ON THE SABBAN FRAME BELONGING TO INVOLUTE-EVOLUTE CURVES

ON THE SABBAN FRAME BELONGING TO

INVOLUTE-EVOLUTE CURVES

by

Suleyman SENYURT a*_{, Yasin ALTUN} a_{, Ceyda CEVAHIR} a_,

and Huseyin KOCAYIGIT b

a _{Faculty of Art and Science, Ordu University, Ordu, Turkey} b _{Faculty of Education, Manisa Celal Bayar University, Manisa,Turkey}

Original scientific paper https://doi.org/10.2298/TSCI181130054S

In this paper, we investigate special Smarandache curves with regard to Sabban frame of involute curve. We create Sabban frame belonging to spherical indicatrix of involute curve. Smarandache curves are explained by Sabban vectors belong-ing to spherical indicatrix. Then, we calculate geodesic curvatures of this Sma-randache curves. The results found for each curve is given depending on evolute curve. The example related to the subject is given and their figures are drawn with MAPPLE program.

Key words: geodesic curvature, involute-evolute curves, Sabban frame, Smarandache curves

Introduction and preliminaries

The involute of the curve is recently known by the mathematicians especially the differential geometry researchers. There are many essential consequences and properties of curves. Involute curves are examined by some authors [1, 2]. Frenet vectors of a curve are tak-en as the position vector and the regular curve drawn by the new vector is idtak-entified. This curve is called the Smarandache curve [3]. Special Smarandache curves are examined by certain au-thors [4-9]. Taskopru and Tosun [10] studied particular Smarandache curves belonging to Sab-ban frame on _S2 _{[10]. Senyurt and Caliskan [11] investigated particular Smarandache curves}

belonging to Sabban frame of spherical indicatrix curves and they gave some characterization of Smarandache curves. Let _{α: I→E}3 _{be a unit speed curve and the quantities { , , , , }T N Bκ τ}

are collectively Frenet-Serret apparatus of this curve. The Frenet formulae are also well known as, as follows [2]:

( ) = ( ) ( ), ( ) = ( ) ( ) ( ) ( ), ( ) = ( ) ( )

T s′ κ s N s N s′ −κ s T s +τ s B s B s′ −τ s N s (1)

Let 3

1: I E

α → be the _C2_{-class differentiable curve and}

1 1 1

{ ( ), ( ), ( ),T s N s B s

1( ), ( )}s 1 s

κ τ is Frenet-Serret apparatus of the α1 involute curve, then [1]:

1= , 1= cos sin , 1= sin cos

T N N − ϕT+ ϕB B ϕT+ ϕB (2)

where ( , ) =W B ϕ. For the curvatures and the torsions we have:

1=_{| |} , 1= ( )₂ | | W c s _{c s W} κτ τκ κ τ κ _κ ′− ′ − ₋ (3) 2 2 1 ₂ 1 ₂ 1 ₂ 2 2 2 sin = , cos = W , = W W W W W ϕ ϕ ϕ ϕ ϕ ϕ ϕ φ ϕ ′   _{′ +} ′ _{′ } ′ _     ′ + ′ + _ ′ + _ (4)

Let _{γ: I→S}2 _{be a unit speed spherical curve. We symbolize s as the arc-length}

pa-rameter of γ. Let’s give:

( ) = ( ), ( ) = ( ),s s t s s d s( ) = ( ) ( )s t s

γ γ γ′ γ ∧ (5)

The { ( ), ( ), ( )}γ s t s d s frame is denominated the Sabban frame of γ on _S2_{. The}

spheri-cal Frenet formulae of γ is:

( ) = ( ),s t s t s( ) = ( )s _g( ) ( ),s d s d s( ) = _g( ) ( )s t s

γ′ ′ −γ +κ ′ −κ (6)

where κ_g is denominated the geodesic curvature of the curve γ on _S2 _{which is in [10]:}

( ) = ( ), ( )

g s t s d s

κ 〈 ′ 〉 (7)

