SMARANDACHE CURVES ACCORDING TO BISHOP FRAME IN EUCLIDEAN 3-SPACE

arXiv:1106.3202v1 [math.GM] 16 Jun 2011

SMARANDACHE CURVES ACCORDING TO BISHOP FRAME IN EUCLIDEAN 3-SPACE MUHAMMED C 存 ETIN, YILMAZ TUNC 存 ER, AND MURAT KEMAL KARACAN Abstract. In this paper, we investigate special Smarandache curves according to Bishop frame in Euclidean 3-space and we give some differential geometric properties of Smarandache curves. Also we find the centers of the osculating spheres and curvature spheres of Smarandache curves.

1. Introduction Special Smarandache curves have been investigated by some differential geometers [1,8]. 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. M. Turgut and S. Yilmaz have defined a special case of such curves and call it Smarandache T B2 Curves in the space E14 [8]. They have dealed with a special Smarandache curves which is defined by the tangent and second binormal vector fields. They have called such curves as Smarandache T B2 Curves. Additionally, they have computed formulas of this kind curves by the method expressed in [10].

A. T. Ali has introduced some special Smarandache curves in the Euclidean space.

He has studied Frenet-Serret invariants of a special case [1].

In this paper, we investigate special Smarandache curves such as Smarandache Curves T N1 , T N2 , N1 N2 and T N1 N2 according to Bishop frame in Euclidean 3-space. Furthermore, we find differential geometric properties of these special curves and we calculate first and second curvature (natural curvatures) of these curves. Also we find the centers of the curvature spheres and osculating spheres of Smarandache curves.

2. Preliminaries 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 (N1 (s), N2 (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 (N1 (s), N2 (s)) depend only on T (s) and not each other we Key words and phrases. Smarandache curves, Bishop frame, Natural curvatures, Osculating sphere, Curvature sphere, Centers of osculating spheres.

2000 AMS Subject Classifications:53A04.

1

2

MUHAMMED C ¸ ETIN, YILMAZ TUNC ¸ ER, AND MURAT KEMAL KARACAN

can make N1 (s) and N2 (s) vary smoothly throughout the path regardless of the curvature [2,6,7].

In addition, suppose the curve α is an arclength-parametrized C 2 curve. Suppose we have C 1 unit vector fields N1 and N2 = T ∧ N1 along the curve α so that hT, N1 i = hT, N2 i = hN1 , N2 i = 0,

i.e., T , N1 , N2 will be a smoothly varying right-handed orthonormal frame as we move along the curve. (To this point, the Frenet frame would work just fine if the curve were C 3 with κ 6= 0) But now we want to impose the extra condition that hN1′ , N2 i = 0. We say the unit first normal vector field N1 is parallel along the curve α. This means that the only change of N1 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 k1 k2 T  N ′1  =  −k1 0 0   N1  .

N ′2 −k2 0 0 N2 One can show that

q k2 dθ(s) 2 2 , k1 6= 0, τ (s) = − κ(s) = k1 + k2 , θ(s) = arctan k1 ds

so that k1 and k2 effectively correspond to a Cartesian coordinate system for the R polar coordinates κ, θ with θ = − τ (s)ds. The orientation of the parallel transport frame includes the arbitrary choice of integration constant θ 0 , which disappears from τ (and hence from the Frenet frame) due to the differentiation [2,6,7].

Bishop curvatures are defined by (2.1)

k1 = κ cos θ,

k2 = κ sin θ

and (2.2) [9].

T =T ,

N1 = N cos θ − B sin θ ,

N2 = N sin θ + B cos θ

3. Smarandache Curves According to Bishop Frame In [1], author gave following definitions: Definition 1. Let α = α(s) be a unit speed regular curve in E 3 and {T, N, B} be its moving Serret-Frenet frame. Smarandache curves T N are defined by 1 β(s∗ ) = √ (T + N ) .

2 Definition 2. Let α = α(s) be a unit speed regular curve in E 3 and {T, N, B} be its moving Serret-Frenet frame. Smarandache curves N B are defined by 1 β(s∗ ) = √ (N + B) .

2 Definition 3. Let α = α(s) be a unit speed regular curve in E 3 and {T, N, B} be its moving Serret-Frenet frame. Smarandache curves T N B are defined by 1 β(s∗ ) = √ (T + N + B) .

