Smarandache curves according to Sabban frame of fixed pole curve belonging to the Bertrand curves pair

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1 Smarandache curves according to Sabban frame of fixed pole curve belonging to the Bertrand curves pair Süleyman Şenyurt, Yasin Altun, and Ceyda Cevahir Citation: AIP Conference Proceedings 76, ; doi: 0.06/ View online: View Table of Contents: Published by the AIP Publishing Articles you may be interested in On characterizations of some special curves of timelike curves according to the Bishop frame of type- in Minkowski -space AIP Conf. Proc. 76, ; 0.06/ A variational characterization and geometric integration for Bertrand curves J. Math. Phys. 54, ; 0.06/ Design of the pole pieces of an electromagnet according to the Garber Henry Hoeve model Rev. Sci. Instrum. 56, ; 0.06/.87 The shapes of pair polarizability curves J. Chem. Phys. 7, ; 0.06/ Investigating the physical nature of the Coriolis effects in the fixed frame Am. J. Phys. 45, 6 977; 0.9/.0780 Reuse of AIP Publishing content is subject to the terms at: IP: On: Sat, Apr 06 05:7:9

2 Smarandache Curves According to Sabban Frame of Fixed Pole Curve Belonging to the Bertrand Curves Pair Süleyman Şenyurt,a, Yasin Altun,b and Ceyda Cevahir,c Faculty of Arts and Sciences, Department of Mathematics, Ordu University, Ordu, Turkey. a Corresponding author: senyurtsuleyman@hotmail.com b yasinaltun85@gmail.com c Ceydacevahir@gmail.com Abstract. In this paper, we investigate the Smarandache curves according to Sabban frame of fixed pole curve which drawn by the unit Darboux vector of the Bertrand partner curve. Some results have been obtained. These results were expressed as the depends Bertrand curve. Mathematics Subject Classification 00. 5A04. Keywords. Bertrand curves pair, Fixed pole curve, Smarandache curves, Sabban frame, Geodesic curvature. INTRODUCTION AND PRELIMINARIES A regular curve in Minkowski space-time, whose position vector is composed by Frenet frame vectors on another regular curve, is called a Smarandache curve []. K. Taşköprü, M. Tosun studied special Smarandache curves according to Sabban frame on S []. Şenyurt and Çalışkan investigated special Smarandache curves in terms of Sabban frame for fixed pole curve and spherical indicatrix and they gave some characterization of Smarandache curves [4, 6]. Let α : I E be a unit speed curve denote by {T, N, B} the moving Frenet frame. For an arbitrary curve α E, with first and second curvature, κ and τ respectively, the Frenet formulae is given by [7, 8] the vector W is called Darboux vector defined by T = κn, N = κt + τb, B = τn W = τt + κb. If we consider the normalization of the Darboux C = W we have, sin ϕ = τ and cos ϕ = κ and [5] W, B = ϕ. C = sin ϕt + cos ϕb Theorem Let α : I E and α : I E be the C -class differentiable unit speed two curves and the amounts of {Ts, Ns, Bs, κs, τs} and {T s, N s, B s, κ s, τ s} are entirely Frenet- serret aparataus of the curves α and the Bertrand partner α, respectively, then T = cos θt sin θb, N = N, B = sin θt + cos θb, T, T = θ, [8]. κ = λκ sin θ λ λκ, τ = sin θ λ τ 4 International Conference on Advances in Natural and Applied Sciences AIP Conf. Proc. 76, ; doi: 0.06/ Published by AIP Publishing /$ Reuse of AIP Publishing content is subject to the terms at: IP: On: Sat, Apr 06 05:7:9

3 Theorem Let α, α be a Bertrand curves pair in E. We have between unit Darboux vectors [], C = C. 5 Theorem Let γ be a unit speed spherical curve. We denote s as the arc-length parameter of γ. Let us denote ts = γ s, and we call ts a unit tangent vector of γ. We now set a vector ds = γs ts along γ. This frame is called the Sabban frame of γ on S Sphere of unit radius. Then we have the following spherical Frenet formulae of γ, [, 6] κ g is called the geodesic curvature of γ on S and γ = t, t = γ + κ g d, d = κ g t 6 κ g = t, d. 7 SMARANDACHE CURVES ACCORDING TO SABBAN FRAME OF FIXED POLE CURVE BELONGING TO THE BERTRAND CURVES PAIR In this section, we investigate Smarandache curves according to the Sabban frame of fixed pole C. Let α C s = C be a unit speed regular spherical curves on S. We denote s C as the arc-lenght parameter of fixed pole C α C s = C s. 8 Differentiating 8, we have and From the equation T C = cos ϕ T sin ϕ B C T C = N. C = sin ϕ T + cos ϕ B, T C = cos ϕ T sin ϕ B, C T C = N 9 is called the Sabban frame of fixed pole curve C. From the 6 κ g = T C, C T C = κ g = W. Then from the 4 we have the following spherical Frenet formulae of C : i. C T C -Smarandache Curves C = T C, T C + W C T C, C T C = W T C. 0 Let S be a unit sphere in E and suppose that the unit speed regular Bertrand partner curve α C s = C s lying fully on S. In this case, C T C - Smarandache curve can be defined by Substituting the equation 9 into equation, we reach β s = C + T C. β s = sin ϕ + cos ϕ T + cos ϕ sin ϕ B. Differentiating, we can write the tangent vector of β -Smarandache curve according to Bertrand partner curve T β = ϕ sin ϕ + W T + W N ϕ + sin ϕ B + W ϕ. + W Reuse of AIP Publishing content is subject to the terms at: IP: On: Sat, Apr 06 05:7:9

