Kinematic mappings for Cayley–Klein geometries via Clifford algebras

Abstract

This paper unifies the concept of kinematic mappings by using geometric algebras. We present a method for constructing kinematic mappings for certain Cayley–Klein geometries. These geometries are described in an algebraic setting by the homogeneous Clifford algebra model. Displacements correspond to Spin group elements. After that Spin group elements are mapped to a kinematic image space. Especially for the group of planar Euclidean displacements $$\mathrm{{SE} }(2)$$ the result is the kinematic mapping of Blaschke and Grünwald. For the group of spatial Euclidean displacements $$\mathrm{{SE} }(3)$$ the result is Study’s mapping. Furthermore, we classify kinematic mappings for Cayley–Klein spaces of dimension $$2$$ and $$3$$ .