Characterization of Tzitzeica Curves Using Positional Adapted Frame

Year 2022, Volume: 10 Issue: 2, 260 - 268, 31.10.2022

Kahraman Esen Özen , Zehra İşbilir , Murat Tosun

Abstract

In this study, Tzitz\'eica curves are taken into consideration in the Euclidean 3-space by using the Positional Adapted Frame (PAF). Such curves are characterized according to PAF elements. Also, some results are obtained on spherical Tzitz\'eica curves. The results obtained in this study are new contributions to the field. It is expected that these results will be useful in various application areas of differential geometry and applied mathematics in the future.

Keywords

Kinematics of a particle, Positional Adapted Frame, Tzitz\, Differential equations

Thanks

The authors would like to thank editors and anonymous referees for their valuable comments and carefully reading.

References

• [1] A. F. Agnew, A. Bobe, W. G. Boskoff and B. D. Suceava, Tzitz´eica curves and surfaces, The Math. J., 12 (2010), 1-18.
• [2] M. E. Aydın and M. Erg¨ut, Non-null curves of Tzitz´eica type in Minkowski 3-space, Rom. J. Math. Comput. Sci., 4(1) (2014), 81-90.
• [3] B. Bayram, E. Tunc¸, K. Arslan and G. O¨ ztu¨rk, On Tzitze´ica curves in Euclidean 3-space E3, Facta Univ. Ser. Math. Inform., 33(3) (2018), 409-416. [4] R. L. Bishop, There is more than one way to frame a curve, Am. Math. Mon., 82(3) (1975), 246-251.
• [5] M. Crˆas¸m˘areanu, Cylindrical Tzitz´eica curves implies forced harmonic oscillators, Balkan J. Geom. Appl., 7(1) (2002), 37-42.
• [6] K. Eren and S. Ersoy, Characterizations of Tzitz´eica curves using Bishop frames, Math. Meth. Appl. Sci., (2021), 1-14. https://doi.org/10.1002/ mma.7483
• [7] H. Goldstein, C. P. Poole and J. L. Safko, Classical Mechanics (Vol. 2) Reading, MA., Addison-Wesley, 1950.
• [8] N. E. G¨urb¨uz, The evolution of an electric field with respect to the type-1 PAF and the PAFORS frames in R31 , Optik, 250(1) (2022), 168285.
• [9] M. K. Karacan and B. B¨ukc¨u, On the elliptic cylindrical Tzitz´eica curves in Minkowski 3-space, Scientia Magna, 5(3) (2009), 44-48.
• [10] K. E. O¨ zen and M. Tosun, A new moving frame for trajectories with non-vanishing angular momentum, J. Math. Sci. Model., 4(1) (2021), 7-18.
• [11] K. E. O¨ zen and M. Tosun, Trajectories generated by special Smarandache curves according to Positional Adapted Frame, Karamanog˘lu Mehmetbey University Journal of Engineering and Natural Sciences, 3(1) (2021), 15-23.
• [12] T. Shifrin, Differential Geometry: A First Course in Curves and Surfaces, University of Georgia, Preliminary Version, 2008.
• [13] M. A. Soliman, N. H. Abdel-All, R. A. Hussien and T. Youssef, Evolution of space curves using type-3 Bishop frame, Caspian J. Math. Sci., 8(1) (2019), 58-73.
• [14] E. M. Solouma, 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.
• [15] M. G. Tzitz´eica, Sur une nouvelle classe de surfaces, Rendiconti del Circolo Matematico di Palermo (1884-1940), 25(1) (1908), 180-187.
• [16] M. G. Tzitz´eica, Sur certaines courbes gouches, Annales Scientifiques de l’ ´ Ecole Normale Sup´erieure, 28 (1911), 9-32.
• [17] S. Yılmaz and M. Turgut, A new version of Bishop frame and an application to spherical images, J. Math. Anal. Appl., 371(2) (2010), 764-776.