Does it make sense to talk about distance between two distinct points at infinity in Projective Geometry?

Question

In projective Geometry I learned that families of parallel lines meet at a point at infinity and therefore, all the possible directions of parallel lines describe a line at infinity. If I understood this correctly there is a line at infinity with two distinct points on it and my question is: is there any mathematical sense in talking about the distance between these two points?

Answer 1

Assuming you are talking about the projective space over a field with characteristic 0, for example the real or complex numbers.

One can define projective space as the quotient of a sphere $ q: S^n \to \mathbb P^n $, in which opposite points are identified.

A sphere has a metric, a notion of the "distance" (or angle) between two points $ d: S^n \times S^n \to \mathbb R^n $.

You can descend this distance metric to projective space by taking the minimum of the distances between the "parent" points: $$ d^\prime(a, b) = \min_{a^\prime \in q^{-1}(a), b^\prime \in q^{-1}(b)} d(a^\prime, b^\prime). $$

I believe that for two points on a "line at infinity", this gives the angle between their two sets of parallel lines, but I am not sure.

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Note that this does not extend the usual euclidean metric on a plane. If you fix a "line at infinity" in $ \mathbb P^n $, and you want to extend the euclidean metric on the "plane that is not at infinity" in a continuous way, it does not really add any information.

This is easy to see by taking two points $ a $ and $ b $ at infinity, and sliding two different points $ c $ and $ d $ in a plane towards them. The distance between $ c $ and $ d $ goes to infinity, so the distance between $ a $ and $ b $ should be $ \infty $. In the same way, the distance between a point at infinity and a point that is not at infinity becomes $ \infty $.

The problem with this construction is that projective space does not have one "line at infinity", but rather, one can choose one line (or hyperplane, for higher-dimensional projective space) to be at infinity and the metric constructed here heavily depends on that choice.

Answer 2

I am not sure you can call such geometry "projective" (because it is a narrow, specific term), but definitely this kind of compactification is valid.

So, you can have a compactification where you differentiate the points at infinity not only by direction, but also by shift.

Moreover, you can invent even more complicated compactifications, for instance, differentiate limits at infinity of different spirals, dependent on their rate of radius growth, orientation, shift, even oscillation properties...