On The Darboux Vector Belonging To Involute Curve A Different View

M ATHEMATICAL S CIENCES AND A PPLICATIONS E-N OTES c 4 (2) 131-138 (2016) MSAEN

On The Darboux Vector Belonging To Involute Curve A Different View Süleyman S¸ ENYURT*, Yasin ALTUN and Ceyda CEVAHIR (Communicated by Bülent ALTUNKAYA)

Abstract In this paper, we investigated special Smarandache curves in terms of Sabban frame drawn on the surface of the sphere by the unit Darboux vector of involute curve. We created Sabban frame belonging to this curve. It was explained Smarandache curves position vector is composed by Sabban vectors belonging to this curve. Then, we calculated geodesic curvatures of this Smarandache curves. Found results were expressed depending on the base curve. We also gave example belonging to the results found.

Keywords: involute curve; Darboux vector; Smarandache curves; Sabban frame; geodesic curvature.

AMS Subject Classification (2010): 53A04. *Corresponding author

1. Introduction and Preliminaries The involute of the curve is well known by the mathematicians especially the differential geometry scientists.

There are many essential consequences and properties of curves. Involute curves have been studied by some authors [3, 7]. Whose position vector is composed by Frenet frame vectors regular curve is called a Smarandache curve [10]. Special Smarandache curves have been studied by some authors [1, 2, 5, 8, 9]. K. Tas.köprü, M. Tosun studied special Smarandache curves according to Sabban frame on S 2 [11]. S¸ enyurt and Çalı¸skan investigated special Smarandache curves in terms of Sabban frame of spherical indicatrix curves and they gave some characterization of Smarandache curves [4]. Let α : I → E 3 be a unit speed curve, we defined the quantities of the Frenet-Serret apparatus, respectively

T (s)

=

α0 (s), N (s) =

α00 (s) , B(s) = T (s) ∧ N (s), kα00 (s)k

κ(s) = kT 0 (s)k, τ (s) = hN 0 (s), B(s)i.

(1.1) (1.2)

we have an orthonormal frame {T, N, B} along α. This frames is called the Frenet frame of α. This curve the Frenet formulae are, respectively, [7]  0  T (s) = κ(s)N (s) (1.3) N 0 (s) = −κ(s)T (s) + τ (s)B(s)   0 B (s) = −τ (s)N (s).

For any unit speed curve α : I → E3 , the vector W is called Darboux vector defined by Received : 11–June–2016, Accepted : 02–November–2016

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Süleyman S¸ ENYURT, Yasin ALTUN & Ceyda CEVAHIR

W = τ T + κB. If we consider the normalization of the Darboux, we have sin ϕ =

τ κ , cos ϕ = kW k kW k

and C = sin ϕT + cos ϕB where ∠(W, B) = ϕ, [6]. Let α : I → E3 unit speed and α∗ : I → E3 be the C 2 − class differentiable two curves. If the tangent vector of the curve α is orthogonal to the tangent vector of the curve α∗ which is called involute of the α. According to definition, if the tangent of the curve α is denoted by T and the tangent of the curve α∗ is denoted

by T ∗ , we can write [7] hT, T ∗ i = 0.

(1.4)

If the curve α∗ is involute of α, then we may write that [7] α∗ (s∗ ) = α(s) + (c − s)T (s).

(1.5)

Let α : I → E 3 and α∗ : I → E 3 be the C 2 -class differentiable unit speed two curves and the amounts of {T (s), N (s), B(s)} and {T ∗ (s∗ ), N ∗ (s∗ ), B ∗ (s∗ )} are entirely Frenet- Serret frame of the curves α and the involute α∗ , respectively, then [3]  ∗  T = N (1.6) N ∗ = − cos ϕT + sin ϕB   ∗ B = sin ϕT + cos ϕB. where ∠(W, B) = ϕ. For the curvatures and the torsions we have κ∗ = where

kW k , |c − s|κ

τ∗ =

(κτ 0 − τ κ0 ) . κ|c − s| kW k2

(1.7)

ds∗ = |c − s|κ. From (1.9) equation, we have ds ∗

sin ϕ = p

0 kW k ϕ0 , cos ϕ = p , ϕ∗ 0 = p ϕ02 + kW k2 ϕ02 + kW k2 ϕ02 + kW k2 ϕ0

∗

p ϕ02 + kW k2 . kW k

(1.8)

Let (α, α∗ ) be a curve pair in E3 . For the vector C ∗ is the direction of the involute curve α∗ we have sin ϕkW k ϕ0 cos ϕkW k T+p N+p B. C∗ = p 0 2 2 0 2 2 (ϕ ) + kW k (ϕ ) + kW k (ϕ0 )2 + kW k2

