Overview & Affine Transformation

3D Rotation

• Cos(theta) in the matrix scaling spots
• Sin(theta) and -Sin(theta) in the matrix sheering spots

2D Rotation $$ \begin{bmatrix} Cos(ϴ) & -Sin(ϴ) \\ Sin(ϴ) & Cos(ϴ) \end{bmatrix} $$

3D Rotation $$ \begin{bmatrix} Cos(ϴ) & -Sin(ϴ) & 0 \\ Sin(ϴ) & Cos(ϴ) & 0 \\\ 0 & 0 & 1 \end{bmatrix} $$

Homogeneous Coordinates

Have an extra collumn in the transformation matrix to represent translation

2D Translation

• Move to a 3D dimension and make the third collumn in the matrix a translation vector

$$ \begin{bmatrix} x+1 \\ y \\ 0 \end{bmatrix}= \begin{bmatrix} x+1\\ y \end{bmatrix}= \begin{bmatrix} 1 & 0 & 1 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \end{bmatrix} * \begin{bmatrix} x \\ y \\ 1 \end{bmatrix} $$