Smarandache Curves of a Spacelike Curve According to the Bishop Frame of Type-2

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1 International J.Math. Combin. Vol.4(016), 9-43 Smarache Curves of a Spacelike Curve According to the Bishop Frame of Type- Yasin Ünlütürk Department of Mathematics, Kırklareli University, Kırklareli, Turkey Süha Yılmaz Dokuz Eylül University, Buca Faculty of Education Department of Elementary Mathematics Education, 35150, Buca-Izmir, Turkey yasinunluturk@klu.edu.tr, suha.yilmaz@deu.edu.tr Abstract: In this study, we introduce new Smarache curves of a spacelike curve according to the Bishop frame of type- in E 3 1. Also, Smarache breadth curves are defined according to this frame in Minkowski 3-space. A third order vectorial differential equation of position vector of Smarache breadth curves has been obtained in Minkowski 3-space. Key Words: Smarache curves, the Bishop frame of type-, Smarache breadth curves, Minkowski 3-space. AMS(010): 53A05, 53B5, 53B30 1. Introduction Bishop frame extended to study canal tubular surfaces [1]. Rotating camera orientations relative to a stable forward-facing frame can be added by various techniques such as that of Hanson Ma []. This special frame also extended to height functions on a space curve [3]. The construction of the Bishop frame is due to L. R. Bishop the advantages of Bishop frame, comparisons of Bishop frame with the Frenet frame in Euclidean 3-space were given by Bishop [4] Hanson [5]. That is why he defined this frame that curvature may vanish at some points on the curve. That is, second derivative of the curve may be zero. In this situation, an alternative frame is needed for non continously differentiable curves on which Bishop (parallel transport frame) frame is well defined constructed in Euclidean its ambient spaces [6,7,8]. A regular curve in Euclidean 3-space, whose position vector is composed of Frenet frame vectors on another regular curve, is called a Smarache curve. M. Turgut S. Yılmaz have defined a special case of such curves call it Smarache TB curves in the space E 4 1 ([9]) Turgut also studied Smarache breadth of pseudo null curves in E 4 1 ([10]). A.T.Ali has introduced some special Smarache curves in the Euclidean space [11]. Moreover, special 1 Received April 3, 016, Accepted November 6, 016.

2 30 Yasin Ünlütürk Süha Yılmaz Smarache curves have been investigated by using Bishop frame in Euclidean space [1]. Special Smarache curves according to Sabban frame have been studied by [13]. Besides, some special Smarache curves have been obtained in E1 3 by [14]. Curves of constant breadth were introduced by L.Euler [15]. Some geometric properties of plane curves of constant breadth were given in [16]. And, in another work [17], these properties were studied in the Euclidean 3-space E 3. Moreover, M Fujivara [18] had obtained a problem to determine whether there exist space curve of constant breadth or not, he defined breadth for space curves on a surface of constant breadth. In [19], these kind curves were studied in four dimensional Euclidean space E 4. In [0], Yılmaz introduced a new version of Bishop frame in E1 3 called it Bishop frame of type- of regular curves by using common vector field as the binormal vector of Serret-Frenet frame. Also, some characterizations of spacelike curves were given according to the same frame by Yılmaz Ünlütürk [1]. A regular curve more than breadths in Minkowski 3-space is called a Smarache breadth curve. In the light of this definition, we study special cases of Smarache curves according to the new frame in E1 3. We investigate position vector of simple closed spacelike curves give some characterizations in case of constant breadth according to type- Bishop frame in E1 3. Thus, we extend this classical topic in E 3 into spacelike curves of constant breadth in E1 3, see [] for details. In this study, we introduce new Smarache curves of a spacelike curve according to the Bishop frame of type- in E1 3. Also, Smarache breadth curves are defined according to this frame in Minkowski 3-space. A third order vectorial differential equation of position vector of Smarache breadth curves has been obtained in Minkowski 3-space.. Preliminaries To meet the requirements in the next sections, here, the basic elements of the theory of curves in the Minkowski 3 space E 3 1 are briefly presented. There exists a vast literature on the subject including several monographs, for example [3,4]. The three dimensional Minkowski space E 3 1 is a real vector space R 3 endowed with the stard flat Lorentzian metric given by, L = dx 1 + dx + dx 3, (x 1, x, x 3 ) is a rectangular coordinate system of E 3 1. This metric is an indefinite one. Let u = (u 1, u, u 3 ) v = (v 1, v, v 3 ) be arbitrary vectors in E1 3, the Lorentzian cross product of u v is defined as i j k u v = det u 1 u u 3. v 1 v v 3 Recall that a vector v E1 3 has one of three Lorentzian characters: it is a spacelike vector if v, v > 0 or v = 0; timelike v, v < 0 null (lightlike) v, v = 0 for v 0. Similarly, an

