Smarandache Curves In Terms of Sabban Frame of Fixed Pole Curve. Key Words: Smarandache Curves, Sabban Frame, Geodesic Curvature, Fixed Pole Curve

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1 Bol. Soc. Paran. Mat. s. v. 4 06: 5 6. c SPM ISSN on line ISSN in press SPM: doi:0.569/bspm.v4i.75 Smarandache Curves In Terms of Sabban Frame of Fixed Pole Curve Süleyman Şenyurt and Abdussamet Çalişkan abstract: In this paper, we study the special Smarandache curve in terms of Sabban frame of Fixed Pole curve and we give some characterization of Smarandache curves. Besides, we illustrate examples of our results. Key Words: Smarandache Curves, Sabban Frame, Geodesic Curvature, Fixed Pole Curve Contents Introduction 5 Preliminaries 54 Smarandache Curves According to Sabban Frame of Fixed Pole Curve 55. CT C -Smarandache Curves T C C T C -Smarandache Curves CT C C T C -Smarandache Curves Example Introduction 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 []. Special Smarandache curves have been studied by some authors. Ahmad T.Ali studied some special Smarandache curves in the Euclidean space.he studied Frenet-Serret invariants of a special case []. M. Çetin, Y. Tunçer and K. Karacan investigated special smarandache curves according to Bishop frame in Euclidean -Space and they gave some differential goematric properties of Smarandache curves [5]. Şenyurt and Çalışkan investigated special Smarandache curves in terms of Sabban frame of spherical indicatrix curves and they gave some characterization of Smarandache curves, [].Also, in their other work, when the unit Darboux vector of the partner curve of Mannheim curve were taken as the position vectors, the curvature and the torsion of Smarandache curve were calculated. These values were expressed depending upon the Mannheim curve, [4]. They defined NC-Smarandache curve, then they calculated the curvature and torsion of NB and TNB- Smarandache curves together with NC-Smarandache curve, [0]. Ö. 000 Mathematics Subject Classification: 5A04. 5 Typeset by B S P M style. c Soc. Paran. de Mat.

2 54 Süleyman Şenyurt and Abdussamet Çalişkan Bektaş and S. Yüce studied some special smarandache curves according to Darboux Frame in E []. M. Turgut and S. Yılmaz studied a special case of such curves and called it smarandache TB curves in the space E 4 []. K. Taṣköprü, M. Tosun studied special Smarandache curves according to Sabban frame on S []. In this paper, the special smarandache curves such asct C,T C C T C,CT C C T C created by Sabban frame, {C,T C,C T C }, that belongs to fixed pole of a α curve are defined. Besides, we have found some results.. Preliminaries The Euclidean -space E be inner product given by, = x +x +x x,x,x E. 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 [6] T = κn N = κt +τb. B = τn. Accordingly, the spherical indicatrix curves of Frenet vectors are T, N and B respectively.these equations of curves are given by [7], [9] α T s = Ts α N s = Ns. α B s = Bs Let γ : I S be a unit speed spherical curve. We denote s as the arc-length parameter of γ. Let us denote by γs = γs ts = γ s. ds = γs ts. We call ts a unit tangent vector of γ. {γ, t, d} frame is called the Sabban frame of γ on S. Then we have the following spherical Frenet formulae of γ : γ = t t = γ +κ g d.4 d = κ g t is called the geodesic curvature of κ g on S and κ g = t,d, [8]..5

3 Smarandache Curves In Terms of Sabban Frame of Fixed Pole Curve 55. Smarandache Curves According to Sabban Frame of Fixed Pole Curve In this section, we investigate Smarandache curves according to the Sabban frame of fixed pole curve C. Let α C s = Cs be a unit speed regular spherical curves on S.We denote s C as the arc-lenght parameter of fixed pole curve C Differentiating., we have α C s = Cs. dα C ds C ds C ds = C s and From the equation. and From the equation. T C ds C ds = cost sinb. T C = cost sinb C T C = N Cs = Cs T C s = cost sinb C T C s = Ns is called the Sabban frame of fixed pole curve C.From the equation.5 κ g = T C,C T C = κ g = Then from the equation.4 we have the following spherical Frenet formulae of C: C = T C T C = C + C T C. C T C = T C.. CT C -Smarandache Curves Definition.. Let S be a unit sphere in E and suppose that the unit speed regular curve α C s = Cs lying fully on S. In this case, CT C - Smarandache curve can be defined by ψs = C +T C..4

4 56 Süleyman Şenyurt and Abdussamet Çalişkan Now we can compute Sabban invariants of CT C - Smarandache curves. Differentiating.4, we have ds = cos sint + N cos+sinb, ds = + Thus, the tangent vector of curve ψ is to be..5 = + cos sint + N cos+sinb..6 Differentiating.6, we get ds = + λ = sin+cos κ cos sin λ T +λ N +λ B.7 + sin+cos λ = + κcos sin+τcos+sin + λ = τ + cos sin cos sin + cos+sin. Substituting the equation.5 into equation.7, we reach T ψ = + λ T +λ N +λ B..8 Considering the equations.4 and.6, it easily seen that C T C ψ = 4+ cos sint +.9 +N + cos+sinb

5 Smarandache Curves In Terms of Sabban Frame of Fixed Pole Curve 57 From the equation.8 and.9, the geodesic curvature of ψs is κ g ψ = T ψ,c T C ψ = + cos sinλ +λ + cos+sinλ... T C C T C -Smarandache Curves Definition.. Let S be a unit sphere in E and suppose that the unit speed regular curve α C s = Cs lying fully on S. In this case, T C C T C - Smarandache curve can be defined by ψs = T C +C T C..0 Now we can compute Sabban invariants of T C C T C - Smarandache curves. Differentiating.0, we have ds = sin cost + N + sin cosb ds = + In that case, the tangent vector of curve ψ is as follows.. = + sin cost +. + N + sin cosb Differentiating., it is obtained that ds = + λ T +λ N +λ B.