(1.9)

where the vector C is the direction of the Darboux vector W of the base curve α, [3]. Let γ : I → S 2 be a unit speed spherical curve. We denote s as the arc-length parameter of γ. Let us denote by γ(s) = γ(s), t(s) = γ 0 (s), d(s) = γ(s) ∧ t(s)

(1.10)

2

{γ(s), t(s), d(s)} frame is called the Sabban frame of γ on S . Then we have the following spherical Frenet formulae of γ γ 0 (s) = t(s), t0 (s) = −γ(s) + κg (s)d(s), d0 (s) = −κg (s)t(s)

(1.11)

where κg is called the geodesic curvature of the curve γ on S 2 which is, [11] κg (s) = ht0 (s), d(s)i.

(1.12)

On The Darboux Vector Belonging To Involute Curve

133

2. On The Darboux Vector Belonging To Involute Curve A Different View In this section, we investigated special Smarandache curves created by Sabban frame {C ∗ , TC ∗ , C ∗ ∧ TC ∗ }, that belongs to drawn on the surface of the sphere by the unit Darboux vector of a α∗ curve are defined. We found some results. These results will be expressed depending on the base curve. Let αC ∗ (sC ∗ ) = C ∗ (s∗ ) be a unit speed regular spherical curves on S 2 . We denote sC ∗ as the arc-length parameter for drawn on the surface of the sphere by the unit Darboux vector of involute (C ∗ ). Sabban frame for (C ∗ ) is αC ∗ (sC ∗ ) = C ∗ (s∗ )

(2.1)

Differentiating (2.1), we found TC ∗

dsC ∗ = ϕ∗ 0 cos ϕ∗ T ∗ − ϕ∗ 0 sin ϕ∗ B ∗ ds∗

and we can write

dsC ∗ = ϕ∗ 0 ds∗

(2.2)

(2.3)

Hence we have TC ∗ = cos ϕ∗ T ∗ − sin ϕ∗ B ∗ and C ∗ ∧ TC ∗ = N ∗ .

From the equation (1.10), we have

∗

C∗

=

sin ϕ∗ T ∗ + cos ϕ∗ B ∗

TC ∗

=

cos ϕ∗ T ∗ − sin ϕ∗ B ∗

C ∧T

C∗

=

(2.4)

∗

N .

Then from the equation (1.11) we have the following spherical Frenet formulae of (C ∗ ) is: C ∗0 (TC ∗ )0 (C ∗ ∧ TC ∗ )0

= TC ∗ kW ∗ k ∗ C ∧ TC ∗ ϕ∗ 0 kW ∗ k = − ∗ 0 TC ∗ . ϕ

= −C ∗ +

(2.5)

From the equation (1.12), we have the following geodesic curvatures of (C ∗ ) is κg = hTC0 ∗ , C ∗ ∧ TC ∗ i =⇒ κg =

kW ∗ k . ϕ∗ 0

(2.6)

β1 -Smarandache curve can be defined by 1 β1 (sC ∗ ) = √ (C ∗ + TC ∗ ) 2

(2.7)

or substituting the equation (2.4) into equation (2.7), we reach

1 β1 (s∗ ) = √ (sin ϕ∗ + cos ϕ∗ )T ∗ + (cos ϕ∗ − sin ϕ∗ )B ∗ .

2

(2.8)

Differentiating (2.7), we can write ϕ∗ 0 (cos ϕ∗ − sin ϕ∗ ) ∗ kW ∗ k ϕ∗ 0 (cos ϕ∗ + sin ϕ∗ ) ∗ Tβ1 (s∗ ) = q T +q N∗ − q B .

2 2 2 2ϕ∗ 0 + kW ∗ k2 2ϕ∗ 0 + kW ∗ k2 2ϕ∗ 0 + kW ∗ k2 Considering the equations (2.8) and (2.9), it easily seen that

(2.9)

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Süleyman S¸ ENYURT, Yasin ALTUN & Ceyda CEVAHIR

β1 ∧ Tβ1 (s∗ ) =

kW ∗ k(cos ϕ∗ + sin ϕ∗ ) ∗ ϕ∗ 0 kW ∗ k(cos ϕ∗ + sin ϕ∗ ) ∗ q T −q N∗ + q B .

