Smarandache Curves and Applications According to Type-2 Bishop Frame in Euclidean 3-Space

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1 International J.Math. Combin. Vol.(6), -5 Smarandache Curves and Applications According to Type- Bishop Frame in Euclidean 3-Space Süha Yılmaz (Dokuz Eylül University, Buca Educational Faculty, 355, Buca-Izmir, Turkey) Ümit Ziya Savcı (Celal Bayar University, Department of Mathematics Education, 459, Manisa-Turkey) Abstract: In this paper, we investigate Smarandache curves according to type- Bishop frame in Euclidean 3- space and we give some differential geometric properties of Smarandache curves. Also, some characterizations of Smarandache breadth curves in Euclidean 3- space are presented. Besides, we illustrate examples of our results. Key Words: Smarandache curves, Bishop frame, curves of constant breadth. AMS(): 53A5, 53B5, 53B3.. Introduction A regular curve in Euclidean 3-space, whose position vector is composed by Frenet frame vectors on another regular curve, is called a Smarandache curve. M. Turgut and S. Yılmaz have defined a special case of such curves and call it Smarandache TB curves in the space E 4 []. Moreover, special Smarandache curves have been investigated by some differential geometric [6]. A.T.Ali has introduced some special Smarandache curves in the Euclidean space []. Special Smarandache curves according to Sabban frame have been studied by [5]. Besides, It has been determined some special Smarandache curves E 3 by []. Curves of constant breadth were introduced by L.Euler [3]. We investigate position vector of curves and some characterizations case of constant breadth according to type- Bishop frame in E 3.. Preliminaries The Euclidean 3-space E 3 proved with the standard flat metric given by <, >= dx + dx + dx 3 Received November 6, 5, Accepted May 6, 6.

2 Süha Yılmaz and Ümit Ziya Savcı (x, x, x 3 ) is rectangular coordinate system of E 3. Recall that, the norm of an arbitrary vector a E 3 given by a = < a, a >. ϕ is called a unit speed curve if velocity vector υ of ϕ satisfied υ = The Bishop frame or parallel transport frame is alternative approach to defining a moving frame that is well defined even when the curve has vanishing second derivative. One can express parallel transport of orthonormal frame along a curve simply by parallel transporting each component of the frame [8]. The type- Bishop frame is expressed as ξ ξ B ε ξ = ε. ξ (.) ε ε B In order to investigate type- Bishop frame relation with Serret-Frenet frame, first we B = τn = ε ξ + ε ξ (.) Taking the norm of both sides, we have κ(s) = dθ(s) ds, τ(s) = ε + ε (.3) Moreover, we may express ε (s) = τ cosθ(s), ε (s) = τ sin θ(s) (.4) By this way, we conclude θ(s) = Arc tan ε. The frame {ξ, ξ, B} is properly oriented, ε and τ and θ(s) = s κ(s)ds are polar coordinates for the curve α(s). We write the tangent vector according to frame {ξ, ξ, B} as and differentiate with respect to s T = sinθ(s)ξ cosθ(s)ξ T = κn = θ (s)(cosθ(s)ξ + sin θ(s)ξ ) + sinθ(s)ξ cosθ(s)ξ (.5) Substituting ξ = ε B and ξ = ε B in equation (.5) we have κn = θ (s)(cos θ(s)ξ + sinθ(s)ξ ) In the above equation let us take θ (s) = κ(s). So we immediately arrive at N = cosθ(s)ξ + sin θ(s)ξ