On the Sabban frame belonging to involute-evolute curves

In this section, we investigated special Smarandache curves created by Sabban frame such as, { , ,T T T1 T₁ 1 ∧TT₁}, { ,N T N T1 N1, 1∧ N1} and { , ,B T B T1 B1 1∧ B1}. We will give some results. These results will be expressed depending on the evolute curve. Let us find results on this Sma-randache curves. Let αT₁( ) = ( )s T s1 , αN₁( ) = ( )s N s1 and αB₁( ) = ( )s B s1 be a regular spherical

curves on _S2_{. The Sabban frames of spherical indicatrix belonging to involute curve are:}

1 1

1= ,1 T = ,1 1 T = 1

T T T N T T∧ B (8)

1 1

1= ,1 N = cos 1 1 sin 1 1, 1 N = sin 1 1 cos 1 1

N N T − ϕT + ϕB N T∧ ϕT + ϕ B (9)

1 1

1= ,1 B = 1, 1 B = 1

B B T −N B T∧ T (10)

From the eq. (6), the spherical Frenet formulae of ( )T1 , ( )N1 , and ( )B1 are, respectively:

1 1 1 1 1 1 1 1 1 1 1 1 1 = ,_{T T} = _T, ( _T) = _T T T T T τ T T T T τ T κ κ ′ ′ − + ∧ ∧ ′ − (11) 1 1 1 1 1 1 1 1 1 1 1 1 1 = _N , _N = _N , ( _N ) = _N N T T N N T N T T W W ϕ′ ϕ′ ′ ′ − + ∧ ∧ ′ − (12) 1 1 1 1 1 1 1 1 1 1 1 1 1 = B, B = B, ( B) = B B T T B κ B T B T κ T τ τ ′ ′ − + ∧ ∧ ′ − (13)

Using the eq. (7), the geodesic curvatures of ( )T1 , ( )N1 , and ( )B1 are:

1 1 1 1 1 1 1 1 1 = , = , = T N B g τ g _Wϕ g κ κ κ κ κ τ ′ (14)

Definition 1. Let ( )T1 by spherical curve of α1 and let T1 and TT1 be Sabban vectors of 1

(

1

)

1( ) = 1 1

2 T

s T T

β + (15)

Theorem 2. The geodesic curvature according to β1-Smarandache curve of the

invo-lute curve is:

(

)

(

)

1 4 1 1 1 1 2 1 3 5 2 _{2 2} 1 1 = 2 2 gβ κ κ τ λ τ λ κ λ κ τ + + + (16) where: 2 2 4 1 1 1 1 1 1 1 1 2 1 1 1 1 1 1 1 3 1 1 1 3 1 1 1 = 2 , = 2 3 = 2 τ τ τ τ τ τ τ λ λ κ κ κ κ κ κ κ τ τ τ λ κ κ κ ′ ′               − −_{ } +_{   } − − _{ } −_{ } −_{   }               ′       + +             (17) Proof. 1/2 ₁ 1( ) = (1/2 )(s T T1 T) β + or by eq. (8), we have:

(

)

1( ) =s 1₂ T N1 1 β + (18)

Taking the derivative of eq. (18), Tβ₁ vector give as:

(

1 1 1 1 1 1

)

1 2 1 1 1 ( ) = 2 T sβ κT κN τ B κ τ+ − + + (19)

Considering the eqs. (18) and (19), we have:

(

)

1 _{1 2 2} 1 1 1 1 1 1 1 1 1 ( )( ) = 2 4 2 Tβ s T N B β τ τ κ κ τ ∧ − + + (20)

Using the eq. (19), Tβ′₁ vector is:

(

)

(

)

4 1 1 1 2 1 3 1 2 1 _{2 2} 1 1 2 ( ) = 2 T sβ κ λT λ N λB κ τ ′ + + + (21)

From the eqs. (20) and (21), 1

gβ

κ geodesic curvature of β1( )s is:

(

)

(

)

