An Alternative Method for Determination of the Position Vector of a Slant Helix

Year 2023, Issue: 44, 97 - 105, 30.09.2023

Gizem Güzelkardeşler , Burak Şahiner

https://doi.org/10.53570/jnt.1356697

Cited By: 2

https://izlik.org/JA85UA26EF

Abstract

In this paper, we provide an alternative method to determine the position vector of a slant helix with the help of an alternative moving frame. We then construct a vector differential equation in terms of the principal normal vector of a slant helix using an alternative moving frame. By solving this vector differential equation, we determine the position vector of the slant helix. Afterward, we obtain parametric representations of some examples of slant helices for chosen curvature and torsion functions as an application of the proposed method. Finally, we discuss the method and whether further research should be conducted or not.

Keywords

Alternative moving frame , intrinsic equations , position vector , slant helix

References

• D. J. Struik, Lectures on Classical Differential Geometry, 2nd Edition, Dover, New York, 1988.
• N. Chouaieb, A. Goriely, J. H. Maddocks, Helices, Proceedings of the National Academy of Sciences 103 (25) (2006) 9398--9403.
• A. A. Lucas, P. Lambin, Diffraction by DNA, Carbon Nanotubes and Other Helical Nanostructures, Reports on Progress in Physics 68 (5) (2005) 1181--1249.
• C. D. Toledo-Suarez, On the Arithmetic of Fractal Dimension Using Hyperhelices, Chaos, Solitons \& Fractals 39 (1) (2009) 342--349.
• J. D. Watson, F. H. C. Crick, Generic Implications of the Structure of Deoxyribonucleic Acid, Nature 171 (1953) 964--967.
• S. Izumiya, N. Takeuchi, New Special Curves and Developable Surfaces, Turkish Journal of Mathematics 28 (2) (2004) 153--163.
• P. Hartman, A. Wintner, On the Fundamental Equations of Differential Geometry, American Journal of Mathematics 72 (4) (1950) 757--774.
• L. P. Eisenhart, A Treatise on Differential Geometry of Curves and Surfaces, Dover, New York, 1960.
• A. T. Ali, Position Vectors of Curves in the Galilean Space $G_3$, Matematički Vesnik 64 (3) (2012) 200--210.
• A. T. Ali, Position Vectors of General Helices in Euclidean 3-Space, Bulletin of Mathematical Analysis and Applications 3 (2) (2011) 198--205.
• A. T. Ali, Position Vectors of Slant Helices in Euclidean 3-Space, Journal of the Egyptian Mathematical Society 20 (1) (2012) 1--6.
• A. T. Ali, S. R. Mahmoud, Position Vector of Spacelike Slant Helices in Minkowski 3-Space, Honam Mathematical Journal 36 (2) (2014) 233--251.
• A. T. Ali, M. Turgut, Position Vector of a Time-like Slant Helix in Minkowski 3-Space, Journal of Mathematical Analysis and Applications 365 (2) (2010) 559--569.
• H. G. Bozok, S. A. Sepet, M. Ergüt, Position Vectors of General Helices According to Type-2 Bishop Frame in $E^3$, Mathematical Sciences and Applications E-Notes 6 (1) (2018) 64--69.
• A. El Haimi, A. O. Chahdi, Parametric Equations of Special Curves Lying on a Regular Surface in Euclidean 3-Space, Nonlinear Functional Analysis and Applications 26 (2) (2021) 225--236.
• A. El Haimi, M. Izid, A. O. Chahdi, Position Vectors of Curves Generalizing General Helices and Slant Helices in Euclidean 3-Space, Tamkang Journal of Mathematics 52 (4) (2021) 467--478.
• H. Öztekin, S. Tatlıpınar, Determination of the Position Vectors of Curves from Intrinsic Equations in $G_3$, Walailak Journal of Science and Technology 11 (12) (2014) 1011--1018.
• T. Şahin, B. C. Dirişen, Position Vectors of Curves with respect to Darboux Frame in the Galilean space $G^3$, Communications Faculty of Sciences University of Ankara Series A1: Mathematics and Statistics 68 (2) (2019) 2079--2093.
• B. Yılmaz, A. Has, New Approach to Slant Helix, International Electronic Journal of Geometry 12 (1) (2019) 111--115.
• B. Uzunoğlu, İ. Gök, Y. Yaylı, A New Approach on Curves of Constant Precession, Applied Mathematics and Computation 275 (2016) 317--323.
• P. D. Scofield, Curves of Constant Precession, American Mathematical Monthly 102 (6) (1995) 531--537.
• B. Şahiner, Ruled Surfaces According to Alternative Moving Frame (2019) 16 pages, https://arxiv.org/abs/1910.06589.