2D Transformations

Why Transformations?

Transformations allow us to:

• Place predefined objects at arbitrary locations, orientations, and sizes
• Build complex scenes from simple primitives
• Animate objects over time

Basic Transformations

Scale (about origin, factor $m$)

$x' = mx, \quad y' = my$

Rotate (about origin, angle $\theta$)

$x' = x\cos\theta - y\sin\theta$ $y' = x\sin\theta + y\cos\theta$

Translate (by vector $(x_o, y_o)$)

$x' = x + x_o, \quad y' = y + y_o$

Shear (parallel to x-axis, factor $a$)

$x' = x + ay, \quad y' = y$

Matrix Representation

2×2 matrices work for scale, rotate, and shear, but not translation.

Scale: $\begin{bmatrix} x' \\ y' \end{bmatrix} = \begin{bmatrix} m & 0 \\ 0 & m \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix}$

Rotate: $\begin{bmatrix} x' \\ y' \end{bmatrix} = \begin{bmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix}$

Identity: $\begin{bmatrix} x' \\ y' \end{bmatrix} = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix}$

The Translation Problem

Translation cannot be represented as 2×2 matrix multiplication: $\begin{bmatrix} x' \\ y' \end{bmatrix} = \begin{bmatrix} x \\ y \end{bmatrix} + \begin{bmatrix} x_o \\ y_o \end{bmatrix}$

This breaks uniformity—other transforms are multiplicative.

Summary

• Scale, rotate, and shear can be expressed as matrix multiplication
• Translation requires an additive term (not multiplicative)
• This motivates homogeneous coordinates