On the Homogeneous Model of Euclidean Geometry

Abstract

We attach the degenerate signature (n,0,1) to the projectivized dual Grassmann algebra over R(n+1). We explore the use of the resulting Clifford algebra as a model for euclidean geometry. We avoid problems with the degenerate metric by constructing an algebra isomorphism between this Grassmann algebra and its dual, that yields non-metric meet and join operators. We review the Cayley-Klein construction of the projective (homogeneous) model for euclidean geometry leading to the choice of the signature (n,0,1). We focus on the cases of n=2 and n=3 in detail, enumerating the geometric products between simple k- and m-vectors. We establish that versor (sandwich) operators provide all euclidean isometries, both direct and indirect. We locate the spin group, a double cover of the direct euclidean group, inside the even subalgebra of the Clifford algebra, and provide a simple algorithm for calculating the logarithm of such elements. We conclude with an elementary account of euclidean rigid body motion within this framework.