3

SMARANDACHE CURVES ACCORDING TO BISHOP FRAME IN EUCLIDEAN 3-SPACE

3

In this section, we investigate Smarandache curves according to Bishop frame in Euclidean 3-space. Let α = α(s) be a unit speed regular curve in E 3 and denote by {Tα , N1α , N2α } the moving Bishop frame along the curve α. The following Bishop formulae is given by ·

·

·

Tα = k1α N1α + k2α N2α , N1α = −k1α Tα , N2α = −k2α Tα

(3.1) with

Tα = N1α ∧ N2α , N1α = −Tα ∧ N2α , N2α = Tα ∧ N1α .

3.1. T N1 −Smarandache Curves.

Definition 4. Let α = α(s) be a unit speed regular curve in E 3 and {Tα , N1α , N2α } be its moving Bishop frame. Tα N1α −Smarandache curves can be defined by 1 β(s∗ ) = √ (Tα + N1α ) .

2

(3.2)

Now, we can investigate Bishop invariants of Tα N1α −Smarandache curves according to α = α(s). Differentiating (3.2) with respect to s, we get ·

(3.3)

β=

dβ ds∗ −1 = √ (k1α Tα − k1α N1α − k2α N2α ) ds∗ ds 2

and Tβ

−1 ds∗ = √ (k1α Tα − k1α N1α − k2α N2α ) ds 2

where ∗

ds = ds

(3.4)

s

2

2

2 (k1α ) + (k2α ) .

2

The tangent vector of curve β can be written as follow, −1 Tβ = q (k1α Tα − k1α N1α − k2α N2α ) .

2 2 α α 2 (k1 ) + (k2 )

(3.5)

Differentiating (3.5) with respect to s, we obtain dTβ ds∗ = ds∗ ds

(3.6) where

1 2 2 (k1α )

·

+

2

2 (k2α )

α α 32 (λ1 Tα + λ2 N1 + λ3 N2 )

4

2

2

4

·

λ1

=

−k1α (k2α ) − 2 (k1α ) − 3 (k1α ) (k2α ) − (k2α ) + k1α k2α k2α

λ2

=

−2 (k1α ) − (k1α ) (k2α ) + k1α (k2α ) − k1α k2α k2α

λ3

=

−2 (k1α ) k2α − k1α (k2α ) + 2 (k1α ) k2α − 2k1α k1α k2α .

4

3

2

2

3

·

·

2

2

·

·

Substituting (3.4) in (3.6), we get √ 2 α α Tβ′ = 2 (λ1 Tα + λ2 N1 + λ3 N2 ) .

2 2 α α 2 (k1 ) + (k2 )

4

MUHAMMED C ¸ ETIN, YILMAZ TUNC ¸ ER, AND MURAT KEMAL KARACAN

Then, the curvature and principal normal vector field of curve β are respectively, √ q 2 2 Îť1 + Îť22 + Îť23 ′

κβ = T β = 2 2 2 2 (k1ι ) + (k2ι )

and

1 Nβ = q (Ν1 Tι + Ν2 N1ι + Ν3 N2ι ) .

2 2 2 Îť1 + Îť2 + Îť3

On the other hand, we express

TÎą

−k1Îą q Bβ = q

2 2 2 2 2

Îą Îą Îť1 2 (k1 ) + (k2 ) Îť1 + Îť2 + Îť3

N1Îą k1Îą Îť2

1

N2Îą k2Îą Îť3

.

So, the binormal vector of curve β is 1 q (Ďƒ 1 TÎą + Ďƒ 2 N1Îą + Ďƒ 3 N2Îą ) Bβ = q 2 2 2 2 2 Îą Îą 2 (k1 ) + (k2 ) Îť1 + Îť2 + Îť3 where

Ďƒ 1 = Îť3 k1Îą − Îť2 k2Îą , Ďƒ 2 = Îť3 k1Îą + Îť1 k2Îą , Ďƒ 3 = −k1Îą (Îť1 + Îť2 ) .

