Gen. Math. Notes, Vol. 31, No. 2, December 2015, pp.1-15 ISSN 2219-7184; Copyright (cid:13)c ICSRS Publication, 2015 www.i-csrs.org Available free online at http://www.geman.in Smarandache Curves in Terms of Sabban Frame of Spherical Indicatrix Curves Su¨leyman S¸enyurt1 and Abdussamet C¸alı¸skan2 1,2Faculty of Arts and Sciences, Department of Mathematics Ordu University, 52100, Ordu, Turkey 1E-mail: [email protected] 2E-mail: [email protected] (Received: 4-11-14 / Accepted: 20-4-15) Abstract In this paper, we investigate special Smarandache curves in terms of Sab- ban frame of spherical indicatrix curves and we give some characterization of Smarandache curves. Besides, we illustrate examples of our results. Keywords: Smarandache Curves, Sabban Frame, Geodesic Curvature, Spherical Indicatrix Curves. 1 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 [5]. Special Smarandache curves have been studied by some authors. AhmadT.AlistudiedsomespecialSmarandachecurvesintheEuclideanspace. ¨ He studied Frenet-Serret invariants of a special case, [2]. Ozcan Bekta¸s and Salim Yu¨ce studied some special smarandache curves according to Darboux Frame in E3, [4]. Muhammed C¸etin, Yılmaz Tuncer and Kemal Karacan in- vestigated special smarandache curves according to Bishop frame in Euclidean 3-Space and they gave some differential geometric properties of Smarandache curves, [3]. MelihTurgutandSu¨haYılmazstudiedaspecialcaseofsuchcurves and called it smarandache TB curves in the space E4, [5]. Nurten Bayrak, 2 1 ¨ Ozcan Bekta¸s and Salim Yu¨ce studied some special smarandache curves in E3, [6]. Kemal Tas.ko¨pru¨ , Murat Tosun studied special Smarandache curves 1 2 Su¨leyman S¸enyurt et al. according to Sabban frame on S2, [7]. In this paper, we study special Smarandache curves such as TT , T T (T∧T ), TT (T∧T ), NT , T (N∧T ), NT (N∧T ), BT , T (B∧T ) T T T T N N N N N B B B and BT (B∧T ) created by Sabban frame, {T,T ,T ∧T }, {N,T ,N ∧T } B B T T N N and {B,T ,B ∧ T }, that belongs to spherical indicatrix of a α curve are B B defined. Besides we have found some results. 2 Problem Formulations The Euclidean 3-space E3 be inner product given by (cid:104),(cid:105) = x2 +x3 +x2 1 2 3 where (x ,x ,x ) ∈ E3. Let α : I → E3 be a unit speed curve denote by 1 2 3 {T,N,B} the moving Frenet frame . For an arbitrary curve α ∈ E3, with first and second curvature, κ and τ respectively, the Frenet formulae is given by [1]  T(cid:48) = κN   N(cid:48) = −κT +τB (1)  B(cid:48) = −τN. Accordingly, the spherical indicatrix curves of Frenet vectors are (T), (N) and (B) respectively. These equations of curves are given by [10]  α (s) = T(s)  T α (s) = N(s) (2) N  α (s) = B(s) B For any unit speed curve α : I → E3, the vector W is called Darboux vector defined by W = τ(s)T(s)+κ(s)B(s). If we consider the normalization of the Darboux c = W we have (cid:107)W(cid:107) κ(s) τ(s) cosϕ = ,sinϕ = (cid:107)W(cid:107) (cid:107)W(cid:107) and c = sinϕT(s)+cosϕB(s) where ∠(W,B) = ϕ. Smarandache Curves in Terms of Sabban Frame... 