Smarandache Curves According to Bishop Frame in Euclidean 3-Space

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1 Gen. Math. otes, Vol. 0, o., February 04, pp ISS 9-784; Copyright c ICSRS Publication, 04 Available free online at Smarache Curves According to Bishop Frame in Euclidean -Space Muhammed Çetin, Yılmaz Tunçer Murat Kemal Karacan Celal Bayar University, Instutition of Science Technology, Manisa-Turkey mat.mcetin@hotmail.com Uşak University, Faculty of Sciences Arts, Department of Mathematics Eylul Campus, 6400, Uşak-Turkey yilmaz.tuncer@usak.edu.tr Uşak University, Faculty of Sciences Arts, Department of Mathematics Eylul Campus, 6400, Uşak-Turkey murat.karacan@usak.edu.tr Received: -- / Accepted: -- Abstract In this paper, we investigate special Smarache curves according to Bishop frame in Euclidean -space we give some differential geometric properties of Smarache curves. Also we find the centers of the osculating spheres curvature spheres of Smarache curves. Keywords: Smarache curves, Bishop frame, atural curvatures, Osculating sphere, Curvature sphere, Centers of osculating spheres. Introduction Special Smarache curves have been investigated by some differential geometers [4,9]. A regular curve in Minkowski space-time, whose position vector is composed by Frenet frame vectors on another regular curve, is called a Smarache Curve. M. Turgut S. Yilmaz have defined a special case of such curves call it Smarache Curves T B in the space E 4 [9]. They have dealed with a special Smarache curves which is defined by the tangent

2 Smarache Curves According to Bishop... 5 second binormal vector fields. They have called such curves as Smarache T B Curves. Additionally, they have computed formulas of this kind curves by the method expressed in [0]. A. T. Ali has introduced some special Smarache curves in the Euclidean space. He has studied Frenet-Serret invariants of a special case [4]. In this paper, we investigate special Smarache curves such as Smarache Curves T, T, T according to Bishop frame in Euclidean -space. Furthermore, we find differential geometric properties of these special curves we calculate first second curvature natural curvatures of these curves. Also we find the centers of the curvature spheres osculating spheres of Smarache curves. The Bishop frame or parallel transport frame is an alternative approach to defining a moving frame that is well defined even when the curve has vanishing second derivative. We can parallel transport an orthonormal frame along a curve simply by parallel transporting each component of the frame. The parallel transport frame is based on the observation that, while T s for a given curve model is unique, we may choose any convenient arbitrary basis s, s for the remainder of the frame, so long as it is in the normal plane perpendicular to T s at each point. If the derivatives of s, s depend only on T s not each other we can make s s vary smoothly throughout the path regardless of the curvature [,,8]. In addition, suppose the curve is an arclength-parametrized C curve. Suppose we have C unit vector fields = T along the curve so that T, = T, =, = 0, i.e., T,, will be a smoothly varying right-hed orthonormal frame as we move along the curve. To this point, the Frenet frame would work just fine if the curve were C with κ 0 But now we want to impose the extra condition that, = 0. We say the unit first normal vector field is parallel along the curve. This means that the only change of is in the direction of T. A Bishop frame can be defined even when a Frenet frame cannot e.g., when there are points with κ = 0. Therefore, we have the alternative frame equations T = 0 k k k 0 0 k 0 0 T. One can show that κs = k + k, θs = arctan k k, k 0, τs = dθs ds so that k k effectively correspond to a Cartesian coordinate system for the polar coordinates κ, θ with θ = τsds. The orientation of the parallel transport frame includes the arbitrary choice of integration constant θ 0, which

3 5 Muhammed Çetin et al. disappears from τ hence from the Frenet frame due to the differentiation [,,8]. Bishop curvatures are defined by k = κ cos θ, k = κ sin θ [7]. T = T, = cos θ B sin θ, = sin θ + B cos θ Smarache Curves According to Bishop Frame In [4], author gave following definitions: Definition. Let = s be a unit speed regular curve in E {T,, B} be its moving Serret-Frenet frame. Smarache curves T are defined by βs = T +. Definition. Let = s be a unit speed regular curve in E {T,, B} be its moving Serret-Frenet frame. Smarache curves B are defined by βs = + B. Definition. Let = s be a unit speed regular curve in E {T,, B} be its moving Serret-Frenet frame. Smarache curves T B are defined by βs = T + + B. In this section, we investigate Smarache curves according to Bishop frame in Euclidean -space. Let = s be a unit speed regular curve in E denote by {T,, } the moving Bishop frame along the curve. The following Bishop formulae is given by T = k + k, = k T, = k T with T =, = T, = T.

