Calculate rotation matrix to align two vectors in 3D space?

Question

I have two separate vectors of 3D data points that represent curves and I\'m plotting these as scatter data in a 3D plot with matplotlib.

Both the vectors start at t

Answer 1

Problem is here:

r = I + k + np.square(k) * ((1 -c)/(s**2))

np.square(k) squares each element of the matrix. You want np.matmul(k,k) or k @ k which is the matrix multiplied by itself.

I'd also implement the side cases (especially s=0) mentioned in the comments of that answer or you will end up with errors for quite a few cases.

Answer 2

Based on Daniel F's correction, here is a function that does what you want:

import numpy as np

def rotation_matrix_from_vectors(vec1, vec2):
    """ Find the rotation matrix that aligns vec1 to vec2
    :param vec1: A 3d "source" vector
    :param vec2: A 3d "destination" vector
    :return mat: A transform matrix (3x3) which when applied to vec1, aligns it with vec2.
    """
    a, b = (vec1 / np.linalg.norm(vec1)).reshape(3), (vec2 / np.linalg.norm(vec2)).reshape(3)
    v = np.cross(a, b)
    c = np.dot(a, b)
    s = np.linalg.norm(v)
    kmat = np.array([[0, -v[2], v[1]], [v[2], 0, -v[0]], [-v[1], v[0], 0]])
    rotation_matrix = np.eye(3) + kmat + kmat.dot(kmat) * ((1 - c) / (s ** 2))
    return rotation_matrix

Test:

vec1 = [2, 3, 2.5]
vec2 = [-3, 1, -3.4]

mat = rotation_matrix_from_vectors(vec1, vec2)
vec1_rot = mat.dot(vec1)
assert np.allclose(vec1_rot/np.linalg.norm(vec1_rot), vec2/np.linalg.norm(vec2))

Answer 3

I think if you do not have rotation axis, the rotation matrix is not unique.