The projective plane is a common way to extend the Euclidean plane to have points (in fact, an entire line) "at infinity".
In the projective plane, one generally doesn't consider distances between pairs of points. It's more useful to select numerical invariants that better relate to the structure of the projective plane.
For example, given any four points on a projective line, one can obtain their cross ratio. One can see this is related to distance because if O,I,P are three points on a line and ∞ is that line's point at infinity1, then the cross ratio of the quadruple (∞, O, I, P) is precisely the ratio of the (signed) lengths OP / OI. (think of ∞ as fixing an affine perspective, O as an "origin" and I as marking off a "unit distance", and then the cross ratio is computing the length of OP)
1: Note that the projective plane has no intrinsic notion of "at infinity"; every point is just like every other. It's only when we pick a Euclidean plane contained within the projective plane that we get a notion of "at infinity", which is the points of the projective plane that aren't contained in the Euclidean one