"""
Tools for 3DCGA (g3c)
3DCGA Tools
==========================================================
Generation Methods
--------------------
.. autosummary::
:toctree: generated/
random_bivector
standard_point_pair_at_origin
random_point_pair_at_origin
random_point_pair
standard_line_at_origin
random_line_at_origin
random_line
random_circle_at_origin
random_circle
random_sphere_at_origin
random_sphere
random_plane_at_origin
random_plane
generate_n_clusters
generate_random_object_cluster
random_translation_rotor
random_rotation_translation_rotor
random_conformal_point
generate_dilation_rotor
generate_translation_rotor
Geometry Methods
--------------------
.. autosummary::
:toctree: generated/
intersect_line_and_plane_to_point
quaternion_and_vector_to_rotor
get_center_from_sphere
get_radius_from_sphere
point_pair_to_end_points
get_circle_in_euc
circle_to_sphere
line_to_point_and_direction
get_plane_origin_distance
get_plane_normal
get_nearest_plane_point
val_convert_2D_polar_line_to_conformal_line
convert_2D_polar_line_to_conformal_line
val_convert_2D_point_to_conformal
convert_2D_point_to_conformal
val_distance_point_to_line
distance_polar_line_to_euc_point_2d
Misc
--------------------
.. autosummary::
:toctree: generated/
meet_val
meet
normalise_n_minus_1
val_apply_rotor
apply_rotor
val_apply_rotor_inv
apply_rotor_inv
euc_dist
mult_with_ninf
val_norm
norm
val_normalised
normalised
val_up
fast_up
val_normalInv
val_homo
val_down
fast_down
dual_func
fast_dual
disturb_object
project_val
Root Finding
--------------------
.. autosummary::
:toctree: generated/
dorst_norm_val
check_sigma_for_positive_root_val
check_sigma_for_positive_root
check_sigma_for_negative_root_val
check_sigma_for_negative_root
check_infinite_roots_val
check_infinite_roots
positive_root_val
negative_root_val
positive_root
negative_root
general_root_val
general_root
val_annihilate_k
annihilate_k
pos_twiddle_root_val
neg_twiddle_root_val
pos_twiddle_root
neg_twiddle_root
square_roots_of_rotor
interp_objects_root
average_objects
rotor_between_objects
val_rotor_between_objects
val_rotor_between_lines
rotor_between_lines
rotor_between_planes
val_rotor_rotor_between_planes
"""
import math
import numba
import numpy as np
from clifford.tools.g3 import quaternion_to_rotor, random_euc_mv, \
random_rotation_rotor, generate_rotation_rotor, val_random_euc_mv
from clifford.g3c import *
import clifford as cf
from clifford import val_get_left_gmt_matrix, grades_present
import warnings
# Allow syntactic alternatives to the standard included in the clifford package
ninf = einf
no = -eo
# Define some useful objects
E = ninf ^ (no)
I5 = e12345
I3 = e123
E0 = ninf ^ -no
niono = ninf ^ no
E0_val = E0.value
I5_val = I5.value
ninf_val = ninf.value
no_val = no.value
I3_val = I3.value
unit_scalar_mv = 1.0 + 0.0*e1
unit_scalar_mv_val = unit_scalar_mv.value
adjoint_func = layout.adjoint_func
gmt_func = layout.gmt_func
omt_func = layout.omt_func
imt_func = layout.imt_func
epsilon = 10*10**(-6)
mask0 = layout.get_grade_projection_matrix(0)
mask1 = layout.get_grade_projection_matrix(1)
mask2 = layout.get_grade_projection_matrix(2)
mask3 = layout.get_grade_projection_matrix(3)
mask4 = layout.get_grade_projection_matrix(4)
mask_2minus4 = mask2 - mask4
def interpret_multivector_as_object(mv):
"""
Takes an input multivector and returns what kind of object it is
-1 -> not a blade
0 -> a 1 vector but not a point
1 -> a euclidean point
2 -> a conformal point
3 -> a point pair
4 -> a circle
5 -> a line
6 -> a sphere
7 -> a plane
"""
g_pres = grades_present(mv, 0.00000001)
if len(g_pres) != 1: # Definitely not a blade
return -1
grade = g_pres[0]
if grade == 1:
# This can now either be a euc mv or a conformal point or something else
if mv(e123) == mv:
return 1 # Euclidean point
elif np.sum(np.abs((mv**2).value)) < 0.00000001:
return 2 # Conformal point
else:
return 0 # Unknown mv
elif mv.isBlade():
if grade == 2:
return 3 # Point pair
elif grade == 3: # Line or circle
if abs(mv[e123]) > epsilon:
return 4 # Circle
else:
return 5 # Line
elif grade == 4: # Sphere or plane
if abs(((mv*I5)|ninf)[0]) > epsilon:
return 7 # Plane
else:
return 6 # Sphere
else:
return -1
def normalise_TR_to_unit_T(TR):
"""
Takes in a TR rotor
extracts the R and T
normalises the T to unit displacement magnitude
rebuilds the TR rotor with the new displacement rotor
returns the new TR and the original length of the T rotor
"""
R_only = TR(e123)
T_only = (TR*~R_only).normal()
t = -2*(T_only|no)
scale = abs(t)
t = t/scale
new_TR = (generate_translation_rotor(t)*R_only).normal()
return new_TR, scale
def scale_TR_translation(TR, scale):
"""
Takes in a TR rotor and a scale
extracts the R and T
scales the T displacement magnitude by scale
rebuilds the TR rotor with the new displacement rotor
returns the new TR rotor
"""
R_only = TR(e123)
T_only = (TR*~R_only).normal()
