Kinematic Mappings for Cayley-Klein Geometries via Clifford Algebras

TL;DR

This paper introduces a unified approach to kinematic mappings in Cayley-Klein geometries using Clifford algebras, providing a systematic construction method and classifying mappings in 2D and 3D spaces.

Contribution

It presents a novel algebraic framework for constructing and classifying kinematic mappings in Cayley-Klein geometries using geometric algebras.

Findings

Unified algebraic construction of kinematic mappings for Cayley-Klein geometries.

Explicit mappings for Euclidean displacement groups SE(2) and SE(3).

Classification of kinematic mappings in 2D and 3D Cayley-Klein spaces.

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 SE(2) the result is the kinematic mapping of Blaschke and Gr\"unwald. For the group of spatial Euclidean displacements SE(3) the result is Study's mapping. Furthermore, we classify kinematic mappings for Cayley-Klein spaces of dimension 2 and 3.