1 4 1 1 1 1 2 1 3 5 2 _{2 2} 1 1 = 2 2 gβ κ κ τ λ τ λ κ λ κ τ + + +

Corollary 3. The geodesic curvature belonging to β1-Smarandache curve of the

evo-lute curve is:

1 6 1 2 3 5 2 _{2 2} 2 = ( ) (2 ) g W _W W β κ φ λ φ λ λ φ ′ + ′ + ′ + (22)

where: 2 2 4 1 2 3 3 = 2 , = 2 3 = 2 W W W W W W W W W W ϕ ϕ ϕ ϕ ϕ ϕ ϕ λ λ ϕ ϕ ϕ λ ′ ′  ′   ′   ′   ′   ′   ′   ′  − −_ _ +_  _{  } − − _ _ −_ _ −_ _{ }  _               ′  ′   ′   ′  + +                   (23)

Proof. From the eqs. (2) and (18), we calculate:

1( ) = 1 ( cos sin )

2

s T N B

β − φ + + φ (24)

If we take derivative of this expression, Tβ₁ vector is:

1 _{2 2 2 2 2 2}

sin cos cos sin

= 2 2 2 W W W T T N B W W W β ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ′ − ′ + − + ′ ′ ′ + + + (25)

If cross product from the eqs. (24) and (25), we have: 1

1 _{2 2 2}

2 2 2

2 sin cos 2 cos sin

= 4 2 4 2 4 2 W W T T N B W W W β ϕ ϕ ϕ φ ϕ ϕ ϕ β ϕ ϕ ϕ ′ + ′ − ∧ − + ′ ′ ′ + + + (26)

From the eq. (25), Tβ′₁ vector is: 1 4 4 3 2 1 2 2 2 2 2 2 4 3 2 2 2 2 ( sin cos ) 2 2 = (2 ) (2 ) ( cos sin ) 2 (2 ) W W T T N W W W _B W β λ ϕ λ ϕ λ ϕ ϕ λ ϕ λ ϕ ϕ − ′ + + ′ ′ + + + + ′ + (27)

If inner product from the eqs. (26) and (27), 1

gβ

κ geodesic curvature is found like eq. (22).

The proofs of the subsequent theorems and corollaries belonging to β2, β3, βς₁, βς₂, 3

ς

β , βξ₁, βξ₂, and βξ₃- Smarandache curves will be similar to Theorem 2 and Corollary 3. Definition 4. Let ( )T1 be spherical curve of α1 and let TT₁ and T T1∧ T1 be Sabban vec-tors of ( )T1 . In the fact β2-Smarandache curve is defined as:

1 1

2( ) =s 1₂(TT T T1 T)

β + ∧ (28)

Theorem 5. Let _1/2 α1 be involute of α. The geodesic curvature belonging to 2( ) = (1/2 )(s N1 B1),

β + β₂-Smarandache curve of involute curve is:

(

)

[

]

2 4 1 1 1 1 2 3 5 2 _{2 2} 1 1 = 2 ( ) 2 gβ κ κ τ ε κ ε ε κ τ + − + + (29) where:

3 2 4 1 1 1 1 1 1 1 1 2 1 1 1 1 1 1 1 2 4 1 1 1 3 1 1 1 = 2 2 , = 1 3 2 = 2 τ τ τ τ τ τ τ ε ε κ κ κ κ κ κ κ τ τ τ ε κ κ κ ′ ′             + _{ } + _{   } − − _{ } − _{ } −_{ }             ′       −_{ } − _{ } +_{ }       (30)

Corollary 6. Let α1 be an involute of α. The geodesic curvature belonging to 1/2

2= (1/2 )[(sin cos )T (sin cos ) ]B

β φ− φ + φ+ φ β₂-Smarandache curve of evolute curve is:

(

)

2 5 2 4 1 2 3 2 2 = 2 2( ) g W W W β κ ϕ ε ε ε ϕ  _{′ + − +}     _{+ ′}    (31) where: 3 2 4 1 2 2 4 3 = 2 , = 1 3 2 = 2 W W W W W W W W W W ϕ ϕ ϕ ϕ ϕ ϕ ϕ ε ε ϕ ϕ ϕ ε ′ ′  ′   ′   ′   ′   ′   ′   ′  + + − − − −                                           ′  ′   ′   ′  −_ _ − _ _ +_ _       (32)