02 02 02 ∗ 2 ∗ ∗ 2 ∗ ∗ 2 ∗ 2kW k + 4ϕ 2kW k + 4ϕ 2kW k + 4ϕ

(2.10)

Differentiating (2.9), where  ∗ 0 ∗ 2 k kW ∗ k k  + kW χ1 = −2 − kW ∗0 ∗0  ϕ ϕ ϕ∗ 0      ∗ 2 ∗ 4 ∗ 0 k k k χ2 = −2 − 3 kW − kW − kW ϕ∗ 0 ϕ∗ 0 ϕ∗ 0       ∗ ∗ 3 ∗ 0  k k k χ3 = 2 kW + kW + kW ϕ∗ 0 ϕ∗ 0 ϕ∗ 0

kW ∗ k ϕ∗ 0

(2.11)

including we can reach,

Tβ0 1 (s∗ )

√ √ √ ∗0 4 2(χ1 cos ϕ∗ − χ2 sin ϕ∗ ) ∗ (ϕ∗ 0 )4 2(χ1 sin ϕ∗ + χ2 cos ϕ∗ ) ∗ χ3 (ϕ∗ 0 )4 2 ∗ (ϕ ) = T + B . (2.12)

2 2 N + 2 kW ∗ k2 + (ϕ∗ 0 )2 kW ∗ k2 + (ϕ∗ 0 )2 kW ∗ k2 + (ϕ∗ 0 )2

From the equation (2.10) and (2.12), κβg 1 geodesic curvature for involute curve β1 (s∗ ) is κβg 1

= hTβ0 1 , β1 ∧ Tβ1 i kW ∗ k

1

= 2+

kW ∗ k ϕ∗ 0

2

5

)2

∗0

ϕ

χ1 −

kW ∗ k χ + 2χ .

2 3 0 ϕ∗

From the equation (1.6) and (1.9), Sabban apparatus of the β1 -Smarandache curve for base curve are

β1 (s)

Tβ1 (s)

=

=

(β1 ∧ Tβ1 )(s)

Tβ0 1 (s)

and

ϕ0 + kW k (kW k − ϕ0 ) cos ϕ (kW k − ϕ0 ) sin ϕ p T−p N+ p B, 2ϕ02 + 2kW k2 2ϕ02 + 2kW k2 2ϕ02 + 2kW k2 p (−kW k − ϕ0 )η sin ϕ − kW k2 + ϕ02 cos ϕ η(ϕ0 − kW k) p p p T+p N kW k2 + ϕ02 1 + 2η 2 kW k2 + ϕ02 1 + 2η 2 p kW k2 + ϕ02 sin ϕ − η(kW k + ϕ0 ) cos ϕ p p + B, kW k2 + ϕ02 1 + 2η 2 p (kW k + ϕ0 ) sin ϕ − 2η kW k2 + ϕ02 cos ϕ ϕ0 − kW k p p p = T−p N 2 + 4η 2 kW k2 + ϕ02 2 + 4η 2 kW k2 + ϕ02 p (kW k + ϕ0 ) cos ϕ + 2η kW k2 + ϕ02 sin ϕ p p + B, 2 + 4η 2 kW k2 + ϕ02

p √ √ (χ1 kW k − χ2 ϕ0 )η 4 2 sin ϕ − χ3 η 4 2kW k2 + 2ϕ02 cos ϕ η 4 2(χ1 ϕ0 − χ2 kW k) p p = T+ N (1 + 2η 2 )2 kW k2 + ϕ02 (1 + 2η 2 )2 kW k2 + ϕ02 p √ (χ1 kW k − χ2 ϕ0 )η 4 2 cos ϕ + χ3 η 4 2kW k2 + 2ϕ02 sin ϕ p + B (1 + 2η 2 )2 kW k2 + ϕ02

(2.13)

On The Darboux Vector Belonging To Involute Curve

κβg 1 =

1 (2 +

1 η2 )

5 2

1 1 χ1 − χ2 + 2χ3 , η η

135

(2.14)

where

0 1 (ϕ∗ )0 ϕ0 = = cos ϕ(c − s) p 2 η kW ∗ k ϕ02 + kW k

(2.15)

and  1 1 1  χ1 = −2 − η2 + η0 η χ2 = −2 − 3 η12 − η14 −   χ3 = 2 η1 + η13 + η10 .

