Salkowski Curves and Their Modified Orthogonal Frames in $\mathbb{E}^{3}$
Abstract
In this study, we examine some properties of Salkowski curves in $\mathbb{E}^{3}$. We then make sense of the angle $(nt)$ in the parametric equation of the Salkowski curves. We provide the relationship between this angle and the angle between the binormal vector and the Darboux vector of the Salkowski curves. Through this angle, we obtain the unit vector in the direction of the Darboux vector of the curve. Finally, we calculate the modified orthogonal frames with both the curvature and the torsion and give the relationships between the Frenet frame and the modified orthogonal frames of the curve.
Keywords
References
- M. P. Do Carmo, Differential Geometry of Curves and Surfaces: Revised and Updated Second Edition, Courier Dover Publications, 2016.
- H. H. Hacısalihoğlu, Differantial Geometry, Ankara University Faculty of Science Press, Ankara, Türkiye, 2000.
- L. Kula, N. Ekmekci, Y. Yayli, K. İlarslan, Characterizations of Slant Helices in Euclidean 3-Space, Turkish Journal of Mathematics 34 (2) (2010) 261-274.
- A. T. Ali, Position Vectors of Slant Helices in Euclidean 3-Space, Journal of the Egyptian Mathematical Society 20 (1) (2012) 1-6.
- E. Salkowski, Zur Transformation Von Raumkurven Mathematische Annalen 66 (4) (1909) 517-557.
- J. Monterde, Salkowski Curves Revisited: A Family of Curves with Constant Curvature and Non-Consant Torsion, Computer Aided Geometric Design 26 (2009) 271-278.
- S. Gur, S. Senyurt, Frenet Vectors and Geodesic Curvatures of Spheric Indicators of Salkowski Curve in E3, Hadronic Journal 33 (5) (2010) 485-512.
- S. Şenyurt, B. Öztürk, Smarandache Curves of Salkowski Curve According to Frenet Frame, Turkish Journal of Mathematics and Computer Science 10 (2018) 190-201.
- T. Sasai, The Fundamental Theorem of Analytic Space Curves and Apparent Singularities of Fuchsian Differential Equations, Tohoku Math Journal 36 (1984) 17-24.
- T. Sasai, Geometry of Analytic Space Curves with Singularities and Regular Singularities of Differential Equations, Funkcialaj Ekvacioj 30 (1987) 283-303.