Advances in Applied Clifford Algebras

, Volume 27, Issue 1, pp 185–208 | Cite as

Geometric Algebras for Euclidean Geometry

  • Charles GunnEmail author


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 nineteenth 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.


Metric geometry Euclidean geometry Cayley–Klein construction Dual exterior algebra Projective geometry Degenerate metric Projective geometric algebra Conformal geometric algebra Duality Homogeneous model Biquaternions Dual quaternions Kinematics Rigid body motion 


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© Springer International Publishing 2016

Authors and Affiliations

  1. 1.Institüt für Mathematik MA 8-3Technische Universität BerlinBerlinGermany

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