Quaternion polar representation with a complex modulus and complex argument inspired by the Cayley-Dickson form

TL;DR

This paper introduces a novel quaternion polar representation inspired by Cayley-Dickson, representing quaternions with a complex modulus and argument, facilitating amplitude and phase calculation from Cartesian form.

Contribution

It proposes a new quaternion polar form using complex modulus and argument inspired by Cayley-Dickson, enabling easier amplitude and phase computation.

Findings

New quaternion polar representation using complex numbers.

Method to compute amplitude and phase from Cartesian quaternion.

Enhanced understanding of quaternion structure in complex form.

Abstract

We present a new polar representation of quaternions inspired by the Cayley-Dickson representation. In this new polar representation, a quaternion is represented by a pair of complex numbers as in the Cayley-Dickson form, but here these two complex numbers are a complex 'modulus' and a complex 'argument'. As in the Cayley-Dickson form, the two complex numbers are in the same complex plane (using the same complex root of -1), but the complex phase is multiplied by a different complex root of -1 in the exponential function. We show how to calculate the amplitude and phase from an arbitrary quaternion in Cartesian form.