homogeneous coordinates

Homogeneous coordinates are a mathematical system used to represent points in space, particularly in computer graphics and geometry. They extend standard coordinate systems by adding an extra coordinate, allowing for a unified way to handle both points and transformations like translations and perspective projections.1

In a standard 2D Cartesian system, a point is represented by (x, y). In homogeneous coordinates, this same point is represented as (x, y, 1). For 3D space, a point (x, y, z) becomes (x, y, z, 1). This extra '1' acts as a scaling factor. Multiplying a homogeneous coordinate by any non-zero scalar results in the same point, meaning (x, y, 1) is equivalent to (kx, ky, k) for any k ≠ 0.1

This system is particularly useful because it allows geometric transformations, such as translation, rotation, and scaling, to be represented as matrix multiplications. This simplifies calculations, especially in computer graphics, where objects are frequently manipulated and projected onto a 2D screen. Homogeneous coordinates also elegantly handle points at infinity, which are crucial for representing perspective in 3D graphics.1