One-parameter plane hyperbolic motions

Abstract

Muller [3], in the Euclidean plane E-2, introduced the one parameter planar motions and obtained the relation between absolute, relative, sliding velocities (and accelerations). Also, Muller [11] provided the relation between the velocities (in the sense of Complex) under the one parameter motions in the Complex plane C := {x + iy vertical bar x, y. R, i(2) = -1}. Ergin [7] considering the Lorentzian plane L-2, instead of the Euclidean plane E2, 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 vertical bar 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 Muller [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.