t = -2*(T_only|no)*scale
new_TR = (generate_translation_rotor(t)*R_only).normal()
return new_TR
def left_gmt_generator():
k_list = layout.k_list
l_list = layout.l_list
m_list = layout.m_list
mult_table_vals = layout.mult_table_vals
gaDims = layout.gaDims
@numba.njit
def get_left_gmt(x_val):
return val_get_left_gmt_matrix(x_val, k_list, l_list,
m_list, mult_table_vals, gaDims)
return get_left_gmt
get_left_gmt_matrix = left_gmt_generator()
def get_line_reflection_matrix(lines, n_power=1):
"""
Generates the matrix that sums the reflection of a point in many lines
"""
mat2solve = np.zeros((32,32), dtype=np.float64)
for Li in lines:
LiMat = get_left_gmt_matrix(Li.value)
tmat = np.matmul(np.matmul(LiMat, mask_2minus4), LiMat)
mat2solve += tmat
mat = np.matmul(mask1,mat2solve)
if n_power != 1:
mat = np.linalg.matrix_power(mat, n_power)
return mat
@numba.njit
def val_get_line_reflection_matrix(line_array, n_power):
"""
Generates the matrix that sums the reflection of a point in many lines
"""
mat2solve = np.zeros((32,32), dtype=np.float64)
for i in range(line_array.shape[0]):
LiMat = get_left_gmt_matrix(line_array[i,:])
tmat = np.matmul(np.matmul(LiMat, mask_2minus4), LiMat)
mat2solve += tmat
mat = np.matmul(mask1,mat2solve)
if n_power != 1:
mat = np.linalg.matrix_power(mat, n_power)
return mat
@numba.njit
def val_truncated_get_line_reflection_matrix(line_array, n_power):
"""
Generates the truncated matrix that sums the
reflection of a point in many lines
n_power should be 1 or a power of 2
"""
mat2solve = np.zeros((32,32), dtype=np.float64)
for i in range(line_array.shape[0]):
LiMat = get_left_gmt_matrix(line_array[i,:])
tmat = np.matmul(np.matmul(LiMat, mask_2minus4), LiMat)
mat2solve += tmat
mat_val = np.matmul(mask1,mat2solve)
mat_val = mat_val[1:6, 1:6].copy()/line_array.shape[0]
if n_power != 1:
c_pow = 1
while c_pow < n_power:
mat_val = np.matmul(mat_val,mat_val)
c_pow=c_pow*2
return mat_val
@numba.njit
def val_get_line_intersection(L3_val,Ldd_val):
"""
Gets the point of intersection of two orthogonal lines that meet
Xdd = Ldd*no*Ldd + no
Xddd = L3*Xdd*L3
Pd = 0.5*(Xdd+Xddd)
P = -(Pd*ninf*Pd)(1)/(2*(Pd|einf)**2)[0]
"""
Xdd = gmt_func(gmt_func(Ldd_val,no_val),Ldd_val) + no_val
Xddd = gmt_func(gmt_func(L3_val,Xdd),L3_val)
Pd = 0.5*(Xdd+Xddd)
P = -gmt_func(gmt_func(Pd,ninf_val),Pd)
imt_value = imt_func(Pd,ninf_val)
P_denominator = 2*(gmt_func(imt_value,imt_value))[0]
return project_val(P/P_denominator,1)
def get_line_intersection(L3,Ldd):
"""
Gets the point of intersection of two orthogonal lines that meet
Xdd = Ldd*no*Ldd + no
Xddd = L3*Xdd*L3
Pd = 0.5*(Xdd+Xddd)
P = -(Pd*ninf*Pd)(1)/(2*(Pd|einf)**2)[0]
"""
return layout.MultiVector(value=val_get_line_intersection(L3.value,Ldd.value))
@numba.njit
def val_midpoint_between_lines(L1_val, L2_val):
"""
Gets the point that is maximally close to both lines
Hadfield and Lasenby AGACSE2018
"""
L3 = val_normalised(L1_val + L2_val)
Ldd = val_normalised(L1_val - L2_val)
S = val_normalised(gmt_func(I5_val,val_get_line_intersection(L3,Ldd)))
return val_normalise_n_minus_1(gmt_func(S,gmt_func(ninf_val,S)))
def midpoint_between_lines(L1, L2):
"""
Gets the point that is maximally close to both lines
Hadfield and Lasenby AGACSE2018
"""
return layout.MultiVector(value=val_midpoint_between_lines(L1.value,L2.value))
def midpoint_of_line_cluster(line_cluster):
"""
Gets an approximate center point of a line cluster
Hadfield and Lasenby AGACSE2018
"""
average_line = average_objects(line_cluster,check_grades=False)
val_point_track = np.zeros(32)
for L2 in line_cluster:
p = val_midpoint_between_lines(average_line.value, L2.value)
val_point_track += p
S = gmt_func(I5_val,val_point_track)
center_point = val_normalise_n_minus_1(project_val(gmt_func(S,gmt_func(ninf_val,S)),1))
return layout.MultiVector(value=center_point)
@numba.njit(parallel=True)
def val_midpoint_of_line_cluster(array_line_cluster):
"""
Gets an approximate center point of a line cluster
Hadfield and Lasenby AGACSE2018
"""
average_line = val_average_objects(array_line_cluster)
val_point_track = np.zeros(32)
for i in range(array_line_cluster.shape[0]):
p = val_midpoint_between_lines(average_line, array_line_cluster[i,:])
val_point_track += p
S = gmt_func(I5_val,val_point_track)
center_point = val_normalise_n_minus_1(project_val(gmt_func(S,gmt_func(ninf_val,S)),1))
return center_point
@numba.njit(parallel=True)
def val_midpoint_of_line_cluster_grad(array_line_cluster):
"""
Gets an approximate center point of a line cluster
Hadfield and Lasenby AGACSE2018
"""
average_line = val_average_objects(array_line_cluster)
val_point_track = np.zeros(32)
for i in range(array_line_cluster.shape[0]):
p = val_midpoint_between_lines(average_line, array_line_cluster[i,:])
val_point_track += p
S = gmt_func(I5_val,val_point_track)
center_point = val_normalise_n_minus_1(project_val(gmt_func(S,gmt_func(ninf_val,S)),1))