4 Differentiating, we get T β = ϕ 4 χ sin ϕ + χ cos ϕ W + T + χ ϕ 4 W + N + ϕ 4 χ cos ϕ χ sin ϕ W + B 4 χ = W χ = W + W W + W + W. Considering the equations and, it easily seen that C T C β = W cos ϕ + sin ϕ W + 4 T, χ = W W 4 W W 5 N + W cos ϕ + sin ϕ B W + 4ϕ W Substituting the and 4 into equation,, 4 and 6, Sabban aparataus of the β -Smarandache curve according to Bertrand curve β s = sin ϕ cos ϕt + cos ϕ+sin ϕb, Tβ = ϕ sin ϕ cos ϕ ϕ + T ϕ + N+ ϕ cos ϕ + sin ϕ ϕ + B, C T C β = cos ϕ sin ϕ T + 4 cos ϕ + sin ϕ N B, + 4ϕ + 4 T β = ϕ 4 χ sin ϕ χ cos ϕ + T χ = χ = + + χ ϕ 4 + N + ϕ 4 χ cos ϕ + χ sin ϕ + B,, χ = Geodesic curvatures of the β s β -Smarandache curve according to Bertrand partner and Bertrand curves, recpectively, κ β g = + W 5 W ii. T C C T C -Smarandache Curves T C C T C -Smarandache curve can be defined by χ W χ + χ, κ β g = + ϕ 5 χ χ + χ. β s = T C + C T C. 8 Solving the above equation by substitution of T C and C T C from 9, and 4, we reach β -Smarandache curve according to Bertrand partner and Bertrand curves, respectively, β s = cos ϕ T + N sin ϕ B, β s = cos ϕt + N + sin ϕb Reuse of AIP Publishing content is subject to the terms at: IP: On: Sat, Apr 06 05:7:9

5 Geodesic curvature of the β s β -Smarandache curve according to Bertrand curve δ = + + κ β g = iii. C T C C T C -Smarandache Curves + 5, δ = C T C C T C -Smarandache curve can be defined by δ δ + δ 4, δ = 4 +. β s = C + T C + C T C. 0 Solving the above equation by substitution of C, T C and C T C from 9, and 4, we reach β -Smarandache curve according to Bertrand partner and Bertrand curves, respectively, β s = sin ϕ + cos ϕ T + N + cos ϕ sin ϕ B, β s = sin ϕ cos ϕt + N + sin ϕ cos ϕb. Geodesic curvature of the β s β -Smarandache curve according to Bertrand curve, κ β g = ϕ 5 ρ 5 + ρ + 5 ρ ρ = + 4 ρ = + ρ = , , ACKNOWLEDGEMENT This work was supported by BAP The Scientific Research Projects Coordination Unit, Ordu University. REFERENCES [] Turgut M. and Yılmaz S., International Journal of Mathematical Combinatorics, [] Taşköprü K. and Tosun M., Boletim da Sociedade Paranaense de Mathematica srie., [] Özgüner Z. and Şenyurt S., University of Ordu Journal of Science and Technology,, [4] Çalışkan A. and Şenyurt, S., Gen. Math. Notes,, 5, 05. [5] Fenchel, W., Bull. Amer. Math. Soc. 57, [6] Çalışkan A. and Şenyurt, S., Boletim da Sociedade Parananse de Mathematica srie. 4, [7] Hacısalihoğlu H.H., Differential Geometry İnönü University Publications, Malatya, 994. [8] Sabuncuoğlu A., Differential GeometryNobel Publications, Ankara, Reuse of AIP Publishing content is subject to the terms at: IP: On: Sat, Apr 06 05:7:9