Homogeneous Coordinates - Alternative Definition

Alternative Definition

Another definition of projective space can be given in terms of equivalence classes. For non-zero element of R3, define (x₁, y₁, z₁) ~ (x₂, y₂, z₂) to mean there is a non-zero λ so that (x₁, y₁, z₁) = (λx₂, λy₂, λz₂). Then ~ is an equivalence relation and the projective plane can be defined as the equivalence classes of R3 − {0}. If (x, y, z) is one of elements of the equivalence class p then these are taken to be homogeneous coordinates of p.

Lines in this space are defined to be sets of solutions of equations of the form ax + by + cz = 0 where not all of a, b and c are zero. The condition ax + by + cz = 0 depends only on the equivalence class of (x, y, z) so the equation defines a set of points in the projective line. The mapping (x, y) → (x, y, 1) defines an inclusion from the Euclidean plane to the projective plane and the complement of the image is the set of points with z=0. This is the equation of a line according to the definition and the complement is called the line at infinity.

The equivalence classes, p, are the lines through the origin with the origin removed. The origin does not really play an essential part in the previous discussion so it can be added back in without changing the properties of the projective plane. This produces a variation on the definition, namely the projective plane is defined as the set of lines in R3 that pass through the origin and the coordinates of a non-zero element (x, y, z) of a line are taken to be homogeneous coordinates of the line. These lines are now interpreted as points in the projective plane.

Again, this discussion applies analogously to other dimensions. So the projective space of dimension n can be defined as the set of lines through the origin in Rn+1.