Geometric Algebras for Euclidean Geometry

Abstract

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 P(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.

Geometric algebra and computer graphics

“…Other geometric algebras have been proposed for doing euclidean geometry, notably conformal geometric algebra (CGA). Interested readers are referred to the comparison article [Gun17b], which should shed light on the choice to base these notes on PGA.…”

“…The elements of E n are points, those of R n are vectors; the motions of E n include translations and rotations, those of R n are rotations preserving the origin O. E n is intrinsically more complex than R n : the tangent space at each point is R n . See [Gun17b], §4, for a deeper analysis of this issue. We will see that euclidean PGA includes both E n and R n in an organic whole.…”

“…PGA's development reflects the fact that much of the existing literature on geometric algebras deals only with nondegenerate metrics, reflecting widespread prejudices regarding degenerate metrics. (See [Gun17b] for a thorough analysis and refutation of these misconceptions.) After long experience we are convinced that the degenerate metric, far from being a liability, is an important part of PGA's success -exactly the degenerate metric models the metric relationships of euclidean geometry faithfully (see [Gun17b], 7 PGA for the euclidean plane: P(R * 2,0,1 )…”

“…See [Hil13]. PGA presents no special challenges in this regard; in fact, it demonstrates clear advantages over other geometric algebra approaches to euclidean geometry in this regard ( [Gun17b]…”

Doing Euclidean Plane Geometry Using Projective Geometric Algebra

“…PGA for euclidean geometry first appeared in the modern literature in [Sel00] and [Sel05] and was extended and developed in [Gun11a], [Gun11b], [Gun11c], and [Gun16b]. Readers unfamiliar with duality or projectivization, or just interested in a fuller, more rigorous treatment, should consult the latter references.…”