3 Let γ : I → S2 be a unit speed spherical curve. We denote s as the arc-length parameter of γ. Let us denote by  γ(s) = γ(s)   t(s) = γ(cid:48)(s) (3)  d(s) = γ(s)∧t(s). We call t(s) a unit tangent vector of γ. {γ,t,d} frame is called the Sabban frame of γ on S2 . Then we have the following spherical Frenet formulae of γ :  γ(cid:48) = t   t(cid:48) = −γ +κ d (4) g  d(cid:48) = −κ t g where is called the geodesic curvature of κ on S2 and g κ = (cid:104)t(cid:48),d(cid:105) [8] (5) g 3 Smarandache Curves in Terms of Sabban Frame of Spherical Indicatrix Curves In this section, we investigate Smarandache curves according to the Sabban frame of Spherical Indicatrix Curves. Let α (s) = T(s) be a unit speed regular spherical curves on S2. We denote T s as the arc-lenght parameter of tangents indicatrix (T) T α (s) = T(s) (6) T Differentiating (6), we have dα ds T T = T(cid:48)(s) ds ds T and ds T T = κN (7) T ds From the equation (7) T = N T and T ∧T = B T 4 Su¨leyman S¸enyurt et al. From the equation (3)  T(s) = T(s)   T (s) = N(s) T  T ∧T (s) = B(s) T is called the Sabban frame of tangents indicatrix (T). From the equation (5) τ κ = (cid:104)T(cid:48),T ∧T (cid:105) =⇒ κ = g T T g κ Then from the equation (4) we have the following spherical Frenet formulae of (T):  T(cid:48) = T  T T(cid:48) = −T + τT ∧T (8) T κ T  (T ∧T )(cid:48) = −τT T κ T Let α (s) = N(s) be a unit speed regular spherical curves on S2.We denote N s as the arc-lenght parameter of principal normals indicatrix (N) N α (s) = N(s) (9) N Differentiating (9), we have T = −cosϕT +sinϕB N and N ∧T = sinϕT +cosϕB. N From the equation (3)  N(s) = N(s)   T (s) = −cosϕT(s)+sinϕB(s) N  N ∧T (s) = sinϕT(s)+cosϕB(s) N is called the Sabban frame of principal normals indicatrix (N). From the equa- tion (5) ϕ(cid:48) κ = g (cid:107)W(cid:107) Then from the equation (4) we have the following spherical Frenet formulae of (N):  N(cid:48) = T  N  T(cid:48) = −N + ϕ(cid:48) (N ∧T ) (10) N (cid:107)W(cid:107) N (N ∧T )(cid:48) = − ϕ(cid:48) T N (cid:107)W(cid:107) N Smarandache Curves in Terms of Sabban Frame... 