Dual Plane and Kinematics

Abstract

Muller, H. R. [2], on the Euclidean plane E-2, introduced the one-parameter planar motion and obtained the relation between absolute, relative, sliding velocities (and accelerations). Also, Muller, H. R. [6] provided the relation between the velocities (in the sense of Complex) under the one-parameter motion on the Complex plane C = {x + iy vertical bar x, y is an element of R, i(2) = -1}. Ergin, A. A. [4] considered the Lorentzian plane L-2, instead of the Euclidean plane E-2, and introduced the one-parameter planar motion on the Lorentzian plane L-2 and also gave the relations between both the velocities and the accelerations. Yuce, S. [12] introduced the relation between the velocities (in the sense of Hyperbolic) under the one-parameter motions on the Hyperbolic plane H = {x + jy vertical bar x, y is an element of R, j(2) = 1}. Yuce, S. [1] considered the Galilean plane G(2), instead of the Euclidean plane E-2 and Lorentzian plane L-2, and introduced the one-parameter planar motion on the Galilean plane G(2) and also gave the relations between both the velocities and accelerations. In analogy with the Complex numbers and Hyperbolic numbers, a system of Dual numbers can be introduced: D = {x + epsilon y vertical bar x, y is an element of R, epsilon(2) = 0}. Complex numbers are related to the Euclidean geometry, the Dual and Hyperbolic systems of numbers are related to the Galilean geometry and Lorentz (or Minkowski) geometry, respectively, [9,10]. In this paper, in analogy with Complex motions as given by Muller, H.R. [6] and Hyperbolic motions as given by Yuce, S. [12], one-parameter motion on the Dual plane are defined. Also the relations between absolute, relative, sliding velocities (and accelerations) and pole lines are discussed.