1 1 η0 η

(2.16)

β2 -Smarandache curve can be defined by 1 β2 (sC ∗ ) = √ (C ∗ + C ∗ ∧ TC ∗ ) 2 or from the equation (1.6), (1.8) and (2.4), we can write

β2 (s) =

p p kW k cos ϕ + ϕ02 + kW k2 sin ϕ kW k sin ϕ − ϕ02 + kW k2 cos ϕ ϕ0 p p T+p N+ B. 2ϕ02 + 2kW k2 2ϕ02 + 2kW k2 2ϕ02 + 2kW k2

(2.17)

(2.18)

Differentiating (2.18), we can write −ϕ0 sin ϕ kW k ϕ0 cos ϕ Tβ2 (s) = p T+p N−p B. 2 02 2 02 kW k + ϕ kW k + ϕ kW k2 + ϕ02

(2.19)

Considering the equations (2.18) and (2.19), with ease seen that p p kW k2 + ϕ02 sin ϕ − kW k cos ϕ −kW k sin ϕ − kW k2 + ϕ02 cos ϕ ϕ0 p p p β2 ∧ Tβ2 (s) = T− N+ B. 2kW k2 + 2ϕ02 2kW k2 + 2ϕ02 2kW k2 + 2ϕ02 (2.20) Differentiating (2.19), we can write

Tβ0 2 (s)

=

p √ √ −η 2kW k sin ϕ − 2kW k2 + 2ϕ02 cos ϕ η 2ϕ0 p p T+ N (η − 1) kW k2 + ϕ02 (η − 1) kW k2 + ϕ02 p √ 2kW k2 + 2ϕ02 sin ϕ − η 2kW k cos ϕ p + B. (η − 1) kW k2 + ϕ02

(2.21)

κβg 2 geodesic curvature for base curve β2 (sβ2 ) is κβg 2 =

1+η . η−1

(2.22)

β3 -Smarandache curve can be defined by 1 β3 (sC ∗ ) = √ (TC ∗ + C ∗ ∧ TC ∗ ) 2 or from the equation (2.4), (1.6) and (1.8), we can write p p −ϕ0 sin ϕ − ϕ02 + kW k2 cos ϕ ϕ02 + kW k2 sin ϕ − ϕ0 cos ϕ kW k p p β3 (s) = T+p N+ B. 2ϕ02 + 2kW k2 2ϕ02 + 2kW k2 2ϕ02 + 2kW k2

(2.23)

(2.24)

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Süleyman S¸ ENYURT, Yasin ALTUN & Ceyda CEVAHIR

Differentiating (2.24), we reach

Tβ3 (s)

p (ϕ0 − ηkW k) sin ϕ − kW k2 + ϕ02 cos ϕ ηϕ0 + kW k p p p T−p N 2 + η 2 kW k2 + ϕ02 2 + η 2 kW k2 + ϕ02 p (ϕ0 − ηkW k) cos ϕ + kW k2 + ϕ02 sin ϕ p p + B. 2 + η 2 kW k2 + ϕ02

=

(2.25)

Considering the equations (2.24) and (2.25), it is easily seen

β3 ∧ Tβ3 (s)

=

p (2kW k + ηϕ0 ) sin ϕ − η kW k2 + ϕ02 cos ϕ 2ϕ0 − ηkW k p p p T+p N 4 + 2η 2 kW k2 + ϕ02 4 + 2η 2 kW k2 + ϕ02 p (2kW k + ηϕ0 ) cos ϕ + η kW k2 + ϕ02 sin ϕ p p + B. 4 + 2η 2 kW k2 + ϕ02

(2.26)

Differentiating (2.25), where σ1 =

1 1 1 1 1 1 1 1 1 1 + 2 3 + 2 0 , σ 2 = −1 − 3 2 − 2 4 − 0 , σ 3 = − 2 − 2 4 + 0 η η η η η η η η η η

(2.27)

including we have

Tβ0 3 (s)

=

p √ √ (σ 2 kW k + σ 1 ϕ0 )η 4 2 sin ϕ − σ 3 η 4 2kW k2 + 2ϕ02 cos ϕ (σ 2 ϕ0 − σ 1 kW k)η 4 2 p p T+ N (2 + η 2 )2 ϕ02 + kW k2 (2 + η 2 )2 ϕ02 + kW k2 0

+

(σ 2 kW k + σ 1 ϕ )η

4

(2.28)

√

4

p

2kW k2

2 cos ϕ + σ 3 η p (2 + η 2 )2 ϕ02 + kW k2

+

2ϕ02

sin ϕ

B.