3 Smarandache Curves and Applications According to Type- Bishop Frame in Euclidean 3-Space 3 Considering the obtained equations, the relation matrix between Serret-Frenet and the type- Bishop frame can be expressed T sin θ(s) cos θ(s) ξ N = cos θ(s) sin θ(s). ξ (.6) B B 3. Smarandache Curves According to Type- Bishop Frame in E 3 Let α = α(s) be a unit speed regular curve in E 3 and denote by {ξ α, ξ α, B α } the moving Bishop frame along the curve α. The following Bishop formulae is given by ξ α = εα Bα, ξ α = εα Bα, B α = ε α ξα + εα ξα 3. ξ ξ -Smarandache Curves Definition 3. Let α = α(s) be a unit speed regular curve in E 3 and {ξ α, ξα, Bα } be its moving Bishop frame. ξ ξ -Smarandache curves can be defined by β(s ) = (ξ α + ξ α ) (3.) Now, we can investigate Bishop invariants of ξ ξ -Smarandache curves according to α = α(s). Differentiating (3..) with respect to s, we get β = dβ ds ds ds = (ε α + ε α )B α T β ds ds = (ε α + εα )Bα (3.) The tangent vector of curve β can be written as follow; Differentiating (3.4) with respect to s, we obtain ds ds = (ε α + ε α ) (3.3) T β = B α = (ε α ξ α + ε α ξ α ) (3.4) dt β ds ds ds = εα ξ α + ε α ξ α (3.5)

4 4 Süha Yılmaz and Ümit Ziya Savcı Substituting (3.3) in (3.5), we get T β = ε α + (ε α εα ξα + εα ξα ) Then, the curvature and principal normal vector field of curve β are respectively, T β N β = = κ β = ε α + εα (ε α ) + (ε α ) (ε α ) + (ε α ) (ε α ξα + εα ξα ) On the other hand, we express B β = det (ε α ) + (ε α ) ξ α ξ α B α ε α ε α. So, the binormal vector of curve β is B β = (ε α ) + (ε α ) (ε α ξα εα ξα ) We differentiate (3.) with respect to s in order to calculate the torsion of curve β β = {[(ε α ) + ε α εα ]ξα + [ε α εα + (εα ) ]ξ α + [ ε α + ε α ]}Bα ] and similarly β = (δ ξ α + δ ξ α + δ 3B α ) δ = 3ε α ε α +ε α ε α +ε α ε α - (ε α ) 3 -(ε α ) ε α δ = ε α εα +εα ε α +3εα ε α -εα (εα ) -(ε α )3 δ 3 = ε α + ε α The torsion of curve β is τ β = ε α +εα 4 [(ε α ) +(ε α ) ] {[(εα + ε α )(ε α ε α + (ε α ) ]δ [(ε α + ε α )((ε α ) + ε α ε α )]δ } 3. ξ B-Smarandache Curves Definition 3. Let α = α(s) be a unit speed regular curve in E 3 and {ξ α, ξα, Bα } be its moving

5 Smarandache Curves and Applications According to Type- Bishop Frame in Euclidean 3-Space 5 Bishop frame. ξ B-Smarandache curves can be defined by β(s ) = (ξ α + Bα ) (3.6) Now, we can investigate Bishop invariants of ξ B-Smarandache curves according to α = α(s). Differentiating (3.6) with respect to s, we get β = dβ ds ds ds = (ε α B α + ε α ξ α + ε α ξ α ) T β ds ds = ( ε α B α + ε α ξ α + ε α ξ α ) ds ds = (ε α ) + (ε α ) The tangent vector of curve β can be written as follow; (3.7) (3.8) T β = (ε α ) + (ε α ) (ε α ξα + εα ξα εα Bα ) (3.9) Differentiating (3.9) with respect to s, we obtain dt β ds ds ds = [ 3 (ε α ) +(ε α )] (µ ξ α +µ ξ α +µ 3 B α ) (3.) µ = ε α ε α ε α + ε α (ε α ) µ = (ε α ) ε α -ε α ε α ε α + (ε α ) ε α -(ε α ) 3 ε α -ε α (ε α ) 3 µ 3 = ε α ε α ε α (ε α ) 4 + (ε α ) (ε α ) ε α (ε α ) Substituting (3.8) in (3.), we have T β = [ (ε α ) + (ε α )] (µ ξ α + µ ξ α + µ 3B α ) Then, the first curvature and principal normal vector field of curve β are respectively T β =κ β = µ [(ε α ) +(ε α ) ] + µ + µ 3 N β = µ + µ + µ 3 (µ ξ α + µ ξ α + µ 3 B α )

