In Fig. 3-3a, segments OA and OK represent equal spacetime intervals. Time dilation is represented by the ratio OB / OK. The invariant hyperbola has the equation w = √ x + k where k = OK, and the red line representing the world line of a particle in motion has the equation w = x / β = xc / v. A bit of algebraic manipulation yields O B = O K / 1 − − v 2 / c 2 .