Higher-Order Accelerations and Poles under the One-Parameter Planar Hyperbolic Motions and Their Inverse Motions

Serdal Şahin,Salim Yüce

doi:10.1155/2014/686509

Abstract

We expressed the higher-order velocities, accelerations, and poles under the one-parameter planar hyperbolic motions and their inverse motions. The higher-order accelerations and poles are also presented by considering the rotation angleφas a parameter of the motion and its inverse motion.

Highlights

First of all, we need to define the set of hyperbolic numbers
We need to define the set of hyperbolic numbers. For this reason let us recall complex numbers, which are the extension of real numbers that include the imaginary unit i and can be written in terms of the standard basis {1, i} as z = x + iy, where x, y ∈ R
X is the real part and y is the imaginary part of complex numbers

Summary

Introduction

We need to define the set of hyperbolic numbers For this reason let us recall complex numbers, which are the extension of real numbers that include the imaginary unit i and can be written in terms of the standard basis {1, i} as z = x + iy, where x, y ∈ R. This extension is performed by adjoining the unipotent (hyperbolic imaginary) j, where j2 = 1 but j ≠ ±1 In this case, the hyperbolic numbers set can be written as follows:. Let z = x + jy and w = u + jV ∈ H; the addition and multiplication of hyperbolic numbers can defined as follows:. Since the sets of hyperbolic numbers and complex numbers are two-dimensional vector spaces over the real numbers field, we can identify z = x + jy with a point or vector (x, y). In analogy to one-parameter planar complex motion which was introduced by Blaschke and Muller [5], Yuce and Kuruoglu [6] introduced the one-parameter planar hyperbolic motion and they obtained the velocities, accelerations, and poles

The Planar Hyperbolic Motion

Rotation Angle as a Parameter