Smarandache curves according to Sabban frame for Darboux vector of Mannheim partner curve

Smarandache curves according to Sabban frame for Darboux vector of Mannheim partner curve SĂźleyman Ĺženyurt, Yasin Altun, and Ceyda Cevahir Citation: AIP Conference Proceedings 1833, 020024 (2017); doi: 10.1063/1.4981672 View online: http://dx.doi.org/10.1063/1.4981672 View Table of Contents: http://aip.scitation.org/toc/apc/1833/1 Published by the American Institute of Physics

Smarandache curves According to Sabban Frame for Darboux vector of Mannheim Partner Curve S¨uleyman S¸enyurt 1,a) , Yasin Altun1,b) and Ceyda Cevahir1,c) 1

Faculty of Arts and Sciences, Department of Mathematics, Ordu University, Ordu, Turkey a)

Corresponding author: senyurtsuleyman@hotmail.com b) yasinaltun2852@gmail.com c) Ceydacevahir@gmail.com

Abstract. In this paper, we investigated special Smarandache curves belonging to Sabban frame drawn on the surface of the sphere

by Darboux vector of Mannheim partner curve. We created Sabban frame belonging to this curve. It was explained Smarandache curves position vector is consisted by Sabban vectors belonging to this curve. Then, we calculated geodesic curvatures of this Smarandache curves. Found results were expressed depending on the Mannheim curve.

Introduction and Preliminaries 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 [8]. K. Tas¸k¨opr¨u, M. Tosun studied special Smarandache curves according to Sabban frame on S 2 [9]. S¸enyurt and C¸alıs¸kan investigated special Smarandache curves in terms of Sabban frame for fixed pole curve and spherical indicatrix and they gave some characterization of Smarandache curves [1, 2].

S¸enyurt et al. investigated special Smarandache curves according to Sabban frame for the curve drawn on the surface of the sphere by the unit Darboux vector of Bertrand partner curve and this gave some characterization of Smarandache curves [10]. 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 [5] T 0 = κN,

N 0 = −κT + τB,

B0 = −τN

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the vector W is called Darboux vector defined by W = τT + κB. If we consider the normalization of the Darboux C = and

1 τ κ W we have, sin ϕ = and cos ϕ = kWk kWk kWk

C = sin ϕT + cos ϕB

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where ∠(W, B) = ϕ, [4]. Let α and α1 be the C -class differentiable two curves and T 1 (s), N1 (s), B1 (s) be the Frenet vectors of α1 . If the binormal vector of the curve α1 is linearly dependent on the principal normal vector of the curve α, then (α) is defined a Mannheim curve and (α1 ) a Mannheim partner curve of (α), [3, 6, 7]. The relations between the Frenet vectors we can write 2

T 1 = sin ϕT + cos ϕB, N1 = − cos ϕT + sin ϕB, B1 = N

(3)

and the curvature and torsion we get κ1 = ϕ0

κ κ , τ1 = . ητkWk ητ

II. International Conference on Advances in Natural and Applied Sciences AIP Conf. Proc. 1833, 020024-1–020024-5; doi: 10.1063/1.4981672 Published by AIP Publishing. 978-0-7354-1503-4/$30.00

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where η =

κ = constant. Let γ : I → S 2 be a unit speed spherical curve. We can write κ2 + τ2 γ(s) = γ(s), t(s) = γ0 (s), d(s) = γ(s) ∧ t(s), [9]

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{γ(s), t(s), d(s)} frame is expressed the Sabban frame of γ on S 2 . Then we have equations, γ0 (s) = t(s), t0 (s) = −γ(s) + κg (s)d(s), d0 (s) = −κg (s)t(s), [9].

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where κg is expressed the geodesic curvature of the curve γ on S 2 which is κg (s) = ht0 (s), d(s)i [9].

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Smarandache curves According to Sabban Frame for Darboux vector of Mannheim Partner Curve Let (C1 ) be a unit speed spherical curve on S 2 . Then we can write C1 = sin ϕ1 T 1 + cos ϕ1 B1 , TC1 = cos ϕ1 T 1 − sin ϕ1 B1 , C1 ∧ TC1 = N1 .

