Minimal Curves on Ruled Surfaces Generated by Legendre Curves

Year 2024, Issue: 49, 43 - 52, 31.12.2024

Yusuf Yaylı , İsmet Gölgeleyen

https://doi.org/10.53570/jnt.1545097

Abstract

In this paper, we study the conditions under which Legendre curves on ruled surfaces are classified as minimal loci. By investigating the scenario where the directive curve is a binormal vector, we establish the criteria for these curves to have minimal loci on B-scroll-ruled surfaces. Furthermore, we present illustrative examples showcasing concrete instances of minimal curves within this context. Finally, we discuss the need for further research.

Keywords

Legendre curves , minimal curves , B-scroll surfaces

References

• M. P. Do Carmo, Differential geometry of curves and surfaces, Prentice-Hall, New Jersey, 1976.
• H. Pottman, A. Asperl, M. Hofer, A. Kilian, Architectural geometry, Bentley Institute Press, Pennsylvania, 2007.
• H. Pottmann, M. Eigensatz, A. Vaxman, J. Wallner, Architectural geometry, Computers & Graphics 47 (2015) 145–164.
• I. Linkeová, V. Zeleny, Application of ruled surfaces in freeform and gear metrology, Acta Polytechnica 61 (SI) (2021) 99-109.
• M. Incesu, The new characterization of ruled surfaces corresponding to dual Bézier curves, Mathematical Methods in the Applied Sciences 45 (18) (2022) 12030-12045.
• E. Karaca, M. Çalışkan, Tangent bundle of unit 2-sphere and slant ruled surfaces, Filomat 37 (2) (2023) 491-503.
• A. Yildirim, On some special ruled surfaces, New Trends in Mathematical Sciences 11 (4) (2023) 17-27.
• A. Alghanemi, A. Asiri, On geometry of ruled surfaces generated by the spherical indicatrices of a regular space curve I, Journal of Computational and Theoretical Nanoscience 13 (8) (2016) 5383-5388.
• A. Alghanemi, On geometry of ruled surfaces generated by the spherical indicatrices of a regular space curve II, International Journal of Algebra 10 (4) (2016) 193-205.
• T. Fukunaga, M. Takahashi, Existence and uniqueness for Legendre curves, Journal of Geometry 104 (2013) 297-307.
• M. Bekar, F. Hathout, Y. Yaylı, Legendre curves and the singularities of ruled surfaces obtained by using rotation minimizing frame, Ukrainian Mathematical Journal 73 (2021) 686-700.
• F. Hathout, M. Bekar, Y. Yaylı, Ruled surfaces and tangent bundle of unit 2-sphere, International Journal of Geometric Methods in Modern Physics 14 (10) (2017) 1750145 14 pages.
• M. Bekar, F. Hathout, Y. Yayli, Singularities of rectifying developable surfaces of Legendre curves on ${UTS}^{2}$, International Journal of Geometry 11 (4) (2022) 20-33.
• K. Derkaoui, F. Hathout, H. M. Dida, Slant curves and Legendre curves in three-dimensional Walker manifolds, Asian-European Journal of Mathematics 16 (05) (2023) 2350087 13 pages.
• Ky Sokphally, On properties of ruled surfaces and their asymptotic curves, Master's Thesis Concordia University (2020) Montreal.
• A. Pressley, Elementary differential geometry, Springer, London, 2010.
• L. K. Graves, Codimension one isometric immersions between Lorentz spaces, Transactions of the American Mathematical Society 252 (1979) 367-392.
• Ş. Kılıçoğlu, H. Hacısalihoğlu, On the fundamental forms of the B-scroll with null directrix and Cartan frame in Minkowskian 3-Space, Applied Mathematical Sciences 9 (80) (2015) 3957-3965.
• H. Balgetir, M. Bektaş, M. Ergüt, On the B-scrolls in the 3-dimensional Lorentzian space $L^{3}$, Kragujevac Journal of Mathematics 27 (2005) 163--174.
• S. Ouarab, A. O. Chahdi, M. Izid, Ruled surface generated by a curve lying on a regular surface and its characterizations, Journal for Geometry and Graphics 24 (2) (2020) 257-267.