Definition 7. Let ( )T1 be spherical curve of α1, T1, TT1, and T T1∧ T1 be Sabban vectors of ( )T1 . In the fact β3-Smarandache curve is defined as:

1 1

3( ) =s 1₃(T T1 T T T1 T)

β + + ∧ (33)

Theorem 8. Let _1/2 α1 be involute of α. The geodesic curvature belonging to 3( ) = (1/3 )(s T N1 1 B1)

β + + β₃-Smarandache curve of involute curve is:

(

)

3 4 1 1 1 1 1 1 2 1 1 3 5 2 _{2 2} 1 1 1 1 = [(2 ) ( ) (2 ) ] 4 2 gβ κ κ τ κ ϕ κ τ ϕ κ τ ϕ κ κ τ τ − − + + − + + (34) where: 2 3 1 1 1 1 1 1 1 1 1 1 1 2 3 4 1 1 1 1 1 1 2 1 1 1 1 1 1 2 3 4 1 1 1 1 3 1 1 1 1 = 2 4 2 2 1 = 2 2 4 2 2 1 = 2 4 4 2 τ τ τ τ τ φ κ κ κ κ κ τ τ τ τ τ τ φ κ κ κ κ κ κ τ τ τ τ φ κ κ κ κ ′           − + _{  }− _ + _{ } +_{  } − _           ′             − + _{ }− _{ } + _{ } − _{ } −_{  } + _                     − + −                 1 1 1 2 1 τ τ κ κ ′     +_{  } − _     (35)

Corollary 9. Let _1/2 α1 be involute of α. The geodesic curvature belonging to 3( ) = (1/3 )[(sins cos )T N (sin cos ) ],B

β ϕ− ϕ + + ϕ+ ϕ β₃-Smarandache curve of evolute

(

)

3 4 1 2 3 5 2 _{2 2} = (2 ) ( ) (2 ) 4 2 g W W W W W W β κ ϕ φ ϕ φ ϕ φ ϕ ϕ  ′− − + ′ + − ′    ′ ′ + + (36) where: 2 3 1 2 3 4 2 2 3 = 2 4 4 2 2 1 = 2 2 4 2 1 = 2 4 W W W W W W W W W W W W W W ϕ ϕ ϕ ϕ ϕ ϕ φ ϕ ϕ ϕ ϕ ϕ ϕ φ ϕ ϕ φ ′  ′   ′   ′   ′   ′   ′  − + _ _+ _ _{ } − _ + _ _ +_ _{ }  _ −             ′  ′   ′   ′   ′   ′   ′  − + _{ }− _ _ +_ _ − _ _ −_  _{ } + _              ′   ′  − +             3 4 4 2 2 1 W W W W ϕ ϕ ϕ ′ ϕ  ′   ′   ′   ′  − + −                ₋          (37)

Definition 10. Let ( )N₁ be spherical curve of α₁ and let N₁ and T_N1 be Sabban vectors of ( )N₁ . In the fact βς₁-Smarandache curve is defined as:

1 1 1 1 ( ) = ( ) 2 N s N T ς β + (38)

Theorem 11. Let _1/2 α1 be involute of α. The geodesic curvature belonging to

1 1 1 1 1 1( ) = (1/2 )( coss T N sin B) ς β − ϕ + + ϕ 1 ς

β -Smarandache curve of involute curve is:

(

)

1 4 1 1 1 2 1 3 5 2 _{2 2} 1 1 = 2 2 ( ) g W W W ς β κ ϕ χ ϕ χ χ ϕ ′ − ′ +  ₊ ′    (39) where: 2 2 4 1 1 1 1 1 1 1 1 2 1 1 1 1 1 1 1 3 1 1 1 3 1 1 1 = 2 , = 2 3 = 2 W W W W W W W W W W ϕ ϕ ϕ ϕ ϕ ϕ ϕ χ χ ϕ ϕ ϕ χ ′ ′  ′   ′   ′   ′   ′   ′   ′  − −_ _ +_  _{  } − − _ _ −_ _ −_ _{ }  _               ′  ′   ′   ′  + +                   (40)

