Review of Geometric algebra for computer science by Leo Dorst, Daniel Fontijne, and Stephen Mann  (Morgan Kaufmann Publishers, 2007)

This paper is an introduction to geometric algebra and geometric calculus, presented in the simplest way I could manage, without worrying too much about completeness or rigor. An understanding of linear algebra and vector calculus is presumed. This should be sufficient to read most of the paper."The principal argument for the adoption of geometric algebra is that it provides a single, simple mathematical framework which eliminates the plethora of diverse mathematical descriptions and techniques it would otherwise be necessary to learn." [15]

Section: Composition Of Boostsmentioning

confidence: 99%

A Survey of Geometric Algebra and Geometric Calculus

Macdonald

2016

“…Projectivities which preserve Q correspond to conformal maps of R n , hence this model is called the conformal model of euclidean geometry, and the associated geometric algebra is called conformal geometric algebra (CGA). It was introduced in its present form in [HLR01] and has developed rapidly since then ( [DL03], [DFM07], [Per09], and [DL11]) . In light of the prolific literature available, we omit a more detailed description here.…”

Section: Geometric Algebras For Euclidean Geometrymentioning

confidence: 99%

“…For example, Chapter 11 of [DFM07], entitled The homogeneous model, describes one such model (which appears in several other textbooks ( [DL03], [Per09]). In §11.1 one reads:…”

Section: Which Homogeneous Model?mentioning

confidence: 99%

Geometric Algebras for Euclidean Geometry

Gunn

2016

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.

“…when P is a simple bivector [15]. In this fully orthogonal plane P * , every direction is orthogonal to the plane P .…”

Section: Bivectors and Orthogonal Componentsmentioning

confidence: 99%

On Cayley’s Factorization of 4D Rotations and Applications

Pérez-Gracia

Thomas

2016

Abstract. A 4D rotation can be decomposed into a left-and a rightisoclinic rotation. This decomposition, known as Cayley's factorization of 4D rotations, can be performed using Elfrinkhof-Rosen method. In this paper, we present a more straightforward alternative approach using the corresponding orthogonal subspaces, for which orthogonal bases can be defined. This yields easy formulations, both in the space of 4×4 real orthogonal matrices representing 4D rotations and in the Clifford algebra C4,0,0.Cayley's factorization has many important applications. It can be used to easily transform rotations represented using matrix algebra to different Clifford algebras. As a practical application of the proposed method, it is shown how Cayley's factorization can be used to efficiently compute the screw parameters of 3D rigid-body transformations.Mathematics Subject Classification (2010). Primary 99Z99; Secondary 15A66.