User:Peh-Jota

$$ T = \begin{bmatrix} 1 & 0 & 0 & t_{x} \\ 0 & 1 & 0 & t_{y} \\ 0 & 0 & 1 & t_{z} \\ 0 & 0 & 0 & 1 \end{bmatrix} $$

$$ S = \begin{bmatrix} s_{x} & 0 & 0 & 0 \\ 0 & s_{y} & 0 & 0 \\ 0 & 0 & s_{z} & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix} $$

$$ R_{x} = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & \cos \theta_x & -\sin \theta_x & 0 \\ 0 & \sin \theta_x & \cos \theta_x & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix} $$

$$R_{y} = \begin{bmatrix} \cos \theta_y & 0 & \sin \theta_y & 0 \\ 0 & 1 & 0 & 0 \\ -\sin \theta_y & 0 & \cos \theta_y & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix} $$

$$ R_{z} = \begin{bmatrix} \cos \theta_z & -\sin \theta_z & 0 & 0 \\ \sin \theta_z & \cos \theta_z & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix} $$

$$ R = R_y R_z R_x $$

$$ R = R_y R_z R_x = \begin{bmatrix} \cos \theta_y \cos \theta_z & \sin \theta_x \sin \theta_y - \cos \theta_x \cos \theta_y \sin \theta_z & \cos \theta_x \sin \theta_y + \sin \theta_x \cos \theta_y \sin \theta_z & 0 \\ \sin \theta_z & \cos \theta_x \cos \theta_z & -\sin \theta_x \cos \theta_z & 0 \\ -\sin \theta_y \cos \theta_z & \cos \theta_x \sin \theta_y \sin \theta_z + \sin \theta_x \cos \theta_y & \cos \theta_x \cos \theta_y - \sin \theta_x \sin \theta_y \sin \theta_z & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix} $$

$$ R = \begin{bmatrix} \cos \theta_y \cos \theta_z & \sin \theta_x \sin \theta_y - \cos \theta_x \cos \theta_y \sin \theta_z & \cos \theta_x \sin \theta_y + \sin \theta_x \cos \theta_y \sin \theta_z & 0 \\ \sin \theta_z & \cos \theta_x \cos \theta_z & -\sin \theta_x \cos \theta_z & 0 \\ -\sin \theta_y \cos \theta_z & \cos \theta_x \sin \theta_y \sin \theta_z + \sin \theta_x \cos \theta_y & \cos \theta_x \cos \theta_y - \sin \theta_x \sin \theta_y \sin \theta_z & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix} $$