Corollary 12. Let α₁ be involute of α. The geodesic curvature Sabban apparatus be-longing to 1 2 2 2 2 2 2 2 2 2 2

sin cos cos sin

( ) = 2 2 2 2 2 2 W W W s T N B W W W ς ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ β ϕ ϕ ϕ ′ − ′ + ′ + ′ + − + ′ + ′ + ′ + 1 ς

β -Smarandache curve of evolute curve is:

(

)

1 _{1 2 3} 5 2 2 1 = ( 2 ) 2 gβς κ ηχ ηχ χ η − + + (41)

where 1 2 2 1 = = cos (c s) W _W ϕ ϕ η ϕ ϕ ′   ′ _ ′ _ −    _{′ +}    and 2 2 4 3 1= 2 , 2 = 2 3 , 3= 2 χ − −η +η η χ′ − − η −η −η η χ′ η η+ +η′ (42)

Definition 13. Let ( )N₁ be spherical curve of α₁, T_N1, and N T₁∧ _N1 be Sabban vectors of ( )N₁ . In the fact

2 ς

β -Smarandache curve is defined as:

1 1 2 1 1 ( ) = ( ) 2 N N s T N T ς β + ∧ (43)

Theorem 14. Let α₁ be involute of α. The geodesic curvature belonging to

1/2

1 1 1 1 1 1

2( ) = (1/2 )[(sins cos )T (sin cos ) ],B ς

β ϕ − ϕ + ϕ + ϕ

2 ς

β -Smarandache curve of involute curve is:

(

)

2 4 1 1 1 2 3 5 1 1 2 _{2 2} 1 1 = 2 2( ) g W W W W ς β κ ϕ φ φ φ ϕ ′ − +  _{+ ′}      (44) where: 2 3 1 1 1 1 1 1 1 1 1 1 1 2 3 4 1 1 1 1 1 1 2 1 1 1 1 1 1 2 3 4 1 1 1 1 3 1 1 1 1 = 2 4 2 2 1 = 2 2 4 2 2 1 = 2 4 4 2 τ τ τ τ τ φ κ κ κ κ κ τ τ τ τ τ τ φ κ κ κ κ κ κ τ τ τ τ φ κ κ κ κ ′           − + _{  }− _ + _{ } +_{  } − _           ′             − + _{ }− _{ } + _{ } − _{ } −_{  } + _                     − + −                 1 1 1 1 2 τ τ κ κ ′     +_{  } − _     (45)

Corollary 15. Let α1 be involute of α. The geodesic curvature belonging to

2 2 2 2 2 2 2 ( )sin ( )cos ( ) = 2 2 2 2 2 2 W W W s T N B W W W ς ϕ ϕ ϕ ϕ ϕ β ϕ ϕ ϕ ′+ ′− ′− + + ′ + ′ + ′ + 2 ς

β -Smarandache curve of evolute curve is:

(

)

2 1 2 3 5 2 2 1 = (2 ) 2 gβς κ ηφ φ φ η − + + (46) where: 3 2 4 2 4 1= 2 2 , 2= 1 3 2 , 3 = 2 φ η+ η + η η φ′ − − η − η −η′ φ −η − η +η′ (47)

Definition 16. Let ( )N1 be spherical curve of α1 and let N1, TN1, and N T1∧ N1 be Sab-ban vectors of ( )N1 . In the fact βς₃-Smarandache curve is defined as:

1 1 3 1 1 1 ( ) = ( ) 3 N N s N T N T ς β + + ∧ (48)

