Journal for Geometry and Graphics 25 (2021), No. 1, 019--031
Copyright Heldermann Verlag 2021
Space Kinematics and Projective Differential Geometry Over the Ring of Dual Numbers
Johannes Siegele
Department of Basic Sciences in Engineering Sciences, University of Innsbruck, Austria
johannes.siegele@uibk.ac.at
Martin Pfurner
Department of Basic Sciences in Engineering Sciences, University of Innsbruck, Austria
martin.pfurner@uibk.ac.at
Hans-Peter Schröcker
Department of Basic Sciences in Engineering Sciences, University of Innsbruck, Austria
hans-peter.schroecker@uibk.ac.at
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.
Keywords: Rational motion, motion polynomial, factorization, vertical Darboux motion, helical motion, osculating line, osculating conic, null cone motion, linkage.
MSC: 16S36; 53A20, 70B10.
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