Transformation

2D

The transformation matrix is Rotate THEN Translate (and scale)

\[ \displaylines{ \begin{bmatrix} \cos\theta & -\sin\theta & t_x \\ \sin\theta & \cos\theta & t_y \\ 0&0&1 \\ \end{bmatrix} = \begin{bmatrix} 1 & 0 & t_x \\ 0 & 1 & t_y \\ 0&0&1 \\ \end{bmatrix} \begin{bmatrix} \cos\theta & -\sin\theta & 0 \\ \sin\theta & \cos\theta & 0 \\ 0&0&1 \\ \end{bmatrix} } \]

Note: We always use \(\mathbf T \mathbf R\) because in this form the translation is applied later and is explicit.

\[ \displaylines{ \mathbf T \mathbf R = \begin{bmatrix}1 & \mathbf t \\ 0 & 1\end{bmatrix} \begin{bmatrix}\mathbf r & 0 \\ 0 & 1\end{bmatrix} = \begin{bmatrix}\mathbf r & \mathbf t \\ 0 & 1\end{bmatrix} \ne \begin{bmatrix}\mathbf r & \mathbf {rt} \\ 0 & 1\end{bmatrix} = \begin{bmatrix}\mathbf r & 0 \\ 0 & 1\end{bmatrix}\begin{bmatrix}1 & \mathbf t \\ 0 & 1\end{bmatrix} = \mathbf R \mathbf T } \]

3D

Main difference from 2D is the three rotation matrices along three axes:

\[ \displaylines{ \mathbf R_x(\alpha) = \begin{bmatrix} 1&0&0&0\\ 0&\cos\alpha & -\sin\alpha &0 \\ 0&\sin\alpha & \cos\alpha &0 \\ 0&0&0 &1 \end{bmatrix} \\ \mathbf R_y(\alpha) = \begin{bmatrix} \cos\alpha &0 & -\sin\alpha &0 \\ 0&1&0&0\\ \sin\alpha &0 & \cos\alpha &0 \\ 0&0&0 &1 \end{bmatrix} \\ \mathbf R_z(\alpha) = \begin{bmatrix} \cos\alpha & -\sin\alpha &0&0 \\ \sin\alpha & \cos\alpha &0&0 \\ 0&0&1&0 \\ 0&0&0&1\\ \end{bmatrix} \\ } \]

With the final form:

\[ \displaylines{ \mathbf R_{xyz}(\alpha) = \mathbf R_x(\alpha)\mathbf R_y(\alpha)\mathbf R_z(\alpha) } \]

Rodrigues' Rotation Formula for rotation along any axis \(\mathbf n\):

\[ \displaylines{ \mathbf R(\mathbf n, \alpha) = \cos\alpha\mathbf I + (1 - \cos\alpha)\mathbf n\mathbf n^T + \sin\alpha \begin{bmatrix} 0 & -n_x &n_y \\ n_z & 0 & -n_x \\ -n_y & n_x & 0 \end{bmatrix} } \]

Decompose 3D transformation

ref: https://math.stackexchange.com/questions/237369/given-this-transformation-matrix-how-do-i-decompose-it-into-translation-rotati/417813

ref: https://nghiaho.com/?page_id=846

Code in python:

from scipy.spatial.transform import Rotation as sciRot

def decompose(M):
    # M: [4, 4], assuming NO scaling.

    # translation 
    T = np.eye(4)
    T[:3, 3] = M[:3, 3]

    # rotation at different axes
    rx = np.arctan2(M[2, 1], M[2, 2])
    ry = np.arctan2(-M[2, 0], np.sqrt(M[2, 1]**2 + M[2, 2]**2))
    rz = np.arctan2(M[1, 0], M[0, 0])

    R = np.eye(4)
    R[:3, :3] = sciRot.from_euler('xyz', [rx, ry, rz], degrees=False).as_matrix()

    M2 = T @ R
    assert np.allclose(M, M2)