6 6 Süha Yılmaz and Ümit Ziya Savcı On the other hand, we get B β = µ +µ +µ 3 (ε α ) + (ε α ) [(µ ε α +µ 3ε α )ξα (µ ξ α +µ 3 ξ α )ξ α +(µ ε α -µ ε α )B α ] We differentiate (3.7) with respect to s in order to calculate the torsion of curve β β = {[- (ε α ) + ε α ]ξα + [ ε α ε α + ε α (ε α ) ]ξ α ε α B α } and similarly β = (Γ ξ α + Γ ξ α + Γ 3B α ) Γ = -6ε α ε α + ε α + (ε α ) 3 Γ = - ε α ε α -ε α ε α + ε α -ε α ε α +ε α (ε α ) - ε α ε α + (ε α ) 3 Γ 3 = - ε α The torsion of curve β is τ β = - [ (εα ) + (ε α ) ] 4 4 (µ +µ +µ 3 ) {[(-εα ε α - ε α +(ε α ) )Γ -((ε α ) ε α )Γ +(-ε α ε α - ε α + (ε α ) )Γ 3 ]ε α -[( ε α (ε α ) )Γ 3 + ε α Γ ]ε α } 3.3 ξ B-Smarandache Curves Definition 3.3 Let α = α(s) be a unit speed regular curve in E 3 and {ξ α, ξ α, B α } be its moving Bishop frame. ξ B-Smarandache curves can be defined by β(s ) = (ξ α + B α ) (3.) Now, we can investigate Bishop invariants of ξ B-Smarandache curves according to α = α(s). Differentiating (3.) with respect to s, we get β = dβ ds ds ds = ( εα B α + ε α ξ α + ε α ξ α ) T β ds ds = (εα ξ α + ε α ξ α ε α B α ) (3.)

7 Smarandache Curves and Applications According to Type- Bishop Frame in Euclidean 3-Space 7 ds ds = (ε α ) + (ε α ) The tangent vector of curve β can be written as follow; (3.3) T β = εα ξα + εα ξα εα Bα (ε α ) + (ε α ) (3.4) Differentiating (3.4) with respect to s, we obtain dt β ds ds ds = [ 3 (ε α ) +(ε α )] (η ξ α +η ξ α +η 3 B α ) (3.5) η = ( ε α (ε α ) -ε α ε α ) η = (ε α ) ε α + (ε α ) ε α -ε α ε α ε α η 3 = (ε α ) ε α + (εα )3 -(ε α )4 - (ε α )4-3 (ε α ) (ε α ) Substituting (3.3) in (3.5), we have T β = [ (ε α ) + (ε α )] (η ξ α + η ξ α + η 3 B α ) Then, the first curvature and principal normal vector field of curve β are respectively T β η =κ β = + η + η 3 [(ε α ) + (ε α )] N β = η + η + (η ξ η α + η ξ α + η 3B α ) 3 On the other hand, we express B β = η +η +η 3(ε α ) + (ε α ) [(η ε α +η 3ε α )ξα (η ξ α +η 3 ξ α )ξ α +(η ε α -η ε α )B α ] We differentiate (3.) with respect to s in order to calculate the torsion of curve β β = {[ε α ξα + ε α (εα ) ]ξ α + [ ε α (εα ) ]ξ α ε α Bα }

