@Exocytosis We really should use this chatroom for extended discussion, not the comments. The mods here often delete comments (or move them to chat) when they develop into an extended discussion. Since we already have this room it'd be a bit silly to create a 2nd room (and mods can't easily transfer comments into this room).

> but that does not mean the euclidean angle is not natural, or does it? Is there a clue as to what is the fundamental nature of velocity?

By definition, velocity is the rate of change of displacement with respect to time. In the context of spacetime, that means velocity is the (relative) slope of a worldline. But that doesn't necessarily imply that velocity is the most natural measure of the quantity of motion.

In Euclidean geometry, we sometimes use slopes directly, other times we use angles. Angles have the useful property that they are additive, slopes are approximately additive when they're small.

In SR, it's more natural to use rapidity than the Euclidean angle because the relationship between time & space is hyperbolic. But check out the links about the Gudermannian function / transformation that robphy just posted in a comment on Rob Jeffries' answer. Coincidentally, I was just about to mention the Gudermannian in response to your latest comment, and then I saw robphy had already mentioned it. :)

There are other important measures of motion, though: principally momentum and kinetic energy. In a very physical sense, momentum is more fundamental than velocity. Of course, both momentum & KE can be expressed in terms of hyperbolic or circular functions. In natural units (where c=1) momentum is mtan(theta) = msinh(phi), where theta is the Euclidean angle and phi is the rapidity, and m is the (rest) mass.

Thank you. I appreciate all the efforts you made to answer my questions. Although this is not much, I am happy to grant you the bounty.