3 Smarache Curves of a Spacelike Curve According to the Bishop Frame of Type- 31 arbitrary curve δ = δ(s) in E1 3 can locally be spacelike, timelike or null (lightlike) if its velocity vector are,respectively, spacelike, timelike or null (lightlike), for every s I R. The pseudo-norm of an arbitrary vector a E1 3 is given by a = a, a. The curve = (s) is called a unit speed curve if its velocity vector is unit one i.e., = 1. For vectors v, w E1 3, they are said to be orthogonal each other if only if v, w = 0. Denote by {T, N, B} the moving Serret-Frenet frame along the curve = (s) in the space E1 3. For an arbitrary spacelike curve = (s) in E1, 3 the Serret-Frenet formulae are given as follows T 0 κ 0 T N = γκ 0 τ. N, (.1) 0 τ 0 B B γ = 1, the functions κ τ are, respectively, the first second (torsion) curvature. T(s) = (s), N(s) = T (s) κ(s), B(s) = T(s) N(s) τ(s) = det(,, ) κ. (s) If γ = 1, then (s) is a spacelike curve with spacelike principal normal N timelike binormal B, its Serret-Frenet invariants are given as κ(s) = T (s), T (s) τ(s) = N (s), B(s). If γ = 1, then (s) is a spacelike curve with timelike principal normal N spacelike binormal B, also we obtain its Serret-Frenet invariants as κ(s) = T (s), T (s) τ(s) = N (s), B(s). The Lorentzian sphere S 1 of radius r > 0 with the center in the origin of the space E3 1 is defined by S 1 (r) = {p = (p 1, p, p 3 ) E 3 1 : p, p = r }. Theorem.1 Let = (s) be a spacelike unit speed curve with a spacelike principal normal. If {Ω 1, Ω, B} is an adapted frame, then we have Ω 1 Ω B = 0 0 ξ ξ. ξ 1 ξ 0 Ω 1 Ω B (.) Theorem. Let {T, N, B} {Ω 1, Ω, B} be Frenet Bishop frames, respectively. There exists a relation between them as T N B = sinh θ(s) cosh θ(s) 0 cosh θ(s) sinh θ(s) Ω 1 Ω B, (.3)

4 3 Yasin Ünlütürk Süha Yılmaz θ is the angle between the vectors N Ω 1. ξ 1 = τ(s)cosh θ(s), ξ = τ(s)sinh θ(s). s The frame {Ω 1, Ω, B} is properly oriented, τ θ(s) = κ(s)ds are polar coordinates for the curve = (s). We shall call the set {Ω 1, Ω, B, ξ 1, ξ } as type- Bishop invariants of the curve = (s) in E Smarache Curves of a Spacelike Curve In this section, we will characterize all types of Smarache curves of spacelike curve = (s) according to type- Bishop frame in Minkowski 3-space E Ω 1 Ω Smarache Curves Definition 3.1 Let = (s) be a unit speed regular curve in E 3 1 {Ω 1, Ω, B } be its moving Bishop frame. Ω 1 Ω Smarache curves are defined by β(s ) = 1 (Ω 1 + Ω ). (3.1) Now we can investigate Bishop invariants of Ω 1 Ω Smarache curves of the curve = (s). Differentiating (3.1) with respect to s gives β = dβ ds.ds ds = 1 (ξ1 ξ )B, (3.) T β. ds ds = 1 (ξ 1 ξ )B, The tangent vector of the curve β can be written as follows Differentiating (3.4) with respect to s, we obtain ds ds = 1 ξ 1 ξ. (3.3) T β = β. (3.4) Substituting (3.3) into (3.5) gives dt β ds.ds ds = (ξ 1 Ω 1 + ξ Ω ). (3.5) T β = ξ 1 ξ (ξ 1 Ω 1 + ξ Ω ).

