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VOL. 78 | 2018 Evolutes of curves in the Lorentz-Minkowski plane
S. Izumiya, M. C. Romero Fuster, M. Takahashi

Editor(s) Shyuichi Izumiya, Goo Ishikawa, Minoru Yamamoto, Kentaro Saji, Takahiro Yamamoto, Masatomo Takahashi


We can use a moving frame, as in the case of regular plane curves in the Euclidean plane, in order to define the arc-length parameter and the Frenet formula for non-lightlike regular curves in the Lorentz-Minkowski plane. This leads naturally to a well defined evolute associated to non-lightlike regular curves without inflection points in the Lorentz-Minkowski plane. However, at a lightlike point the curve shifts between a spacelike and a timelike region and the evolute cannot be defined by using this moving frame. In this paper, we introduce an alternative frame, the lightcone frame, that will allow us to associate an evolute to regular curves without inflection points in the Lorentz-Minkowski plane. Moreover, under appropriate conditions, we shall also be able to obtain globally defined evolutes of regular curves with inflection points. We investigate here the geometric properties of the evolute at lightlike points and inflection points.


Published: 1 January 2018
First available in Project Euclid: 4 October 2018

zbMATH: 07085109
MathSciNet: MR3839951

Digital Object Identifier: 10.2969/aspm/07810313

Primary: 53A35
Secondary: 53C50, 53D35

Rights: Copyright © 2018 Mathematical Society of Japan


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