Translation \(T\), rotation \(R\), scaling \(S\) can be composed as matrix products representation

ex. \(\mathrm{M}=T_0\,R_0\,S_0\,T_1\,R_1\,S_1\,\dots\;\;\;\;\) \(\small \mathrm{M}= \left( \begin{array}{cc|c} \color{red}{m_{00}} & \color{red}{m_{01}} & \color{blue}{t_x} \\ \color{red}{m_{10}} & \color{red}{m_{11}} & \color{blue}{t_y} \\ \hline 0 & 0 & 1 \end{array} \right) \).

\(\color{red}{m_{ij}}\):linear part (rotation and scaling); \(\color{blue}{t_{x/y}}\):translation part

\(p=(x,y,z,1)\) - represents 3D position

\( \small \mathrm{M}= \left( \begin{array}{ccc|c} \color{red}{m_{00}} & \color{red}{m_{01}} & \color{red}{m_{02}} & \color{blue}{t_x} \\ \color{red}{m_{10}} & \color{red}{m_{11}} & \color{red}{m_{12}} & \color{blue}{t_y} \\ \color{red}{m_{20}} & \color{red}{m_{21}} & \color{red}{m_{22}} & \color{blue}{t_z} \\ \hline 0 & 0 & 0 & 1 \end{array} \right) \) - represents 3D affine transformation (rotation, scaling, translation)