A new approach to curve couples with Bishop frame

Year 2024, Volume: 73 Issue: 3, 674 - 683, 27.09.2024

Faik Babadağ , Ali Atasoy

https://doi.org/10.31801/cfsuasmas.1329210

Abstract

This paper presents a detailed study of a new generation of the Bishop frame with components including three orthogonal unit vectors, which are tangent vector, normal vector and binormal vector. It is a frame field described on a curve in Euclidean space, which is an alternative to the Frenet frame. It is useful for curves for which the second derivative is not available. Moreover, the conditions which the Bishop frame of one curve coincides with the Bishop frame of another curve are defined. It would be valuable to replicate similar approaches in the Bishop frame of one curve coincides with the Bishop frame of another curve.

Keywords

Bishop frame, Frenet frame, curvature, regular curve, position vector

References

• Bishop, L. R., There is more than one way to frame a curve, The American Mathematical Monthly, 82(3) (1975), 246-251. https://doi.org/10.2307/2319846
• Chen, B. Y., Dillen, F., Rectifying curves as centrodes and extremal curves, Bull. Inst. Math. Academia Sinica, 33(2) (2005), 77-90.
• Kuhnel, W., Differential Geometry: Curves-Surfaces-Manifolds, Wiesbaden, Germany, 2003.
• Millman, R. S., Parker, G. D., Elements of Differential Geometry, Prentice-Hall, New Jersey, 1977.
• Chen, B. Y., When does the position vector of a space curve always lie an its rectifying plane?, The American Mathematical Monthly, 110 (2003), 147-152. https://doi.org/10.2307/3647775
• Azak, A. Z., Masal, M., Bertrand curves and Bishop frame in the 3-Dimensional Euclidean Space, Sakarya Universty Journal of Science 21(6) (2017), 1140-1145. https://doi.org/10.16984/saufenbilder.267557
• Bükcü, B., Karacan, M. K., The Bishop Darboux rotation axis of the spacelike curve in Minkowski 3-space, Journal of the Faculty of Science, Ege University, 30 (2007), 1-5.
• Bükcü, B., Karacan, M. K., On the slant helices according to Bishop frame of the timelike curve in Lorentzian space, Tamkang Journal of Mathematics, 39(3) (2008), 255-262. https://doi.org/10.5556/j.tkjm.39.2008.18
• Bükcü, B., Karacan, M. K., The slant helices according to Bishop frame, International Journal of Mathematical and Computational Sciences, 3(11) (2009), 67-70.
• Babadağ, F., On similar partner curves in Bishop frames with variable transformations, International Journal of New Technology and Research (IJNTR), 2(4) (2016), 59-64.
• Izumiya, S., Takeuchi, N., New special curves and developable surfaces, Turkish Journal of Mathematics, 28(2) (2004), 153-164.
• Babadağ, F., On dual similar partner curves in dual 3-space, British Journal of Mathematics and Computer Science, 17(4) (2016), 1-8. https://doi.org/10.9734/BJMCS/2016/26242
• İlarslan, K., Nesovic, E., Some characterizations of rectifying curves in the Euclidean space $E^4$, Turkish Journal of Mathematics, 32(1) (2008), 21-30.
• Spivak, M., A Comprehensive Introduction to Differential Geometry, Publish or Perish Inc., 1999.
• Carmo, M., Differential Geometry of Curves and Surfaces, New Jersey, NJ, USA Prentice Hall Inc., 1976.
• Karakuş, S. Ö., İlarslan, K, Yaylı, Y., A new approach characterization of curve couples in Euclidean 3-space, Honam Mathematical Journal, 36(1) (2014), 113-129. https://doi.org/10.5831/HMJ.2014.36.1.113