magazinelogo

Journal of Applied Mathematics and Computation

ISSN Print: 2576-0645 Downloads: 152862 Total View: 1835376
Frequency: quarterly ISSN Online: 2576-0653 CODEN: JAMCEZ
Email: jamc@hillpublisher.com
Article Open Access http://dx.doi.org/10.26855/jamc.2021.03.007

Rotation Minimizing Frame and Rectifying Curves in E_1^n

Özgür Keskin*, Yusuf Yayli

Faculty of Science, Department of Mathematics, Ankara University, Ankara, Turkey.

*Corresponding author: Özgür Keskin

Published: March 17,2021

Abstract

In this paper, some applications of a Rotation minimizing frame (RMF) are studied in E_1^4 and in E_1^n for timelike, spacelike curves. Firstly, in E_1^4, a Rotation minimizing frame (RMF) is obtained on the timelike and spacelike direction curves ∫ N(s) ds. The features of this Rotation minimizing frame are expressed. Secondly, it is determined when the timelike and spacelike curves can be rectifying curves. In addition, it has been investigated the conditions under which timelike and spacelike curves can be sphere calcurves. Then, a new characterization of rectifying curves is given, similar to the characterization of spherical curves. Finally, this Rotation minimizing frame is generalized in E_1^n for timelike, spacelike curves. In E_1^n, the conditions being a spherical curve and arectifying curve are given thank to this frame for timelike and spacelike curves. Also, a relationship between the spherical curve and the rectifying curve is stated. It is shown that the coefficients used in obtaining rectifying curves are constant numbers.

References

[1]    Bishop, L. R. (1975). There is more than one way to frame a curve. Amer. Math. Monthly, 82(3): 246-251.

[2] Etayo, F. (2016). Rotation Minimizing Vector Fields and Frames in Riemannian Manifolds. Geometry, Algebra and Applications: From Mechanics to Cryptography, 161, 91-100.

[3] Etayo, F. (2018). Geometric Properties of RM vector field along curves in Riemannian Manifolds. Turkish Journal of Mathematics, 42, 121-130.

[4] Jüttler, B. (1998). Rotation Minimizing Spherical Motions. Advances in Robot Kinematics: Analysis and Control, pp. 413-422.

[5] Wang, W., Jüttler, B., Zheng, D., and Liu, Y. (2008). Computation of Rotation Minimizing Frame. ACM Transactions on Graphics, 27(1), Article No. 2: 18 pages.

[6] Cambie, S., Geomans, W., and Bussche, I. W. D. (2006). Rectifying curves in the n-dimensional Euclidean space. Turk. J. Math., 40: 210-223.

[7] Ilarslan, K., Nešović, E., and Petrović-Torgav ̌ev, M. (2003). Some characterizations of rectifying curves in the Minkowski 3-space. Novi Sad J. Math., Vol. 2003, 33(2), 23-32.

[8] López, R. (2014). Differential Geometry of Curves and Surfaces in Lorentz-Minkowski Space. Int. Electron. J. Geom., 7(1), 44-107.

[9] Walrave, J. (1995). Curves and Surfaces in Minkowski Space. K.U. LEUVEN Faculteit Der Wetenschappen (PHD Thesis). P. 147.

[10] Gökçelik, F., Bozkurt, Z., Gök, I., Ekmekci, F. N., and Yaylı, Y. (2014). Parallel Transport Frame in 4-dimensional Euclidean Space E^4. Caspian Journal of Mathematical Sciences (CJMS), 3(1), 91-103.

[11] Ahmad, T. A. and Önder, M. (2012). Some Characterizations of Rectifying Spacelike Curves in the Minkowski Space-Time. Global Journal of Science Frontier Research Mathematics & Decision Sciences, 12(1), p. 9.

[12] Wong, Yung-Chow. (1972). On explicit characterization of spherical curves. Proceedings of the American Mathematical Society, 34(1): 239-242.

[13] Petrović-Torgav ̌ev, M. and Šućurović, E. (2001). Some Characterizations of the Lorentzian Spherical Timelike and Null Curves. Matematıqkı Vesnık, 53, 21-27.

[14] Petrović-Torgav ̌ev, M. and Šućurović, E. (2000). Some Characterizations of the Lorentzian Spherical Spacelike Curves with the Timelike and the Null Principal Normal. Mathematica Moravıca., 4, 83-92.

[15] Chen, B. Y. (2003). When does the position vector of a space curve always lie in its rectifying plane? Amer. Math. Monthly, 110(2): 147-152.

[16] Ilarslan, K. and Nešović, E. (2008). Some characterizations of rectifying curves in the Euclidean space E^4. Turk. J. Math., 32, 21-30.

How to cite this paper

Rotation Minimizing Frame and Rectifying Curves in E_1^n

How to cite this paper: Özgür Keskin, Yusuf Yayli. (2021) Rotation Minimizing Frame and Rectifying Curves in E_1^n. Journal of Applied Mathematics and Computation5(1), 56-67.

DOI: http://dx.doi.org/10.26855/jamc.2021.03.007