A Different Polar Representation for Generalized and Generalized Dual Quaternions

Abstract

In this paper, we present a new polar representation for generalized quaternions and generalized dual quaternions. In this different polar form representation, as in the Cayley–Dickson form, a generalized quaternion and a generalized dual quaternion are represented by a pair of complex numbers and dual complex numbers, respectively. But here, for a generalized quaternion, these two complex numbers coincide with a complex modulus and a complex argument which are calculated from an arbitrary generalized quaternion and for a generalized dual quaternion these two dual complex numbers coincide with a dual complex modulus and a dual complex argument which are calculated from an arbitrary generalized dual quaternion in Cayley–Dickson form. It is known that a unit generalized quaternion and unit generalized dual quaternion correspond to a rotation and a screw operator in 4-dimensional generalized space \(E_{\alpha \beta }^{4}\), respectively. By the help of this kind of polar form of a generalized quaternion and a generalized dual quaternion, a generalized rotation and a screw operator in 4-dimensional generalized space can be written in a product of two generalized rotation operators and screw operators in 2 and 3 dimensional generalized spaces, respectively. One of these operators is in the 2-dimensional generalized space \(E_{\alpha \beta }^{2}\) spanned by the vectors 1 and i and the other is in the 3-dimensional space \(E_{\alpha \beta }^{3}\) spanned by the vectors 1, j and k.