Kinematic Mappings for Spin Groups

Abstract

In this chapter we define Cayley-Klein spaces and show how to describe certain Cayley-Klein spaces within the homogeneous Clifford algebra model. This construction is accomplished in detail for the threedimensional Euclidean space. Furthermore, the kinematic mapping of Study and the mapping of Blaschke and Grünwald are constructed in a unified method. Matrices of the collineations in the image and pre-image space are derived. The construction is accomplished in detail for the Euclidean spaces of dimension two and three. After that, we give an overview of possible kinematic mappings for Cayley-Klein spaces of dimension two and three. Moreover, the mapping for the four-dimensional Euclidean space is presented. This chapter is already published, see [41].