A New Approach For Smarandache Curves

Year 2022, Volume: 14 Issue: 1, 155 - 165, 30.06.2022

Semra Kaya Nurkan , İlkay Güven

https://doi.org/10.47000/tjmcs.1004423

Cited By: 2

Abstract

In this paper, we introduce new adjoint curves which are associated curves in Euclidean space of three dimension. They are generated with the help of integral curves of special Smarandache curves. We attain some connections between Frenet apparatus of these new adjoint curves and main curve. We characterize these curves in which conditions they are general helix and slant helix. Finally, we exemplify them with figures.

Keywords

Adjoint curve, helix, slant helix

References

• Abdel-Aziz, H.S., Khalifa Saad, M., Computation of Smarandache curves according to Darboux frame in Minkowski 3-space, J. of Egyptian Math. Society, 25(4), (2017), 382-390.
• Ali, A.T., Special Smarandache curves in the Euclidean space, International J. Math. Combin., 2(2010), 30-36.
• Barros, M., General helices and a theorem of Lancret, Proc. Am. Math. Society, 125(5)(1997), 1503-1509.
• Camcı, Ç. , İlarslan, K., Kula, L., Hacısalihoğlu, H.H., Harmonic curvatures and generalized helices in $E^{n}$, Chaos-Solutions and Fractals, 40(5)(2009), 2590-2596.
• Choi, J.H., Kim, Y.H., Associated curves of a Frenet curve and their applications, Applied Math. and Comp., 218(2012), 9116-9124.
• Choi, J.H., Kim, Y.H., Ali, A.T., Some associated curves of Frenet non-lightlike curves in $E_{1}^{3}$ , J. Math. Anal. Appl., 394(2012), 712-723.
• Deshmukh, S., Chen, B.Y., Algehanemi, A., Natural Mates of Frenet Curves in Euclidean 3-space, Turk. J. of Math., 42(2018), 2826-2840.
• Elzawy, M., Mosa, S., Smarandache curves in the Galilean 4-space $G_{4}$ , J. of Egyptian Math. Society, 25(1)(2017), 53-56.
• Hayden, H.A., On a general helix in Riemannian n-space, Proc. London Math. Society, 32(2)(1931), 37-45.
• Izumiya, S., Takeuchi, N., Special curves and ruled surfaces, Beitrage zur Alg. und Geo. Contributions to Alg. and Geo., 44(1)(2003), 203-212.
• Izumiya, S., Takeuchi, N., New special curves and developable surfaces, Turk J. Math., 28(2)(2004), 153-163.
• Kahraman, T., Ugurlu, H.H., Dual Smarandache curves and Smarandache ruled surfaces, Mathematical Sciences and Applications E-Notes, 2(1)(2014), 83-98.
• Millman, R.S., Parker, G.D., Elements of Differential Geometry, Englewood Cliffs, NJ, USA, Prentice Hall, 1977.
• Nurkan, S.K., Güven, I..A., Karacan, M.K., Characterizations of adjoint curves in Euclidean 3-space, Proceedings of the National Academy of Sciences, India Section A: Physical Sciences, 89(1)(2019), 155-161.
• Solouma, E.M., Special equiform Smarandache curves in Minkowski space-time, J. of Egyptian Math. Society, 25(3)(2017), 319-325.
• Şenyurt, S., Çalışkan, A., Smarandache curves of Mannheim curve couple according to Frenet frame , Mathematical Sciences and Applications E-Notes, 5(1)(2017), 122-136.
• Taşköprü, K., Tosun, M., Smarandache curves on $S^{2} $, Bol. Soc. Paran. Mat., 32(1)(2014), 51-59.
• Turgut, M., Yılmaz, S., Smarandache curves in Minkowski space time, International J. Math. Combin., 3(2008), 51-55.