# Take a derivative of the cost function at this point
grad = np.zeros(32)
for i in range(array_line_cluster.shape[0]):
l_val = array_line_cluster[i, :]
grad += (gmt_func(gmt_func(l_val, center_point), l_val))
grad = val_normalise_n_minus_1(project_val(grad, 1))
s_val = gmt_func(I5_val, project_val(center_point + grad, 1))
center_point = val_normalise_n_minus_1(gmt_func(gmt_func(s_val, ninf_val), s_val))
return center_point
[docs]def get_circle_in_euc(circle):
""" Extracts all the normal stuff for a circle """
Ic = (circle^ninf).normal()
GAnormal = get_plane_normal(Ic)
inPlaneDual = circle*Ic
mag = float((inPlaneDual|ninf)[0])
inPlaneDual = -inPlaneDual/mag
radius_squared = (inPlaneDual*inPlaneDual)[0]
radius = math.sqrt(abs(radius_squared))
if radius_squared < 0:
# We have an imaginary circle, return it as a negative radius as our signal
radius = -radius
GAcentre = down(inPlaneDual*(1+0.5*inPlaneDual*ninf))
return [GAcentre,GAnormal,radius]
[docs]def circle_to_sphere(C):
"""
returns the sphere for which the input circle is the perimeter
"""
Ic = (C ^ einf).normal()
sphere = C * Ic * I5
return sphere
[docs]def line_to_point_and_direction(line):
"""
converts a line to the conformal nearest point to the origin and a
euc direction vector in direction of the line
"""
L_star = line*I5
T = L_star|no
mhat = -(L_star - T*ninf)*I3
p = (T^mhat)*I3
return [p,mhat]
[docs]def get_plane_origin_distance(plane):
""" Get the distance between a given plane and the origin """
return float(((plane*I5)|no)[0])
[docs]def get_plane_normal(plane):
""" Get the normal to the plane """
return (plane*I5 - get_plane_origin_distance(plane)*ninf)
[docs]def get_nearest_plane_point(plane):
""" Get the nearest point to the origin on the plane """
return get_plane_normal(plane)*get_plane_origin_distance(plane)
[docs]def disturb_object(mv_object, maximum_translation=0.01, maximum_angle=0.01):
""" Disturbs an object by a random rotor """
r = random_rotation_translation_rotor(maximum_translation=maximum_translation, maximum_angle=maximum_angle)
return (r*mv_object*~r).normal()
[docs]def generate_n_clusters( object_generator, n_clusters, n_objects_per_cluster ):
""" Creates n_clusters of random objects """
object_clusters = []
for i in range(n_clusters):
cluster_objects = generate_random_object_cluster(n_objects_per_cluster, object_generator,
max_cluster_trans=0.5, max_cluster_rot=np.pi / 16)
object_clusters.append(cluster_objects)
all_objects = [item for sublist in object_clusters for item in sublist]
return all_objects, object_clusters
[docs]def generate_random_object_cluster(n_objects, object_generator, max_cluster_trans=1.0, max_cluster_rot=np.pi/8):
""" Creates a cluster of random objects """
ref_obj = object_generator()
cluster_objects = []
for i in range(n_objects):
r = random_rotation_translation_rotor(maximum_translation=max_cluster_trans, maximum_angle=max_cluster_rot)
new_obj = apply_rotor(ref_obj, r)
cluster_objects.append(new_obj)
return cluster_objects
[docs]def random_translation_rotor(maximum_translation=10.0):
""" generate a random translation rotor """
return generate_translation_rotor(random_euc_mv(maximum_translation))
[docs]def random_rotation_translation_rotor(maximum_translation=10.0, maximum_angle=np.pi):
""" generate a random combined rotation and translation rotor """
return (random_translation_rotor(maximum_translation)*random_rotation_rotor(maximum_angle)).normal()
@numba.njit
def project_val(val, grade):
""" fast grade projection """
output = np.zeros(32)
if grade == 0:
output[0] = val[0]
elif grade == 1:
output[1:6] = val[1:6]
elif grade == 2:
output[6:16] = val[6:16]
elif grade == 3:
output[16:26] = val[16:26]
elif grade == 4:
output[26:31] = val[26:31]
elif grade == 5:
output[31] = val[31]
return output
[docs]def generate_dilation_rotor(scale):
"""
Generates a rotor that performs dilation about the origin
"""
if abs(scale - 1.0) < 0.00001:
u = np.zeros(32)
u[0] = 1.0
return cf.MultiVector(layout, u)
gamma = math.log(scale)
return math.cosh(gamma/2) + math.sinh(gamma/2)*(ninf^no)
[docs]def generate_translation_rotor(euc_vector_a):
"""
Generates a rotor that translates objects along the euclidean vector euc_vector_a
"""
return (1 + ninf * euc_vector_a / 2)
@numba.njit
def meet_val(a_val, b_val):
"""
The meet algorithm as described in "A Covariant Approach to Geometry"
I5*((I5*A) ^ (I5*B))
"""
return dual_func(omt_func(dual_func(a_val), dual_func(b_val)))
[docs]def meet(A, B):
"""
The meet algorithm as described in "A Covariant Approach to Geometry"
I5*((I5*A) ^ (I5*B))
"""
return cf.MultiVector(layout, meet_val(A.value, B.value))
@numba.njit
def val_intersect_line_and_plane_to_point(line_val, plane_val):
"""
Returns the point at the intersection of a line and plane
"""
m = meet_val(line_val, plane_val)
if gmt_func(m, m)[0] < 0.000001:
return np.array([-1.])