5 Let α (s) = B(s) be a unit speed regular spherical curves on S2.We denote B s as the arc-lenght parameter of indicatrix (B) B α (s) = B(s) (11) B Differentiating (11), we have T = −N B and B ∧T = T B From the equation (3)  B(s) = B(s)   T (s) = −N(s) B  (B ∧T )(s) = T(s) B is called the Sabban frame of binormals indicatrix (B). From the equation (5) κ κ = g τ Then from the equation (4) we have the following spherical Frenet formulae of (B):  B(cid:48) = T  B T (cid:48) = −B + κ(B ∧T ) (12) B τ B  (B ∧T )(cid:48) = −κT B τ B i-) TT -Smarandache Curves T Let S2 be a unit sphere in E3 and suppose that the unit speed regular curve α (s) = T(s) lying fully on S2. In this case, TT - Smarandache curve can be T T defined by 1 β (s∗) = √ (T +T ). (13) 1 T 2 Now we can compute Sabban invariants of TT - Smarandache curves. Differ- T entiating (13), we have ds∗ 1 τ T = √ (−T +N + B), β1 ds 2 κ where (cid:114) ds∗ 2+(τ)2 = κ . (14) ds 2 Thus, the tangent vector of curve β is to be 1 1 τ T = (−T +N + B). (15) β1 (cid:112)2+(τ)2 κ κ 6 Su¨leyman S¸enyurt et al. Differentiating (15), we get ds∗ 1 T(cid:48) = (λ T +λ N +λ B) (16) β1 ds (cid:0)2+(τ)2(cid:1)32 1 2 3 κ where τ τ τ (cid:0) (cid:1)(cid:48) (cid:0) (cid:1)2 λ = − −2 1 κ κ κ τ τ τ τ (cid:0) (cid:1)(cid:48) (cid:0) (cid:1)4 (cid:0) (cid:1)2 λ = − − −3 −2 2 κ κ κ κ τ τ τ (cid:0) (cid:1) (cid:0) (cid:1)3 λ = 2 + +2 . 3 κ κ κ Substituting the equation (15) into equation (16), we reach √ 2 T(cid:48) = (λ T +λ N +λ B). (17) β1 (2+(τ)2)2 1 2 3 κ Considering the equations (13) and (15), it easily seen that 1 τ τ (T ∧T ) = ( T − N +2B). (18) T β1 (cid:112)4+2(τ)2 κ κ κ From the equation (17) and (18), the geodesic curvature of β (s∗) is 1 κ β1 = 1 (λ τ −λ τ +2λ ). g (2+(τ)2)25 1κ 2κ 3 κ ii-) T (T ∧T )-Smarandache Curves T T Similarly, T (T ∧T ) - Smarandache curve can be defined by T T 1 β (s∗) = √ (T +T ∧T ). (19) 2 T T 2 In that case, the tangent vector of curve β is as follows 2 1 τ τ T = (−T − N + B). (20) β2 (cid:112)1+2(τ)2 κ κ κ Differentiating (20), it is obtained that √ 2 T(cid:48) = (λ T +λ N +λ B) (21) β2 (1+2(τ)2)2 1 2 3 κ where Smarandache Curves in Terms of Sabban Frame... 7 τ τ τ τ (cid:0) (cid:1)3 (cid:0) (cid:1)(cid:0) (cid:1)(cid:48) λ = +2 +2 1 κ κ κ κ τ τ τ (cid:0) (cid:1)4 (cid:0) (cid:1)2 λ = −2 −3 − −1 2 κ κ κ τ τ τ (cid:0) (cid:1)(cid:48) (cid:0) (cid:1)4 (cid:0) (cid:1)2 λ = −2 − . 3 κ κ κ Using the equations (19) and (20), we easily find 1 τ (T ∧T ) = (2 T −N +B). (22) T β2 (cid:112)2+4(τ)2 κ κ So, the geodesic curvature of β (s∗) is as follows 2 1 τ κ β2 = (2λ −λ +λ ). g (1+2(τ)2)52 1κ 2 3 κ iii-) TT (T ∧T )-Smarandache Curves T T TT T ∧T - Smarandache curve can be defined by T T 1 β (s∗) = √ (T +T +T ∧T ). (23) 3 T T 3 Differentiating (23), we have the tangent vector of curve β is 3 1 τ τ T = (−T +(1− )N + B). (24) β3 (cid:112)2(1− τ +(τ)2) κ κ κ κ Differentiating (24), it is obtained that √ 3 T(cid:48) = (λ T +λ N +λ B). (25) β3 4(1− τ +(τ)2)2 1 2 3 κ κ where τ τ τ τ τ (cid:0) (cid:1)(cid:48)(cid:0) (cid:1) (cid:0) (cid:1)3 (cid:0) (cid:1)2 λ = 2 −1 +2 −4 +4 −2 1 κ κ κ κ κ τ τ τ τ τ τ (cid:0) (cid:1)(cid:48)(cid:0) (cid:1) (cid:0) (cid:1)4 (cid:0) (cid:1)3 (cid:0) (cid:1)2 (cid:0) (cid:1) λ = − +1 −2 +2 −4 +2 −2 2 κ κ κ κ κ κ τ τ τ τ τ τ (cid:0) (cid:1)(cid:48)(cid:0) (cid:1) (cid:0) (cid:1)4 (cid:0) (cid:1)3 (cid:0) (cid:1)2 (cid:0) (cid:1) λ = 2− −2 +4 −4 +2 . 