5 Smarache Curves of a Spacelike Curve According to the Bishop Frame of Type- 33 Then the first curvature the principal normal vector field of β are, respectively, computed as T β = κ β = (ξ ξ1 ξ 1 ) + (ξ ), N β = 1 (ξ 1 ) + (ξ ) (ξ 1 Ω 1 + ξ Ω ). On the other h, we express B β = 1 (ξ 1 ) + (ξ ) Ω 1 Ω β ξ 1 ξ 0, So the binormal vector of β is computed as follows B β = 1 (ξ 1 ) + (ξ ) (ξ Ω 1 + ξ 1 Ω ). Differentiating (3.) with respect to s in order to calculate the torsion of the curve β, we obtain β = 1 [ (ξ 1 + ξ )ξ 1 Ω 1 (ξ 1 + ξ )ξ Ω + ( ξ 1 + ξ )B ], similarly The torsion of the curve β is found... β = 1 [( 3ξ 1 ξ 1 ξ 1 ξ ξ 1 ξ ξ 1 ξ 1 )Ω 1 +( ξ 1 ξ ( ξ ) ξ 1 ξ ξ ξ )Ω +( ξ 1 + ξ (ξ 1 )3 (ξ 1 ) ξ ξ 1 (ξ ) + (ξ ) )B ]. τ β = 1 4 (ξ1 ξ ) (ξ1 ) + (ξ ) [(ξ 1 + ξ )K (s) (ξ1 + ξ )ξ K 1 (s)], K 1 (s) = 3ξ 1 ξ 1 ξ 1 ξ ξ 1 ξ ξ 1 ξ 1, K (s) = ξ 1 ξ ( ξ ) ξ 1 ξ ξ ξ, K 3 (s) = ξ 1 + ξ (ξ 1 ) 3 (ξ 1 ) ξ ξ 1 (ξ ) + (ξ ). 3. Ω 1 B Smarache Curves Definition 3. Let = (s) be a unit speed regular curve in E1 3 {Ω 1, Ω, B } be its moving Bishop frame. Ω 1 B Smarache curves are defined by β(s ) = 1 (Ω 1 + B ). (3.6)

6 34 Yasin Ünlütürk Süha Yılmaz Now we can investigate Bishop invariants of Π 1 B Smarache curves of the curve = (s). Differentiating (3.6) with respect to s, we have β = dβ ds.ds ds = 1 (ξ 1 B ξ 1 Ω 1 ξ Ω ), (3.7) T β. ds ds = 1 (ξ 1 B ξ 1 Ω 1 ξ Ω ), The tangent vector of the curve β can be written as follows ds ds = ξ. (3.8) T β = ( ξ1 Ω 1 ξ Ω + ξ 1 B ). (3.9) ξ Differentiating (3.9) with respect to s gives dt β ds.ds ds = ξ (L 1 (s)ω 1 + L (s)ω + L 3 (s)b ), (3.10) Substituting (3.8) into (3.10) gives L 1 (s) = ξ 1 (ξ 1 ) + ξ 1 ξ ξ L 3 (s) = (ξ 1 ) + (ξ ) + ξ 1 ξ 1 ξ ξ, L (s) = ξ 1 ξ, T β = (ξ ) (L 1(s)Ω 1 + L (s)ω + L 3(s)B ), then the first curvature the principal normal vector field of β are, respectively, T β = κ β = L (ξ 1 (s) + L ) (s) L 3 (s),. N β = 1 L 1 (s) + L (s) L 3 (s)(l 1(s)Ω 1 + L (s)ω + L 3 (s)b ). On the other h, we have B β = ξ L 1 (s) + L (s) L 3 (s)[(ξ 1 L (s) + ξ L 3(s))Ω 1 +(ξ1 L 1(s) + ξ1 L 3(s))Ω + (ξ L 1(s) ξ1 L (s))b ]. Differentiating (3.7) with respect to s in order to calculate the torsion of the curve β, we

