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quat_utils.py
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quat_utils.py
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import numpy as np
def flip_up(q):
if q[0] < 0:
q = -q
return q
def multiply(r, a):
b0 = r[0] * a[0] - r[1] * a[1] - r[2] * a[2] - r[3] * a[3]
b1 = r[0] * a[1] + r[1] * a[0] + r[2] * a[3] - r[3] * a[2]
b2 = r[0] * a[2] - r[1] * a[3] + r[2] * a[0] + r[3] * a[1]
b3 = r[0] * a[3] + r[1] * a[2] - r[2] * a[1] + r[3] * a[0]
return flip_up(np.array([b0, b1, b2, b3]))
def random_quaternions(n):
x0 = np.random.uniform(0, 1, n)
x1 = np.random.uniform(0, 2 * np.pi, n)
x2 = np.random.uniform(0, 2 * np.pi, n)
r1, r2 = np.sqrt(1 - x0), np.sqrt(x0)
s1, c1 = np.sin(x1), np.cos(x1)
s2, c2 = np.sin(x2), np.cos(x2)
return np.array([s1 * r1, c1 * r1, s2 * r2, c2 * r2]).T
def multiply_first_component(r, a):
return r[0] * a[0] - r[1] * a[1] - r[2] * a[2] - r[3] * a[3]
def rotate_into_fundamental_zone(q, generators):
index = np.argmax([abs(multiply_first_component(q, g)) for g in generators])
return multiply(q, generators[index])
def map_points_out(basis_points, basis_weights, superset, subset, map_indices):
assert( len(map_indices) * len(subset) == len(superset) )
superset = np.array(superset)
subset = np.array(subset)
mapped_points = []
mapped_weights = []
for g in superset[map_indices]:
for b, w in zip(basis_points, basis_weights):
r = multiply(b, g)
r = rotate_into_fundamental_zone(r, subset)
mapped_points += [r]
mapped_weights += [w]
return np.array(mapped_points), np.array(mapped_weights)
def quaternion_to_rotation_matrix(q):
a, b, c, d = q
u0 = a*a + b*b - c*c - d*d
u1 = 2*b*c - 2*a*d
u2 = 2*b*d + 2*a*c
u3 = 2*b*c + 2*a*d
u4 = a*a - b*b + c*c - d*d
u5 = 2*c*d - 2*a*b
u6 = 2*b*d - 2*a*c
u7 = 2*c*d + 2*a*b
u8 = a*a - b*b - c*c + d*d
return np.array([[u0, u1, u2], [u3, u4, u5], [u6, u7, u8]])
def rodrigues_to_quaternion(r):
s = 1 / np.sqrt(1 + np.linalg.norm(r)**2)
return np.array([s, s * r[0], s * r[1], s * r[2]])
def rotation_matrix_to_quaternion(u):
r11, r12, r13 = u[0]
r21, r22, r23 = u[1]
r31, r32, r33 = u[2]
q0 = (1.0 + r11 + r22 + r33) / 4.0
q1 = (1.0 + r11 - r22 - r33) / 4.0
q2 = (1.0 - r11 + r22 - r33) / 4.0
q3 = (1.0 - r11 - r22 + r33) / 4.0
q = np.array([q0, q1, q2, q3])
q = np.sqrt(np.maximum(0, q))
i = np.argmax(q)
if i == 0:
q[1] *= np.sign(r32 - r23)
q[2] *= np.sign(r13 - r31)
q[3] *= np.sign(r21 - r12)
elif i == 1:
q[0] *= np.sign(r32 - r23)
q[2] *= np.sign(r21 + r12)
q[3] *= np.sign(r13 + r31)
elif i == 2:
q[0] *= np.sign(r13 - r31)
q[1] *= np.sign(r21 + r12)
q[3] *= np.sign(r32 + r23)
elif i == 3:
q[0] *= np.sign(r21 - r12)
q[1] *= np.sign(r31 + r13)
q[2] *= np.sign(r32 + r23)
return q / np.linalg.norm(q)
#euler angle conventions are taken from EMSoft
def quaternion_to_euler(q):
qq = q**2
q03 = qq[0] + qq[3]
q12 = qq[1] + qq[2]
chi = np.sqrt(q03 * q12)
if chi == 0:
if q12 == 0:
phi = 0
phi2 = 0
phi1 = np.arctan2(-2 * q[0] * q[3], qq[0] - qq[3])
else:
phi = np.pi
phi2 = 0
phi1 = np.arctan2(2 * q[1] * q[2], qq[1] - qq[2])
else:
phi = np.arctan2(2 * chi, q03 - q12)
phi1 = np.arctan2((-q[0] * q[2] + q[1] * q[3]) / chi, (-q[0] * q[1] - q[2] * q[3]) / chi)
phi2 = np.arctan2((+q[0] * q[2] + q[1] * q[3]) / chi, (-q[0] * q[1] + q[2] * q[3]) / chi)
res = np.array([phi1, phi, phi2]) % (2 * np.pi)
res[1] %= np.pi
return res
#euler angle conventions are taken from EMSoft
def euler_to_quaternion(e):
ee = np.array(e) / 2
cphi = np.cos(ee[1])
sphi = np.sin(ee[1])
cm = np.cos(ee[0] - ee[2])
sm = np.sin(ee[0] - ee[2])
cp = np.cos(ee[0] + ee[2])
sp = np.sin(ee[0] + ee[2])
res = np.array([cphi * cp, -sphi * cm, -sphi * sm, -cphi * sp])
return flip_up(res)