3 κ κ κ κ κ κ Using the equations (23) and (24), we have 8 Su¨leyman S¸enyurt et al. (2τ −1)T +(−1− τ)N +(2− τ)B (T ∧T ) = κ √ κ κ · (26) T β3 6(cid:112)1− τ +(τ)2 κ κ So, the geodesic curvature of β (s∗) is 3 λ (2τ −1)+λ (−1− τ)+λ (2− τ) κ β3 = 1 κ √ 2 κ 3 κ · g 4 2(1− τ +(τ)2)52 κ κ iv-) NT -Smarandache Curves N NT - Smarandache curve can be defined by N 1 ς (s∗) = √ (N +T ). (27) 1 N 2 Differentiating (27), we have the tangent vector of curve ς is 3 (−cosϕ+ ϕ(cid:48) sinϕ)T −N +(sinϕ+ ϕ(cid:48) cosϕ)B (cid:107)W(cid:107) (cid:107)W(cid:107) T = · (28) ς1 (cid:113)2+( ϕ(cid:48) )2 (cid:107)W(cid:107) Differentiating (28), we get 1 T(cid:48) = ((λ sinϕ−λ cosϕ)T +λ N +(λ sinϕ+λ cosϕ)B). ς1 (2+( ϕ(cid:48) )2)2 3 2 1 2 3 (cid:107)W(cid:107) (29) where ϕ(cid:48) ϕ(cid:48) ϕ(cid:48) (cid:0) (cid:1)(cid:0) (cid:1)(cid:48) (cid:0) (cid:1)2 λ = − −2 1 (cid:107)W(cid:107) (cid:107)W(cid:107) (cid:107)W(cid:107) ϕ(cid:48) ϕ(cid:48) ϕ(cid:48) ϕ(cid:48) (cid:0) (cid:1)(cid:0) (cid:1)(cid:48) (cid:0) (cid:1)4 (cid:0) (cid:1)2 λ = − − −3 −2 2 (cid:107)W(cid:107) (cid:107)W(cid:107) (cid:107)W(cid:107) (cid:107)W(cid:107) ϕ(cid:48) ϕ(cid:48) ϕ(cid:48) (cid:0) (cid:1)(cid:48) (cid:0) (cid:1)3 (cid:0) (cid:1) λ = 2 + +2 . 3 (cid:107)W(cid:107) (cid:107)W(cid:107) (cid:107)W(cid:107) Considering the equations (27) and (28), it easily seen that (cid:0)2sinϕ+ ϕ(cid:48) cosϕ)T + ϕ(cid:48) N +(2cosϕ− ϕ(cid:48) sinϕ)B (cid:107)W(cid:107) (cid:107)W(cid:107) (cid:107)W(cid:107) (N ∧T ) = · (30) N ς1 (cid:113)4+2( ϕ(cid:48) )2 (cid:107)W(cid:107) The geodesic curvature of ς (s∗) is 1 Smarandache Curves in Terms of Sabban Frame... 9 (cid:0)λ ϕ(cid:48) −λ ϕ(cid:48) +2λ (cid:1) κ ς1 = 1(cid:107)W(cid:107) 2(cid:107)W(cid:107) 3 · g 5 (2+( ϕ(cid:48) )2)2 (cid:107)W(cid:107) v-) T (N ∧T )-Smarandache Curves N N T (N ∧T ) - Smarandache curve can be defined by N N 1 ς (s∗) = √ (T +N ∧T ). (31) 2 N N 2 Differentiating (31), the tangent vector of curve ς is 2 ( ϕ(cid:48) (sinϕ+cosϕ))T −N +( ϕ(cid:48) (cosϕ−sinϕ)B) (cid:107)W(cid:107) (cid:107)W(cid:107) T = · (32) ς2 (cid:113)1+2( ϕ(cid:48) )2 (cid:107)W(cid:107) Differentiating (32), it is obtained that √ 2 (cid:2) (cid:3) T(cid:48) = (λ sinϕ−λ cosϕ)T +λ N +(λ sinϕ+λ cosϕ)B ς2 (1+2( ϕ(cid:48) )2)2 3 2 1 2 3 (cid:107)W(cid:107) (33) where ϕ(cid:48) ϕ(cid:48) ϕ(cid:48) ϕ(cid:48) (cid:0) (cid:1) (cid:0) (cid:1)3 (cid:0) (cid:1)(cid:0) (cid:1)(cid:48) λ = +2 +2 1 (cid:107)W(cid:107) (cid:107)W(cid:107) (cid:107)W(cid:107) (cid:107)W(cid:107) ϕ(cid:48) ϕ(cid:48) ϕ(cid:48) (cid:0) (cid:1)4 (cid:0) (cid:1)4 (cid:0) (cid:1) λ = −2 −3 − −1 2 (cid:107)W(cid:107) (cid:107)W(cid:107) (cid:107)W(cid:107) ϕ(cid:48) ϕ(cid:48) ϕ(cid:48) (cid:0) (cid:1)(cid:48) (cid:0) (cid:1)4 (cid:0) (cid:1)2 λ = −2 − . 