(8)

where ∠(C1 , B1 ) = ϕ1 . Then from the equation 6 we have the following equations of (C1 ) are C1 0 = TC1 , TC0 = −C1 + 1

kW k kW1 k C ∧ TC1 , (C1 ∧ TC1 )0 = − 10 TC1 . ϕ1 0 1 ϕ1

(9)

From the equation 7, we have the following geodesic curvature of (C1 ) is κg = hTC0 , C1 ∧ TC1 i =⇒ κg = 1

kW1 k . ϕ1 0

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Definition 1 Let (C1 ) be a spherical curve on S 2 . C1 and TC1 be unit vector belonging to (C1 ). In this case, β1 Smarandache curve can be defined by 1 (11) β1 (s) = √ (C1 + TC1 ).

2 Theorem 2 curve is

The expression according to the Mannheim curve of geodesic curvature belonging to β1 -Smarandache 1

1 1 κgβ1 = λ − λ + 2λ (12) 3 , 5 σ 1 σ 2 2 + σ12 2

where coefficients are

0 ητ √κ2 + τ2 1 kWk = p σ ϕ0 ϕ02 + kWk2

and λ1 = −2 −

1 1 1 1 1 1 1 1 1 1 + , λ2 = −2 − 3 2 − 4 − 0 , λ3 = 2 + 3 + 2 0 σ σ σ σ σ σ2 σ0 σ σ σ

Proof. Substituting the equation 8 into equation 11 we obtain

1 β1 (s) = √ (sin ϕ1 + cos ϕ1 )T 1 + (cos ϕ1 − sin ϕ1 )B1 .

2

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If equation 11 derivative is taken, we can write T β1 =

ϕ1 0 (cos ϕ1 − sin ϕ1 ) kW1 k ϕ 0 (cos ϕ1 + sin ϕ1 ) T1 + p N1 − 1p B1 . p 2ϕ1 02 + kW1 k2 2ϕ1 02 + kW1 k2 2ϕ1 02 + kW1 k2

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Considering the equations 15 and 16 we get kW k(cos ϕ1 + sin ϕ1 ) ϕ1 0 kW1 k(cos ϕ1 + sin ϕ1 ) β1 ∧ T β1 = q1 2 T 1 − q 2 N1 + q 2 B1 .

2kW1 k2 + 4 ϕ1 0 2kW1 k2 + 4 ϕ1 0 2kW1 k2 + 4 ϕ1 0

(17)

If equation 16 derivative is taken, where coefficients are λ1 = −2 −

kW1 k ϕ1 0

!2 +

kW1 k ϕ1 0

!0

! ! ! ! ! ! ! ! kW1 k kW1 k 2 kW1 k 4 kW1 k 0 kW1 k kW1 k kW1 k 3 kW1 k 0 , λ = −2 − 3 − − , λ = 2 + + 2 3 ϕ1 0 ϕ1 0 ϕ1 0 ϕ1 0 ϕ1 0 ϕ1 0 ϕ1 0 ϕ1 0

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impending T β0 1 is, we reach T β0 1

4 √ 4 √ 4 √ ϕ1 0 2(λ1 sin ϕ1 + λ2 cos ϕ1 ) 2 2(λ1 cos ϕ1 − λ2 sin ϕ1 ) λ3 ϕ1 0 ϕ1 0 = T + N + B1

1 1 2 2 2 2 2 2 kW1 k2 + ϕ1 0 kW1 k2 + ϕ1 0 kW1 k2 + ϕ1 0

From the equations 17 and 19, κgβ1 geodesic curvature according to Mannheim partner curve of the β1 is

kW k kW1 k 1 1 κgβ1 = kW k 2 5 ϕ 0 λ1 − ϕ 0 λ2 + 2λ3 .

1 1 2 + ϕ 10 ) 2

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1

From the equations 3 and 4, Sabban apparatuses according to Mannheim curve of the β1 -Smarandache curve are (ϕ0 + kWk) sin ϕ ϕ0 − kWk (ϕ0 + kWk) cos ϕ T− p N+ p B, p 2ϕ0 2 + 2kWk2 2ϕ0 2 + 2kWk2 2ϕ0 2 + 2kWk2 p σ(ϕ0 + kWk) (ϕ0 − kWk)σ sin ϕ − kWk2 + ϕ0 2 cos ϕ T− p N T β1 (s) = p √ √ kWk2 + ϕ0 2 1 + 2σ2 kWk2 + ϕ0 2 1 + 2σ2 p kWk2 + ϕ0 2 sin ϕ − (kWk − ϕ0 )σ cos ϕ + B, p √ kWk2 + ϕ0 2 1 + 2σ2 p (kWk − ϕ0 ) sin ϕ − 2σ kWk2 + ϕ0 2 cos ϕ ϕ0 + σkWk β1 ∧ T β1 (s) = T+ √ N p p √ 2 + 4σ2 kWk2 + ϕ0 2 2 + 4σ2 kWk2 + ϕ0 2 p (kWk + ϕ0 ) cos ϕ + 2σ kWk2 + ϕ0 2 sin ϕ B, + p √ 2 + 4σ2 kWk2 + ϕ0 2 1

