Geometric Algebras for Euclidean Geometry

Abstract

The discussion of how to apply geometric algebra to euclidean \(n\)-space as 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 \(\mathbf{P}(\mathbb{R}^*_{n,0,1})\) (PGA) as the most promising homogeneous (1-up) candidate for euclidean geometry. We compare 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 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 PGA provides a natural transition, both scientifically and pedagogically, between vector space models and the more complex and powerful CGA.