Transformation Composition

Concatenation

Multiple transformations combine by multiplying their matrices:

$\mathbf{M}_{\text{combined}} = \mathbf{M}_n \cdot \mathbf{M}_{n-1} \cdots \mathbf{M}_2 \cdot \mathbf{M}_1$

Transformations are applied right-to-left.

Standard Order: SRT

Scale → Rotate → Translate

Written as: $\mathbf{M} = \mathbf{T} \cdot \mathbf{R} \cdot \mathbf{S}$

The rightmost matrix ($\mathbf{S}$) is applied first.

Example: Shear then Scale

$\begin{bmatrix} x'' \\ y'' \\ w'' \end{bmatrix} = \begin{bmatrix} m & 0 & 0 \\ 0 & m & 0 \\ 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} 1 & a & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} x \\ y \\ w \end{bmatrix} = \begin{bmatrix} m & ma & 0 \\ 0 & m & 0 \\ 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} x \\ y \\ w \end{bmatrix}$

Non-Commutativity

Transformations generally do not commute.

Example: Scale by 2 along x, then rotate 45° $\neq$ Rotate 45°, then scale by 2 along x.

Transformations About Arbitrary Points

To transform about point $(x_o, y_o)$:

1. Translate $(x_o, y_o)$ to origin
2. Apply transformation
3. Translate back to $(x_o, y_o)$

Scale about $(x_o, y_o)$: $\mathbf{M} = \mathbf{T}(x_o, y_o) \cdot \mathbf{S}(m) \cdot \mathbf{T}(-x_o, -y_o)$

Efficiency

Concatenate matrices once, then apply to all vertices:

• One matrix multiplication per concatenated transform
• More efficient than applying each matrix sequentially

Summary

• Matrices multiply right-to-left
• Standard order: Scale, Rotate, Translate (SRT)
• Transformations are non-commutative
• Arbitrary pivots require translate-transform-translate