We differentiate (3.3) with respect to s in order to calculate the torsion ¡

¡ ¡ ¡¡ −1 Îą Îą 2 Îą 2 Îą Îą 2 Îą Îą Îą Îą Îą (k1 + (k1 ) + (k2 ) )TÎą − (k1 − (k1 ) )N1 − (k2 − k1 k2 )N2 , β=√ 2 and similarly ¡¡¡ 1 β = √ (Ď 1 TÎą + Ď 2 N1Îą + Ď 3 N2Îą ) , 2 where Ď 1

¡¡

¡

¡

3

2

= −k1Îą − 3k1Îą k1Îą − 3k2Îą k2Îą + (k1Îą ) + k1Îą (k2Îą )

Ď 2

=

Ď 3

=

¡ −3k1Îą k1Îą ¡ −2k1Îą k2Îą

3

2

2

3

− (k1Îą ) − k1Îą (k2Îą ) +

¡¡ k1ι

¡

¡¡

− (k1Îą ) k2Îą − (k2Îą ) − k1Îą k2Îą + k2Îą .

The torsion of curve β is

τβ =

  ¡ ¡ √  ((k1Îą )2 − k1Îą )(Ď 3 k1Îą + Ď 1 k2Îą ) + k1Îą (k2Îą − k1Îą k2Îą )(Ď 1 + Ď 2 )  2 ¡ 2 2   −(k Îą + (k Îą ) + (k Îą ) )(Ď k Îą − Ď k Îą ) 1

¡ (k1ι k2ι

−

¡ k1ι k2ι )2

+

¡ (k1ι k2ι

−

1

2 2 (k1Îą )

k2Îą −

2 ¡ k1ι k2ι

2 2

−

3 1

3 (k2Îą ) )2

3

2

+ (2 (k1Îą ) + k1Îą (k2Îą ) )2

The first and second normal field of curve β are as follow. Then, from (2.2) we obtain  

√ Âľ cos θβ Îť1 − sin θβ Ďƒ 1 TÎą   1 √ + Âľ cos θβ Îť2 − sin θβ Ďƒ 2 N1Îą , N1β = √ √  ¾Ρ  + Âľ cos θβ Îť3 − sin θβ Ďƒ 3 N2Îą and

.

SMARANDACHE CURVES ACCORDING TO BISHOP FRAME IN EUCLIDEAN 3-SPACE

5



√ µ sin θβ λ1 + cos θβ σ 1 Tα  1 √ + µ sin θ β λ2 + cos θβ σ 2 N1α N2β = √ √  µη  + µ sin θ β λ3 + cos θβ σ 3 N2α q R 2 2 where θβ = − τ β (s)ds and µ = 2 (k1α ) + (k2α ) , η = λ21 + λ22 + λ23 Now, we can calculate natural curvatures of curve β, so from (2.1) we get q 2(λ21 + λ22 + λ23 ) cos θβ β k1 = 2

2 2 2 (k1α ) + (k2α )  

and

q 2(λ21 + λ22 + λ23 ) sin θ β β k2 = 2 .

2 2 2 (k1α ) + (k2α )

3.2. T N2 −Smarandache Curves.

Definition 5. Let α = α(s) be a unit speed regular curve in E 3 and {Tα , N1α , N2α } be its moving Bishop frame. Tα N2α −Smarandache curves can be defined by 1 (3.7) β(s∗ ) = √ (Tα + N2α ) .

2

Now, we can investigate Bishop invariants of Tα N2α −Smarandache curves according to α = α(s). Differentiating (3.7) with respect to s, we get · dβ ds∗ −1 (3.8) β= ∗ = √ (k2α Tα − k1α N1α − k2α N2α ) ds ds 2 and −1 ds∗ = √ (k2α Tα − k1α N1α − k2α N2α ) Tβ ds 2 where s 2 2 (k1α ) + 2 (k2α ) ds∗ = .

(3.9) ds 2 The tangent vector of curve β can be written as follow, −1 (k2α Tα − k1α N1α − k2α N2α ) .

