Involutes of Pseudo-Null Curves in Lorentz–Minkowski 3-Space

Rafael López, Željka Milin Šipuš, Ljiljana Primorac Gajčić and Ivana Protrka
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Rafael López: Departamento de Geometría y Topología, Universidad de Granada, 18071 Granada, Spain
Željka Milin Šipuš: Department of Mathematics, Faculty of Science, University of Zagreb, Bijenička 30, 10000 Zagreb, Croatia
Ljiljana Primorac Gajčić: Department of Mathematics, J. J. Strossmayer University of Osijek, Trg Ljudevita Gaja 6, 31000 Osijek, Croatia
Ivana Protrka: Faculty of Mining, Geology and Petroleum Engineering, University of Zagreb, Pierottijeva 6, 10000 Zagreb, Croatia

Mathematics, 2021, vol. 9, issue 11, 1-14

Abstract: In this paper, we analyze involutes of pseudo-null curves in Lorentz–Minkowski 3-space. Pseudo-null curves are spacelike curves with null principal normals, and their involutes can be defined analogously as for the Euclidean curves, but they exhibit properties that cannot occur in Euclidean space. The first result of the paper is that the involutes of pseudo-null curves are null curves, more precisely, null straight lines. Furthermore, a method of reconstruction of a pseudo-null curve from a given null straight line as its involute is provided. Such a reconstruction process in Euclidean plane generates an evolute of a curve, however it cannot be applied to a straight line. In the case presented, the process is additionally affected by a choice of different null frames that every null curve allows (in this case, a null straight line). Nevertheless, we proved that for different null frames, the obtained pseudo-null curves are congruent. Examples that verify presented results are also given.

Keywords: Lorentz–Minkowski 3-space; pseudo-null curve; involute; null curve (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2021
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