4 Smarache Curves According to Bishop Smarache Curves T Definition.4 Let = s be a unit speed regular curve in E {T,, } be its moving Bishop frame. Smarache curves T can be defined by βs = T +. 4 We can investigate Bishop invariants of Smarache curves T according to = s. Differentiating 4 with respect to s, we get β = dβ ds ds ds = k T k k 5 T β ds ds = k T k k ds ds = k + k. 6 The tangent vector of curve β can be written as follow, T β = k + k k T k k. 7 Differentiating 7 with respect to s, we obtain dt β ds ds ds = k + k λ T + λ + λ 8 λ = k k k 4 k k k 4 + k k λ = k 4 k k + k k k k k λ = k k k k + k k k k k. k Substituting 6 in 8, we get T β = k + k λ T + λ + λ. Then, the curvature principal normal vector field of curve β are respectively, λ κ β = + λ + λ k + k

5 54 Muhammed Çetin et al. β = λ T + λ λ + λ + λ + λ. Thus the binormal vector of curve β is B β = k + k σ T + σ λ + λ + λ + σ σ = λ k λ k, σ = λ k + λ k, σ = k λ + λ. We differentiate 5 with respect to s in order to calculate the torsion β = { k + k + k T k k } k k k... β = ρ T + ρ + ρ ρ = k k k k k + k + k k ρ = k k k k k + k ρ = k k k k k k k + k. The torsion of curve β is { k k ρ k + ρ k + k } k k k ρ + ρ k + k + k ρ k ρ k τ β = k k k k + k k k k k k k + k + k k. The first the second normal vectors of curve β are as follow. Then, from we obtain β = { µ cos θβ λ sin θ β σ T + } µ cos θβ λ sin θ β σ µη + µ cos θβ λ sin θ β σ β = µη { µ sin θβ λ + cos θ β σ T + µ sin θβ λ + cos θ β σ + µ sin θβ λ + cos θ β σ } θ β = τ β sds, µ = k + k η = λ + λ + λ

6 Smarache Curves According to Bishop We can calculate natural curvatures of curve β, so from we get k β = λ + λ + λ cos θ β k + k k β = λ + λ + λ sin θ β k + k.. Smarache Curves T Definition.5 Let = s be a unit speed regular curve in E {T,, } be its moving Bishop frame. Smarache curves T can be defined by βs = T +. 9 We can get Bishop invariants of Smarache curves T = s. From 9, we obtain according to β = dβ ds ds ds = k T k k 0 T β ds ds = k T k k ds ds = k + k. We can write the tangent vector of curve β as follow, T β = k + k k T k k. From, we have dt β ds ds ds = k + k λ T + λ + λ λ = k k k 4 k k k 4 + k k k λ = k k k k + k k k k k λ = k k k 4 + k k k k k.

7 56 Muhammed Çetin et al. By using, we obtain T β = k + k λ T + λ + λ. Then, the first curvature the principal normal vector field of curve β are respectively, λ κ β = + λ + λ k + k β = λ T + λ λ + λ + λ + λ. The binormal vector of curve β is B β = k + k σ T + σ λ + λ + λ + σ σ = λ k λ k, σ = λ + λ k, σ = λ k + λ k. From 0 we can calculate the torsion of curve β β = { k + k + k T k k k } k k... β = ρ T + ρ + ρ, ρ = k k k k k + k k + k ρ = k k k k k k k + k ρ = k k k k k + k. The torsion of curve β is { k k k k ρ + ρ + k k } ρ k + ρ k k + k + k ρ k ρ k τ β = k k k k + k k k + k k k k + k k + k. The first the second normal vectors of curve β are as follow. Then, by using we get

8 Smarache Curves According to Bishop β = µη { µ cos θβ λ sin θ β σ T + µ cos θβ λ sin θ β σ + µ cos θβ λ sin θ β σ } β = µη { µ sin θβ λ + cos θ β σ T + µ sin θβ λ + cos θ β σ + µ sin θβ λ + cos θ β σ } θ β = τ β sds, µ = k + k η = λ + λ + λ. We can calculate natural curvatures of curve β, so from we get k β = λ + λ + λ cos θ β k + k k β = λ + λ + λ sin θ β k + k.. Smarache Curves Definition.6 Let = s be a unit speed regular curve in E {T,, } be its moving Bishop frame. Smarache curves can be defined by βs = +. 4 We can obtain Bishop invariants of Smarache curves to = s. From 9, we get according β = dβ ds ds ds = k + k T 5 ds T β ds = k + k T ds ds = k + k. 6 The tangent vector of curve β can be written as follow, T β = T. 7

