A NOTE ON SOME CHARACTERIZATIONS OF CURVES DUE TO BISHOP FRAME IN EUCLIDEAN PLANE E2

Year 2016, Volume: 2 Issue: 2, 109 - 119, 25.12.2016

Süha Yılmaz , Yasin Ünlütürk

Abstract

In this paper, we first obtain the differential equation characterizing position vector of a regular

curve in Euclidean plane 2 E . Then we study the special curves such as Smarandache curves,

curves of constant breadth due to the Bishop frame in Euclidean plane 2 E . We give some

characterizations of these special curves due to the Bishop frame in Euclidean plane 2 E .

AMS Subject Classification: 53A35, 53A40, 53B25

Keywords

A regular plane curve, Euclidean plane, Bishop frame, Smarandache curves

References

• [1] A.T. Ali, Special Smarandache curves in the Euclidean space. Int J Math Comb 2:30-36 2010.
• [2] Bishop LR(1975) There is more than one way to frame a curve. Am Math Mon 82:246-251
• [3] M. Çetin, Y. Tuncer Y and M.K. Karacan, Smarandache curves according to Bishop frame in Euclidean 3-space, Gen. Math. Notes, 2014; 20: 50-56.
• [4] B.Y. Chen, When does the position vector of a space curve always lie in its rectifying plane?, Amer. Math. Monthly 110 (2003), 147-152.
• [5] L. Euler, De curvis triangularibus, Acta Acad. Petropol., 3-30, 1778 (1780).
• [6] Fujivara M (1914) On space curves of constant breadth. Tohoku Math J 5:179-184.
• [7] C. G. Gibson, Elementary geometry of differentiable curves. An undergraduate introduction. Cambridge University Press, Cambridge, 2001.
• [8] A. Gray, E. Abbena, S. Salamon, Modern differential geometry of curves and surfaces with Mathematica Chapman and Hall/CRC, Boca Raton, FL, (2006). 1
• [9] S. Izumiya, N. Takeuchi, New special curves and developable surfaces, Turkish J. Math. 28(2), 2004,531-537.
• [10] S. Izumiya, D. Pei, T. Sano, E. Torii, Evolutes of hyperbolic plane curves, Acta Math. Sin.(Engl. Ser.), 20 (2004), 543--550.
• [11] M.K. Karacan, B. Bükçü, Parallel curve (offset) in Euclidean plane, Erciyes Üniversitesi Fen Bilimleri Enstitüsü Dergisi 24 (1-2) 334- 345 (2008)
• [12] Köse Ö (1984) Some properties of ovals and curves of constant width in a plane. Doğa Turk J Math (8) 2:119-126
• [13] Köse Ö (1986) On space curves of constant breadth. Doğa Turk J Math (10)1:11--14
• [14] R. Lopez, The theorem of Schur in the Minkowski plane, Jour Geom Phys 61 (2011) 342-- 346
• [15] A. Mağden, Ö. Köse, On the curves of constant breadth in space, Turk. J. of Mathematics, 21(3) (1997), 277-284.
• [16] M. Turgut, S. Y lmaz, Smarandache curves in Minkowski space-time, International J. Math. Combin. 2008; 3,: 51-55.
• [17] Turgut (2009) Smarandache breadth pseudo null curves in Minkowski space-time. Int J Math Comb 1:46-49.