Direction curves and construction of developable surfaces in Lorentz 3-space

Year 2025, Volume: 74 Issue: 1, 47 - 55, 08.03.2025

Esma Demir Çetin , Çağla Ramis , Yusuf Yaylı

https://doi.org/10.31801/cfsuasmas.1395601

https://izlik.org/JA94ZH67KE

Abstract

In this work we investigate singularities for the three types of developable surfaces, introduced by Izumiya and Takeuchi, in Lorentz 3 space and give a local classification in terms of k-order frame. Moreover we search the necessary conditions of being a geodesic for principal direction curves of the rectifying developable surface.

Keywords

Developable surfaces , singular points , special curves

References

• Ali, A. T., New special curves and their spherical indicatrix, Global Journal of Advanced Research on Classical and Modern Geometries, 1(2) (2012), 28-38. https://doi.org/10.48550/arXiv.0909.2390
• Ali, A. T., Lopez, R., Slant helices in Minkowski 3-space, J. Korean Math. Soc., 48(1) (2011), 159-167. https://doi.org/10.4134/JKMS.2011.48.1.159
• Brander, D., Singularities of spacelike constant mean curvature surfaces in Lorentz- Minkowski space, Cambridge Philosophical Society, Mathematical Proceedings, 150 (2011), 527-556. https://doi.org/10.1017/S0305004111000077
• Fujimori, S., Saji, K., Umehara, M., Yamada, K., Singularities of maximal surfaces, Math. Z., 259 (2008), 827-848. https://doi.org/10.1007/s00209-007-0250-0
• Ishikawa, G., Yamashita, T., Singularities of tangent surfaces to directed curves, Topology and its Applications, 234 (2018), 198-208. https://doi.org/10.1016/j.topol.2017.11.018
• Izumiya, S., Takeuchi, N., New special curves and developable surfaces, Turk J. Math., 28 (2004), 153-163. https://journals.tubitak.gov.tr/math/ vol28/iss2/6
• Izumiya, S., Katsumi, H., Yamasaki, T., The rectifying developable and the spherical Darboux image of a space curve, Geometry and Topology of Caustics-Caustics ’98-Banach Center Publications, (1999), 137-149. https://doi.org/10.4064/-50-1-137-149
• Kokubu, M., Rossman, W., Saji, K., Umehara, M., Yamada, K., Singularities of flat fronts in hyperbolic space, Pacific J. Math., 221 (2005), 303-351. https://doi.org/10.2140/PJM.2005.221.303
• Murata, S., Umehara, S., Flat surfaces with singularities in Euclidean 3-space, J. Differential Geometry, 82 (2009), 279-316. https://doi.org/10.4310/jdg/1246888486
• Ramis, Ç., Uzunoglu, B., Yaylı, Y., New associated curves k-principle direction curves and $N_k$-slant helix, Hagia Sophia Journal of Geometry, 4(2) (2022), 19-27. https://doi.org/10.48550/arXiv.1404.7369
• Uzunoglu, B., Ramis, Ç., Yaylı, Y., On curves of $N_k$ slant helix and $N_k$ constant precession in Minkowski 3–space, Journal of Dynamical Systems and Geometric Theories, 12(2) (2014), 175-189. https://doi.org/10.1080/1726037X.2014.988933
• Zhao, Q., Pei, D., Wang, Y., Singularities for one-parameter developable surfaces of curves, Symmetry, 11(108) (2019). https://doi.org/10.3390/sym11010108