7 Smarache Curves of a Spacelike Curve According to the Bishop Frame of Type- 35 find β = 1 [ ((ξ 1 ) + ξ 1 )Ω 1 + ( ξ 1 ξ ξ )Ω ( ξ 1 (ξ 1 ) + (ξ ) )B ], similarly... β = 1 [( ξ 1 ξ 1 ξ 1 )Ω 1 + ( ξ 1 ξ ξ ξ 1 ξ )Ω The torsion of the curve β is +( (ξ 1 )3 ξ 1 ξ 1 + ξ 1 ξ ξ ξ ξ )B ]. τ β = (ε )4 16 [ ξ 1 M 1 (s) ξ M (s) ξ 1 M 3 (s)], M 1 (s) = 3ξ 1 ξ 1 ξ 1 ξ ξ 1 ξ ξ 1 ξ 1, M (s) = ξ 1 ξ ( ξ ) ξ 1 ξ ξ ξ, M 3 (s) = ξ 1 + ξ (ξ 1 )3 (ξ 1 ) ξ ξ 1 (ξ ) + (ξ ). 3.3 Ω B Smarache Curves Definition 3.3 Let = (s) be a unit speed regular curve in E 3 1 {Ω 1, Ω, B } be its moving Bishop frame. Ω B Smarache curves are defined by β(s ) = 1 (Ω + B ). (3.11) Now we can investigate Bishop invariants of Ω 1 B Smarache curves of the curve = (s). Differentiating (3.11) with respect to s, we have β = dβ ds.ds ds = 1 ( ξ B ξ1 Ω 1 ξ Ω ), (3.1) T β. ds ds = 1 ( ξ 1 Ω 1 ξ Ω ξ B ), The tangent vector of the curve β can be written as follows ds (ξ ds = ) (ξ1 ). (3.13) T β = ξ 1 Ω 1 ξ Ω ξ B. (ξ ) (ξ1 (3.14) ) Differentiating (3.14) with respect to s gives dt β ds.ds ds = (N 1(s)Ω 1 + N (s)ω + N 3 (s)b ), (3.15)

8 36 Yasin Ünlütürk Süha Yılmaz N 1 (s) = 1 (4ξ ξ ξ 1 ξ 1 )((ξ ) (ξ 1 ) ) 3 ξ 1 ((ξ ) (ξ 1 ) ) 1 ξ 1 + ((ξ ) (ξ 1 ) ) 1 ξ 1 ξ, N (s) = 1 (4ξ ξ ξ 1 ξ 1 )((ξ ) (ξ 1 ) ) 3 ξ 1 ((ξ ) (ξ 1 ) ) 1 ξ 1 + ((ξ ) (ξ 1 ) )(ξ ), N 3 (s) = 1 (4ξ ξ ξ ξ 1 1 )((ξ ) (ξ1 ) ) 3 ((ξ ) (ξ 1 ) ) 1 ((ξ ) (ξ 1 ) ξ ). Substituting (3.13) into (3.15) gives T β = (ξ ) (ξ1 (N 1(s)Ω ) 1 + N (s)ω + N 3(s)B ), then the first curvature the principal normal vector field of β are, respectively, found as follows κ β = T β = N (ξ ) (ξ1 1 (s) + N ) (s) + N 3 (s), N β = On the other h, we have 1 N 1 (s) + N (s) + N 3 (s)(n 1(s)Ω 1 + N (s)ω + L 3 (s)b ). (3.16) B β = 1 (ξ ) (ξ 1 ) N 1 (s) + N (s) + N 3 (s)[( ξ N 3(s) + ξ N (s))ω 1 +( ξ N 3(s) + ξ N 1(s))Ω + (ξ 1 N ξ N 1(s))B ]. (3.17) Differentiating (3.1) with respect to s in order to calculate the torsion of the curve β, we obtain β = 1 [(ξ ξ 1 + ξ 1 )Ω 1 + ((ξ ) ξ )Ω + ( ξ + ξ (ξ 1 ) )B ], similarly... β = 1 [( ξ ξ 1 + ξ ξ 1 ξ 1 ξ ξ 1 + (ξ 1 )3 )Ω 1 +(3ξ ξ ξ (ξ ) (ξ1 ) ξ )Ω +((ξ1 ) ξ (ξ )3 + ξ ξ ξ + ξ 3ξ ξ 1 1 )B ]. The torsion of the curve β is τ β = (ξ ) (ξ1 ) 4 {[P 3 (s)((ξ ) ξ ) P (s)( ξ + ξ (ξ1 ) )]ξ1 +[P 3 (s)(ξ ξ 1 ξ 1 ) P 1(s)( ξ + ξ (ξ 1 ) )]ξ +[P (s)(ξ ξ1 ξ 1 ) P 1 (s)((ξ ) ξ )]ξ3 },

