Summary: 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.
For the entire collection see [Zbl 1234.68004].