Theorem 17. Let _1/2 α1 be involute of α. The geodesic curvature belonging to

1 1 1 1 1 1 1

3( ) = (1/3 )[(sins cos )T N (sin cos ) ]B ς

β ϕ − ϕ + + ϕ + ϕ , βς₃-Smarandache curve of

invo-lute curve is:

3 1 1 1 1 2 3 1 1 1 5 2 2 1 1 1 1 2 1 1 2 = 4 2 1 g W W W W W ς β ϕ _ρ ϕ _ρ ϕ _ρ κ ϕ ϕ  _′   _′   _′   ₋  _{+ − −}  ₊ ₋                     _{ }  ′ ′  _{− + }   _{ }   _{ }    (49) where: 2 3 1 1 1 1 1 1 1 1 1 1 1 2 3 4 1 1 1 1 1 1 2 1 1 1 1 1 1 = 2 4 2 2 1 = 2 2 4 2 2 1 W W W W W W W W W W W ϕ ϕ ϕ ϕ ϕ ρ ϕ ϕ ϕ ϕ ϕ ϕ ρ ′  _′   _′   _′   _′   _′            − + _{  }− _ + _{ } +_{  } − _                     ′  _′   _′   _′   _′   _′   _′              − + _{ }− _{ } + _{ } − _{ } −_{  } +                        2 3 4 1 1 1 1 1 1 3 1 1 1 1 1 1 = 2 4 4 2 2 W W W W W W ϕ ϕ ϕ ϕ ϕ ϕ ρ  ′  _′   _′   _′   _′   _′   _′   ₋   ₊   ₋   ₊   ₋                                      (50)

Corollary 18. Let α1 be involute of α. The geodesic curvature belonging to 2 2 3 2 2 2 2 2 2 2 2 ( )sin cos ( ) = 3 3 3 3 ( )cos sin 3 3 W W W s T N W W W W B W ς ϕ ϕ ϕ ϕ ϕ β φ ϕ ϕ ϕ ϕ ϕ ϕ ′+ − ′ + ′ − + + ′ + ′ + ′+ + ′ + + ′ + 3 ς

β -Smarandache curve of evolute curve is:

(

)

3 1 2 3 5 2 2 (2 1) ( 1 ) (2 ) = 4 2 1 gβς η ρ η ρ η ρ κ η η − + − − + − − + (51)

where: 2 3 2 3 4 1 2 2 3 4 3 = 2 4 4 2 2 (2 1), = 2 2 4 2 2 (1 ) = 2 4 4 2 (2 ) ρ η η η η η ρ η η η η η η ρ η η η η η η ′ ′ − + − + + − − + − + − − + ′ − + − + − (52)

Definition 19. Let ( )B1 be spherical curve of α1 and let B1 and TB1 be Sabban vectors of ( )B₁ . In the fact βξ₁-Smarandache curve is defined as:

1 1 1 1 ( ) = ( ) 2 B s B T ξ β + (53)

Theorem 20. Let α₁ be involute of α. The geodesic curvature belonging to

1/2 1 1 1( ) = (1/2 )(s N B) ξ β − + 1 ξ

β -Smarandache curve of involute curve is:

(

)

1 4 1 1 1 1 2 1 3 5 2 _{2 2} 1 1 = ( 2 ) 2 gβξ τ κ κ ω κ ω τ ω τ κ − + + (54) where: 2 2 4 1 1 1 1 1 1 1 1 2 1 1 1 1 1 1 1 3 1 1 1 3 1 1 1 = 2 , = 2 3 = 2 κ κ κ κ κ κ κ ω ω τ τ τ τ τ τ τ κ κ κ ω τ τ τ ′ ′               − −_{ } +_{   } − − _{ } −_{ } −_{   }               ′       + +             (55)

Corollary 21. Let α1 be involute of α. The geodesic curvature belonging to 1

1

( ) = [(cos sin ) (cos sin ) ]

2 s T B ξ β ϕ+ ϕ + ϕ− ϕ 1 ξ

β -Smarandache curve of evolute curve is:

(

)

(

)

