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 $\mathbf{P}({\mathbb{R}}_{n,0,1}^{\ast })$ (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.

25 pages, 4 figures in Advances in Applied Clifford Algebras, pages 1--24, 2016, online at link.springer.com

References in corpus (1)

• Geometric Algebras for Euclidean Geometry

Cited by in corpus (11)

• Geometric Algebras for Euclidean Geometry
• Doing euclidean plane geometry using projective geometric algebra
• Course notes Geometric Algebra for Computer Graphics, SIGGRAPH 2019
• Geometric Algebra for Optimal Control with Applications in Manipulation Tasks
• On Some Lie Groups in Degenerate Clifford Geometric Algebras
• Projective geometric algebra: A new framework for doing euclidean geometry
• A Note on Centralizers and Twisted Centralizers in Clifford Algebras
• Extending the Cooperative Dual-Task Space in Conformal Geometric Algebra
• Projective metric geometry and Clifford algebras
• Rational trigonometry via projective geometric algebra
• On Lie Groups Preserving Subspaces of Degenerate Clifford Algebras