Constructing $k$-Slant Curves in Three Dimensional Euclidean Spaces

Year 2025, Issue: 50, 98 - 115, 28.03.2025

Çetin Camcı

https://doi.org/10.53570/jnt.1647509

Abstract

Helices and constant procession curves are special examples of slant curves. However, there is no example of a $k$-slant curve for a positive integer $k\geq 2$ in three dimensional Euclidean spaces. Furthermore, the position vector of a $k$-slant curve for a positive integer $k\geq 2$ has not been known thus far. In this paper, we propose a method for constructing $k$-slant curves in three dimensional Euclidean spaces. We then show that spherical $k$-slant curves and $N_{k}$-constant procession curves can be derived from circles, for $k \in \mathbb{N}$, the set of all nonnegative integers. In addition, we provide a new proof of the spherical curve characterization and define a curve in the sphere called a spherical prime curve. Afterward, we apply $k$-slant curves to magnetic curves. Finally, we discuss the need for further research.

Keywords

General helices, spherical curves, slant curves

References

• R. Blum, A remarkable class of Mannheim curves, Canadian Mathematical Bulletin 9 (1966) 223–228.
• S. Izumiya, N. Takeuchi, New special curves and developable surfaces, Turkish Journal of Mathematics 28 (2004) 153–163.
• L. Kula, Y. Yaylı, On slant helix and its spherical indicatrix, Applied Mathematics and Computation 169 (1) (2005) 600–607.
• M. Anton, Characterization of the slant helix as successor curve of the general helix, International Electronic of Geometry 7 (2) (2014) 84–91.
• Ç. Camcı, L. Kula, M. Altınok, On spherical slant helices in euclidean 3-space (2013), https://arxiv.org/abs/1308.5532, Accessed 31 Jan 2025.
• A. T. Ali, Position vectors of slant helices in Euclidean $3$-space, Journal of the Egyptian Mathematical Society 20 (1) (2012) 1–6.
• T. Takahashi, N. Takeuchi, Clad helices and developable surfaces, Bulletin of Tokyo University 66 (2014) 1–9.
• P. D. Scofield, Curves of constant precession, American Mathematical Monthly 102 (1995) 531–537.
• B. Uzunoğlu, İ. Gök, Y. Yaylı, A new approach on curves of constant precession, Applied Mathematics and Computation 275 (2016) 317–323.
• F. Ates, I. Gök, N. F. Ekmekci, A new kind of slant helix in Lorentzian (n+2)-spaces}, Kyungpook Mathematical Journal 56 (3) (2016) 1003–1016.
• J. E. Lee, On slant curves in Sasakian Lorentzian 3-manifolds, International Electronic Journal of Geometry 13 (2) (2020) 108–115.
• S. Uddin, M. S. Stankovic, M. Iqbal, S. K. Yadav, M. Aslam, Slant helices in Minkowski 3-space $E_1^3$ with Sasai’s modified frame fields, Filomat 36 (1) (2022) 151–164.
• A. Zhou, K. Yao, D. Pei, k-type hyperbolic framed slant helices in hyperbolic 3-space, Filomat 38 (11) (2024) 3839–3850.
• S. H. Khan, M. Jamali, C. Singh, Partially null and pseudo null slant helices of (k,m)-type in semi Euclidean space $R_2^4$, Palestine Journal of Mathematics 13 (3) (2024) 208–214.
• O. Ateş, İ. Gök, Y. Yaylı, A new representation for slant curves in Sasakian 3-manifolds, International Electronic Journal of Geometry 17 (1) (2024) 227–289.
• D. J. Struik, Lectures on classical differential geometry, Dover Publications, 1988.
• S. Izumiya, N. Takeuchi, Generic properties of helices and Bertrand curves, Journal of Geometry 74 (2002) 97–109.
• A. T. Ali, New special curves and their spherical indicatrices (2009), https://arxiv.org/abs/0909.2390}, Accessed 31 Jan 2025.
• C. Ramis, B. Uzunoglu, Y. Yaylı, New associated curve $k$-principle direction curves and $N_{k}$-slant helix (2014), https://arxiv.org/abs/1404.7369, Accessed 31 Jan 2025.
• W. Blaschke, Bemerkungen Über allgemeine schraubenlinien, Monatshefte für Mathematik und Physik 19 (1908) 188–204.
• E. Kreyszig, Differential geometry, Dover Publications, 1991.
• S. Breuer, D. Gottlieb, Explicit characterization of spherical curves, Proceedings of the American Mathematical Society 27 (1971) 126–127.
• Y. C. Wong, On an explicit characterization of spherical curves}, Proceedings of the American Mathematical Society 34 (1) (1972) 239–242.
• W. W. Bell, Special functions for scientists and engineers, Dover Publications, 2004.
• D. E. Blair, Contact manifolds in Riemannian geometry, Springer, 1976.
• Ç. Camcı, Extended cross product in a 3-dimensional almost contact metric manifold with applications to curve theory, Turkish Journal of Mathematics 35 (2011) 1–14.
• M. Barros, A. Romero, J. L. Cabrerizo, M. Fernandez, The Gauss-Landau-Hall problem on Riemannian surfaces, Journal of Mathematical Physics 46 (2005) 112905.
• J. L. Cabrerizo, Magnetic fields in 2D and 3D sphere, Journal of Nonlinear Mathematical Physics 20 (3) (2013) 440–450.
• Z. Bozkurt, İ. Gök, Y. Yaylı, F. N. Ekmekci, A new approach for magnetic curves in 3D Riemannian manifolds, Journal of Mathematical Physics 55 (2014) 053501.