Geometrization of the Relativistic Velocity Addition Formula

If we define the proper-time component, $vτ$⁠, of the coordinate velocity to be $vτ = cdτ/dt$⁠, the expression for the invariant interval between two infinitesimally close events on the world line of a particle leads to an equation $vx2+vy2+vz2+vτ2 = c2$⁠. Allowing only motion in the $x, y$ plane, we see that this equation is that of a sphere of radius $c$ in velocity space. Each possible velocity in the $x, y$ plane is represented by a different point on the surface of a sphere. If an object is observed from a set of inertial frames, all in relative motion with respect to one another in a given fixed direction, the locus of representative points on the sphere forms a great circle. A method is given for relating the relative velocity between two Lorentz frames to the angle between two points on the surface of the sphere for the special case in which all motion lies in the $x$ direction.