Space Kinematics and Projective Differential Geometry Over the Ring of Dual Numbers

TL;DR

This paper explores the geometric and kinematic properties of rigid body motions using dual quaternions and projective differential geometry, revealing phenomena with clear physical interpretations and applications in linkage design.

Contribution

It introduces a novel geometric framework linking dual quaternions with differential geometry, and applies this to analyze and construct complex mechanical linkages.

Findings

Existence of non-straight curves with a continuum of osculating tangents.

Geometric methods for selecting osculating conics and their motions.

Construction of overconstrained linkages based on factorizability of motions.

Abstract

We study an isomorphism between the group of rigid body displacements and the group of dual quaternions modulo the dual number multiplicative group from the viewpoint of differential geometry in a projective space over the dual numbers. Some seemingly weird phenomena in this space have lucid kinematic interpretations. An example is the existence of non-straight curves with a continuum of osculating tangents which correspond to motions in a cylinder group with osculating vertical Darboux motions. We also suggest geometrically meaningful ways to select osculating conics of a curve in this projective space and illustrate their corresponding motions. Furthermore, we investigate factorizability of these special motions and use the obtained results for the construction of overconstrained linkages.