B-Lift curves and involute curves in Lorentzian 3-space

Year 2024, Volume: 73 Issue: 2, 474 - 485, 21.06.2024

Anıl Altınkaya , Mustafa Çalışkan

https://doi.org/10.31801/cfsuasmas.1338814

Abstract

The involute of a curve is often called the perpendicular trajectories of the tangent vectors of a unit speed curve. Furthermore, the B-Lift curve is the curve acquired by combining the endpoints of the binormal vectors of a unit speed curve. In this study, we investigate the correspondences between the Frenet vectors of a curve’s B-lift curve and its involute. We also give an illustration of a helix that resembles space in Lorentzian 3-space and show how to visualize these curves by deriving the B-Lift curve and its involute.

Keywords

Involute curve, B-lift curve, Minkowski space

References

• Izumiya, S., Romero-Fuster, M. C., Takahashi, M., Evolutes of curves in the Lorentz-Minkowski plane, Advanced Studies in Pure Mathematics, 78 (2018) 313-330. https://doi.org/10.2969/aspm/07810313
• Kula, L., Ekmekci, N., Yaylı, Y., İlarslan, K., Characterizations of slant helices in Euclidean 3-space, Turk J Math, 34 (2010) 261-273. https://doi.org/10.3906/mat-0809-17
• Lopez, R., Sipus, Z. M., Gajcic, L. P., Protrka, I., Involutes of pseudonull curves in Lorentz–Minkowski 3-space, Mathematics, 9(11) (2021), 1256. https://doi.org/10.3390/math9111256
• Hanif, M., Hou, Z. H., Generalized involute and evolute curve-couple in Euclidean space, Int. J. Open Problems Compt. Math., 11 (2018) 28-39. https://doi.org/10.12816/0049059
• Deshmukh, S., Chen, B., Alshammari, S. H., On rectifying curves in Euclidean 3-space, Turk. J. Math., 42 (2018) 609-620. https://doi.org/10.3906/mat-1701-52
• Izumiya, S., Takeuchi, N., Generic properties of helices and Bertrand curves, Journal of Geometry, 74 (2002) 97-109. https://doi.org/10.1007/PL00012543
• Ekmekci, N., Okuyucu, O. Z., Yaylı, Y., Characterization of speherical helices in Euclidean 3-space, An. St. Univ. Ovidius Constanta, 22 (2014) 99-108. https://doi.org/10.2478/auom-2014-0035
• Zhang, C., Pei, D., Generalized Bertrand curves in Minkowski 3-space, Mathematics, 8(12) (2020). https://doi.org/10.3390/math8122199
• Lopez, R., Differential geometry of curves and surfaces in Lorentz-Minkowski space, International Electronic Journal of Geometry, 7 (2014) 44-107. https://doi.org/10.36890/iejg.594497
• Millman, R. S., Parker, G. D., Elements of Differential Geometry, Prentice-Hall, Englewood Cliffs, NJ, 1977.
• Hacısalihoğlu, H. H., Differential Geometry, Inonu University, Malatya, 1983.
• Çalışkan, M., Bilici, M., Some characterizations for the pair of involute-evolute curves in Euclidean space E3, Bulletin of Pure and Applied Sciences, 21 (2012), 289-294.
• Bilici, M., Çalışkan, M., On the involutes of the spacelike curve with a timelike binormal in Minkowski 3-space, International Mathematical Forum, 31 (2009) 1497-1509.
• Thorpe, J. A., Elemantary Topics in Differential Geometry, Springer-Verlag, New York, 1979.
• Karaca, E., Çalışkan, M., Ruled surfaces and tangent bundle of unit 2-sphere of natural lift curves, Gazi University Journal of Science, 33 (2020), 751-759. https://doi.org/10.35378/gujs.588496
• Ergün, E., Bayram, E., Surface family with a common natural geodesic lift, International J. Math. Combin., 1 (2016) 34-41.
• Ergün, E., Çalışkan, M., The natural lift curve of the spherical indicatrix of a nonnull curve in Minkowski 3-space, International Mathematical Forum, 7 (2012) 707-717. https://doi.org/10.32513/asetmj/19322008226
• Ergün, E., Çalışkan, M., On natural lift of a curve, Pure Mathematical Sciences, 2 (2012) 81-85.
• Ergün, E., Çalışkan, M., On the natural lift curve and the involute curve, Journal of Science and Arts, 45 (2018) 869-890.
• O’Neill, B., Semi Riemann Geometry, Academic Press, New York and London, 1983.
• Önder, M., Uğurlu, H. H., Frenet frames and invariants of timelike ruled surfaces, Ain Shams Engineering Journal, 4 (2013) 507-513.
• Walrave, J., Curves and Surfaces in Minkowski Space, K. U. Leuven Faculteit, Der Wetenschappen, 1995.
• Ratcliffe, J. G., Foundations of Hyperbolic Manifolds, Springer-Verlag, New York, 1994.
• Altınkaya, A., Çalışkan, M., B-lift curves in Lorentzian 3-space, Journal of Science and Arts, 22 (2022), 5-14. https://doi.org/10.46939/J.Sci.Arts-22.1-a01