In Fig. 3-3b, segments OA and OK represent equal spacetime intervals. Length contraction is represented by the ratio OB / OK. The invariant hyperbola has the equation x = √ w + k, where k = OK, and the edges of the blue band representing the world lines of the endpoints of a rod in motion have slope 1/ β = c / v. Event A has coordinates (x, w) = (γk, γβk). Since the tangent line through A and B has the equation w = (x − OB)/ β, we have γβk = (γk − OB)/ β and