Characterizations of Adjoint Curves According to Alternative Moving Frame

Abstract

In this paper, the adjoint curve is defined by using the alternative moving frame of a unit speed space curve in 3-dimensional Euclidean space. The relationships between Frenet vectors and alternative moving frame vectors of the curve are used to offer various characterizations. Besides, ruled surfaces are constructed with the curve and its adjoint curve, and their properties are examined. In the last section, there are examples of the curves and surfaces defined in the previous sections.

Keywords

• Adjoint curve
• Alternative moving frame
• Distribution parameter
• Ruled surface

References

1. [1] A. T. Ali, Position vectors of slant helices in Euclidean space E3, J. Egypt Math. Soc., 20 (2012), 1-6.
2. [2] L. Kula, Y. Yaylı, On slant helix and its spherical indicatrix, Appl. Math. Comput., 169 (2005), 600-607.
3. [3] M. Babaarslan, Y. A. Tando˘gan, Y. Yaylı, A note on Bertrand curves and constant slope surfaces according to Darboux frame, J. Adv. Math. Stud., 5 (2012), 87-96.
4. [4] A. T. Ali, New special curves and their spherical indicatrices, Glob. J. Adv. Res. Class. Mod. Geom., 1 (2012), 28-38.
5. [5] J. H. Choi, Y. H. Kim, Associated curves of a Frenet curve and their applications, Appl. Math. Comput., 218 (2012), 9116-9124.
6. [6] S. K. Nurkan, ˙I. A. Güven, M. K. Karacan, Characterizations of adjoint curves in Euclidean 3-space, Prc. Natl. Acad. Sci. India Sect. A Phys. Sci., 89(1) (2019), 155-161.
7. [7] N. Macit, M. Düldül, Some new associated curves of a Frenet curve in E3 and E4, Turk J. Math., 38 (2014), 1023-1037.
8. [8] B. Uzunoğlu, I. Gök, Y. Yaylı, A new approach on curves of constant precession, Appl. Math. Comput., 275 (2016), 317-323.

9. [9] L. Buse, M. Elkadi, A. Galligo, A computational study of ruled surfaces, J. Symb. Comput., 44(3) (2009), 232-241.
10. [10] E. Damar, N. Yüksel, A. T. Vanlı, The ruled surfaces according to type-2 Bishop frame in E3, Int. Math. Forum, 12(3) (2017), 133-143.
11. [11] F. Güler, The timelike ruled surfaces according to type-2 Bishop frame in Minkowski 3-space, J. Sci. Arts, 18 (243) (2018), 323-330.
12. [12] Ş. Kılıçoğlu, H. H. Hacısaliho˘glu, On the ruled surfaces whose frame is the Bishop frame in the Euclidean 3-space, Int. Electron. J. Geom., 6(2) (2013), 110-117.
13. [13] Y. Liu, H. Pottmann, J. Wallner, Y. Yang, W. Wang, Geometric modeling with conical meshes and developable surfaces, ACM Trans. Graph., 25(3) (2006), 681-689.
14. [14] Y. Yu, H. Liu, S. D. Jung, Structure and characterization of ruled surfaces in Euclidean 3-space, Appl. Math. Comp., 233 (2014),252-259.
15. [15] B. O’Neill, Elementary Differential Geometry, Academic Press, New York, 1966.
16. [16] D. J. Struik, Lectures on Classical Differential Geometry, Dover Publication. Newyork, 1988.
17. [17] B. Şahiner, Ruled surfaces according to alternative moving frame, (2019), arXiv:1910.06589 [math.DG].
18. [18] S. Ouarab, A. O. Chahdi, M. Izid, Ruled surfaces with alternative moving frame in Euclidean 3-space, Int. J. Math. Sci. Eng. Appl., 12(2) (2018), 43-58.