D- Smarandache Curves According to the Sabban Frame of the Spherical Indicatrix Curve

Turk. J. Math. Comput. Sci. 9(2018) 39–49 c MatDer http://dergipark.gov.tr/tjmcs http://tjmcs.matder.org.tr MATDER D-Smarandache Curves According to The Sabban Frame of The Spherical Indicatrix Curve Süleyman Şenyurt Department of Mathematics, Faculty of Arts and Sciences, Ordu University, 52200, Ordu, Turkey Received: 17-09-2018 • Accepted: 22-11-2018 Abstract. In this study, we first gave the Darboux vector according to the alternative frame. Then we formed a Sabban frame of spherical indicatrix curve of D-alternative vector defined by a differentiable curve. Then the geodesic curvature of this vector is calculated according to this frame. Finally we defined Smarandache curves generated by the Sabban frame and give some characterizations of them. 2010 AMS Classification: 53A04. Keywords: Sabban frame, Smarandache curve, Alternative frame, Spherical indicatrix curve. 1. Introduction In differential geometry, special curves have an important role. One of these curves is a Smarandache curve. Smarandache curves are first defined by M. Turgut and S. Yılmaz in 2008 [6]. Special Smarandache curves also have been studied by some authors [1,2]. Let α = α(s) be a regular unit speed curve in E 3 . The Frenet frame and alternative frame of this curve are and {N, C, W}, respectively. Here, N is normal vector, W is unit Darboux vector and C = W ∧ N [4]. In this paper, we created the Smarandache curves according to the alternative frame of the unit speed curve. Then we introduced alternative frame and its properties. Finally we calculated geodesic curvature of these curves according to alternative frame. 2. Preliminaries Let α = α(s) be a regular curve with unit speed. Then the Frenet apparatus of the curve (α) [3] α00 (s) , B(s) = T (s) ∧ N(s), k α00 (s) k hα0 (s) ∧ α00 (s), α000 (s)i = k α00 (s) k, τ(s) = , k α0 ∧ α00 k2 = κN, N 0 = −κT + τB, B0 = −τN. T (s) = κ(s) T0 α0 (s), N(s) = In Euclidean 3-space any regular curve α(s) depending on the Frenet vectors moves around the axis of Darboux vector and the Darboux vector and defining a unit vector field are given as [4] Email address: D-Smarandache Curves According to The Sabban Frame of The Spherical Indicatrix Curve W= √ τ κ2 + τ2 T+ √ κ κ 2 + τ2 B, 40 κ τ C =W∧N =−√ T+ √ B. 2 2 2 κ +τ κ + τ2 So build another orthonormal moving frame along the curve α(s). This frame defined as alternative frame and is represented by {N, C, W}. The derivative formulae of the alternative frame is given by [4] N 0 = βC, C 0 = −βN + γW, W 0 = −γC, β= √ κ2 + τ2 , γ= τ
0 κ2 . 2 2 κ +τ κ The relationship between the Frenet frame and alternative frame is            T  0 −κ̄ τ̄  N   N   0 1 0  T   C  = −κ̄ 0 τ̄ N  or N  = 1 0 0  C  , κ̄ = κ , τ̄ = τ .             β β 0 τ̄ κ̄ W B τ̄ 0 κ̄ B W Principal normal vector N is common both frames. Let γ : I → S 2 be a unit speed spherical curve and s arc-length 0 parameter of γ.Let us denote t(s) = γ (s) and d(s) = γ(s) ∧ t(s). This frame is called the Sabban frame of γ on S 2 . Then we have the following spherical Frenet formulae of γ γ0 (s) = t(s), t0 (s) = −γ(s) + κg (s)d(s), d0 (s) = −κg (s)t(s), κg (s) = ht0 (s), d(s)i where κg (s) is the geodesic curvature of the curve of γ on S 2 [5]. (2.1) (2.2) 3. Smarandache Curves of Alternative Frame Theorem 3.1. Let α(s) be unit speed curve and alternative