A Representation of the Euclidean Group by Spin Groups, and Spatial Kinematics Mappings

A new derivation of the spin and biquaternion representation of the Euclidean group is presented. The derivation is based upon the even Clifford algebra representation of the orientation preserving orthogonal automorphisms of nondegenerate orthogonal spaces, also called spin representation. Embedding the degenerate orthogonal space IR^1,0,3 into the nondegenerate orthogonal space IR^1,4, and imposing certain conditions on the orthogonal automorphisms of IR^1,4, one obtains a subgroup of the spin group. The action of this subgroup, on a subspace of IR^1,4, is isomorphic to IR^1,0,3, is precisely a Euclidean motion. The conditions imposed on the orthogonal automorphisms of IR^1,4 lead to the biquaternion representation. Furthermore, the invariants of the representations are easily obtained. The derivation also allows the spin representation to be related to the action of the representation over an element of a three-dimensional vector space proposed by Porteous, and used by Selig. As a byproduct, the derivation provides an insightful interpretation of the dual unit used in both the spin representation and the biquaternion representation.