends(tet, (0, 1, 2, 3)),
t3m.E02: _compute_edge_ends(tet, (0, 2, 1, 3)),
t3m.E12: _compute_edge_ends(tet, (2, 1, 0, 3)),
t3m.E03: _compute_edge_ends(tet, (0, 3, 1, 2)),
t3m.E13: _compute_edge_ends(tet, (1, 3, 0, 2)),
t3m.E23: _compute_edge_ends(tet, (2, 3, 0, 1))
}
def _check_consistency(self):
for tet in self.mcomplex.Tetrahedra:
for F in t3m.TwoSubsimplices:
for V in t3m.ZeroSubsimplices:
if V & F:
v0 = tet.O13_matrices[F] * vector(tet.R13_vertices[V])
v1 = tet.Neighbor[F].R13_vertices[tet.Gluing[F].image(V)]
err = R13_dot(v0, v1)
if err > 1e-10 or err < -1e-10:
print("PROBLEM")
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]
def _check_consistency(self):
for tet in self.mcomplex.Tetrahedra:
for F in t3m.TwoSubsimplices:
for V in t3m.ZeroSubsimplices:
if V & F:
v0 = tet.O13_matrices[F] * vector(tet.R13_vertices[V])
v1 = tet.Neighbor[F].R13_vertices[tet.Gluing[F].image(
V)]
if abs(R13_dot(v0, v1) - (-1.0)) > 1e-6:
print("Inconsistency ", tet.Index, F)
print(v0)
print(v1)
def _compute_tet_vertices(self):
c = vector(self.RF, [1, 0, 0, 0])
def _compute_vertex(tet, perm):
m = tet.permutahedron_matrices[perm]
return GL2C_to_O13(_adjoint(m)) * c
for tet in self.mcomplex.Tetrahedra:
tet.R13_vertices = {
t3m.V0: _compute_vertex(tet, (0, 1, 3, 2)),
t3m.V1: _compute_vertex(tet, (1, 0, 2, 3)),
t3m.V2: _compute_vertex(tet, (2, 0, 3, 1)),
t3m.V3: _compute_vertex(tet, (3, 0, 1, 2))
}
def _compute_planes(self):
c = vector(self.RF, [0.0, 0.0, 0.0, -1.0])
def _compute_plane(tet, perm):
m = tet.permutahedron_matrices[perm]
v = c * GL2C_to_O13(m)
return vector([-v[0], v[1], v[2], v[3]])
for tet in self.mcomplex.Tetrahedra:
tet.R13_planes = {
t3m.F0: _compute_plane(tet, (2, 3, 1, 0)),
t3m.F1: _compute_plane(tet, (0, 3, 2, 1)),
t3m.F2: _compute_plane(tet, (0, 1, 3, 2)),
t3m.F3: _compute_plane(tet, (0, 2, 1, 3))
}
def complex_to_pair(z):
return vector([z.real(), z.imag()])
def compute_translation_and_inverse_from_pick_point(
self, size, frag_coord, depth):
# Depth value emitted by shader is tanh(distance from camera)
# Limit the maximal depth for orbiting to avoid numeric issues
depth = min(depth, _max_depth_for_orbiting)
fov = self.ui_uniform_dict['fov'][1]
isIdeal = self.ui_parameter_dict['perspectiveType'][1]
# Reimplement functionality from fragment shader
# See get_ray_eye_space
fov_scale = 2.0 * math.tan(fov / 360.0 * math.pi)
# Reimplement computation of xy from fragment coordinate
x = (frag_coord[0] - 0.5 * size[0]) / size[0]
y = (frag_coord[1] - 0.5 * size[1]) / size[0]
# Reimplement get_ray_eye_space to determine end point of
# ray. The end point is encoded as pair distance to origin
# direction to origin.
if isIdeal:
scaled_x = 0.5 * fov_scale * x
scaled_y = 0.5 * fov_scale * y
# Use "parabolic transformation magic by Saul"
# to determine the start point and direction of ray.
# Then compute end point using depth value.
foo = 0.5 * (scaled_x * scaled_x + scaled_y * scaled_y)
rayEnd = R13_normalise(
vector([
RF((foo + 1.0) + depth * foo),
RF(scaled_x + depth * scaled_x),
RF(scaled_y + depth * scaled_y),
RF(foo + depth * (foo - 1.0))
]))
# Distance of rayEnd from origin
dist = math.acosh(rayEnd[0])
# Direction from origin to rayEnd
dir = vector([rayEnd[1], rayEnd[2], rayEnd[3]])
else:
scaled_x = fov_scale * x
scaled_y = fov_scale * y
# Camera is assumed to be at origin.
dist = math.atanh(depth)
# Reimplemented from get_ray_eye_space
dir = vector([RF(scaled_x), RF(scaled_y), RF(-1.0)])
# Normalize direction
dir = dir.normalized()
# Compute the circumference of a circle of radius dist
#
# Do this by using a concentric circle in the Poincare disk
# model
poincare_dist = math.tanh(dist / 2.0)
hyp_circumference_up_to_constant = (
poincare_dist / (1.0 - poincare_dist * poincare_dist))
speed = min(
_max_orbit_speed, _max_linear_camera_speed /
max(1e-10, hyp_circumference_up_to_constant))
# Compute translation along direction by distance.
# And inverse.
return (unit_3_vector_and_distance_to_O13_hyperbolic_translation(
dir, dist),
unit_3_vector_and_distance_to_O13_hyperbolic_translation(
dir, -dist), speed)
def _compute_plane(tet, perm):
m = tet.permutahedron_matrices[perm]
v = c * GL2C_to_O13(m)
return vector([-v[0], v[1], v[2], v[3]])