On the Homogeneous Model of Euclidean Geometry

We attach the degenerate signature (n,0,1) to the dual Grassmann algebra of projective space to obtain a real Clifford algebra which provides a powerful, efficient model for euclidean geometry. We avoid problems with the degenerate metric by constructing an algebra isomorphism between the Grassmann algebra and its dual that yields non-metric meet and join operators. We focus on the cases of n=2 and n=3 in detail, enumerating the geometric products between k-blades and m-blades. We identify sandwich operators in the algebra that 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 group elements. We conclude with an elementary account of euclidean kinematics and rigid body motion within this framework.