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Evolutes of curves in the Lorentz-Minkowski plane. (English) Zbl 1428.53020
Izumiya, Shyuichi (ed.) et al., Singularities in generic geometry. Proceedings of the 4th workshop on singularities in generic geometry and applications (Valencia IV), Kobe, Japan, June 3–6, 2015 and Kyoto, Japan, June 8–10, 2015. Tokyo: Mathematical Society of Japan (MSJ). Adv. Stud. Pure Math. 78, 313-330 (2018).
For any regular curve in the Lorentz-Minkowski plane \(\mathbb{R}_1^2\), basic notions like arc length, moving Frenet frame, and curvature have been defined in analogy to the Euclidean plane under the assumption that the given curve contains no timelike points. However, at a light-like point of a curve the definition of the Frenet frame fails and therefore several notions are no longer available.
In order to overcome this shortcoming, the authors introduce an alternative moving frame, composed (at each point of the regular curve) of the light-like vectors \((1,1)\) and \((-1,1)\), and they call it the lightcone frame. This frame allows them to define an evolute of a regular curve without inflection points irrespective of the existence of light-like points on the curve. They also introduce evolutes of regular curves with inflection points under an extra condition. The article contains several nicely illustrated examples, which show the differences between these newly introduced evolutes and the conventional evolutes in the Euclidean plane.
For the entire collection see [Zbl 1407.58001].
53A35 Non-Euclidean differential geometry
53A04 Curves in Euclidean and related spaces