Clifford algebra and the projective model of Elliptic spaces

TL;DR

This paper uses Clifford algebra to model elliptic spaces in 1, 2, and 3 dimensions, simplifying geometric transformations and avoiding complex trigonometry, with a focus on geometric structures and computational methods.

Contribution

It applies Clifford algebra to elliptic geometry, providing explicit constructions and uniform representations of transformations in spaces up to three dimensions.

Findings

Clifford algebra simplifies geometric transformations in elliptic spaces.

Explicit construction of Clifford parallels and their properties.

Uniform representation of Clifford translations via geometric multiplication.

Abstract

I apply the algebraic framework developed in arXiv:1101.4542 to study geometry of elliptic spaces in 1, 2, and 3 dimensions. The background material on projectivised Clifford algebras and their application to Cayley-Klein geometries is described in arXiv:1307.2917. The use of Clifford algebra largely obviates the need for spherical trigonometry as elementary geometric transformations such as projections, rejections, reflections, and rotations can be accomplished with geometric multiplication. Furthermore, the same transformations can be used in 3-dimensional elliptic space where effective use of spherical trigonometry is problematic. I give explicit construction of Clifford parallels and discuss their properties in detail. Clifford translations are represented in a uniform fashion by geometric multiplication as well. The emphasis of the exposition is on geometric structures and computation rather than proofs.