1 1 κgβ1 = λ − λ + 2λ 3 , 5 σ 1 σ 2 2 + σ12 2 β1 (s) =

where

0 λτ √κ2 + τ2 ϕ1 0 1 kWk = = p σ kW1 k ϕ0 ϕ02 + kWk2

impending coefficients are λ1 = −2 −

1 1 1 1 1 1 1 1 1 1 + , λ2 = −2 − 3 2 − 4 − 0 , λ3 = 2 + 3 + 2 0 σ σ σ σ σ σ2 σ0 σ σ σ

Definition 3 Let (C1 ) be a spherical curve on S 2 . C1 and C1 ∧ TC1 be unit vector of (C1 ). In this case, β2 - Smarandache curve can be defined by 1 β2 (s) = √ (C1 + C1 ∧ TC1 ).

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Theorem 4 The expressions according to the Mannheim curve of Sabban apparatuses belonging to β2 Smarandache curve are p p ϕ0 kWk sin ϕ − ϕ0 2 + kWk2 cos ϕ ϕ0 2 + kWk2 sin ϕ + kWk cos ϕ T+ p N+ B, β2 (s) = p p 2ϕ0 2 + 2kWk2 2ϕ0 2 + 2kWk2 2ϕ0 2 + 2kWk2 T β2 (s) = β2 ∧ T β2 (s) = κgβ2 (s) =

ϕ0 sin ϕ p

kWk2 + ϕ0 2

T− p

kWk kWk2 + ϕ0 2

N+ p

ϕ0 cos ϕ kWk2 + ϕ0 2

B,

p p ϕ0 kWk2 + ϕ0 2 sin ϕ − kWk cos ϕ −kWk sin ϕ − kWk2 + ϕ0 2 cos ϕ T− p N+ B, p p 2kWk2 + 2ϕ0 2 2kWk2 + 2ϕ0 2 2kWk2 + 2ϕ0 2 1+σ , 1−σ

(22)

Proof. The proof is similar to the proof of Theorem 2.

Definition 5 Let (C1 ) be a spherical curve on S 2 . TC1 and C1 ∧ TC1 be unit vector of (C1 ). In this case, β3 Smarandache curve can be defined by 1 β3 (s) = √ (TC1 + C1 ∧ TC1 ).

(23) 2 Theorem 6 The expressions according to the Mannheim curve of Sabban apparatuses belonging to β3 Smarandache curve are p p kWk kWk2 + ϕ0 2 sin ϕ + ϕ0 cos ϕ ϕ0 sin ϕ − kWk2 + ϕ0 2 cos ϕ T+ p N+ B, β3 (s) = p p 2ϕ0 2 + 2kWk2 2ϕ0 2 + 2kWk2 2ϕ0 2 + 2kWk2 p kWk − σϕ0 − kWk2 + ϕ0 2 cos ϕ − (ϕ0 + σkWk) sin ϕ T+ √ N T β3 (s) = p p √ 2 + σ2 kWk2 + ϕ0 2 2 + σ2 kWk2 + ϕ0 2 p kWk2 + ϕ0 2 sin ϕ − (σkWk + ϕ0 ) cos ϕ + B, p √ 2 + σ2 kWk2 + ϕ0 2 p (2kWk − σϕ0 ) sin ϕ − σ kWk2 + ϕ0 2 cos ϕ 2ϕ0 − σkWk β3 ∧ T β3 (s) = T+ √ N p p √ 4 + 2σ2 kWk2 + ϕ0 2 4 + 2σ2 kWk2 + ϕ0 2 p (2kWk − σϕ0 ) cos ϕ + σ kWk2 + ϕ0 2 sin ϕ + B, p √ 4 + 2σ2 kWk2 + ϕ0 2

1 κgβ3 (s) = 2σ5 41 − σ4 42 + σ4 43 , (24) 5 (2 + σ2 ) 2 where coefficients are 41 =

1 1 1 1 1 1 1 1 1 1 + 2 3 + 2 0 , 42 = −1 − 3 2 − 2 4 − 0 , 43 = − 2 − 2 4 + 0 . σ σ σ σ σ σ σ σ σ σ

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Proof. The proof is similar to the proof of Theorem 2.

