Involutes of Pseudo-Null Curves in Lorentz–Minkowski 3-Space
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. T...
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MDPI AG
2021-05-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/9/11/1256 |
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| author | Rafael López Željka Milin Šipuš Ljiljana Primorac Gajčić Ivana Protrka |
| author_facet | Rafael López Željka Milin Šipuš Ljiljana Primorac Gajčić Ivana Protrka |
| author_sort | Rafael López |
| collection | DOAJ |
| description | 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. |
| first_indexed | 2024-03-10T10:51:57Z |
| format | Article |
| id | doaj.art-b2870b9bdbc444eba05e4b1dc57cce34 |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-10T10:51:57Z |
| publishDate | 2021-05-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-b2870b9bdbc444eba05e4b1dc57cce342023-11-21T22:10:36ZengMDPI AGMathematics2227-73902021-05-01911125610.3390/math9111256Involutes of Pseudo-Null Curves in Lorentz–Minkowski 3-SpaceRafael López0Željka Milin Šipuš1Ljiljana Primorac Gajčić2Ivana Protrka3Departamento de Geometría y Topología, Universidad de Granada, 18071 Granada, SpainDepartment of Mathematics, Faculty of Science, University of Zagreb, Bijenička 30, 10000 Zagreb, CroatiaDepartment of Mathematics, J. J. Strossmayer University of Osijek, Trg Ljudevita Gaja 6, 31000 Osijek, CroatiaFaculty of Mining, Geology and Petroleum Engineering, University of Zagreb, Pierottijeva 6, 10000 Zagreb, CroatiaIn 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.https://www.mdpi.com/2227-7390/9/11/1256Lorentz–Minkowski 3-spacepseudo-null curveinvolutenull curve |
| spellingShingle | Rafael López Željka Milin Šipuš Ljiljana Primorac Gajčić Ivana Protrka Involutes of Pseudo-Null Curves in Lorentz–Minkowski 3-Space Mathematics Lorentz–Minkowski 3-space pseudo-null curve involute null curve |
| title | Involutes of Pseudo-Null Curves in Lorentz–Minkowski 3-Space |
| title_full | Involutes of Pseudo-Null Curves in Lorentz–Minkowski 3-Space |
| title_fullStr | Involutes of Pseudo-Null Curves in Lorentz–Minkowski 3-Space |
| title_full_unstemmed | Involutes of Pseudo-Null Curves in Lorentz–Minkowski 3-Space |
| title_short | Involutes of Pseudo-Null Curves in Lorentz–Minkowski 3-Space |
| title_sort | involutes of pseudo null curves in lorentz minkowski 3 space |
| topic | Lorentz–Minkowski 3-space pseudo-null curve involute null curve |
| url | https://www.mdpi.com/2227-7390/9/11/1256 |
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