9 Smarache Curves of a Spacelike Curve According to the Bishop Frame of Type- 37 P 1 (s) = ξ ξ 1 + ξ ξ 1 ξ 1 ξ 1 ξ 1 + ξ 1 ξ + (ξ 1 )3, P (s) = 3 ξ ξ ξ (ξ ) (ξ 1 ) ξ, P 3 (s) = (ξ 1 ) ξ (ξ )3 + ξ ξ ξ + ξ 3 ξ 1 ξ Ω 1 Ω B Smarache Curves Definition 3.4 Let = (s) be a unit speed regular curve in E 3 1 {Ω 1, Ω, B } be its moving Bishop frame. Ω 1 Ω B Smarache curves are defined by β(s ) = 1 3 (Ω 1 + Ω + B ). (3.18) Now we can investigate Bishop invariants of Ω 1 Ω B Smarache curves of the curve = (s). Differentiating (3.18) with respect to s, we have β = dβ ds.ds ds = 1 ( ξ1 Ω 1 ξ Ω + (ξ1 ε )B ), (3.19) 3 T β. ds ds = 1 3 ( ξ 1 Ω 1 ξ Ω + (ξ 1 ξ )B ), ds (ξ ds = 1 ξ ) + (ξ ) (ξ1 ). (3.0) 3 The tangent vector of the curve β is found as follows T β = 1 (ξ 1 ξ ) + (ξ ) (ξ 1 ) ( ξ 1 Ω 1 ξ Ω + (ξ 1 ξ )B ). (3.1) Differentiating (3.1) with respect to s, we find dt β ds.ds ds = [ Q(s) ξ 1 Q(s)(ξ 1 ) + Q(s)ξ 1 ξ Q (s)ξ 1 ]Ω 1 +[ Q(s) ξ Q(s)ξ 1 ξ + Q(s)(ξ ) Q (s)ξ ]Ω +[Q(s)(ξ 1 ξ ) -Q(s)(ξ 1 ) +Q (s)(ξ 1 ξ )]B, (3.) Q(s) = 1 (ξ 1 ξ ) + (ξ ) (ξ 1 ). Substituting (3.0) into (3.) by using (3.3) gives T β = 3 K(s) (M 1(s)Ω 1 + M (s)ω + M 3 (s)b ),

10 38 Yasin Ünlütürk Süha Yılmaz R 1 (s) = Q(s) ξ 1 Q(s)(ξ 1 ) + Q(s)ξ 1 ξ Q (s)ξ 1, R (s) = Q(s) ξ Q(s)ξ 1 ξ + Q(s)(ξ ) Q (s)ξ, (3.3) R 3 (s) = Q(s)(ξ 1 ξ ) Q(s)(ξ 1 ) + Q (s)(ξ 1 ξ ). Then the first curvature the principal normal vector field of β are, respectively, obtained as follows κ β = T β 3 = R K(s) 1 (s) + R (s) + R 3 (s), B β = 1 K(s) R 1 (s) + R (s) + R 3 (s)[(m (ξ 1 ξ ) + M 3 ξ )Ω 1 +(M 1 (ξ 1 ξ ) + M 3ξ )Ω + (ξ M 1(s) ξ 1 M (s))b ]. (3.4) Differentiating (3.19) with respect to s in order to calculate the torsion of the curve β, we obtain β = 1 [( ξ 1 3 (ξ 1 ) + ξ1 ξ )Ω 1 +( ξ + (ξ ) ξ 1 ξ + (ξ ) )Ω + ( ξ 1 ξ (ξ 1 ) )B ], similarly... β = 1 3 [( ξ 1 ξ 1 ξ 1 + ξ 1 ξ + ξ 1 ξ )Ω 1 +( ξ + 4ξ ξ ξ 1 ξ ξ 1 ξ )Ω +(ξ ξ + ξ1 (ξ ) (ξ1 ) (ξ1 ) 3 + (ξ1 ) ξ )B ]. The torsion of the curve β is τ β = 1 K (s) 9 M1 (s) + M (s) + M 3 (s) {[Q 3(s)( ξ + (ξ ) ξ1 ξ ) Q (s)(ξ1 ξ ξ1 ξ )]ξ1 + [Q 3 (s)( ξ 1 (ξ1 ) + ξ1 ξ ) Q 1 (s)( ξ + (ξ ) ξ1 ξ )]ξ [Q (s)( ξ1 (ξ 1 ) + ξ1 ξ ) Q 1 (s)( ξ + (ξ ) ξ1 ξ )](ξ1 ξ )}, Q 1 (s) = ξ 1 (ξ 1 ) + ξ 1 ξ, Q (s) = ξ 1 ξ + ξ ξ ξ, Q 3 (s) = ξ 1 (ξ ) (ξ 1 ) (ξ 1 ) 3 + (ξ 1 ) ξ + ξ ξ. 3.5 Example Example 3.1 Next, let us consider the following unit speed curve w = w(s) in E 3 1 as follows w(s) = (s, ln(sec h(s)), arctan(sinh(s))). (3.5)

