Homogeneous Coordinates

Homogeneous Coordinates

Given a point $(x,y)$ on the Euclidean plane, for any non-zero real number $Z$, the triple $(xZ,yZ,Z)$ is called a set of homogeneous coordinates for the point. Thus a single point can be represented by infinitely many homogeneous coordinates.

Proposition

Multiplying the three homogeneous coordinates by a common, non-zero factor gives a new set of homogeneous coordinates for the same point. The original Cartesian coordinates are recovered by dividing the first two positions by the third.

e.g. In particular, $(x,y,1$) is such a system of homogeneous coordinates for the point $(x,y)$. For example, the Cartesian point $(1,2)$ can be represented in homogeneous coordinates as $(1,2,1)$ or $(2,4,2)$.

Remark

A point in Cartesian coordinates is a ray in homogeneous coordinates.