8 8 Süha Yılmaz and Ümit Ziya Savcı and similarly β = (η ξ α + η ξ α + η 3B α ) η = - ε α ε α 5 ε α ε α + ε α +(ε α ) ε α + (ε α ) 3 The torsion of curve β is η = -4 ε α ε α + ε α + ε α η 3 = - ε α τ β = - [(εα ) + (ε α ) ] 4 4 (η +η +η 3 ) {[ ε α η + (ε α (ε α ) )η 3 ]ε α +[ (ε α ) η +(ε α ε α - ε α +(ε α ) )η +(-ε α ε α + ε α )η 3 ]ε α } 3.4 ξ ξ B-Smarandache Curves Definition 3.4 Let α = α(s) be a unit speed regular curve in E 3 and {ξ α, ξα, Bα } be its moving Bishop frame. ξ α ξ B-Smarandache curves can be defined by β(s ) = 3 (ξ α + ξα + Bα ) (3.6) Now, we can investigate Bishop invariants of ξ α ξ B-Smarandache curves according to α = α(s). Differentiating (3.6) with respect to s, we get β = dβ ds ds ds = 3 [(ε α + εα )Bα ε α ξα εα ξα ] T β ds ds = 3 [(ε α + εα )Bα ε α ξα εα ξα )] ds ds = [(ε α ) + ε α εα + (εα ) ] 3 The tangent vector of curve β can be written as follow; T β = εα ξ α + ε α ξ α (ε α + ε α )B α [(ε α ) + ε α εα + (εα ) ] (3.7) (3.8) (3.9)

9 Smarandache Curves and Applications According to Type- Bishop Frame in Euclidean 3-Space 9 Differentiating (3.9) with respect to s, we get dt β ds ds ds = (λ ξ α+λ ξ α+λ 3B α ) [ 3 (ε α ) + ε α εα + (εα )] (3.) λ = [ ε α - (εα ) -ε α εα ]u(s)-εα [εα ε α + ε α εα +εα εα + ε α εα ] λ = [ ε α - (εα ) -ε α εα ]u(s)-εα [ ε α + εα ε α + εα ε α ] λ 3 = [-ε α - ε α ]u(s)+εα [εα ε α + 3ε α εα +εα ε α + ε α εα ] + ε α [ ε α (εα ) + (ε α ) ] Substituting (3.8) in (3.), we have T β = 3(λ ξ α +λ ξ α +λ 3 B α ) 4 [(ε α ) + ε α εα + (εα )] Then, the first curvature and principal normal vector field of curve β are respectively T β N β = 3 λ =κ β = + λ + λ 3 4 [(ε α ) + ε α εα + (εα )] λ + λ + λ 3 (λ ξ α + λ ξ α + λ 3B α ) (3.) On the other hand, we express B β = [(ε α ) +ε α εα +(εα ) ] λ det +λ +λ 3 ξ α ξ α B α ε α ε α -(ε α +ε α ) λ λ λ 3 So, the binormal vector field of curve β is B β = [(ε α ) +ε α εα + (εα ) ] λ {[(ε α + ε α )λ +λ +λ 3 ε α λ 3]ξ α +[ εα λ 3-(ε α + εα )]ξα +[εα λ -ε α λ ]B α } We differentiate (3.) with respect to s in order to calculate the torsion of curve β β = - 3 {[ (ε α ) +ε α ξ α - ε α ]ξ α +[ (ε α ) +ε α ε α - ε α ]ξ α + [ε α + ε α ]B α }

10 Süha Yılmaz and Ümit Ziya Savcı and similarly β = - 3 (σ ξ α + σ ξ α + σ 3B α ) η = 4 ε α ε α +3 ε α ε α - ε α -(ε α ) 3 - (ε α ) ε α η = 5 ε α ε α + ε α ε α +ε α ε α - ε α -(ε α ) 3 -ε α (ε α ) η 3 = ε α +ε α The torsion of curve β is τ β = - 6[(εα ) +ε α εα +(εα ) ] 9 {[( (ε α 3 λ ) +ε α ε α - ε α )σ +(- ε α - (ε α ) -ε α ε α )σ +λ +λ 3 +( (ε α ) +ε α εα - ε α )σ 3]ε α +[- ε α - ε α +(εα ) +ε α εα )σ +(- (ε α ) -ε α ε α + ε α )σ + ( (ε α ) +ε α ε α - ε α )σ 3 ]ε α }. 4. Smarandache Breadth Curves According to Type- Bishop Frame in E 3 A regular curve with more than breadths in Euclidean 3-space is called Smarandache breadth curve. Let α = α(s) be a Smarandache breadth curve. Moreover, let us suppose α = α(s) simple closed space-like curve in the space E 3. These curves 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 the opposite point P. We consider a curve in the class Γ as in having parallel tangents ξ and ξ opposite directions at opposite points α and α of the curves. A simple closed curve having parallel tangents in opposite directions at opposite points can be represented with respect to type- Bishop frame by the equation α (s) = α(s) + λξ + ϕξ + ηb (4.) λ(s), ϕ(s) and η(s) are arbitrary functions also α and α are opposite points. Differentiating both sides of (4.) and considering type- Bishop equations, we have dα ds ds =ξ ds = (dλ ds + ηε + )ξ + ( dϕ ds + ηε )ξ + ( λε ϕε + dη ds )B (4.)

