Homogeneous Coordinates

The Problem

Translation cannot be represented as a 2×2 matrix multiplication on 2D vectors. We need an additive term.

The Solution

Add a third coordinate $w$:

$\text{2D point } (x, y) \equiv \text{Homogeneous } (x, y, 1)$

More generally: $(x, y, w) \equiv \left(\frac{x}{w}, \frac{y}{w}\right)$

Properties

An infinite number of homogeneous coordinates map to each 2D point:

• $(1, 2, 1) \equiv (2, 4, 2) \equiv (3, 6, 3)$

When $w = 0$: Point at infinity (direction vector).

Transformation Matrices

Scale (factor $m$)

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

Rotate (angle $\theta$)

$\begin{bmatrix} x' \\ y' \\ w' \end{bmatrix} = \begin{bmatrix} \cos\theta & -\sin\theta & 0 \\ \sin\theta & \cos\theta & 0 \\ 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} x \\ y \\ w \end{bmatrix}$

Translate (by $(x_o, y_o)$)

$\begin{bmatrix} x' \\ y' \\ w' \end{bmatrix} = \begin{bmatrix} 1 & 0 & x_o \\ 0 & 1 & y_o \\ 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} x \\ y \\ w \end{bmatrix}$

Shear (factor $a$)

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

3D Homogeneous Coordinates

A 3D point $(x, y, z)$ is represented as $(x, y, z, w)$: $(x, y, z, w) \equiv \left(\frac{x}{w}, \frac{y}{w}, \frac{z}{w}\right)$

Points vs Vectors

Type	$w$	Affected by Translation?
Point	1	Yes
Vector	0	No

Vectors represent directions and should not be translated.

Summary

• Homogeneous coordinates add $w$ to represent translation as matrix multiplication
• Points: $w = 1$; Vectors: $w = 0$
• All affine transforms are now uniform matrix multiplications