output = np.zeros(32)
A = val_normalised(m)
if A[15] < 0:
output[1] = A[8]
output[2] = A[11]
output[3] = A[14]
else:
output[1] = -A[8]
output[2] = -A[11]
output[3] = -A[14]
output = val_up(output)
return output
[docs]def intersect_line_and_plane_to_point(line, plane):
"""
Returns the point at the intersection of a line and plane
If there is no intersection it returns None
"""
ans = val_intersect_line_and_plane_to_point(line.value, plane.value)
if ans[0] == -1.:
return None
return layout.MultiVector(value=ans)
@numba.njit
def val_normalise_n_minus_1(mv_val):
"""
Normalises a conformal point so that it has an inner product of -1 with einf
"""
scale = imt_func(mv_val,ninf_val)[0]
if scale != 0.0:
return -mv_val/scale
else:
raise ZeroDivisionError('Multivector has 0 einf component')
[docs]def normalise_n_minus_1(mv):
"""
Normalises a conformal point so that it has an inner product of -1 with einf
"""
return layout.MultiVector(value=val_normalise_n_minus_1(mv.value))
[docs]def quaternion_and_vector_to_rotor(quaternion, vector):
"""
Takes in a quaternion and a vector and returns a conformal rotor that
implements the transformation
"""
rotation = quaternion_to_rotor(quaternion)
translation = generate_translation_rotor(
vector[0] * e1 + vector[1] * e2 * vector[3] * e3)
return translation * rotation
[docs]def get_center_from_sphere(sphere):
"""
Returns the conformal point at the centre of a sphere by reflecting the
point at infinity
"""
center = sphere * ninf * sphere
return center
[docs]def get_radius_from_sphere(sphere):
"""
Returns the radius of a sphere
"""
dual_sphere = sphere * I5
dual_sphere = dual_sphere / (-dual_sphere | ninf)[0]
return math.sqrt(abs(dual_sphere * dual_sphere))
@numba.njit
def val_point_pair_to_end_points(T):
"""
Extracts the end points of a point pair bivector
"""
beta = np.sqrt(abs(gmt_func(T, T)[0]))
F = T / beta
P = 0.5*F
P[0] += 0.5
P_twiddle = -0.5*F
P_twiddle[0] += 0.5
A = val_normalise_n_minus_1(-gmt_func(P_twiddle , imt_func(T, ninf_val)))
B = val_normalise_n_minus_1(gmt_func(P,imt_func(T, ninf_val)))
output = np.zeros((2,32))
output[0,:]=A
output[1,:]=B
return output
[docs]def point_pair_to_end_points(T):
"""
Extracts the end points of a point pair bivector
"""
output = val_point_pair_to_end_points(T.value)
A = layout.MultiVector(value=output[0,:])
B = layout.MultiVector(value=output[1,:])
return A, B
[docs]def euc_dist(conf_mv_a, conf_mv_b):
""" Returns the distance between two conformal points """
dot_result = (conf_mv_a|conf_mv_b)[0]
if dot_result < 0.0:
return math.sqrt(-2.0*dot_result)
else:
return 0.0
@numba.jit
def dorst_norm_val(sigma_val):
""" Square Root of Rotors - Implements the norm of a rotor"""
sigma_4 = project_val(sigma_val, 4)
sqrd_ans = sigma_val[0] ** 2 - gmt_func(sigma_4,sigma_4)[0]
return math.sqrt(sqrd_ans)
@numba.njit
def check_sigma_for_positive_root_val(sigma_val):
""" Square Root of Rotors - Checks for a positive root """
return (sigma_val[0] + dorst_norm_val(sigma_val)) > 0
[docs]def check_sigma_for_positive_root(sigma):
""" Square Root of Rotors - Checks for a positive root """
return check_sigma_for_positive_root_val(sigma.value)
@numba.njit
def check_sigma_for_negative_root_val(sigma_value):
""" Square Root of Rotors - Checks for a negative root """
return (sigma_value[0] - dorst_norm_val(sigma_value)) > 0
[docs]def check_sigma_for_negative_root(sigma):
""" Square Root of Rotors - Checks for a negative root """
return check_sigma_for_negative_root_val(sigma.value)
@numba.njit
def check_infinite_roots_val(sigma_value):
""" Square Root of Rotors - Checks for a infinite roots """
return (sigma_value[0] + dorst_norm_val(sigma_value)) < 0.0000000001
[docs]def check_infinite_roots(sigma):
""" Square Root of Rotors - Checks for a infinite roots """
return check_infinite_roots_val(sigma.value)
@numba.njit
def positive_root_val(sigma_val):
"""
Square Root of Rotors - Evaluates the positive root
Square Root of Rotors - Evaluates the positive root
"""
norm_s = dorst_norm_val(sigma_val)
denominator = (math.sqrt(2) * math.sqrt(sigma_val[0] + norm_s))
result = sigma_val/denominator
result[0] += norm_s/denominator
return result
@numba.njit
def negative_root_val(sigma_val):
"""
Square Root of Rotors - Evaluates the positive root
"""
norm_s = dorst_norm_val(sigma_val)