11 Smarandache Curves and Applications According to Type- Bishop Frame in Euclidean 3-Space Since ξ = ξ rewriting (4.) we have dλ ds = dϕ ds = ϕε ηε ds ds dη ds = λε + ϕε (4.3) If we call θ as the angle between the tangent of the curve (C) at point α(s) with a given direction and consider dθ = κ, we have (4.3) as follow: ds dλ dθ = ηε κ f(θ) dϕ dθ = ϕε κ (4.4) dη dθ = λε κ + ϕε κ f(θ) = δ + δ, δ = κ, δ = κ denote the radius of curvature at α and α respectively. And using system (4.4), we have the following differential equation with respect to λ as d 3 λ dθ 3 [ κ d ε dθ (ε λ κ )]d dθ +[ ε κ -ε κ - d dθ ( κ ) d ε dθ (ε κ ) κ d ε dθ (ε κ )]dλ dθ + [ε d κ dθ (ε κ ) ε ε κ ]λ+ +[- κ ε - κ ε d dθ (ε κ )]d f dθ [ κ ε + κ ε d dθ (ε κ )]df dθ (4.5) [ ε ε κ +ε ε + d dθ ( κ ε ) d dθ (ε κ )+ κ ε d dθ (ε κ )]f(θ) = Equation (4.5) is characterization for α. If the distance between opposite points of (C) and (C ) is constant, then we can write that Hence, we write Considering system (4.4) we obtain α α = λ + ϕ + η = l = constant (4.6) λ dλ dθ + ϕdϕ dθ + ηdη dθ = (4.7) λ f(θ) = (4.8)

12 Süha Yılmaz and Ümit Ziya Savcı We write λ = or f(θ) =. Thus, we shall study in the following subcases. Case. λ =. Then we obtain and θ η = κ ε f(θ)dθ, ϕ = θ θ ( η ε κ dθ)ε κ dθ (4.9) d f dθ df dθ [(τ κ ) sin3 θ cosθ τ cosθ]f = (4.) κ General solution of (4.) depends on character of τ. Due to this, we distinguish following κ subcases. Subcase. f(θ) =. then we obtain λ = θ ϕ = θ η = θ η ε κ dθ η ε κ dθ λ ε κ dθ + θ ϕ ε κ dθ (4.) Case. Let us suppose that λ, ϕ,η and λ, ϕ, η constant. Thus the equation (4.4) we obtain ε κ = and ε κ =. Moreover, the equation (4.5) has the form d3 λ dθ 3 = The solution (4.) is λ = L θ + L θ + L 3 L, L and L 3 real numbers. And therefore we write the position vector ant the curvature α = α + A ξ + A ξ + A 3 B A = λ, A = ϕ and A 3 = η real numbers. And the distance between the opposite points of (C) and (C ) is α α = A + A + A 3 = constant 5. Examples In this section, we show two examples of Smarandache curves according to Bishop frame in E 3. Example 5. First, let us consider a unit speed curve of E 3 by β(s)= ( 5 36 sin(9s) 9 85 sin(5s), 5 36 cos(9s) cos(5s), 5 36 sin(8s))