6 58 Süleyman Şenyurt and Abdussamet Çalişkan λ = cos+sin+ cos sin λ = κ sin cos κ + sin κ cos+ sin cos τ sin cos sin cos+ λ = sin+ τ + cos+τ + cos+sin+ sin+ cos. Substituting the equation. into equation., we get T ψ = + λ T +λ N +λ B.4 Using the equations.0 and., we easily find C T C ψ = +4 sin cost +.5 +N + cos+sinb So, the geodesic curvature of ψs is as follows κ g ψ = T ψ,c T C ψ = + sin cosλ +λ + cos+sinλ... CT C C T C -Smarandache Curves Definition.. Let S be a unit sphere in E and suppose that the unit speed regular curve α C s = Cs lying fully on S. In this case, CT C C T C - Smarandache curve can be defined by ψs = C +T C +C T C..6

7 Smarandache Curves In Terms of Sabban Frame of Fixed Pole Curve 59 Let us calculate Sabban invariants of CT C C T C - Smarandache curves. Differentiating.6, we have ds = cos sin cost + + N + sin cos+ sinb ds = +..7 Thus, the tangent vector of curve ψ is = + cos sin cost.8 + N + sin cos+ sinb Differentiating.8, it is obtained that λ = ds = + λ T +λ N +λ B.9 + sin+cos+sin κ + cos sin+ cos+ sin λ = τ sin+cos λ = + κ cos sin cos + cos τ + sin cos+cos + sin+ sin+ cos sin+cos. Substituting the equation.7 into equation.9, we reach = 4 + λ T +λ N +λ B..0

8 60 Süleyman Şenyurt and Abdussamet Çalişkan Using the equations.6 and.8, we have C T C ψ = 6 + sin cos. cost + N +sin+ cos + sinb From the equation.0 and., the geodesic curvature of ψs is κ ψ g = T ψ,c T C ψ =.4. Example 4 [λ sin + cos cos+λ + λ sin+ cos+ sin]. Let us consider the unit speed spherical curve: αs = { 9 08 sin6s 7 sin6s, 9 08 cos6s+ 7 cos6s, 6 65 sin0s}. It is rendered in Figure. Figure : Fixed Pole curve T In terms of definitions, we obtain Smarandache curves according to Sabban frame on S, see Figures - 4.

9 Smarandache Curves In Terms of Sabban Frame of Fixed Pole Curve 6 Figure : CT C - Smarandache Curve Figure : T C C T C - Smarandache Curve Figure 4: CT C C T C - Smarandache Curve

10 6 Süleyman Şenyurt and Abdussamet Çalişkan References. Ali A. T., Special Smarandache Curves in the Euclidean Space, International Journal of Mathematical Combinatorics, Vol., pp.0-6, 00.. Bektaş Ö. and Yüce S., Special Smarandache Curves According to Darboux Frame in Euclidean - Space,Romanian Journal of Mathematics and Computer Science, Vol., Issue, pp.48-59, 0.. Çalışkan A. and Şenyurt, S., Smarandache Curves In Terms of Sabban Frame of Spherical Indicatrix Curves, Gen. Math. Notes, in press, Çalışkan A. and Şenyurt, S., N C Smarandache Curves of Mannheim Curve Couple According to Frenet Frame, International Journal of Mathematical Combinatorics, Vol:, pp. -5, Çetin M., Tunçer Y. and Karacan M. K., Smarandache Curves According to Bishop Frame in Euclidean - Space, Gen. Math. Notes, Vol. 0, pp.50-66, Do Carmo, M. P., Differential Geometry of Curves and Surfaces, Prentice Hall, Englewood Cliffs, NJ, Hacısalihoğlu H.H., Differantial Geometry, Academic Press Inc. Ankara, Koenderink J., Solid Shape, MIT Press, Cambridge, MA, O Neill B., Elemantary Differential Geometry, Academic Press Inc. New York, Senyurt S. and Sivas S., An Application of Smarandache Curve, University of Ordu journal of science and technology, Vol:, pp.46-60, 0.. Taşköprü K. and Tosun M., Smarandache Curves on S, Boletim da Sociedade Paranaense de Matemtica Srie. Vol., no:, pp.5-59, 04.. Turgut M. and Yılmaz S., Smarandache Curves in Minkowski Space-time, International J. Math. Combin., Vol., pp.5-55, 008. Süleyman Şenyurt Corresponding author Faculty of Arts and Sciences, Department of Mathematics, Ordu University, 5000, Ordu/Turkey address: senyurtsuleyman@hotmail.com and Abdussamet Çalişkan Faculty of Arts and Sciences, Department of Mathematics, Ordu University, 5000, Ordu/Turkey address: abdussamet65@gmail.com