κβg 3 geodesic curvature for base curve β3 (sβ3 ) is κβg 3 =

1 (2 +

1 52 η2 )

1 2 σ1 − σ2 + σ3 . η

(2.29)

β4 -Smarandache curve can be defined by 1 β4 (sC ∗ ) = √ (C ∗ + TC ∗ + C ∗ ∧ TC ∗ ) 3

(2.30)

or from the equation (1.6), (1.8) and (2.4), we can write

β4 (s)

p (kW k − ϕ0 ) sin ϕ − ϕ02 + kW k2 cos ϕ ϕ0 + kW k p = T+p N 3ϕ02 + 3kW k2 3ϕ02 + 3kW k2 p (kW k − ϕ0 ) cos ϕ + ϕ02 + kW k2 sin ϕ p + B. 3ϕ02 + 3kW k2

(2.31)

Differentiating (2.31), we reach

Tβ4 (s)

=

p

(1 − η)ϕ0 − ηkW k sin ϕ − kW k2 + ϕ02 cos ϕ (η − 1)kW k − ηϕ0 p p p T+p N 2(1 − η + η 2 ) kW k2 + ϕ02 2(1 − η + η 2 ) kW k2 + ϕ02 (2.32) p

(η − 1)ϕ − ηkW k cos ϕ + kW k2 + ϕ02 sin ϕ p p + B. 2(1 − η + η 2 ) kW k2 + ϕ02 0

On The Darboux Vector Belonging To Involute Curve

137

Considering the equations (2.31) and (2.32), it is easily seen

β4 ∧ Tβ4 (s)

=

p ((2 − η)kW k + (1 + η)ϕ0 ) sin ϕ − (2η − 1) kW k2 + ϕ02 cos ϕ (2 − η)ϕ0 − (1 + η)kW k p p p T+p N 6 − 6η + 6η 2 kW k2 + ϕ02 6 − 6η + 6η 2 kW k2 + ϕ02 (2.33) p 0 2 02 ((2 − η)kW k + (1 + η)ϕ ) cos ϕ + (2η − 1) kW k + ϕ sin ϕ p p + B. 6 − 6η + 6η 2 kW k2 + ϕ02

Differentiating (2.32), where 

1 1 1 1 1  ρ1 = −2 + 4 η − 4 η2 + 2 η3 + 2 η0 2 η − 1

ρ2 = −2 + 2 η1 − 4 η12 + 2 η13 − 2 η14 − η10 1 + η1 

 ρ3 = 2 η1 − 4 η12 + 4 η13 − 2 η14 + η10 2 − η1

(2.34)

including we can write

Tβ0 4 (s)

=

p √ √ (ρ1 kW k − ρ2 ϕ0 )η 4 3 sin ϕ − ρ3 η 4 3kW k2 + 3ϕ02 cos ϕ (ρ1 ϕ0 + ρ2 kW k)η 4 3 p p T+ N 4(1 − η + η 2 )2 ϕ02 + kW k2 4(1 − η + η 2 )2 ϕ02 + kW k2 (2.35)

√

p (ρ1 kW k − ρ2 ϕ )η 3 cos ϕ + ρ3 3kW k2 + 3ϕ02 sin ϕ p B. + 4(1 − η + η 2 )2 ϕ02 + kW k2 0

4

κβg 4 geodesic curvature for base curve β4 (sβ4 ) is κβg 4 =

(2 η1 − 1)ρ1 + (−1 − η1 )ρ2 + (2 − η1 )ρ3 . √ 5 4 2(1 − η + η 2 ) 2

(2.36)

Example. Let us consider the unit speed spherical curve: α(s) = {

2 1 2 1 4 sin (2 s) − sin (8 s) , − cos (2 s) + cos (8 s) , sin (3 s)} 5 40 5 40 15

in the context of definitions, we reach (C ∗ ) curve (see Figure 1) and Smarandache curves according to Sabban frame on S 2 . β1 , β2 , β3 and β4 (see Figure 2).

Figure 1. (C ∗ )-curve

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Süleyman S¸ ENYURT, Yasin ALTUN & Ceyda CEVAHIR

Figure 2. Smarandache curves

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Affiliations S ÜLEYMAN S¸ ENYURT A DDRESS : Ordu University, Faculty of Arts and Sciences, Department of Mathematics, 52200, Ordu-Turkey.

E - MAIL : senyurtsuleyman@hotmail.com YASIN ALTUN A DDRESS : Ordu University, Faculty of Arts and Sciences, Department of Mathematics, 52200, Ordu-Turkey.

E - MAIL : yasinaltun2852@gmail.com C EYDA CEVAHIR A DDRESS : Ordu University, Faculty of Arts and Sciences, Department of Mathematics, 52200, Ordu-Turkey.

E - MAIL : Ceydacevahir@gmail.com