13 Smarandache Curves and Applications According to Type- Bishop Frame in Euclidean 3-Space 3 Fig. The curve β = β(s) See the curve β(s) in Fig.. One can calculate its Serret-Frenet apparatus as the following T = ( 5 cos9s cos5s, sin 9s 9 5 sin5s, 7 cos8s) N = ( csc8s(sin 9s sin 5s), csc 8s(cos9s cos5s), 7 ) B = ( (5 sin9s 9 sin 5s), (5 cos9s + 9 cos5s), 5 7 sin 8s) κ = 5 sin8s and τ = 5 cos8s In order to compare our main results with Smarandache curves according to Serret-Frenet frame, we first plot classical Smarandache curve of β Fig.. Now we focus on the type- Bishop trihedral. In order to form the transformation matrix (.6), let us express θ(s) = Since, we can write the transformation matrix s 5 sin(8s)ds = 5 8 cos(8s) T sin( 5 8 cos8s) cos(5 8 cos8s) ξ N = cos( 5 8 cos8s) sin(5 8 cos8s). ξ B B

14 4 Süha Yılmaz and Ümit Ziya Savcı Fig. ξ ξ Smarandache curve By the method of Cramer, one can obtain type- Bishop frame of β as follows ξ = (sin θ( 5 cos9s 9 5 cos5s) + cosθ csc8s(sin 9s sin5s), sin θ( 5 sin 9s 9 5 sin 5s) cosθ csc 8s(cos9s cos5s), 5 7 sin θ cos8s cosθ) ξ = ( cosθ( 5 cos9s 9 5 cos5s) + sin θ csc 8s(sin9s sin 5s), - cosθ( 5 sin 9s 9 5 sin 5s) sin θ csc8s(cos9s cos5s), 5 7 cosθ cos8s sinθ) B = ( (5 sin9s 9 sin5s), (5 cos9s + 9 cos5s), 5 7 sin 8s) θ = 5 8 cos(8s). So, we have Smarandache curves according to type- Bishop frame of the unit speed curve β = α(s), see Fig.-4 and Fig.5. Fig.3 ξ B Smarandache curve

15 Smarandache Curves and Applications According to Type- Bishop Frame in Euclidean 3-Space 5 Fig.4 ξ B Smarandache curve Fig.5 ξ ξ B Smarandache curve References [] A. Mağden and Ö. Köse, On the curves of constant breadth in E4 space, Tr. J. of Mathematics, (997), [] A.T. Ali, Special Smarandache curves in The Euclidean space, Math. Cobin. Bookser, (), [3] L. Euler, De curvis trangularibus, Acta Acad Petropol, (778), 3-3. [4] L.R. Bishop, There is more than one way to frame a curve, Amer. Math. Monthly, 8 (975), [5] K. Taşköprü, M.Tosun, Smarandache curves on S, Bol. Soc. Paran. Mat., 3() (4), [6] M. Çetin, Y.Tuncer and M. K. Karacan, Smarandache curves according to Bishop frame in Euclidean 3-space, Gen. Math. Notes, (4), [7] M.K. Karacan, B.Bükcü, An alternative moving frame for tubular surface around the spacelike curve with a spacelike binormal in Minkowski 3-space, Math. Morav., (7), [8] M. Özdemir, A.A Ergin, Parallel frames of non-lightlike curves, Missouri J. Math. Sci., () (8), -. [9] M.Petroviç-Torgasev and E. Nesoviç, Some characterizations of the space-like, the timelike and the null curves on the pseudohyperbolic space H in E3, Kragujevac J. Math., (), 7-8. [] M.Turgut, S. Yılmaz, Smarandache curves in Minkowski space-time, International J. Math. Combin., 3 (8), [] M.Turgut, Smarandache breadth pseudo null curves in Minkowski space-time, International J. Math. Combin., (9), [] N.Bayrak, Ö.Bekt as and S. Yüce, Special Smarandache curves in E3, Mathematical Phy. Sics. and [3] S. Yılmaz, M. Turgut, A new version of Bishop frame and an application to spherical images, Journal of Mathematical Analysis and Applications, 37() (),