1 4 1 2 3 5 2 2 2 = 2 2 g W W ξ β ϕ κ ω ω ϕ ω ϕ ′ _{ ′ } − +   ′ + (56) where: 2 2 4 1 2 3 3 = 2 , = 2 3 = 2 2 W W W W W W W W W W ω ω ϕ ϕ ϕ ϕ ϕ ϕ ϕ ω ϕ ϕ ϕ ′ ′               − − _′  + _′   _′  − −  _′  − _′  − _′   _′                ′       + +  _′   _′   _′        (57)

Definition 22. Let ( )B1 be spherical curve of α1 and let TB1 and B T1∧ B1 be Sabban vectors of ( )B₁ . In the fact β_ξ2-Smarandache curve is defined as:

1 1 1 2 1 ( ) = ( ) 3 B B s T B T ξ β + ∧ (58)

Theorem 23. Let _1/2 α1 be involute of α. The geodesic curvature belonging to 1 1 2( ) = (1/2 )(s T N ), ξ β − 2 ξ

β -Smarandache curve of involute curve is:

(

)

2 4 1 1 1 1 2 1 3 5 2 _{2 2} 1 1 = (2 ) 2 gβξ τ κ κ ψ τ ψ τ ψ τ κ − + + (59) where: 3 2 4 1 1 1 1 1 1 1 1 2 1 1 1 1 1 1 1 2 4 1 1 1 3 1 1 1 = 2 2 , = 1 3 2 = 2 κ κ κ κ κ κ κ ψ ψ τ τ τ τ τ τ τ κ κ κ ψ τ τ τ ′ ′             + _{ } + _{   } − − _{ } − _{ } −_{ }             ′       −_{ } − _{ } +_{ }       (60)

Corollary 24. Let _1/2 α1 be involute of α. The geodesic curvature belonging to 2( ) = (1/2 )(coss T N sin B),

ξ

β ϕ + − ϕ

2 ξ

β -Smarandache curve of evolute curve is:

(

)

(

)

2 4 1 2 3 5 2 2 2 = 2 2 g W W ξ β ϕ κ ψ ϕ ψ ϕ ψ ϕ ′ _{′ ′} − + ′ + (61) where: 3 2 4 1 2 2 4 3 = 2 , = 1 3 2 = 2 W W W W W W W W W W ψ ψ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ψ ϕ ϕ ϕ ′ ′               + + − − − −  _′   _′   _′   _′   _′   _′   _′                ′       −_{ } − _{ } +_{ } ′ ′ ′       (62)

Definition 25. Let ( )B₁ be spherical curve of α₁ and let B₁, T_B1, and B T₁∧ _B1 be Sabban vectors of ( )B₁ . In the fact βξ₃-Smarandache curve is defined as:

1 1 1 1 3 1 ( ) = ( ) 3 B B s B T B T ξ β + + ∧ (63)

Theorem 26. Let α1 be involute of α. The geodesic curvature belonging to 1/2 1 1 1 3 = (1/3 )(T N B), ξ β − + 3 ξ

β -Smarandache curve of involute curve is:

(

)

[

]

3 4 1 1 1 1 1 1 2 1 1 3 5 2 _{2 2} 1 1 1 1 = (2 ) ( ) (2 ) 4 2 gβξ τ κ κ τ ζ τ κ ζ τ κ ζ τ κ τ κ − − + + − + + (64) where: 2 3 1 1 1 1 1 1 1 1 1 1 1 2 3 4 1 1 1 1 1 1 2 1 1 1 1 1 1 = 2 4 2 2 1 = 2 2 4 2 2 1 κ κ κ κ κ ζ τ τ τ τ τ κ κ κ κ κ κ ζ τ τ τ τ τ τ ′           − + _{  }− _ + _{ } +_{  } − _           ′             − + _{ }− _{ } + _{ } − _{ } −_{  } + _             (65)

2 3 4 1 1 1 1 1 1 3 1 1 1 1 1 1 = 2 κ 4 κ 4 κ 2 κ κ 2 κ ζ τ τ τ τ τ τ ′             − + − + −                         (65)

