Homogeneous Coordinates - Homogeneity

Homogeneity

Homogeneous coordinates are not uniquely determined by a point, so a function defined on the coordinates, say f(x, y, z), does not determine a function defined on points as with Cartesian coordinates. But a condition f(x, y, z) = 0 defined on the coordinates, as might be used to describe a curve, determines a condition on points if the function is homogeneous. Specifically, suppose there is a k such that

If a set of coordinates represent the same point as (x, y, z) then it can be written (λx, λy, λz) for some non-zero value of λ. Then

A polynomial g(x, y) of degree k can be turned into a homogeneous polynomial by replacing x with x/z, y with y/z and multiplying by zk, in other words by defining

The resulting function f is a polynomial so it makes sense to extend its domain to triples where z = 0. The process can be reversed by setting z = 1, or

The equation f(x, y, z) = 0 can then be thought of as the homogeneous form of g(x, y) = 0 and it defines the same curve when restricted to the Euclidean plane. For example, the homogeneous form of the equation of the line ax + by + c = 0 is ax + by + cz = 0.