9 58 Muhammed Çetin et al. Differentiating 7 with respect to s, we get Substituting 6 in 8, we get dt β ds ds ds = k k 8 T β = k + k k + k. The first curvature the principal normal vector field of curve β are respectively, k + k β = κ β = k + k k + k k + k. The binormal vector of curve β is B β = k + k k k. We differentiate 5 with respect to s, we get the torsion of curve β β = { k + k T + k + k k + k k + k }... β = ρ T + ρ + ρ, ρ = k + k k k k k k k ρ = k k + k k + k k ρ = k k + k k + k k. The torsion of curve β is τ β = { ρ k + k k ρ k k + k } { k k + k k + k + k + k k }.

10 Smarache Curves According to Bishop The first the second normal vectors of curve β are as follow. From we get β = k + k {k sin θ β k cos θ β k cos θ β + k sin θ β }, β = k + k { k sin θ β + k cos θ β + k cos θ β k sin θ β } θ β = τ β sds. We can calculate the natural curvatures of curve β, so from we have k β = k β = k + k cos θ β k + k k + k sin θ β. k + k.4 Smarache Curves T Definition.7 Let = s be a unit speed regular curve in E {T,, } be its moving Bishop frame. Smarache curves T can be defined by βs = T ac- We can investigate Bishop invariants of Smarache curves T cording to = s. By using 9, we get β = dβ ds ds ds = k + k T k k 0 T β ds ds = k + k T k k ds ds = k + k k + k.

11 60 Muhammed Çetin et al. The tangent vector of curve β is as follow, T β = { k + k k + k } k + k T k k. Differentiating with respect to s, we get dt β ds ds ds = { k + k k + k } λ T + λ + λ λ = k k k k + k k k k 4 k k 4 k k k k k 4 + k k k + k k λ = k 4 4 k k 4 k k k k + k k k +k k k k k k k λ = k k 4 k k 4k k k 4 + k k + k k k k k k k k. Substituting in, we get T β = 4 { k + k k + k } λ T + λ + λ. The first curvature the principal normal vector field of curve β are respectively, λ κ β = + λ + λ 4 { k + k k + k } β = Hence, the binormal vector of curve β is B β = λ T + λ λ + λ + λ + λ. 4 k + k k + k σ T + σ λ + λ + λ + σ 5 σ = λ k λ k, σ = λ k + k + λ k, σ = λ k + k + λ k. Differentiating 0 with respect to s, we can calculate the torsion of curve β, β = { k + k + k + k T k k } k k k k k k

12 Smarache Curves According to Bishop β = ρ T + ρ + ρ, ρ = k k k k k k + k + k k + k k + k ρ = k k k k k k k k k + k ρ = k k k k k k k k k + k. The torsion of curve β is τ β = k k k k ρ k ρ k ρ k + k k k k ρ k + ρ k + ρ k k + k + k + k ρ k + ρ k { k k k k + k k k k k k k k k + k + k k + k k k k + k k }. The first the second normal vectors of curve β are as follow. By using we obtain β = µη { µ cos θβ λ sin θ β σ T + µ cos θ β λ sin θ β σ + µ cos θ β λ sin θ β σ β = µη { µ sin θβ λ + cos θ β σ T + µ sin θ β λ + cos θ β σ + µ sin θ β λ + cos θ β σ } }, θ β = τ β sds, µ = k + k k + k η = λ + λ + λ. We can calculate the natural curvatures of curve β, so from we get k β = λ + λ + λ cos θ β 4k + k k + k k β = λ + λ + λ sin θ β 4k + k k + k.

