Geometric Algebras for Euclidean Geometry

TL;DR

This paper clarifies conceptual misunderstandings in applying geometric algebra to Euclidean space, introduces a new algebraic model (euclidean PGA), and compares it with existing models like CGA, highlighting its advantages for geometry and mechanics.

Contribution

The paper introduces euclidean PGA as a minimal, structure-preserving geometric algebra for Euclidean geometry and compares it with CGA, demonstrating its practical and pedagogical benefits.

Findings

euclidean PGA is the smallest structure-preserving Euclidean GA

euclidean PGA and CGA exhibit similar features for flat primitives

euclidean PGA offers a natural pedagogical transition to CGA

Abstract

The discussion of how to apply geometric algebra to euclidean $n$-space has been clouded by a number of conceptual misunderstandings which we first identify and resolve, based on a thorough review of crucial but largely forgotten themes from $19^{th}$ century mathematics. We then introduce the dual projectivized Clifford algebra $\mathbf{P}(\mathbb{R}^*_{n,0,1})$ (euclidean PGA) as the most promising homogeneous (1-up) candidate for euclidean geometry. We compare euclidean PGA and the popular 2-up model CGA (conformal geometric algebra), restricting attention to flat geometric primitives, and show that on this domain they exhibit the same formal feature set. We thereby establish that euclidean PGA is the smallest structure-preserving euclidean GA. We compare the two algebras in more detail, with respect to a number of practical criteria, including implementation of kinematics and rigid body mechanics. We then extend the comparison to include euclidean sphere primitives. We conclude that euclidean PGA provides a natural transition, both scientifically and pedagogically, between vector space models and the more complex and powerful CGA.