The Characterizations of The Curve Generated by a Curve with Constant Torsion

Year 2023, Volume: 11 Issue: 1, 1 - 7, 30.04.2023

Nural Yüksel , Burçin Saltık , Murat Kemal Karacan

https://izlik.org/JA46RK87AL

Abstract

In this article, a curve generated by the curve which has a constant torsion is examined in 3-dimensional Euclidean space. And, the characterizations of this curve have been made and some important theorems have been given. It is seen that the $\overline{\alpha}$ is a curve with constant curvature and the relationships between the two curves $\alpha$ and $\overline{\alpha}$ are revealed. In addition, the conditions for this obtained curve to be helix, slant helix, Bertrand and Salkowski curves are given.

Keywords

constant torsion , Salkowski curve , Bertrand curve , slant helix

References

• [1] O’Neill, B., Semi Riemannian geometry with applications to relativity, Academic Press, Inc. New York, 1983.
• [2] L. M. Bates and O. M. Melko, On curves of constant torsion I, J. Geom. Vol:104, No.2 (2013), 213-227.
• [3] J.M. Bertrand, Memoire sur la theorie des courbes a double courbure, Journal de Mathematiques Pures et Appliquees Vol:15 (1850), 332-350.
• [4] Do Carmo, M. P., Differential Geometry of Curves and Surfaces, Prentice-Hall Inc., Englewood Cliffs, N.J., 1976.
• [5] Hacısaliho˘glu, H.H., Diferensiyel Geometri, Ertem Matbaası, Ankara, 2000.
• [6] T. A. Ivey, Minimal curves of constant torsiont, Proceedings of the American Mathematical Society Vol:128, No.7 (2000), 2095–2103.
• [7] S.Izumiya and N. Takeuchi, Generic properties of helices and Bertrand curves, Journal of Geometry Vol:74 (2002), 97-109.
• [8] S.Izumiya and N. Takeuchi, New special curves and developable surfaces, Turk. J. Math. Vol:28 (2004), 531-537.
• [9] D. Kazaras and I. Sterlingan, Explicit formula for spherical curves with constant torsion, Pasific Journal of Mathematics Vol:259 (2012), 361-372.
• [10] J. Monterde, Salkowski curves revisited: A family of curves with constant curvature and non-constant torsion, Computer Aided Geometric Design Vol:26 (2009), 271-278.
• [11] E. Salkowski, Zur transformation von raumkurven, Mathematische Annalen Vol:4, No.66 (1909), 517-557.
• [12] J. Weiner, Closed curves of constant torsion II, Proc. Amer. Math. Soc. Vol:67, No.2 (1997), 306-308.