import numpy as np
def rotation_matrix_from_points(m0, m1):
"""Returns a rigid transformation/rotation matrix that minimizes the
RMSD between two set of points.
m0 and m1 should be (3, npoints) numpy arrays with
coordinates as columns::
(x1 x2 x3 ... xN
y1 y2 y3 ... yN
z1 z2 z3 ... zN)
The centeroids should be set to origin prior to
computing the rotation matrix.
The rotation matrix is computed using quaternion
algebra as detailed in::
Melander et al. J. Chem. Theory Comput., 2015, 11,1055
"""
v0 = np.copy(m0)
v1 = np.copy(m1)
# compute the rotation quaternion
R11, R22, R33 = np.sum(v0 * v1, axis=1)
R12, R23, R31 = np.sum(v0 * np.roll(v1, -1, axis=0), axis=1)
R13, R21, R32 = np.sum(v0 * np.roll(v1, -2, axis=0), axis=1)
f = [[R11 + R22 + R33, R23 - R32, R31 - R13, R12 - R21],
[R23 - R32, R11 - R22 - R33, R12 + R21, R13 + R31],
[R31 - R13, R12 + R21, -R11 + R22 - R33, R23 + R32],
[R12 - R21, R13 + R31, R23 + R32, -R11 - R22 + R33]]
F = np.array(f)
w, V = np.linalg.eigh(F)
# eigenvector corresponding to the most
# positive eigenvalue
q = V[:, np.argmax(w)]
# Rotation matrix from the quaternion q
R = quaternion_to_matrix(q)
return R
def quaternion_to_matrix(q):
"""Returns a rotation matrix.
Computed from a unit quaternion Input as (4,) numpy array.
"""
q0, q1, q2, q3 = q
R_q = [[q0**2 + q1**2 - q2**2 - q3**2,
2 * (q1 * q2 - q0 * q3),
2 * (q1 * q3 + q0 * q2)],
[2 * (q1 * q2 + q0 * q3),
q0**2 - q1**2 + q2**2 - q3**2,
2 * (q2 * q3 - q0 * q1)],
[2 * (q1 * q3 - q0 * q2),
2 * (q2 * q3 + q0 * q1),
q0**2 - q1**2 - q2**2 + q3**2]]
return np.array(R_q)
[docs]def minimize_rotation_and_translation(target, atoms):
"""Minimize RMSD between atoms and target.
Rotate and translate atoms to best match target. For more details, see::
Melander et al. J. Chem. Theory Comput., 2015, 11,1055
"""
p = atoms.get_positions()
p0 = target.get_positions()
# centeroids to origin
c = np.mean(p, axis=0)
p -= c
c0 = np.mean(p0, axis=0)
p0 -= c0
# Compute rotation matrix
R = rotation_matrix_from_points(p.T, p0.T)
atoms.set_positions(np.dot(p, R.T) + c0)