def tangProj(m, z): # preshapes
'''Orthogonally project z onto tangent space at m (considering the ambient space)'''
if np.abs(her(m, z)) < 1e-2:
return z
ta = theta(z, m)
v_proj = exp(1j * ta) * (z - her(m, z) * m)
return v_proj
def d_P(z, w): # preshapes
'''Partial distance between [z] and [w]
min_{theta} ||e^{i theta} z - w||'''
aux = 2 * np.abs(her(z, w))
if aux > 2: # same possibility as for d_F
return 0.0
return sqrt(2 - aux)
def d_F(z, w): # preshapes
'''Full distance between [z] and [w]
min_{alpha,theta} ||alpha e^{i theta} z - w||
= || w - P_z w || = ||z - P_w z|| \in [0,1] '''
aux = np.abs(her(z, w))**2
if aux > 1: # due to numerical approximations, but mathematically should be <= 1
return 0.0
return sqrt(1 - aux)
def proj_S(D_c): # columns of D_c are general configurations in C^N
'''converts non-zero columns of D_c to preshapes, by projecting them on the subspace
rthogonal to u {d | u.T.conj() Phi d = 0)}
and then normalizing them. Those which are zero are left as they are.
'''
from settings import uu
J = D_c.shape[1]
for j in range(J):
d = D_c[:, j]
no = norm(d)
if no > 1e-8:
orth_d = her(uu, d) * uu
d = d - orth_d
d = d / norm(d)
D_c[:, j] = d
return D_c
def theta(z, w): # preshapes
'''Computes the optimal angle theta(z,w) = arg(z* Phi w)
solution to min_{theta} ||e^{i theta} z - w||'''
return np.angle(her(z, w))
def rho(z, w): # preshapes
'''Geodesic distance between [z] and [w]. '''
aux = np.abs(her(z, w))
if aux > 1: # happens sometimes due to numerical errors
aux = 1
return arccos(aux)