Bernsteinn polynomials approach to determine timelike curves of constant breadth in Minkowski 3-space

Year 2018, Volume: 3 Issue: 2, 9 - 22, 30.08.0208

Tuba Agirman Aydin , Mehmet Sezer Huseyin Kocayigit

Abstract

In this study, we first show that the system of Frenet-like differential equation characterizing timelike curves of constant
breadth is equivalent to a third order, linear, differential equation with variable coefficients. Then, by using a rational approximation
based on Bernstein polynomials, we obtain the set of solution of the mentioned differential equation under the given initial conditions.
Furthermore, we discuss that the obtained results are useable to determine timelike curves of constant breadth in Minkowski 3-space
E₁
³.

Keywords

Timelike curve of constant breadth, Bernstein polynomials, Linear differential equations

References

• [1] A. Magden, O. Köse, ¨ On The Curves Of Constant Breadth In E4 Space, Turk J. Math.,(21), (1997), 277-284.
• [2] A.P. Mellish, Notes On Differential Geometry, Ann Of Math. (2)32, no.1, (1931), 181-190.
• [3] F. Reuleaux, The Kinematics Of Machinery, Trans. By Kennedy A.B.W., Dover Pub. (1963), New York.
• [4] H. Gluck. Higher Curvatures of Curves In Euclidean Space, Proc. Amer. Math. Montly,(73)(1966),699-704.
• [5] H.H. Hacisalihoglu, Diferensiyel Geometri, Ankara Universitesi Fen Fak¨ultesi, (1993), Ankara, 269s.
• [6] J. Walrave, Curves and Surfaces In Minkowski Space, Ph. D. Thesis (1995), K. U. Leuven, Faculty Of Sciences, Leuven. [7] L. Euler, De Curvis trangularibis, Acta Acad. Petropol, (1778, 1780), 3-30.
• [8] M. Fujivara, On Space Curves of Constant Breadth, Thoku Math. J.(5), (1914), 179-184.
• [9] M. Onder, H. Kocayi˘git, E. Candan, Differential Equations Characterizing Timelike and Spacelike Curves of Constant Breadth ˙In Minkowski 3-Space E. J. Korean Math. Soc.(48), no.4, (2011), 849-866.
• [10] M. Sezer, Differential Equations Characterizing Space Curves of Constant Breadth and A Criterion For These Curves, Doga TU J. Math., 13 (2), (1989), 70-78.
• [11] M. Sezer, Integral Characterizations For A System of Frenet Like Differential Equations and Applications, E. U. Faculity of Science, Series Of Scientific Meetings, (1), (1991), 435-444.
• [12] M.I. Bhatti, B. Brocken, Solutions of Differential Equations ˙In A Bernstein Polynomial Basis. Journal of Computational And Applied Mathematics. (205), (2007), 272-280.
• [13] O.R. Isik, M. Sezer and Z. Güney, A rational approximation based on Bernstein polynomials for high order initial and boundary values problems, Applied Mathematics and Computation, 217, (2011), 9438-9450.
• [14] O. Köse, Düzlemde Ovaller ve Sabit Genislikli Egrilerin bazı özellikleri, Doga Bilim Dergisi, Seri B, (2), (1984), 119-126. [15] O. Köse, On Space Curve of Constant Breadth, Doga TU J. Math.,(1), (1986), 11-14.
• [16] T.A. Aydin, Differential Equations Characterizing Curves of Constant Breadth And Spherical Curves In En-Space and Their Solutions, Ph. D. Thesis (2014), Mugla Sıtkı Koc¸man Universty, Mugla.
• [17] V. Dannon, Integral Characterizations And The Theory of Curves, Proc. Amer. Math. Soc.,(4), (1981), 600-602.
• [18] Z. Akdogan, A. Magden, Some Characterization of Curves of Constant Breadth In En Space, Turk J. Math.,(25), (2001), 433-444.