3 (cid:107)W(cid:107) (cid:107)W(cid:107) (cid:107)W(cid:107) Using the equations (31) and (32), we easily find (sinϕ+cosϕ)T +2 ϕ(cid:48) N +(cosϕ−sinϕ)B (cid:107)W(cid:107) (N ∧T ) = · (34) N ς2 (cid:113)2+4( ϕ(cid:48) )2 (cid:107)W(cid:107) So, the geodesic curvature of ς (s∗) is as follows 2 (cid:0) ϕ(cid:48) λ −λ +λ (cid:1) κ ς2 = (cid:107)W(cid:107) 1 2 3 · g 5 (1+2( ϕ(cid:48) )2)2 (cid:107)W(cid:107) 10 Su¨leyman S¸enyurt et al. vi-) NT (N ∧T )-Smarandache Curves N N NT N ∧T - Smarandache curve can be defined by N N 1 ς (s∗) = √ (N +T +N ∧T ). (35) 3 N N 3 Differentiating (35), the tangent vector of curve ς is 2 (cid:0)−cosϕ+ ϕ(cid:48) (cosϕ+sinϕ)(cid:1)T −N +(cid:0)sinϕ+ ϕ(cid:48) (cosϕ−sinϕ)(cid:1)B (cid:107)W(cid:107) (cid:107)W(cid:107) T = · ς3 (cid:114) (cid:16) (cid:17) 2 1− ϕ(cid:48) +(cid:0) ϕ(cid:48) (cid:1)2 (cid:107)W(cid:107) (cid:107)W(cid:107) (36) Differentiating (36), it is obtained that √ 3 (cid:2) (cid:3) T(cid:48) = (−λ cosϕ+λ sinϕ)T +λ N +(λ cosϕ+λ sinϕ)B . ς3 4(cid:16)1− ϕ(cid:48) +( ϕ(cid:48) )2(cid:17)2 2 3 1 3 2 (cid:107)W(cid:107) (cid:107)W(cid:107) (37) where ϕ(cid:48) ϕ(cid:48) ϕ(cid:48) ϕ(cid:48) ϕ(cid:48) (cid:0) (cid:1)(cid:48)(cid:0) (cid:1) (cid:0) (cid:1)3 (cid:0) (cid:1)2 (cid:0) (cid:1) λ = 2 −1 +2 −4 +4 −2 1 (cid:107)W(cid:107) (cid:107)W(cid:107) (cid:107)W(cid:107) (cid:107)W(cid:107) (cid:107)W(cid:107) ϕ(cid:48) ϕ(cid:48) ϕ(cid:48) ϕ(cid:48) ϕ(cid:48) ϕ(cid:48) (cid:0) (cid:1)(cid:48)(cid:0) (cid:1) (cid:0) (cid:1)4 (cid:0) (cid:1)3 (cid:0) (cid:1)2 (cid:0) (cid:1) λ = − +1 −2 +2 −4 +2 −2 2 (cid:107)W(cid:107) (cid:107)W(cid:107) (cid:107)W(cid:107) (cid:107)W(cid:107) (cid:107)W(cid:107) (cid:107)W(cid:107) ϕ(cid:48) ϕ(cid:48) ϕ(cid:48) ϕ(cid:48) ϕ(cid:48) ϕ(cid:48) (cid:0) (cid:1)(cid:48)(cid:0) (cid:1) (cid:0) (cid:1)4 (cid:0) (cid:1)3 (cid:0) (cid:1)2 (cid:0) (cid:1) λ = 2− −2 +4 −4 +2 . 3 (cid:107)W(cid:107) (cid:107)W(cid:107) (cid:107)W(cid:107) (cid:107)W(cid:107) (cid:107)W(cid:107) (cid:107)W(cid:107) Using the equations (35) and (36), we have 1 (cid:16) (N ∧TN)ς3 = √6(cid:113)1− ϕ(cid:48) +( ϕ(cid:48) )2 (2sinϕ+cosϕ (38) (cid:107)W(cid:107) (cid:107)W(cid:107) ϕ(cid:48) ϕ(cid:48) + (cosϕ−sinϕ))T +(−1+2 )N (cid:107)W(cid:107) (cid:107)W(cid:107) ϕ(cid:48) (cid:17) +(2cosϕ−sinϕ− (cosϕ−sinϕ))B (cid:107)W(cid:107) The geodesic curvature of ς (s∗) is 3