Transformation

Transformation

2D

The transformation matrix is Rotate THEN Translate (and scale)

$$\begin{array}{c}\left[\begin{array}{ccc}\cos \theta & -\sin \theta & t_x\\ \sin \theta & \cos \theta & t_y\\ 0& 0& 1\end{array}\right]=\left[\begin{array}{ccc}1& 0& t_x\\ 0& 1& t_y\\ 0& 0& 1\end{array}\right]\left[\begin{array}{ccc}\cos \theta & -\sin \theta & 0\\ \sin \theta & \cos \theta & 0\\ 0& 0& 1\end{array}\right]\end{array}$$

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

$$\begin{array}{c}\mathbf{T}\mathbf{R}=\left[\begin{array}{cc}1& \mathbf{t}\\ 0& 1\end{array}\right]\left[\begin{array}{cc}\mathbf{r}& 0\\ 0& 1\end{array}\right]=\left[\begin{array}{cc}\mathbf{r}& \mathbf{t}\\ 0& 1\end{array}\right]\ne \left[\begin{array}{cc}\mathbf{r}& \mathbf{rt}\\ 0& 1\end{array}\right]=\left[\begin{array}{cc}\mathbf{r}& 0\\ 0& 1\end{array}\right]\left[\begin{array}{cc}1& \mathbf{t}\\ 0& 1\end{array}\right]=\mathbf{R}\mathbf{T}\end{array}$$

3D

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

$$\begin{array}{c}{\mathbf{R}}_x(\alpha )=\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& \cos \alpha & -\sin \alpha & 0\\ 0& \sin \alpha & \cos \alpha & 0\\ 0& 0& 0& 1\end{array}\right]\\ {\mathbf{R}}_y(\alpha )=\left[\begin{array}{cccc}\cos \alpha & 0& -\sin \alpha & 0\\ 0& 1& 0& 0\\ \sin \alpha & 0& \cos \alpha & 0\\ 0& 0& 0& 1\end{array}\right]\\ {\mathbf{R}}_z(\alpha )=\left[\begin{array}{cccc}\cos \alpha & -\sin \alpha & 0& 0\\ \sin \alpha & \cos \alpha & 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]\end{array}$$

With the final form:

$$\begin{array}{c}{\mathbf{R}}_{xyz}(\alpha )={\mathbf{R}}_x(\alpha ){\mathbf{R}}_y(\alpha ){\mathbf{R}}_z(\alpha )\end{array}$$

Rodrigues' Rotation Formula for rotation along any axis $\mathbf{n}$:

$$\begin{array}{c}\mathbf{R}(\mathbf{n},\alpha )=\cos \alpha \mathbf{I}+(1-\cos \alpha )\mathbf{n}{\mathbf{n}}^T+\sin \alpha \left[\begin{array}{ccc}0& -n_x& n_y\\ n_z& 0& -n_x\\ -n_y& n_x& 0\end{array}\right]\end{array}$$

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)