Rapidity

In relativity, we define the rapidity $\phi$ in terms of the speed $v$ in some reference frame

Rapidity can be easier to work with than velocity directly. Rapidities add linearly between reference frames, unlike velocities which must be either Lorentz transformed or added specially.

The hyperbolic tangent in the rapidity relates to the structure of Minkowski space. This hyperbolic geometry emerges from the Minkowski metric.