Simultaneous characterizations of alternative partner-ruled surfaces

Kemal Eren; Soley Ersoy; Mohammad Nazrul Islam Khan; Kemal Eren; Soley Ersoy; Mohammad Nazrul Islam Khan

doi:10.3934/math.2025407

AIMS Mathematics

2025, Volume 10, Issue 4: 8891-8906. doi: 10.3934/math.2025407

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Research article

Simultaneous characterizations of alternative partner-ruled surfaces

1.
Picode Software, Education Training Consultancy Research and Development and Trade Ltd. Co., Sakarya-54050, Turkey
2.
Department of Mathematics, Faculty of Sciences, Sakarya University, Sakarya-54050, Turkey
3.
Department of Computer Engineering, College of Computer, Qassim University, Buraydah-51452, Saudi Arabia

Received: 07 February 2025 Revised: 17 March 2025 Accepted: 28 March 2025 Published: 18 April 2025
MSC : 53A04, 53A05

In this study, we present partner-ruled surfaces generated by the vectors of the alternative frame of a space curve in Euclidean 3-space. First, each pair of the alternative partner-ruled surfaces to be simultaneously developable and minimal is investigated based on the alternative frame by partial differential equations. Then, simultaneous characterizations of the coordinate curves of these surfaces to be asymptotic, geodesic, and lines of curvature are obtained and explicated. Finally, to illustrate the concepts, the study concludes with an example of alternative partner-ruled surfaces, featuring graphical representations.
- partner-ruled surfaces,
- alternative frame,
- developable and minimal surfaces,
- differential equations,
- geodesic curves,
- partial differential equations,
- asymptotic curves
Citation: Kemal Eren, Soley Ersoy, Mohammad Nazrul Islam Khan. Simultaneous characterizations of alternative partner-ruled surfaces[J]. AIMS Mathematics, 2025, 10(4): 8891-8906. doi: 10.3934/math.2025407

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Abstract

In this study, we present partner-ruled surfaces generated by the vectors of the alternative frame of a space curve in Euclidean 3-space. First, each pair of the alternative partner-ruled surfaces to be simultaneously developable and minimal is investigated based on the alternative frame by partial differential equations. Then, simultaneous characterizations of the coordinate curves of these surfaces to be asymptotic, geodesic, and lines of curvature are obtained and explicated. Finally, to illustrate the concepts, the study concludes with an example of alternative partner-ruled surfaces, featuring graphical representations.

References

[1]	L. Busé, M. Elkadi, A. Galligo, A computational study of ruled surfaces, J. Symb. Comput., 44 (2009), 232–241. https://doi.org/10.1016/j.jsc.2007.04.005 doi: 10.1016/j.jsc.2007.04.005
[2]	Y. Liu, H. Pottmann, J. Wallner, Y. Yang, W. Wang, Geometric modeling with conical meshes and developable surfaces, ACM Trans. Graph., 25 (2006), 681–689. https://doi.org/10.1145/1141911.1141941 doi: 10.1145/1141911.1141941
[3]	S. Ouarab, A. O. Chahdi, M. Izid, Ruled surfaces with alternative moving frame in Euclidean 3-space, Int. J. Math. Sci. Eng. Appl., 12 (2018), 43–58.
[4]	B. O'Neill, Elementary differential geometry, Revised 2nd Edition, Academic Press, 2006.
[5]	S. Zhang, A highly accurate RBF quasi-interpolation method for approximating the derivatives, Comp. Appl. Math., 40 (2021), 224. https://doi.org/10.1007/s40314-021-01623-2 doi: 10.1007/s40314-021-01623-2
[6]	S. Izumiya, N. Takeuchi, New special curves and developable surfaces, Turk. J. Math., 28 (2004), 153–163.
[7]	Y. Yuan, H. Liu, Binormal surfaces for space curves, J. Northeast. Univ., 33 (2012), 1517–1520.
[8]	B. Uzunoğlu, İ. Gök, Y. Yaylı, A new approach on curves of constant precession, Appl. Math. Comput., 275 (2016), 317–323. https://doi.org/10.1016/j.amc.2015.11.083 doi: 10.1016/j.amc.2015.11.083
[9]	G. Güzelkardeşler, B. Şahiner, An alternative approach to find the position vector of a general helix, Celal Bayar Univ. Fen Bilim. Derg., 20 (2024), 54–60. https://doi.org/10.18466/cbayarfbe.1479066 doi: 10.18466/cbayarfbe.1479066
[10]	Y. Tuncer, Vectorial moments of curves in Euclidean 3-space, Int. J. Geom. Methods Mod. Phys., 14 (2017), 1750020. https://doi.org/10.1142/S0219887817500207 doi: 10.1142/S0219887817500207
[11]	O. Kaya, M. Önder, New partner curves in the Euclidean 3-space $E^3$, Int. J. Geomech., 6 (2017), 41–50.
[12]	S. Ouarab, NC-Smarandache ruled surface and NW-Smarandache ruled surface according to alternative moving frame in $E^3$, J. Math., 2021 (2021), 9951434. https://doi.org/10.1155/2021/9951434 doi: 10.1155/2021/9951434
[13]	Y. Li, K. Eren, K. H. Ayvacı, S. Ersoy, Simultaneous characterizations of partner-ruled surfaces using Flc frame, AIMS Mathematics, 7 (2022), 20213–20229. https://doi.org/10.3934/math.20221106 doi: 10.3934/math.20221106
[14]	Y. Li, K. Eren, S. Ersoy, On simultaneous characterizations of partner-ruled surfaces in Minkowski 3-space, AIMS Mathematics, 8 (2023), 22256–22273. https://doi.org/10.3934/math.20231135 doi: 10.3934/math.20231135
[15]	S. Ersoy, K. Eren, A. Çalışkan, Characterizations of spatial quaternionic partner-ruled surfaces, Axioms, 13 (2024), 612. https://doi.org/10.3390/axioms13090612 doi: 10.3390/axioms13090612
[16]	S. Ouarab, Simultaneous developability of partner-ruled surfaces according to Darboux frame in $E^3$, Abstr. Appl. Anal., 2021 (2021), 3151501. https://doi.org/10.1155/2021/3151501 doi: 10.1155/2021/3151501
[17]	A. Savić, K. Eren, S. Ersoy, V. Baltić, Alternative view of inextensible flows of curves and ruled surfaces via alternative frame, Mathematics, 12 (2024), 2015. https://doi.org/10.3390/math12132015 doi: 10.3390/math12132015

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