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There is a lot of work done on projective spaces, over real, complex numbers or over an abstract field. But I do not find a reference for similar theory where the vectors are projected to the same "infinite" point only if one is a non-negative multiple of the other. Can you advise some?

In an abstract point of view, it is definitely more delicate since the ordering (possibly linear) has to be considered. But at least for problems over real numbers it is not big obstacle.

The theory which I am asking about would be connected with theory of convex sets and mainly "convex cones". My question is motivated by some situations in Information theory, probability or measure theory (ergodic theory) which I encountered, where one-directional rays have better meaning than "both-directional".

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    $\begingroup$ spherical geometry might be what you are looking for. $\endgroup$
    – Dima Pasechnik
    Commented May 9, 2022 at 10:00
  • $\begingroup$ Yes, spherical geometry seems to be close to what I am looking for. For normed real space, points can be projected on the set of unit vectors. But still, I wonder if there is a theory, and terminology that recognize this projection and have it as the core. Anyway, I will follow your advise and look on the spherical geometry. Thanks. $\endgroup$
    – Michal Kupsa
    Commented May 9, 2022 at 10:14

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