Corollary 27. Let _1/2 α1 be involute of α. The geodesic curvature belonging to 3( ) = (1/3 )[(sins cos )T N (cos sin ) ],B

ξ

β ϕ+ ϕ + + ϕ− ϕ

3 ξ

β -Smarandache curve of evolute curve is: 3 4 1 2 3 5 2 5 2 = (2 ) ( ) (2 ) 4 2 g W W W W W ξ β ϕ κ ϕ ζ ϕ ζ ϕ ζ ϕ ϕ ′ _{ ′ ′ ′ } − − + + −    _{′ + ′ +}    (66) where: 2 3 1 2 3 4 2 2 3 4 3 = 2 4 4 2 2 1 = 2 2 4 2 1 = 2 4 4 2 W W W W W W W W W W W W W W W W W ζ ϕ ϕ ϕ ϕ ϕ ϕ ζ ϕ ϕ ϕ ϕ ϕ ϕ ζ ϕ ϕ ϕ ϕ ϕ ′             − + _{ }+ _{  }− _ + _{ } +_{  } − _ ′ ′ ′ ′ ′ ′             ′             − +  _′ −  _′  + _′  −  _′  − _′   + _′                       − + − +  _′   _′   _′   _′  _′         2 W ϕ ′   −    _′      (67)

Example. Let us consider the unit speed evolute and involute curve, respectively:

( )

( )

( )

( )

( )

2 1 2 1 4

( ) = sin 2 sin 8 , cos 2 cos 8 , sin 3

5 40 5 40 15 s t t t t t α _ − − + _  

( )

( )

(

) ( )

1 2 1 4 2

( ) = sin 2 sin 8 1 cos 5 , ,

5 40 5 5

s s s s s

α _ − + − −

( )

1

( )

4

(

) ( )

4

( )

3 3

cos 2 cos 8 1 sin 5 , sin 3

40 5 15 5 5

s + s + −s s s − + s_

The Frenet vectors belonging to involute curve α₁ are found:

1= ₅4cos(5 ), sin(5 ),4₅ 3₅

T _ s s − _

 

( )

( )

( )

( )

( )

( )

1= 1₅cos 8 4₅cos 2 sin 3 4₅sin 2 1₅sin 8 cos 3

N _ s − s _ s −_ s + s _ s

   



( )

( )

( )

( )

( )

( )

4_{sin 2} 1_{sin 8 sin 3 cos 3} 4_{cos 2} 1_{cos 8 ,0}

5 s 5 s s s 5 s 5 s  _{− +}  ₊  ₊           

( )

( )

( )

( )

( )

( )

1= cos 3 4₅cos 2 1₅cos 8 sin 3 4₅sin 2 1₅sin 8

B _ s _ s − s _− s _ s + s _

   



( )

4

( )

1

( )

( )

4

( )

1

( )

4

cos 3 sin 2 sin 8 sin 3 cos 2 cos 8 ,

5 5 5 5 5

s _ s − s _+ s _ s + s _{ }

According to the definitions, we reach specific Smarandache curves belonging to Sab-ban frame of this curve. β₁, β₂, β₃, βς₁, βς₂, βς₃, βξ₁, βξ₂, and βξ₃, see figs. 1-3.

Figure 1. (a) β1-curve, (b) β2-curve, and (c) β3-curve

(a) (b) (c)

Figure 2. (a) βς₁-curve, (b) βς₂-curve, and (c) βς₃-curve

(a) (b) (c)

Figure 3. (a) βξ₁-curve, (b) βξ₂-curve, and (c) βξ₃-curve

(a) (b) (c)

Conclusion

In this paper, we reviewed the well-known involute and evolute curves in the litera-ture. We created the Sabban frames on the unit sphere of the involute and evolute curves. We gived Smarandache curves from the Sabban frame and calculated the geodesic curvature of these curves. Finally, we gave an example and draw their figures in the MAPPLE program.

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Paper submitted: November 30, 2018 Paper revised: December 26, 2018 Paper accepted: January 12, 2019

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