Spherical-vectors and geometric interpretation of unit quaternions

TL;DR

This paper introduces spherical-vectors as a new way to interpret unit quaternions geometrically, establishing their algebraic structure and providing a novel polar form and exponential representation.

Contribution

It defines spherical-vectors, shows they form a non-abelian group isomorphic to unit quaternions, and develops a new polar and exponential form for quaternions with geometric insights.

Findings

Spherical-vectors form a non-abelian additive group.

Unit quaternions can be represented on the sphere in R^3.

A new polar form and exponential representation for quaternions are introduced.

Abstract

In this article, we introduce and study the concept of $\textit{spherical-vectors}$, which can be perceived as a natural extension of the arguments of complex numbers in the context of quaternions. We initially establish foundational properties of these distinct vectors, followed by a demonstration, through the transfer of structure, that spherical-vectors constitute a non-abelian additive group, isomorphic to the group of unit quaternions. This identification facilitates the presentation of a novel polar form of quaternions, highlighting its algebraic properties, as well as the algebraic properties of the exponential writing. Furthermore, it enables the depiction of unit quaternions on the unit sphere of $\mathbb{R}^3$, allowing for a geometric interpretation of their multiplication.