Orbits of Quaternionic M\"obius Transformations

TL;DR

This paper explores quaternionic Möbius transformations, revealing their structure as rotations of the Riemann sphere, and provides formulas for their axes and angles, connecting them to Lie theory.

Contribution

It introduces a detailed analysis of quaternionic Möbius transformations as rotation groups, offering explicit formulas for axes and angles, and linking them to Lie theory.

Findings

Quaternionic Möbius transformations form a subgroup isomorphic to rotations.

Formulas for axes and rotation angles of transformations are derived.

Transformations can be viewed as continuous orbits of rotations sharing a common axis.

Abstract

M\"obius transformations of the extended complex plane are at the crossroads of many interesting topics, e.g., they form a group under composition, are the simplest form of rational function, and are a path to Lie theory. Quaternionic transformations are a subgroup of M\"obius transformations isomorphic to rotations of the Riemann sphere, which also represent quaternion conjugation. These representations yield formulas for the axis and radians of rotation, and thereby portray each particular quaternionic transformation as part of a continuous orbit of rotations sharing a common axis.