Geometric Algebras for Euclidean Geometry
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 (1up) candidate for euclidean geometry. We compare euclidean PGA and the popular 2up 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 structurepreserving 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.
 Publication:

arXiv eprints
 Pub Date:
 November 2014
 DOI:
 10.48550/arXiv.1411.6502
 arXiv:
 arXiv:1411.6502
 Bibcode:
 2014arXiv1411.6502G
 Keywords:

 Mathematics  General Mathematics;
 51FXX
 EPrint:
 25 pages, 4 figures in Advances in Applied Clifford Algebras, pages 124, 2016, online at link.springer.com