Smarandache Curves In Terms of Sabban Frame of Fixed Pole Curve

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 [12]. 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 [1]. M. Çetin , Y. Tunçer and K. Karacan investigated special smarandache curves according to Bishop frame in Euclidean 3-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, [3].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 TNBSmarandache curves together with NC-Smarandache curve, [10]. Ö.


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 [11]. 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 [1]. M. Ç etin , Y. Tunçer and K. Karacan investigated special smarandache curves according to Bishop frame in Euclidean 3-Space and they gave some differential goematric properties of Smarandache curves [5]. Şenyurt and Ç alışkan investigated special Smarandache curves in terms of Sabban frame of spherical indicatrix curves and they gave some characterization of Smarandache curves, [4].Ö. Bektaş and S. Yüce studied some special smarandache curves according to Darboux Frame in E 3 [2]. M. Turgut and S. Yılmaz studied a special case of such curves and called it smarandache T B 2 curves in the space E 4 1 [11]. N. Bayrak ,Ö. Bektaş and S. Yüce studied some special smarandache curves in E 3 1 [3]. K. Tas . köprü , M. Tosun studied special Smarandache curves according to Sabban frame on S 2 [10].
In this paper, the special smarandache curves such as CT 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 3-space E 3 be inner product given by Let α : I → E 3 be a unit speed curve denote by {T, N, B} the moving Frenet frame . For an arbitrary curve α ∈ E 3 , with first and second curvature, κ and τ respectively, the Frenet formulae is given by [ (2.1) Accordingly, the spherical indicatrix curves of Frenet vectors are (T ), (N ) and (B) respectively.These equations of curves are given by [7] Let γ : I → S 2 be a unit speed spherical curve. We denote s as the arc-length parameter of γ. Let us denote by We call t(s) a unit tangent vector of γ. {γ, t, d} frame is called the Sabban frame of γ on S 2 . Then we have the following spherical Frenet formulae of γ : where is called the geodesic curvature of κ g on S 2 and κ g = ⟨t ′ , d⟩ [8] (2.5)

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) = C(s) be a unit speed regular spherical curves on S 2 .We denote s C as the arc-lenght parameter of fixed pole curve (C) From the equation (3.2) is called the Sabban frame of fixed pole curve (C) .From the equation (2.5) Then from the equation (2.4) we have the following spherical Frenet formulae of (C):

CT C -Smarandache Curves
Now we can compute Sabban invariants of CT C -Smarandache curves. Differentiating (3.4) , we have Thus, the tangent vector of curve ψ is to be where Substituting the equation (3.5) into equation (3.7) , we reach Considering the equations (3.4) and (3.6) , it easily seen that From the equation (3.8) and (3.9) , the geodesic curvature of ψ(s * ) is ) .

T C (C ∧ T C )-Smarandache Curves
Definition 3.2 Let S 2 be a unit sphere in E 3 and suppose that the unit speed regular curve α C (s) = C(s) lying fully on S 2 . In this case, T C (C ∧ T C ) -Smarandache curve can be defined by Now we can compute Sabban invariants of T C (C ∧T C ) -Smarandache curves. Differentiating (3.10), we have In that case, the tangent vector of curve ψ is as follows Differentiating (3.12), it is obtained that Substituting the equation (3.11) into equation (3.13) , we get Using the equations (3.10) and (3.12) , we easily find So, the geodesic curvature of ψ(s * ) is as follows

CT C (C ∧ T C )-Smarandache Curves
Thus, the tangent vector of curve ψ is Differentiating (3.18), it is obtained that
It is rendered in Figure 1.