Projective metric geometry and Clifford algebras

TL;DR

This paper explores the relationship between Clifford algebras, Lipschitz groups, and projective metric geometry, providing an algebraic framework for kinematic mappings derived from quotient groups of the Lipschitz group.

Contribution

It introduces a novel quotient of the Lipschitz group that is meaningful in projective metric geometry and offers an algebraic description of kinematic mappings.

Findings

Identification of a meaningful quotient of the Lipschitz group in projective geometry

Representation of this quotient as a point set in projective space

Algebraic characterization of kinematic mappings under certain conditions

Abstract

Each vector space that is endowed with a quadratic form determines its Clifford algebra. This algebra, in turn, contains a distinguished group, known as the Lipschitz group. We show that only a quotient of this group remains meaningful in the context of projective metric geometry. This quotient of the Lipschitz group can be viewed as a point set in the projective space on the Clifford algebra and, under certain restrictions, leads to an algebraic description of so-called kinematic mappings.