Bertrand partner P-trajectories in the Euclidean 3-space $E^3$

Year 2023, Volume: 72 Issue: 1, 216 - 228, 30.03.2023

Zehra İşbilir , Kahraman Esen Özen , Murat Tosun

https://doi.org/10.31801/cfsuasmas.1094170

https://izlik.org/JA85UC75JL

Abstract

The concept of a pair of curves, called as Bertrand partner curves, was introduced by Bertrand in 1850. Bertrand partner curves have been studied widely in the literature from past to present. In this study, we take into account of the concept of Bertrand partner trajectories according to Positional Adapted Frame (PAF) for the particles moving in 3-dimensional Euclidean space. Some characterizations are given for these trajectories with the aid of the PAF elements. Then, we obtain some special cases of these trajectories. Moreover, we provide a numerical example.

Keywords

Kinematics of a particle , Bertrand curves , positional adapted frame

References

• Bertrand, J., Memoire sur la theorie des courbes `a double courbure, J. Math. Pures Appl., (1850) 332–350.
• Bishop, R. L., There is more than one way to frame a curve, Amer. Math. Monthly, 82(3) (1975) 246–251. https://doi.org/10.2307/2319846
• Burke, J. F., Bertrand curves associated with a pair of curves, Math. Mag., 34(1) (1960), 60–62. https://doi.org/10.2307/2687860
• Casey, J., Siacci’s resolution of the acceleration vector for a space curve, Meccanica, 46 (2011), 471–476. https://doi.org/10.1007/s11012-010-9296-x
• da Silva, L. C., Differential Geometry of Rotation Minimizing Frames, Spherical Curves, and Quantum Mechanics of a Constrained Particle, Ph.D Thesis, 2017, (2018), arXiv preprint, arXiv:1806.08830.
• Darboux, G., Le¸cons Sur La Theorie Generale Des Surfaces I-II-III-IV, Gauthier Villars, Paris, 1896.
• Dede, M., Ekici, C., Directional Bertrand curves, Gazi Univ. J. Sci., 31(1) (2018), 202–211.
• Dede, M., Aslan, M. Ç., Ekici, C., On a variational problem due to the B-Darboux frame in Euclidean 3-space, Math. Methods Appl. Sci., 44(17) (2021), 12630–12639. https://doi.org/10.1002/mma.7567
• Goldstein, H., Poole, C. P., Safko, J. L., Classical Mechanics, Vol. 2, Reading, MA: Addison-Wesley, 1950.
• Görgülü, A., A generalization of the Bertrand curves as general inclined curves in $E^n$, Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat., 35(01.02) (1986), 54–60. https://doi.org/10.1501/Commua1 0000000254
• Gürbüz, N. E., The evolution of an electric field with respect to the type-1 PAF and the PAFORS frames in $R^3_1$, Optik, 250(1) (2022), 168285. https://doi.org/10.1016/j.ijleo.2021.168285
• Izumiya, S., Takeuchi, N., Generic properties of helices and Bertrand curves, J. Geom., 74 (2002), 97–109. https://doi.org/10.1007/PL00012543
• İlarslan, K., Nesovic, E., Some characterizations of osculating curves in the Euclidean spaces, Demonstr. Math., 41(4) (2008), 931–939. https://doi.org/10.1515/dema-2008-0421
• Kazaz, M., Uğurlu, M. M., Önder, M., Oral, S., Bertrand partner D-curves in the Euclidean 3-space $E^3$, Afyon Kocatepe Üniversitesi Fen ve Mühendislik Bilimleri Dergisi, 16(1) (2016), 76–83. https://doi.org/10.5578/fmbd.25270
• Özen, K. E., Tosun, M., A new moving frame for trajectories with non-vanishing angular momentum, J. Math. Sci. Model., 4(1) (2021), 7–18. https://doi.org/10.33187/jmsm.869698
• Özen, K. E., Tosun, M., Trajectories generated by special Smarandache curves according to Positional Adapted Frame, Karamanoğlu Mehmetbey University Journal of Engineering and Natural Sciences, 3(1) (2021), 15–23.
• Papaioannou, S. G., Kiritsis, D., An application of Bertrand curves and surfaces to CADCAM, Computer-Aided Design, 17(8) (1985), 348–352. https://doi.org/10.1016/0010-4485(85)90025-9
• Shifrin T., Differential Geometry: A First Course in Curves and Surfaces, University of Georgia, Preliminary Version, 2008.
• Solouma, E. M., Characterization of Smarandache trajectory curves of constant mass point particles as they move along the trajectory curve via PAF, Bulletin of Mathematical Analysis and Applications, 13(4) (2021), 14–30. https://doi.org/10.54671/bmaa-2021-4-2
• Whittemore, J. K., Bertrand curves and helices, Duke Math. J., 6(1) (1940), 235–245. https://doi.org/10.1215/S0012-7094-40-00618-4
• Yerlikaya, F., Karaahmetoğlu, S., Aydemir, İ., On the Bertrand B-pair curve in 3-dimensional Euclidean space, Journal of Science and Arts, 3(36) (2016), 215–224.