Notes on dual elliptic quaternions and their new polar forms

Abstract

An elliptic quaternion is formed by Q = q0e0 + q1e1 + q2e2 + q3e3 where q0,q1,q2,q3 is an element of R and e0, e1, e2, e3 are elliptic quaternion units which have the relations e1e2=Delta gamma e3=-e2e1 , e2e3=Delta alpha e1=-e3e2 , e3e1=Delta beta e2=-e1e3 for alpha,beta,gamma is an element of R+ . The purpose of the paper is to extend the concept of elliptic quaternions by substituting real coefficients in Q with dual coefficients, thereby constructing what we refer to as dual elliptic quaternions. Hence, any dual elliptic quaternion can be represented in the form Q = q0e0 + q1e1 + q2e2 + q3e3, or equivalently, Q = Q + Q*epsilon, where q0, q1, q2, q3 are dual numbers, and Q, Q*are elliptic quaternions. Subsequently, we explore the fundamental algebraic properties of the dual elliptic quaternion numbers. Additionally, we provide Euler, de Moivre formulas and roots for dual elliptic quaternions. Finally, we investigate new polar representations of an (dual) elliptic quaternions as product of a (dual) generalized complex number and a truncated (dual) elliptic quaternion. To verify our findings, we include several related numerical examples. Furthermore, we demonstrate that the elliptic parameters alpha, beta, and gamma naturally induce a leaf parametrization (foliation) of the kinematic space. We show that the coupling of generalized screw motions on different manifolds can be algebraically synthesized using a framework analogous to Clebsch-Gordan coefficients. Finally, the practical utility of the proposed algebra is illustrated through a high-fidelity kinematic model of Tokamak magnetic flux surfaces, representing a novel application of dual elliptic quaternions to plasma physics.