denominator = (math.sqrt(2) * math.sqrt(sigma_val[0] - norm_s))
result = sigma_val/denominator
result[0] -= norm_s/denominator
return result
[docs]def positive_root(sigma):
"""
Square Root of Rotors - Evaluates the positive root
"""
res_val = positive_root_val(sigma.value)
return cf.MultiVector(layout, res_val)
[docs]def negative_root(sigma):
""" Square Root of Rotors - Evaluates the negative root """
res_val = negative_root_val(sigma.value)
return cf.MultiVector(layout, res_val)
@numba.njit
def general_root_val(sigma_value):
"""
Square Root and Logarithm of Rotors
in 3D Conformal Geometric Algebra
Using Polar Decomposition
Leo Dorst and Robert Valkenburg
"""
if check_sigma_for_positive_root_val(sigma_value):
k = positive_root_val(sigma_value)
output = np.zeros((2, sigma_value.shape[0]))
output[0, :] = k.copy()
return output
elif check_sigma_for_negative_root_val(sigma_value):
k = positive_root_val(sigma_value)
k2 = negative_root_val(sigma_value)
output = np.zeros((2, sigma_value.shape[0]))
output[0, :] = k.copy()
output[1, :] = k2.copy()
return output
elif check_infinite_roots_val(sigma_value):
output = np.zeros((2, sigma_value.shape[0]))
output[:, 0] = 1.0
return output
else:
raise ValueError('No root exists')
[docs]def general_root(sigma):
""" The general case of the root of a grade 0,4 multivector """
output = general_root_val(sigma.value)
return [cf.MultiVector(layout, output[0, :].copy()), cf.MultiVector(layout, output[1, :].copy())]
@numba.njit
def val_annihilate_k(K_val, C_val):
""" Removes K from C = KX via (K[0] - K[4])*C """
k_4 = -project_val(K_val, 4)
k_4[0] += K_val[0]
return val_normalised(gmt_func(k_4, C_val))
[docs]def annihilate_k(K, C):
""" Removes K from C = KX via (K[0] - K[4])*C """
return cf.MultiVector(layout, val_annihilate_k(K.value, C.value))
@numba.njit
def pos_twiddle_root_val(C_value):
"""
Square Root and Logarithm of Rotors
in 3D Conformal Geometric Algebra
Using Polar Decomposition
Leo Dorst and Robert Valkenburg
"""
sigma_val = gmt_func(C_value, adjoint_func(C_value))
k_value = general_root_val(sigma_val)
output = np.zeros((2,32))
output[0, :] = val_annihilate_k(k_value[0, :], C_value)
output[1, :] = val_annihilate_k(k_value[1, :], C_value)
return output
@numba.njit
def neg_twiddle_root_val(C_value):
"""
Square Root and Logarithm of Rotors
in 3D Conformal Geometric Algebra
Using Polar Decomposition
Leo Dorst and Robert Valkenburg
"""
sigma_val = -gmt_func(C_value, adjoint_func(C_value))
k_value = general_root_val(sigma_val)
output = np.zeros((2, 32))
output[0, :] = val_annihilate_k(k_value[0, :], C_value)
output[1, :] = val_annihilate_k(k_value[1, :], C_value)
return output
[docs]def pos_twiddle_root(C):
"""
Square Root and Logarithm of Rotors
in 3D Conformal Geometric Algebra
Using Polar Decomposition
Leo Dorst and Robert Valkenburg
"""
output = pos_twiddle_root_val(C.value)
return [cf.MultiVector(layout, output[0, :]), cf.MultiVector(layout, output[1, :])]
[docs]def neg_twiddle_root(C):
"""
Square Root and Logarithm of Rotors
in 3D Conformal Geometric Algebra
Using Polar Decomposition
Leo Dorst and Robert Valkenburg
"""
output = neg_twiddle_root_val(C.value)
return [cf.MultiVector(layout, output[0, :]), cf.MultiVector(layout, output[1, :])]
[docs]def square_roots_of_rotor(R):
"""
Square Root and Logarithm of Rotors
in 3D Conformal Geometric Algebra
Using Polar Decomposition
Leo Dorst and Robert Valkenburg
"""
return pos_twiddle_root(1 + R)
def n_th_rotor_root(R, n):
"""
Takes the n_th root of rotor R
n must be a power of 2
"""
if not (((n & (n - 1)) == 0) and n != 0):
raise ValueError('n is not a power of 2')
if n == 1:
return R
else:
return n_th_rotor_root(square_roots_of_rotor(R)[0], int(n/2))
[docs]def interp_objects_root(C1, C2, alpha):
"""
Hadfield and Lasenby, Direct Linear Interpolation of Geometric Objects, AGACSE2018
Directly linearly interpolates conformal objects
Return a valid object from the addition result C
"""
C = (1 - alpha) * C1 + alpha*C2
C3 = normalised(neg_twiddle_root(C)[0])
if cf.grade_obj(C1, 0.00001) != cf.grade_obj(C3, 0.00001):
raise ValueError('Created object is not same grade')
return C3
from scipy.interpolate import interp1d
def general_object_interpolation(object_alpha_array, object_list, new_alpha_array, kind='linear'):
"""
Hadfield and Lasenby, Direct Linear Interpolation of Geometric Objects, AGACSE2018