13 6 Muhammed Çetin et al. Curvature Spheres Osculating Spheres of Smarache Curves Definition. Let : a, b R n be a regular curve let F : R n R be a differentiable function. We say that a parametrically defined curve an implicity defined hypersurface F 0 have contact of order k at t 0 provided we have F ot 0 = F o t 0 =... = F o k t 0 = 0 but F o k+ t 0 0 [5]. If any circle which has a contact of first-order with a planar curve, it is called the curvature circle. If any sphere which has a contact of second-order with a space curve, It is called the curvature sphere. Furthermore, if any sphere which has a contact of third-order with a space curve, it is called the osculating sphere. Theorem. Let βs be a unit speed Smarache curve of s with first curvature k β second curvature k β, then the centres of curvature spheres of βs according to Bishop frame are cs = βs + kβ k β Γ β 4k β + kβ ± k β Γ 4k β k β Γ = r k β k β. β Proof: Assume that cs is the centre of curvature sphere of βs, then radius vector of curvature sphere is c β = δ T β + δ β + δ β. Let we define the function F such that F = cs βs, cs βs r. First second derivatives of F are as follow. F = δ F = k β δ + k β δ. The condition of contact of second-order requires F = 0, F = 0 F = 0. Then, coefficients δ, δ, δ are obtained δ = 0, δ = kβ kβ Γ, δ 4k β = k β ±kβ Γ 4k β kβ.

14 Smarache Curves According to Bishop... 6 Thus, centres of the curvature spheres of Smarache curve β are cs = βs + kβ k β Γ β 4k β + kβ ± k β Γ β 4k β k β. Let δ = λ, then cs = βs + δ β + λ β is a line passing the point βs + δ β parallel to line β. Thus we have the following corollary. Corollary. Let βs be a unit speed Smarache curve of s then each centres of the curvature spheres of β are on a line. Theorem.4 Let βs be a unit speed Smarache curve of s with first curvature k β second curvature k β, then the centres of osculating spheres of βs according to Bishop frame are cs = βs + k β k β k β k β β k β k β k β k β β. Proof: Assume that cs is the centre of osculating sphere of βs, then radius vector of osculating sphere is cs βs = δ T β + δ β + δ β. Let we define the function F such that F = cs βs, cs βs r. First, second third derivatives of F are as follow. F = δ F = k β δ + k β δ F = k β δ k β δ + k β δ k β δ. The condition of contact of third-order requires F = 0, F = 0, F = 0 F = 0. Then, coefficients δ, δ, δ are obtained δ = 0, δ = k β k β k β k β, δ = kβ k β k β k β. Thus, centres of the osculating spheres of Smarache curve β can be written as follow cs = βs k β + k β k β β k β k β k β k β β. k β

15 64 Muhammed Çetin et al. Corollary.5 Let βs be a unit speed Smarache curve of s with first curvature k β second curvature k β, then the radius of the osculating sphere of βs is k β + k β rs = k β k β k β. Example.6 Let t be the Salkowski curve given by t = t, t, t = n + n sin + n t + m 4 + n 4 n sin n t sin t = = n + m 4 + n cos nt 4m + m cos + n t + + n 4 n cos n t + cos t n = m +m. t can be expressed with the arc lenght parameter. We get unit speed regular curve by using the parametrization t = n n arcsin + m s. Furthermore, graphics of special Smarache curves are as follow. Here m =.

16 Smarache Curves According to Bishop References [] A. Gray, Modern Differential Geometry of Curves Surfaces with Mathematica nd Edition, CRC Press, 998. [] A.J. Hanson H. Ma, Parallel transport approach to curve framing, Indiana University, Techreports-TR45, January 995. [] A.J. Hanson H. Ma, Quaternion frame approach to streamline visualization, IEEE Transactions on Visualization Computer Graphics, I June 995, [4] A.T. Ali, Special Smarache curves in the Euclidean space, International Journal of Mathematical Combinatorics, 00, 0-6. [5] B. Bükçü M.K. Karacan, On the slant Helices according to Bishop frame, International Journal of Computational Mathematical Sciences, Spring 009, [6] B. Bükçü M.K. Karacan, Special Bishop motion Bishop Darboux rotation axis of the space curve, Journal of Dynamical Systems Geometric Theories, 6008, 7-4. [7] E. Turhan T. Körpınar, Biharmonic slant Helices according to Bishop frame in E, International Journal of Mathematical Combinatorics, 00,

17 66 Muhammed Çetin et al. [8] L.R. Bishop, There is more than one way to frame a curve, Amer. Math. Monthly, 8 975, [9] M. Turgut S. Yilmaz, Smarache curves in Minkowski space-time, International Journal of Mathematical Combinatorics, 008, [0] S. Yilmaz M. Turgut, On the differential geometry of the curves in Minkowski space-time I, Int. J. Contemp. Math. Sciences, 7 008, 4-49.