Tubular Surfaces of Adjoint Curves According to the Modified Orthogonal Frame

Year 2025, Volume: 8 Issue: 1, 206 - 213, 15.01.2025

Esra Damar , Burçin Saltık Baek , Nural Yüksel , Nurdan Oğraş

https://doi.org/10.34248/bsengineering.1582756

Abstract

In this paper, we study tubular surfaces defined by adjoint curves which have a wide range of applications. In three-dimensional Euclidean space, we consider tubular surfaces in modified orthogonal frame generated by a curve β whose center curve is adjoint of a curve α. We give some characterizations for tubular surfaces constructed according to with curvature and with torsion modified orthogonal frames. Through these characterizations we obtain some important results. We also study asymptotic and geodesic curves as well as flat, minimal, Weingarten and linear-Weingarten surfaces using conventional differential geometry techniques. Finally, we present case examples for both versions of the frame to validate our theoretical results.

Keywords

Tubular surface, Adjoint curve, Modified orthogonal frame

Ethical Statement

Ethics committee approval was not required for this study because of there was no study on animals or humans.

References

• Arıkan M, Nurkan KS. 2020. Adjoint curve according to modified orthogonal frame with torsion in 3-space. Uşak Üniv Fen Doğa Bilim Derg, 4: 54-64.
• Bükcü B, Karacan M. K. 2016. On the modified orthogonal frame with curvature and torsion in 3-space. Math Sci Applicat E-Notes, 4: 184-188.
• Cakmak A, Şahin V. 2022. Characterizations of adjoint curves according to alternative moving frame. Fundamen J Math Applicat, 5: 42-50.
• Karacan MK, Es H, Yayli Y. 2006. Singular points of tubular surfaces in Minkowski 3-space. Sarajevo J Math, 2: 73-82.
• Karacan MK, Yayli Y. 2008. On the geodesics of tubular surfaces in Minkowski 3-space. Bullet Malaysian Math Sci Soc Second Series, 31: 1-10.
• Karacan MK, Tuncer Y. 2013. Tubular surfaces of Weingarten types in Galilean and pseudo-Galilean. Bull Math Anal Appl, 5: 87-100.
• Kim Y. H, Liu H, Qian J. 2016. Some characterizations of canal surfaces. Bull Korean Math Soc, 53: 461-477.
• Kühnel W, Hunt B. 2005. Differential geometry: curves-surfaces-manifolds. Providence, Rhode Island: American Mathematical Society 3rd ed., New York, USA, pp: 402.
• López R. 2009. Linear Weingarten surfaces in Euclidean and hyperbolic space. arXiv preprint arXiv: 0906.3302.
• Mazlum GS, Şenyurt S, Bektaş M. 2022. Salkowski curves and their modified orthogonal frames in J New Theory, 40: 12-26.
• Nurkan KS, Güven Aİ, Karacan M. K. 2019. Characterizations of adjoint curves in Euclidean 3-space. Proc National Acad Sci, India Section A: Phys Sci, 89: 155-161.
• Nurkan KS. Güven Aİ. 2022. A new approach for Smarandache curves. Turkish J Math Comput Sci, 14: 155-165.
• O’Neill B. 1996. Elementary differential geometry. Academic Press, Inc, New York, USA, pp: 154.
• Saad MK, Yüksel N, Oğraş N, Alghamdi F, Abdel-Salam AA. 2024. Geometry of tubular surfaces and their focal surfaces in Euclidean 3-space. AIMS Math, 9: 12479-12493.
• Sasai T. 1984. The fundamental theorem of analytic space curves and apparent singularities of Fuchsian differential equations. Tohoku Math J, Second Series, 36(1):17-24.
• Yuksel N, Tuncer Y, Karacan MK. 2011. Tabular surfaces with Bishop frame of Weingarten types in Euclidian 3-Space. Acta Univer Apulensis, 27: 39-50.
• Yüksel N, Saltık B, Damar E. 2022. Parallel curves in Minkowski 3-space. Gümüşhane Üniv Fen Bilim Derg, 12: 480-486.
• Xu Z, Feng R, Sun J. G. 2006. Analytic and algebraic properties of canal surfaces. J Comput Appl Math, 195: 220-228.