One-parameter plane hyperbolic motions

Abstract

Müller [3], in the Euclidean plane 2 , introduced the one parameter planar motions and obtained the relation between absolute, relative, sliding velocities (and accelerations). Also, Müller [11] provided the relation between the velocities (in the sense of Complex) under the one parameter motions in the Complex plane ℂ := {x + iy | x, y ∈ ℝ, i 2 = -1}. Ergin [7] considering the Lorentzian plane 2 , instead of the Euclidean plane 2 , and introduced the one-parameter planar motion in the Lorentzian plane and also gave the relations between both the velocities and accelerations. In analogy with the Complex numbers, a system of hyperbolic numbers can be introduced: ℍ := {x + jy | x, y ∈ ℝ, j 2 = 1}. Complex numbers are related to the Euclidean geometry, the hyperbolic system of numbers are related to the pseudo-Euclidean plane geometry (space-time geometry), [5,15]. In this paper, in analogy with Complex motions as given by Müller [11], one parameter motions in the hyperbolic plane are defined. Also the relations between absolute, relative, sliding velocities (and accelerations) and pole curves are discussed. © 2008 Birkhauser Verlag Basel/Switzerland. © 2019 Elsevier B.V., All rights reserved.