This is a general interpolation through the
"""
obj_array = np.transpose(ConformalMVArray(object_list).value)
f = interp1d(object_alpha_array, obj_array, kind=kind)
new_value_array = np.transpose(f(new_alpha_array))
new_conf_array = ConformalMVArray.from_value_array(new_value_array)
return [normalised(neg_twiddle_root(C)[0]) for C in new_conf_array]
@numba.njit
def val_average_objects_with_weights(obj_array, weights_array):
"""
Hadfield and Lasenby, Direct Linear Interpolation of Geometric Objects, AGACSE2018
Directly averages conformal objects
Return a valid object from the addition result C
"""
C_val = np.zeros(32)
for i in range(obj_array.shape[0]):
C_val += obj_array[i, :]*weights_array[i]
C3 = val_normalised(neg_twiddle_root_val(C_val)[0, :])
return C3
@numba.njit
def val_average_objects(obj_array):
"""
Hadfield and Lasenby, Direct Linear Interpolation of Geometric Objects, AGACSE2018
Directly averages conformal objects
Return a valid object from the addition result C
"""
C_val = np.zeros(32)
for i in range(obj_array.shape[0]):
C_val += obj_array[i,:]
C_val = C_val / obj_array.shape[0]
C3 = val_normalised(neg_twiddle_root_val(C_val)[0, :])
return C3
[docs]def average_objects(obj_list, weights=[], check_grades=True):
"""
Hadfield and Lasenby, Direct Linear Interpolation of Geometric Objects, AGACSE2018
Directly averages conformal objects
Return a valid object from the addition result C
"""
if len(weights) == len(obj_list):
C = sum([o * w for o, w in zip(obj_list, weights)])
else:
C = sum(obj_list) / len(obj_list)
C3 = normalised(neg_twiddle_root(C)[0])
if check_grades:
if cf.grade_obj(obj_list[0], 0.00001) != cf.grade_obj(C3, 0.00001):
raise ValueError('Created object is not same grade')
return C3
[docs]def rotor_between_objects(X1, X2):
"""
Lasenby and Hadfield AGACSE2018
For any two conformal objects X1 and X2 this returns a rotor that takes X1 to X2
Return a valid object from the addition result 1 + gamma*X2X1
"""
return cf.MultiVector(layout,
val_rotor_between_objects_root(val_normalised(X1.value),
val_normalised(X2.value)))
def calculate_S_over_mu(X1, X2):
"""
Lasenby and Hadfield AGACSE2018
For any two conformal objects X1 and X2 this returns a factor that corrects
the X1 + X2 back to a blade
"""
gamma1 = (X1 * X1)[0]
gamma2 = (X2 * X2)[0]
M12 = X1 * X2 + X2 * X1
K = 2 + gamma1 * M12
mu = (K[0]**2 - K(4)**2)[0]
if sum(np.abs(M12(4).value)) > 0.0000001:
lamb = (-(K(4) * K(4)))[0]
mu = K[0] ** 2 + lamb
root_mu = np.sqrt(mu)
if abs(lamb) < 0.0000001:
beta = 1.0 / (2 * np.sqrt(K[0]))
else:
beta_sqrd = 1 / (2 * (root_mu + K[0]))
beta = np.sqrt(beta_sqrd)
S = -gamma1/(2*beta) + beta*M12(4)
return S/np.sqrt(mu)
else:
S = np.sqrt(abs(K[0]))
return S/np.sqrt(mu)
@numba.njit
def val_rotor_between_objects_root(X1, X2):
"""
Lasenby and Hadfield AGACSE2018
For any two conformal objects X1 and X2 this returns a rotor that takes X1 to X2
Uses the square root of rotors for efficiency and numerical stability
"""
X21 = gmt_func(X2, X1)
X12 = gmt_func(X1, X2)
X_SUM = X21 + X12
X_SUM[0] -= 2.0
if np.sum(np.abs(X_SUM)) > 0.0000001:
gamma = gmt_func(X1, X1)[0]
C_val = gamma*gmt_func(X2, X1)
C_val[0] += 1
else:
C_val = X21
C_val[0] += 1
return val_normalised(C_val)
return val_normalised(pos_twiddle_root_val(C_val)[0, :])
@numba.njit
def val_rotor_between_objects_explicit(X1, X2):
"""
Lasenby and Hadfield AGACSE2018
For any two conformal objects X1 and X2 this returns a rotor that takes X1 to X2
Implements an optimised version of:
gamma1 = (X1 * X1)[0]
gamma2 = (X2 * X2)[0]
M12 = X1*X2 + X2*X1
K = 2 + gamma1*M12
if np.sum(np.abs(K.value)) < 0.0000001:
return 1 + 0*e1
if sum(np.abs(M12(4).value)) > 0.0000001:
lamb = (-(K(4) * K(4)))[0]
mu = K[0]**2 + lamb
root_mu = np.sqrt(mu)
if abs(lamb) < 0.0000001:
beta = 1.0/(2*np.sqrt(K[0]))
else:
beta_sqrd = 1/(2*(root_mu + K[0]))
beta = np.sqrt(beta_sqrd)
R = ( beta*K(4) - (1/(2*beta)) )*(1 + gamma1*X2*X1)/(root_mu)
return R
else:
return (1 + gamma1*X2*X1)/(np.sqrt(abs(K[0])))
"""
gamma1 = gmt_func(X1, X1)[0]
gamma2 = gmt_func(X2, X2)[0]
# if abs(abs(gamma1) - 1.0) > 0.00001:
# raise ValueError('X1 must be normalised to give abs(X1*X1) == 1')
# elif abs(abs(gamma2) - 1.0) > 0.00001:
# raise ValueError('X1 must be normalised to give abs(X1*X1) == 1')
# elif abs(gamma1 - gamma2) > 0.00001:
# raise ValueError('X1 and X2 must square to the same value')
M12_val = gmt_func(X1, X2) + gmt_func(X2, X1)
