Cauchy Formulas for Enveloping Curves in the Lorentzian Plane and Lorentzian Kinematics

Abstract

In the Lorentzian plane, we give Cauchy-length formulas to the envelope of a family of lines. Using these, we prove the length of the enveloping trajectories of non-null lines under the planar Lorentzian motions and give the Holditch-type theorems for the length of the enveloping trajectories. Furthermore, Holditch-type theorem for the orbit areas of three collinear points which is given by Yuce and Kuruoglu [8] is generalized to three non-collinear points.