(3.10) Tβ = q 2 2 (k1α ) + 2 (k2α ) Differentiating (3.10) with respect to s, we obtain dTβ ds∗ 1 α α (3.11) = 32 (λ1 Tα + λ2 N1 + λ3 N2 ) ds∗ ds (k1α )2 + 2 (k2α )2 where

2

·

4

2

2

·

4

λ1

= − (k1α ) k2α − (k1α ) − 3 (k1α ) (k2α ) − 2 (k2α ) + k1α k1α k2α

λ2

= − (k1α )3 k2α − 2k1α (k2α )3 + 2k1α (k2α )2 − 2k1α k2α k2α

λ3

= − (k1α ) (k2α ) − 2 (k2α ) + (k1α ) k2α − k1α k1α k2α .

·

2

2

4

·

2

·

·

6

MUHAMMED C ¸ ETIN, YILMAZ TUNC ¸ ER, AND MURAT KEMAL KARACAN

Substituting (3.9) in (3.11), we get √ 2 Îą Îą ′ Tβ = 2 (Îť1 TÎą + Îť2 N1 + Îť3 N2 ) .

2 2 Îą Îą (k1 ) + 2 (k2 )

Then, the first curvature and the principal normal vector field of curve β are respectively, √ q 2 2 Îť1 + Îť22 + Îť23 ′

κβ = T β = 2 2 2 (k1ι ) + 2 (k2ι ) and

1 Nβ = q (Ν1 Tι + Ν2 N1ι + Ν3 N2ι ) .

2 2 2 Îť1 + Îť2 + Îť3

On the other hand, we express

TÎą

1

−k2Îą q Bβ = q

2 2 2 2 2

Îą Îą Îť1 (k1 ) + 2 (k2 ) Îť1 + Îť2 + Îť3

N1Îą k1Îą Îť2

N2Îą k2Îą Îť3

.

So, the binormal vector of curve β is 1 q Bβ = q (Ďƒ 1 TÎą + Ďƒ 2 N1Îą + Ďƒ 3 N2Îą ) 2 2 2 2 2 Îą Îą (k1 ) + 2 (k2 ) Îť1 + Îť2 + Îť3 where

Ďƒ 1 = Îť3 k1Îą − Îť2 k2Îą , Ďƒ 2 = (Îť3 + Îť1 ) k2Îą , Ďƒ 3 = − (Îť2 k2Îą + Îť1 k1Îą ) .

We differentiate (3.8) with respect to s in order to calculate the torsion of curve β ¡

¡ ¡ ¡¡ −1 2 2 2 (k2Îą + (k1Îą ) + (k2Îą ) )TÎą − (k1Îą − k1Îą k2Îą )N1Îą − (k2Îą − (k2Îą ) )N2Îą , β=√ 2 and similarly ¡¡¡ 1 β = √ (Ď 1 TÎą + Ď 2 N1Îą + Ď 3 N2Îą ) , 2 where ¡¡

¡

¡

2

3

Ď 1

= −k2Îą − 3k1Îą k1Îą − 3k2Îą k2Îą + (k1Îą ) k2Îą + (k2Îą )

Ď 2

= −2k1Îą k2Îą − (k1Îą )3 − k1Îą (k2Îą )2 − k1Îą k2Îą + k1Îą

Ď 3

= −3k2Îą k2Îą − (k1Îą ) k2Îą − (k2Îą ) + k2Îą .

¡

¡

¡

2

3

¡¡

¡¡

The torsion of curve β is

τβ =

  ¡ ¡ √  k2Îą (k1Îą k2Îą − k1Îą )(Ď 3 + Ď 1 ) + (k2Îą − (k2Îą )2 )(Ď 2 k2Îą + Ď 1 k1Îą )  2 ¡ 2 2   −(k Îą + (k Îą ) + (k Îą ) )(Ď k Îą − Ď k Îą ) 2

¡ (k1ι k2ι

−

¡ k1ι k2ι )2

1

2

+ (−2 (k2Îą )3 − (k1Îą )2 k2Îą )2 +

2 2

3 1 ¡ (2k1Îą (k2Îą )2 − k1Îą k2Îą

¡

+ k1Îą k2Îą + (k1Îą )3 )2

The first and second normal field of curve β are as follow. Then, from (2.2) we obtain

.