K_val = gamma1 * M12_val
K_val[0] = K_val[0] + 2
if np.sum(np.abs(K_val)) < 0.0000001:
return unit_scalar_mv_val
if np.sum(np.abs(project_val(M12_val, 4))) > 0.00001:
K_val_4 = project_val(K_val, 4)
lamb = -gmt_func(K_val_4, K_val_4)[0]
mu = K_val[0] ** 2 + lamb
root_mu = np.sqrt(mu)
beta_sqrd = 1 / (2 *root_mu + 2 *K_val[0])
beta = np.sqrt(beta_sqrd)
temp_1 = beta * K_val_4
temp_1[0] = temp_1[0] - (1 / (2 * beta))
temp_2 = gamma1*gmt_func(X2,X1)
temp_2[0] = temp_2[0] + 1.0
R_val = gmt_func(temp_1, temp_2) / root_mu
return R_val
else:
temp_1 = gamma1 * gmt_func(X2, X1)
temp_1[0] += 1.0
return temp_1/(np.sqrt(abs(K_val[0])))
sparse_line_gmt = layout.gmt_func_generator(grades_a=[3], grades_b=[3])
@numba.njit
def val_norm(mv_val):
""" Returns sqrt(abs(~A*A)) """
return np.sqrt(np.abs(gmt_func(adjoint_func(mv_val), mv_val)[0]))
[docs]def norm(mv):
""" Returns sqrt(abs(~A*A)) """
return val_norm(mv.value)
@numba.njit
def val_normalised(mv_val):
""" Returns A/sqrt(abs(~A*A)) """
return mv_val/val_norm(mv_val)
[docs]def normalised(mv):
""" fast version of the normal() function """
return cf.MultiVector(layout, val_normalised(mv.value))
@numba.njit
def val_rotor_between_lines(L1_val, L2_val):
""" Implements a very optimised rotor line to line extraction """
L21_val = sparse_line_gmt(L2_val, L1_val)
L12_val = sparse_line_gmt(L1_val, L2_val)
K_val = L21_val + L12_val
K_val[0] += 2.0
beta_val = project_val(K_val, 4)
alpha = 2 * K_val[0]
denominator = np.sqrt(alpha / 2)
numerator_val = -beta_val/alpha
numerator_val[0] += 1.0
normalisation_val = numerator_val/denominator
output_val = L21_val
output_val[0] += 1
return gmt_func(normalisation_val, output_val)
[docs]def rotor_between_lines(L1, L2):
""" return the rotor between two lines """
return cf.MultiVector(layout, val_rotor_between_lines(L1.value, L2.value))
[docs]def rotor_between_planes(P1, P2):
""" return the rotor between two planes """
return cf.MultiVector(layout, val_rotor_rotor_between_planes(P1.value, P2.value))
@numba.njit
def val_rotor_rotor_between_planes(P1_val, P2_val):
""" return the rotor between two planes """
P21_val = -gmt_func(P2_val, P1_val)
P21_val[0] += 1
return val_normalised(P21_val)
[docs]def random_bivector():
"""
Creates a random bivector on the form described by R. Wareham in
Mesh Vertex Pose and Position Interpolation using Geometric Algebra.
$$ B = ab + c*n_{\inf}$$ where $a, b, c \in \mathcal(R)^3$
"""
a = random_euc_mv()
c = random_euc_mv()
return a * I3 + c * ninf
[docs]def standard_point_pair_at_origin():
""" Creates a standard point pair at the origin """
return (up(-0.5*e1)^up(0.5*e1)).normal()
[docs]def random_point_pair_at_origin():
"""
Creates a random point pair bivector object at the origin
"""
mv_a = random_euc_mv()
plane_a = (mv_a*I3).normal()
mv_b = plane_a*mv_a*plane_a
pp = (up(mv_a) ^ up(mv_b)).normal()
return pp
[docs]def random_point_pair():
"""
Creates a random point pair bivector object
"""
mv_a = random_euc_mv()
mv_b = random_euc_mv()
pp = (up(mv_a) ^ up(mv_b)).normal()
return pp
[docs]def standard_line_at_origin():
return (standard_point_pair_at_origin()^einf).normal()
[docs]def random_line_at_origin():
"""
Creates a random line at the origin
"""
pp = (random_point_pair_at_origin()^einf).normal()
return pp
[docs]def random_line():
"""
Creates a random line
"""
mv_a = random_euc_mv()
mv_b = random_euc_mv()
line_a = ((up(mv_a) ^ up(mv_b) ^ ninf)).normal()
return line_a
[docs]def random_circle_at_origin():
"""
Creates a random circle at the origin
"""
mv_a = random_euc_mv()
mv_r = random_euc_mv()
r = generate_rotation_rotor(np.pi/2, mv_a, mv_r)
mv_b = r*mv_a*~r
mv_c = r * mv_b * ~r
pp = (up(mv_a) ^ up(mv_b) ^ up(mv_c) ).normal()
return pp
[docs]def random_circle():
"""
Creates a random circle
"""
mv_a = val_random_euc_mv()
mv_b = val_random_euc_mv()
mv_c = val_random_euc_mv()
return layout.MultiVector(value=val_normalised(omt_func(omt_func(val_up(mv_a), val_up(mv_b)), val_up(mv_c))))
[docs]def random_sphere_at_origin():
"""
Creates a random sphere at the origin
"""
C = random_circle_at_origin()
sphere = circle_to_sphere(C)
return sphere
[docs]def random_sphere():
"""
Creates a random sphere
"""
mv_a = random_euc_mv()
mv_b = random_euc_mv()