11 Smarache Curves of a Spacelike Curve According to the Bishop Frame of Type- 39 It is rendered in Figure 1, as follows Figure 1 The curvature function Serret-Frenet frame of the curve w(s) is expressed as T = (1, tanh(s), sech(s)), N = (0, sech(s), tanh(s)), B = (, tanh(s), sec h(s)), (3.6) κ = sech(s), θ = s sech(s)ds = arctan(sinh(s)). (3.7) 0 Figure Ω 1 Ω -Smarache curve Figure 3 Ω 1 B-Smarache curve Also the Bishop frame is computed as Ω 1 = ( sinhθ, sinhθ tanh(s) coshθ sech(s), sinhθ sech(s) coshθ tanh(s)), (3.8) Ω = (coshθ, coshθ tanh(s) + sinhθ sec h(s), coshθ sec h(s) sinhθ tanh(s)), (3.9)

12 40 Yasin Ünlütürk Süha Yılmaz B = (, tanh(s), sec h(s)). (3.30) Let us see the graphs which belong to all versions of Smarache curves according to the Bishop frame in E 3 1. The parametrizations plottings of Ω 1 Ω, Ω 1 B, Ω B Ω 1 Ω B Smarache curves are, respectively, given in Figures -5. Figure 4 Ω B-Smarache curve Figure 5 Ω 1 Ω B-Smarache curve 4. Smarache Breadth Curves According to the Bishop Frame of Type- in E 3 1 A regular curve more than breadths in Minkowski 3-space is called a Smarache breadth curve. Let = (s) be a Smarache breadth curve, also suppose that = (s) is a simple closed curve in E 3 1. This curve will be denoted by (C). The normal plane at every point P on the curve meets the curve at a single point Q other than P. We call the point Q as the opposite point of P We consider a curve = (s ),in the class Γ, which has parallel tangents ζ ζ at opposite directions at the opposite points of the curve. A simple closed curve having parallel tangents in opposite directions at opposite points can be represented with respect to Bishop frame by the equation (s ) = (s) + λω 1 + µω + ηb, (4.1) λ(s), µ(s) η(s) are arbitrary functions, are opposite points. Differentiating both sides of (4.1) considering Bishop equations, we have d ds = ds Ω 1 ds = (dλ ds ηξ 1 + 1)Ω 1 + ( dµ ds ηξ )Ω +( dη ds + λξ 1 µξ )B. (4.)

