Homogeneous Coordinates

Homogeneous Coordinates

Given a point on the Euclidean plane, for any non-zero real number , the triple 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, ) is such a system of homogeneous coordinates for the point . For example, the Cartesian point can be represented in homogeneous coordinates as or .

Remark

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