Definition 7 Let (C1 ) be spherical curve on S 2 . C1 , TC1 and C1 ∧ TC1 be unit vector of (C1 ). In this case, β4 Smarandache curve can be defined by β4 (s) =

1 (C + TC1 + C1 ∧ TC1 ).

3 1

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Theorem 8 The expressions according to the Mannheim curve of Sabban apparatuses belonging to β4 Smarandache curve are β4 (s) =

(ϕ0 + kWk) sin ϕ −

p ϕ0 2 + kWk2 cos ϕ

p 3ϕ0 2 + 3kWk2

ϕ0 − kWk

p ϕ0 2 + kWk2 sin ϕ + (ϕ0 + kWk) cos ϕ T+ p N+ B, p 3ϕ0 2 + 3kWk2 3ϕ0 2 + 3kWk2

p (σ − 1)ϕ0 − σkWk sin ϕ − kWk2 + ϕ0 2 cos ϕ σϕ0 + (σ − 1)kWk T β4 (s) = T+ p N p p p 2(1 − σ + σ2 ) kWk2 + ϕ0 2 2(1 − σ + σ2 ) kWk2 + ϕ0 2

p kWk2 + ϕ0 2 sin ϕ − σkWk + (1 − σ)ϕ0 cos ϕ + B, p p 2(1 − σ + σ2 ) kWk2 + ϕ0 2 p (2 − σ)ϕ0 + (1 + σ)kWk ((2 − σ)kWk − (1 + σ)ϕ0 ) sin ϕ − (2σ1) kWk2 + ϕ0 2 cos ϕ β4 ∧ T β4 (s) = T+ √ N p p √ 6 − 6σ + 6σ2 kWk2 + ϕ0 2 6 − 6σ + 6σ2 kWk2 + ϕ0 2 p ((2 − σ)kWk + (1 + σ)ϕ0 ) cos ϕ + (2σ − 1) kWk2 + ϕ0 2 sin ϕ + B, p √ 6 − 6σ + 6σ2 kWk2 + ϕ0 2 κgβ4 (s) =

(2σ4 − σ5 )κ1 − (σ5 + σ4 )κ2 + (2σ5 − σ4 )κ3 , √ 5 4 2(1 − σ + σ2 ) 2

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where coefficients are 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 κ1 = −2+4 −4 2 +2 3 +2 0 2 −1 , κ2 = −2+2 −4 2 +2 3 −2 4 − 0 1+ , κ3 = 2 −4 2 +4 3 −2 4 + 0 2− σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ (28)

Proof. The proof is similar to the proof of Theorem 2.

REFERENCES [1] [2] [3] [4] [5] [6] [7] [8] [9] [10]

A.T. Ali, International Journal of Mathematical Combinatorics, 2, 30-36 (2010).

A. C¸alıs¸kan and S. S¸enyurt, Gen. Math. Notes, 31(2), 1-15 (2015).

A. C¸alıs¸kan and S. S¸enyurt, International Journal of Mathematical Combinatorics, 1, 1-13 (2015).

W. Fenchel, Bulletin of the American Mathematical Society, 57, 44-54 (1951).

H.H. Hacısaliho˘glu, Differential geometry(in Turkish), Academic Press Inc. Ankara, 1994.

H. Liu, F. Wang, Journal of Geometry, 88(1-2), 120-126 (2008).

K. Orbay and E. Kasap, International Journal of Physical Sciences, 4(5), 261-264 (2009).

M. Turgut and S. Yılmaz, International Journal of Mathematical Combinatorics, 3, 51-55 (2008).

K. Tas¸k¨opr¨u and M. Tosun, Boletim da Sociedade Paranaense de Matematica 3 Srie. 32(1), 51-59 (2014).

S. S¸enyurt,Y. Altun, C. Cevahir, AIP Conference Proceedings 1726, 020045, http://dx.doi.org/10.1063/1.4945871, 2016.

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