Quaternion polar representation with a complex modulus and complex argument inspired by the Cayley-Dickson 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.
Cite
@article{arxiv.0802.0852,
title = {Quaternion polar representation with a complex modulus and complex argument inspired by the Cayley-Dickson form},
author = {Stephen J. Sangwine and Nicolas Le Bihan},
journal= {arXiv preprint arXiv:0802.0852},
year = {2010}
}
Comments
Version 2 has some additional text in Theorem 1 to cover degenerate cases such as q=k, where alpha=0. There is also an extra numerical example in section 3 to illustrate this