frame {N, C, W}. The alternative Darboux vector D̄ of the curve α is given by D̄ = γN + βW. Proof. The alternative Darboux vector of the curve D̄ can be written as follow (Figure 1) D̄ = aN + bC + cW. Taking the cross product of (3.1) and N, we get N 0 = D̄ ∧ N ⇒ βC = (aN + bC + cW) ∧ N ⇒ βC = −bW + cC ⇒ b = 0, c = β. Taking the cross product of (3.1) and C, we get C 0 = D̄ ∧ C ⇒ −βN + γW = (aN + bC + cW) ∧ C ⇒ −βN + γW = aW − cN ⇒ a = γ, c = β. Taking the cross product of (3.1) and W, we get W 0 = D̄ ∧ W ⇒ −γC = (aN + bC + cW) ∧ W ⇒ −γC = −aC − bN ⇒ a = γ, b = 0. Thus, alternative Darboux vector D̄ is obtained as D̄ = γN + βW and the unit Darboux vector D is given by (3.1) S. Şenyurt, Turk. J. Math. Comput. Sci., 9(2018), 39–49 D = pN + qW, 41 p= p γ γ2 + β2 , β q= p . 2 γ + β2 (3.2) Figure 1. Alternative Darboux vector  Let D = D(s) and αD (s) = D(s) be a unit speed regular spherical curves on S , {D, T D , (D ∧ T D )} and {DαD , T DαD , (D ∧ T D )αD } be the Sabban frame of these curves, respectively. If we take the derivative of the equation αD (s) = D(s), then T D vector is 2 Figure 2. {D, T D , D ∧ T D } Sabban Frame TD. d s∗ ds = p0 N + q0 W, q0 d s∗ N+ p W, = 0 2 0 2 0 2 0 2 ds (p ) + (q ) (p ) + (q ) Considering the D(s) and T D vectors we can write, p0 q − pq0 D ∧ TD = p C. (p0 )2 + (q0 )2 | {z } TD = p0 p =1 q (p0 )2 + (q0 )2 . D-Smarandache Curves According to The Sabban Frame of The Spherical Indicatrix Curve 42 If we take the derivative of this equation then (D ∧ T D )0 vector is d s∗ ds = −βN + γW, (D ∧ T D )0 = p (D ∧ T D )0 = (D ∧ T D )0 . −β (p0 )2 + (q0 )2 N+ p γ (p0 )2 + (q0 )2 (3.3) W,  p0 β − q0 γ    q0 −p0 − p N+ p W . p (p0 )2 + (q0 )2 (p0 )2 + (q0 )2 (p0 )2 + (q0 )2 | {z } | {z } κgD TD Accordingly, the {D, T D , (D ∧ T D )} Sabban frame is obtained from the D vector.If we take the derivative of the equation (3.1), then T D0 vector is T D0 . d s∗ ds = T D0 . d s∗ ds = T D0 = T D0 =    0   p0 q0 q0 N+ p N0 + p W+ p W 0, (p0 )2 + (q0 )2 (p0 )2 + (q0 )2 (p0 )2 + (q0 )2 (p0 )2 + (q0 )2  0  p0 β − q0 γ   0 p0 q0 N+ p C+ p W, p (p0 )2 + (q0 )2 (p0 )2 + (q0 )2 (p0 )2 + (q0 )2  0 0 p 0 2   q0 (q ) p0 β − q0 γ 0 0 q N − p W + .C , ((p0 )2 + (q0 )2 )2 (p0 )2 + (q0 )2  0  0 0 p 0 2 2 γ q0 (q ) β β γN + βW p0 β − q0 γ + 02 C. − p 2 2 + γ2 (p ) + (q0 )2 β 0 2 0 2 2 2 (p ) + (q ) + (β + γ ) | {z } | {z } κgD D | {z } p0  0 p (3.4) =1 From the equation (2.2), (3.3) and (3.4) the geodesic curvature of αD (s) = D(s) is κgD (s) = p0 β − q0 γ . (p0 )2 + (q0 )2 If we take the derivative of this equation (3.2) then p0 and q0 are p= γ β(γ0 β − β0 γ) 0 ⇒ p =
3 , γ2 + β2 γ2 + β2 2 q= β γ(γ0 β − β0 γ) 0 ⇒ q = −
3 γ2 + β2 γ2 + β2 2 (3.5) and (γ0 β − β0 γ)2 (p0 )2 + (q0 )2 =  2 , γ2 + β2 (γ0 β − β0 γ) p0 β − q0 γ = p . γ2 + β2 (3.6) Using the equation (2.2), (3.5) and (3.6) we can write κgD geodesic curvature is κgD =
3 γ2 + β2 2 . γ0 β − β0 γ (3.7) Then from the equation (2.1), (3.3) and (3.4) we have the following spherical Sabban formulae of αD (s), D0 = T D , T D0 = −D + κgD (D ∧ T D ), (D ∧ T D )0 = −κgD T D . S. Şenyurt, Turk. J. Math. Comput. Sci., 9(2018), 39–49 43 Definition 3.2. Let (D) be a spherical curve of α(s), D and T D be Sabban vectors of (D). Then DT D -Smarandache curve can be identified as 1 αDT D = √ (D + T D ) or 2

β+γ N+ β−γ W . αDT D = √ p 2 γ2 + β2 (3.8) Theorem 3.3. The geodesic curvature accordin (...truncated)