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

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 nonstraight curves with a continuum of osculating tangents which correspond to motions in a cylinder group with osculating vertical Darboux motions. We also look at the set of osculating conics of a curve in projective space, suggest geometrically meaningful examples and briefly discuss and illustrate their corresponding motions.

And yet it moves: Paradoxically moving linkages in kinematics

“…The algebra DH of dual quaternions was invented by Clifford [10]. Modern references for dual quaternions in kinematics are [37,42]; here we list the relevant definitions and properties.…”