One-Parameter Plane Hyperbolic Motions

Abstract.

Müller [3], in the Euclidean plane ${{\mathbb{E}}}^2$, 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 ${\mathbb{C}} := \{x + iy | x, y \in {\mathbb{R}}, i^2 = -1\}$.

Ergin [7] considering the Lorentzian plane ${{\mathbb{L}}}^2$, instead of the Euclidean plane ${{\mathbb{E}}}^2$, 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: ${\mathbb{H}} := \{x + jy | x, y \in {\mathbb{R}}, j^2 = 1\}$. 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.