On The Curves $N-T^{\ast }N^{\ast }$ in $E^3$

Abstract

In this paper we have defined and examined the new kind curves, with the principal normal vector of the first curve and the vector lying on the osculator plane of the second curve are linearly dependent. As a result we have called these new curves as $N-T^{\ast }N^{\ast }$ curves. Also similiar to the other offset curves under the spesific condition, we give Frenet apparatus of the second curve based on the Frenet apparatus of the first curve.

Keywords

• Associated curves
• Mannheim curves
• Bertrand pairs

References

1. Gray, A., Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, p. 205, 1997.
2. Hacisalihoğlu, H.H., Diferensiyel Geometri, Cilt 1, İnönü Üniversitesi Yayinlari, Malatya 1994.
3. İlarslan, K., Nesovic, E., Some characterizations of osculating curves in the Euclidean spaces, Demonstratio Mathematica, 16(4)(2008), 931--939.
4. Kılıçoğlu, Ş., Şenyut, S., An examination on NP* curves in $E^3$, Turk. J. Math. Comput. Sci, 12(1)(2020), 26--30.
5. Körpınar, T., Sarıaydın, M.T., Turhan, E., Associated curves according to Bishop frame in Euclidean 3-space, AMO, 15(2015), 71.
6. Lipschutz, M.M., Diferential Geometry, Schaum's Outlines.
7. Liu, H., Wang, F., Mannheim partner curves in 3-space, Journal of Geometry, 88(1)(2008), 120--126.
8. Schief, W.K., On the integrability of Bertrand curves and Razzaboni surfaces, Journal of Geometry and Physics, 45(1-2)(2003), 130--150.