Space kinematics and projective differential geometry over the ring of dual numbers. (English) Zbl 1489.53017

Summary: 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.


53A20 Projective differential geometry
70B10 Kinematics of a rigid body
16S99 Associative rings and algebras arising under various constructions
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