3D Transformation Matrices

Identity

$\begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix}$

Scale (uniform, factor $m$)

$\begin{bmatrix} m & 0 & 0 & 0 \\ 0 & m & 0 & 0 \\ 0 & 0 & m & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix}$

For non-uniform scaling, use different values on the diagonal.

Translate (by $(t_x, t_y, t_z)$)

$\begin{bmatrix} 1 & 0 & 0 & t_x \\ 0 & 1 & 0 & t_y \\ 0 & 0 & 1 & t_z \\ 0 & 0 & 0 & 1 \end{bmatrix}$

Rotation Matrices

Rotate about x-axis

$\mathbf{R}_x(\theta) = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & \cos\theta & -\sin\theta & 0 \\ 0 & \sin\theta & \cos\theta & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix}$

Rotate about y-axis

$\mathbf{R}_y(\theta) = \begin{bmatrix} \cos\theta & 0 & \sin\theta & 0 \\ 0 & 1 & 0 & 0 \\ -\sin\theta & 0 & \cos\theta & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix}$

Rotate about z-axis

$\mathbf{R}_z(\theta) = \begin{bmatrix} \cos\theta & -\sin\theta & 0 & 0 \\ \sin\theta & \cos\theta & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix}$

Mnemonic

The axis being rotated about has 1 on the diagonal and 0 in that row/column. The other entries form a 2D rotation matrix.

Non-Commutativity

3D rotations are non-commutative:

• $\mathbf{R}_x(90°) \cdot \mathbf{R}_z(90°) \neq \mathbf{R}_z(90°) \cdot \mathbf{R}_x(90°)$

Summary

• Translation uses the fourth column
• Rotation matrices are orthogonal (determinant = 1)
• 3D rotations are non-commutative