A Complex Quaternion Model for Hyperbolic 3-Space

TL;DR

This paper develops a novel complex quaternion model for hyperbolic 3-space, extending Macfarlane's hyperbolic quaternions, providing new computational tools for studying hyperbolic isometries and generalizing to other quaternion algebras.

Contribution

It introduces a complex quaternion framework for hyperbolic 3-space, enabling new computational methods and extending the model to various quaternion algebras.

Findings

New quaternionic representation of hyperbolic 3-space points and isometries

Enhanced computational tools for hyperbolic geometry analysis

Generalization of the model to other quaternion algebras

Abstract

In 1900, Macfarlane proposed a hyperbolic variation on Hamilton's quaternions that closely resembles Minkowski spacetime. Viewing this in a modern context, we expand upon Macfarlane's idea and develop a model for real hyperbolic 3-space in which both points and isometries are expressed as complex quaternions, analogous to Hamilton's famous theorem on Euclidean rotations. We use this to give new computational tools for studying isometries of hyperbolic 2- and 3-space. We also give a generalization to other quaternion algebras.