Space kinematics and projective differential geometry over the ring of dual numbers. (English) Zbl 1466.53011

Schröcker, Hans-Peter; Pfurner, Martin; Siegele, Johannes

Cheng, Liang-Yee (ed.), ICGG 2020 – Proceedings of the 19th international conference on geometry and graphics, São Paulo, Brazil, January 18–22, 2021. Cham: Springer. Adv. Intell. Syst. Comput. 1296, 15-24 (2021).

The aim of this paper is to 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. The authors 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.
This paper is organized as follows: Section 1 is an introduction to the subject. Section 2 deals with some preliminaries. In Section 3 the authors demonstrate that a helical motion and a vertical Darboux motion can have second-order contact at any parameter value. Since vertical Darboux motions correspond to straight lines in \(\mathbb{P}^3(\mathbb{D})\) this amounts to saying that a curve corresponding to a helical motion has an osculating tangent at any point. This is a remarkable difference to classical differential geometry over the real numbers where this property characterizes straight lines. Section 4 deals with osculating conics. In this section the authors turn their attention to conic sections in \(\mathbb{P}^3(\mathbb{D})\) and study them as rational curves of degree two.
Generically, there exists a four-dimensional set of osculating conics in every curve point. Among them they find the well-known Bennett motions and suggest another type of osculating conic with geometric significance. Its construction is based on the construction of osculating circles in elliptic geometry. The conclusion Section 5 discusses and interprets the results obtained.
For the entire collection see [Zbl 1458.00031].

Reviewer: Ahmed Lesfari (El Jadida)

MSC:

53A20	Projective differential geometry
53A17	Differential geometric aspects in kinematics

Keywords:

projective differential geometry; space kinematics; group of rigid body displacements; group of dual quaternions