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I have done a R package which implements Ungar's approach to hyperbolic geometry, for the hyperboloid model. In this theory, there is a parameter $s>0$ which controls the curvature of the hyperbolic segments:

Ungar's theory is not very well-known. An intro and a reference can be found on my blog.

I'm wondering whether this parameter $s$ has a devoted name? Would you know? I know Ungar's theory but I'm ignorant in classical hyperbolic geometry in the hyperboloid model, maybe this parameter also appears in the classical theory?

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  • $\begingroup$ Perhaps the paper arXiv 1908.10389 is of some interest. E.g. a formula for the hyperbolic midpoint of two points in the unit disk is given there. $\endgroup$
    – user478769
    Commented Mar 14, 2022 at 17:56

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In the context of relativistic mechanics, the parameter $s$ is the speed of light, see for example The Intrinsic Beauty, Harmony and Interdisciplinarity in Einstein Velocity Addition Law: Gyrogroups and Gyrovector Spaces

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  • $\begingroup$ Thanks. But "speed of light" is not for the case of the Einstein gyrovector space only? $\endgroup$
    – Stéphane Laurent
    Commented May 11, 2022 at 19:13

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