Some Characterizations on Geodesic, Asymptotic and Slant Helical Trajectories According to PAFORS

Abstract

In this paper, we study the geodesic, asymptotic and slant helical trajectories according to PAFORS in three-dimensional Euclidean space and give some characterizations on them. Also, we explain how we determine the helix axis for slant helical trajectories (according to PAFORS). Moreover, we develop a method which enables us to find the slant helical trajectory (if exists) lying on a given implicit surface which accepts a given fixed unit direction as an axis and a given angle as the constant angle. This method also gives information when the desired slant helical trajectory does not exist. The results obtained here involve some differential and partial differential equations or they are based on these equations. The aforementioned results are new contributions to the field and they may be useful in some specific applications of particle kinematics and differential geometry.

Keywords

• kinematics of a particle
• regular surfaces
• slant helix
• differential equations
• initial value problem

References

1. B. O'Neil, Elemantary differential geometry, Academic Press, Newyork, 1966.
2. F. Doğan and Y. Yaylı, Tubes with Darboux frame, Int. J. Contemp. Math. Sci., 7(16) (2012), 751-758.
3. S. Kızıltuğ and Y. Yaylı, Timelike tubes with Darboux frame in Minkowski 3-space, International Journal of Physical Sciences, 8(1) (2013), 31-36.
4. Ö. Bektaş and S. Yüce, Smarandache curves according to Darboux frame in E3, Romanian Journal of Mathematics and Computer Science, 3(1) (2013), 48-59.
5. B. Altunkaya and F. K. Aksoyak, Curves of constant breadth according to Darboux frame, Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, 66(2) (2017), 44-52.
6. G. Y. Şentürk and S. Yüce, Bertrand offsets of ruled surfaces with Darboux frame, Results in Mathematics, 72(3) (2017), 1151-1159.
7. T. Körpınar and Y. Ünlütürk, An approach to energy and elastic for curves with extended Darboux frame in Minkowski space, AIMS Mathematics, 5(2) (2020), 1025-1034.
8. K. E. Özen and M. Tosun, A new moving frame for trajectories on regular surfaces, Accepted for publication in Ikonion Journal of Mathematics (2021).

9. T. Shifrin, Differential geometry: A first course in curves and surfaces, University of Georgia, Preliminary Version, 2008.
10. S. Izumiya and N. Takeuchi, New special curves and developable surfaces. Turk. J. Math., 28(2) (2004), 153-163.
11. B. Bükcü and M. K. Karacan, The slant helices according to Bishop frame, Int. J. Comput. Math. Sci., 3(2) (2009), 67-70.
12. A. T. Ali and M. Turgut, Some characterizations of slant helices in the Euclidean space En. Hacet. J. Math. Stat., 39(3) (2010), 327-336.
13. O. Z. Okuyucu, İ. Gök, Y. Yaylı and N. Ekmekci, Slant helices in three dimensional Lie groups, Appl. Math. Comput., 221 (2013), 672-683.
14. P. Lucas and J. A. Ortega-Yagües, Helix surfaces and slant helices in the three-dimensional anti-De Sitter space. RACSAM, 111(4) (2017), 1201-1222.
15. N. Macit and M. Düldül, Relatively normal-slant helices lying on a surface and their characterizations, Hacet. J. Math. Stat., 46(3) (2017), 397-408.
16. K. E. Özen and M. Tosun, A new moving frame for trajectories on regular surfaces, Accepted for publication in Journal of Mathematical Sciences and Modelling (2021).