I would suggest John Barrett's essay on The Hyperbolic Theory of Special Relativity as a comprehensive answer.
The principle of relativity corresponds to the hypothesis that the kinematic space is a space of constant negative curvature. The value of the radius of curvature is the speed of light. The relativistic law of combination of velocities can be interpreted as the triangle of velocities in hyperbolic space. New formulations are given in optics relating hyperbolic velocity with logarithmic redshift and in dynamics including a reformulation of Newton's 2nd law in terms of hyperbolic acceleration.