13 Smarache Curves of a Spacelike Curve According to the Bishop Frame of Type- 41 Since Ω 1 = Ω 1, rewriting (4.) we obtain respectively dλ ds = ηξ 1 1 ds ds, dµ ds = ηξ, dη ds = λξ 1 + µξ. (4.3) If we call θ as the angle between the tangent of the curve (C) at the point (s) with a given direction consider dθ = τ, (4.3) turns into the following form: ds dλ dθ = ηξ 1 τ 1 ds (1 + τ ds ), dµ dθ = ηξ τ, dη dθ = λ τ ξ 1 + µ τ ξ, (4.4) ds ds = ds dθ.dθ ds = 1 ds τ dθ, 1 + ds = f(θ), τ 0. ds Using system (4.4), we have the following vectorial differential equation with respect to λ as follows d 3 λ dθ 3 + { τ ξ1 ξ f(θ) ( ξ 1ξ τ 3 (ξ 1 τ ) ) τ } d λ ξ 1 dθ + {(ξ 1 τ ) +[ ξ 1 τ (ξ 1 τ ) ] (1 λ) ( τ ) [ ξ 1ξ ξ 1 τ 3 (ξ 1 τ )] ( τ ) ξ 1 [ ξ 1ξ τ 3 (ξ 1 τ ) ] [ ξ τ 3 (ξ 1 τ ) ] + [ 1 ( τ )( τ ) f(θ)]} dλ ξ 1 ξ ξ dθ +{[ ξ 1ξ τ 3 (ξ 1 τ ) ] ( τ )( τ ) }( dλ ξ 1 ξ ξ dθ ) + { 1 + τ } dλ d λ ξ 1 ξ 1 dθ dθ +{( ξ 1 τ ) + [ ξ 1ξ τ 3 (ξ 1 τ ) ] 1 f(θ) + ( ξ 1 1) 1 f(θ)}λ ξ 1 ξ ξ 1 +{[( ξ 1 τ ) + 1 f(θ)][ ξ 1ξ ξ 1 τ 3 (ξ 1 τ ) ] [ τ ( 1 ) f(θ) + τ f (θ) + ξ 1 ]} ξ ξ 1 ξ 1 ξ ξ +{ ξ 1ξ τ 3 (ξ 1 τ ) [( 1 ) f(θ) ( 1 )f (θ) + ξ 1 ξ 1 ξ 1 ξ 1 τ f(θ)] +( 1 τ ) f(θ) + 1 τ f (θ)} = 0. The equation (4.5) is a characterization for. If the distance between opposite points of (C) (C ) is constant, then we can write that hence, we write (4.5) = λ + µ + η = l = const., (4.6) Considering the system (4.4) together with (4.7), we obtain From system (4.4) we have λ dλ dθ + µdµ dθ + ηdη = 0. (4.7) dθ λf(θ) = η( ξ 1 τ τηξ ξ 1 ). (4.8) η = τ ξ 1 dλ dθ + 1 ξ 1 f(θ). (4.9)

14 4 Yasin Ünlütürk Süha Yılmaz Substituting (4.9) into (4.8) gives λf(θ) = (τ dλ dθ + f(θ))(ξ 1 τ τηξ ξ 1 ) or τ dλ λf(θ) + f(θ) = dθ G(θ), (4.10) G(θ) = ξ 1 τ τηξ ξ 1, τ 0. Thus we find λ = θ 0 f(θ) ( λ τ G(λ) 1)dθ, (4.11) also from (4.4), (4.9) (4.4) 1 we obtain µ = θ ( dλ dθ + f(θ) )ξ dθ. (4.1) τ 0 η = θ 0 [ τ ( dλ ξ 1 dθ + f(θ) )]dθ. (4.13) τ References [1] Karacan MK, Bükcü B (008), An alternative moving frame for a tubular surface around a spacelike curve with a spacelike normal in Minkowski 3-space, Rend Circ Mat Palermo, 57 (): [] Hanson AJ, Ma H (1995), Quaternion frame approach to streamline visualization, IEEE T Vis Comput Gr., 1(): [3] Liu H., Pei D (013), Singularities of a space curve according to the relatively parallel adapted frame its visualization, Math Probl Eng. Art, ID 5100, 1 pp. [4] Bishop LR(1975), There is more than one way to frame a curve, Am. Math. Mon., 8: [5] Hanson AJ (005), Visualizing Quaternions, Morgan Kaufmann, San Francisco. [6] Bükçü B, Karacan MK (009), The slant helices according to bishop frame of the spacelike curve in Lorentzian space, J. Interdiscipl. Math., 1(5): [7] Özdemir M, Ergin AA (008), Parallel frames of non-lightlike curves, Mo. J. Math. Sci., 0(): [8] Yılmaz S (010), Bishop spherical images of a spacelike curve in Minkowski 3-space, Int. J. Phys. Sci., 5(6): [9] Turgut M, Yılmaz S (008), Smarache curves in Minkowski space-time, Int. J. Math. Comb., 3: [10] Turgut (009), Smarache breadth pseudo null curves in Minkowski space-time, Int. J. Math. Comb., 1: [11] Ali AT (010), Special Smarache curves in the Euclidean space, Int. J. Math. Comb., :30-36.

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