mv_c = random_euc_mv()
mv_d = random_euc_mv()
sphere = ((up(mv_a) ^ up(mv_b) ^ up(mv_c) ^ up(mv_d))).normal()
return sphere
[docs]def random_plane_at_origin():
"""
Creates a random plane at the origin
"""
c = random_circle_at_origin()
plane = (c ^ einf).normal()
return plane
[docs]def random_plane():
"""
Creates a random plane
"""
c = random_circle()
plane = (c ^ ninf).normal()
return plane
@numba.njit
def val_apply_rotor(mv_val, rotor_val):
""" Applies rotor to multivector in a fast way - JITTED """
return gmt_func(rotor_val, gmt_func(mv_val, adjoint_func(rotor_val)))
[docs]def apply_rotor(mv_in, rotor):
""" Applies rotor to multivector in a fast way """
return cf.MultiVector(layout, val_apply_rotor(mv_in.value, rotor.value))
@numba.njit
def val_apply_rotor_inv(mv_val, rotor_val, rotor_val_inv):
""" Applies rotor to multivector in a fast way takes pre computed adjoint"""
return gmt_func(rotor_val, gmt_func(mv_val, rotor_val_inv))
[docs]def apply_rotor_inv(mv_in, rotor, rotor_inv):
""" Applies rotor to multivector in a fast way takes pre computed adjoint"""
return cf.MultiVector(layout, val_apply_rotor_inv(mv_in.value, rotor.value, rotor_inv.value))
@numba.njit
def mult_with_ninf(mv):
""" Convenience function for multiplication with ninf """
return gmt_func(mv, ninf_val)
#@numba.njit
@numba.njit
def val_up(mv_val):
""" Fast jitted up mapping """
temp = np.zeros(32)
temp[0] = 0.5
return mv_val - no_val + omt_func(temp, gmt_func(gmt_func(mv_val, mv_val), ninf_val))
[docs]def fast_up(mv):
""" Fast up mapping """
return cf.MultiVector(layout, val_up(mv.value))
@numba.njit
def val_normalInv(mv_val):
""" A fast, jitted version of normalInv """
Madjoint_val = adjoint_func(mv_val)
MadjointM = gmt_func(Madjoint_val,mv_val)[0]
return Madjoint_val / MadjointM
@numba.njit
def val_homo(mv_val):
""" A fast, jitted version of homo() """
return gmt_func(mv_val, val_normalInv(imt_func(-mv_val, ninf_val)))
@numba.njit
def val_down(mv_val):
""" A fast, jitted version of down() """
return gmt_func(omt_func(val_homo(mv_val), E0_val), E0_val)
[docs]def fast_down(mv):
""" A fast version of down() """
return cf.MultiVector(layout, val_down(mv.value))
[docs]def val_distance_point_to_line(point, line):
"""
Returns the euclidean distance between a point and a line
"""
return float(abs( cf.MultiVector(layout, omt_func(point,line) ) ))
@numba.njit
def val_convert_2D_point_to_conformal(x, y):
""" Convert a 2D point to conformal """
mv_val = np.zeros(32)
mv_val[1] = x
mv_val[2] = y
return val_up(mv_val)
[docs]def distance_polar_line_to_euc_point_2d(rho, theta, x, y):
""" Return the distance between a polar line and a euclidean point in 2D """
point = val_convert_2D_point_to_conformal(x, y)
line = val_convert_2D_polar_line_to_conformal_line(rho, theta)
return val_distance_point_to_line(point, line)
dual_gmt_func = layout.gmt_func_generator(grades_a=[5], grades_b=[0, 1, 2, 3, 4, 5])
@numba.njit
def dual_func(a_val):
"""
Fast dual
"""
return dual_gmt_func(I5_val, a_val)
[docs]def fast_dual(a):
"""
Fast dual
"""
return cf.MultiVector(layout, dual_func(a.value))
class ConformalMVArray(cf.MVArray):
"""
This class is for storing arrays of conformal multivectors
"""
def up(self):
"""
Up mapping
"""
return v_up(self)
def down(self):
"""
Down mapping
"""
return v_down(self)
def dual(self):
"""
Dualisation
"""
return v_dual(self)
def apply_rotor(self, R):
"""
Application of a rotor
"""
R_inv = ~R
return v_apply_rotor_inv(self, R, R_inv)
def apply_rotor_inv(self, R, R_inv):
"""
Application of a rotor with precomputed inverse
"""
return v_apply_rotor_inv(self, R, R_inv)
@property
def value(self):
"""
Return an np array of the values of multivectors
"""
return np.array([mv.value for mv in self])
@staticmethod
def from_value_array(value_array):
"""
Constructs an array of mvs from a value array
"""
return ConformalMVArray(v_new_mv(value_array))
v_dual = np.vectorize(fast_dual, otypes=[ConformalMVArray])
v_new_mv = np.vectorize(lambda v: cf.MultiVector(layout, v), otypes=[ConformalMVArray], signature='(n)->()')
v_up = np.vectorize(fast_up, otypes=[ConformalMVArray])
v_down = np.vectorize(fast_down, otypes=[ConformalMVArray])
v_apply_rotor_inv = np.vectorize(apply_rotor_inv, otypes=[ConformalMVArray])
v_meet = np.vectorize(meet, otypes=[ConformalMVArray], signature='(),()->()')