SMARANDACHE CURVES ACCORDING TO BISHOP FRAME IN EUCLIDEAN 3-SPACE

7



√ µ cos θβ λ1 − sin θβ σ 1 Tα  1 √ + µ cos θβ λ2 − sin θβ σ 2 N1α N1β = √ , √  µη  + µ cos θβ λ3 − sin θβ σ 3 N2α  

and



√ µ sin θ β λ1 + cos θβ σ 1 Tα  1 √ + µ sin θβ λ2 + cos θβ σ 2 N1α N2β = √ , √  µη  + µ sin θβ λ3 + cos θβ σ 3 N2α q R 2 2 where θβ = − τ β (s)ds and µ = (k1α ) + 2 (k2α ) , η = λ21 + λ22 + λ23 .

Now, we can calculate natural curvatures of curve β, so from (2.1) we get q 2(λ21 + λ22 + λ23 ) cos θβ β k1 = 2

2 2 (k1α ) + 2 (k2α )  

and

q 2(λ21 + λ22 + λ23 ) sin θ β k2β = 2 . (k1α )2 + 2 (k2α )2

3.3. N1 N2 −Smarandache Curves.

Definition 6. Let α = α(s) be a unit speed regular curve in E 3 and {Tα , N1α , N2α } be its moving Bishop frame. N1α N2α −Smarandache curves can be defined by 1 (3.12) β(s∗ ) = √ (N1α + N2α ) .

2 Now, we can investigate Bishop invariants of N1α N2α −Smarandache curves according to α = α(s). Differentiating (3.12) with respect to s, we get (3.13)

·

β=

dβ ds∗ − (k1α + k2α ) √ Tα = ds∗ ds 2

and Tβ

ds∗ − (k1α + k2α ) √ Tα = ds 2

where kα + kα ds∗ = 1√ 2 . ds 2 The tangent vector of curve β can be written as follow,

(3.14)

(3.15)

Tβ = −Tα .

Differentiating (3.15) with respect to s, we obtain dTβ ds∗ = −k1α N1α − k2α N2α ds∗ ds Substituting (3.14) in (3.16), we get √ − 2 ′ (k α N α + k2α N2α ) .

Tβ = α k1 + k2α 1 1 (3.16)

8

MUHAMMED C ¸ ETIN, YILMAZ TUNC ¸ ER, AND MURAT KEMAL KARACAN

Then, the first curvature and the principal normal vector field of curve β are respectively, √ q Îą 2 2 (k1 ) + (k2Îą )2 ′ κβ = Tβ = k1Îą + k2Îą and −1 Nβ = q (k1Îą N1Îą + k2Îą N2Îą ) .

2 2 Îą Îą (k1 ) + (k2 ) On the other hand, we express

TÎą N1Îą

−1 0 Bβ = q

2 2

Îą Îą 0 −k1Îą (k1 ) + (k2 ) 1

N2Îą 0 −k2Îą

.

So, the binormal vector of curve β is −1 Bβ = q (k2Îą N1Îą − k1Îą N2Îą ) .

2 2 (k1Îą ) + (k2Îą )

We differentiate (3.13) with respect to s in order to calculate the torsion of curve β

¡ ¡ ¡¡ −1 Îą Îą Îą Îą Îą 2 Îą 2 Îą Îą Îą Îą k1 + k2 TÎą + (k1 ) + k1 k2 N1 + k1 k2 + (k2 ) N2 , β= √ 2 and similarly ¡¡¡ −1 β = √ (Ď 1 TÎą + Ď 2 N1Îą + Ď 3 N2Îą ) , 2 where Ď 1

=

Ď 2

=

Ď 3

=

¡¡

¡¡

3

2

2

3

k1Îą + k2Îą − (k1Îą ) − (k1Îą ) k2Îą − k1Îą (k2Îą ) − (k2Îą )

¡ 3k1ι k1ι ¡ 2k1ι k2ι

+ +

¡ 2k1ι k2ι ¡ 3k2ι k2ι

+ +

¡ k1ι k2ι ¡ k1ι k2ι .

The torsion of curve β is o √ n 2 2 − 2 Ď 3 ((k1Îą ) + k1Îą k2Îą ) − Ď 2 (k1Îą k2Îą + (k2Îą ) ) n o. τβ = 2 2 (k1Îą + k2Îą ) (k1Îą k2Îą + (k2Îą ) )2 + ((k1Îą ) + k1Îą k2Îą )2

The first and second normal field of curve β are as follow. Then, from (2.2) we obtain 1 {(k2Îą sin θβ − k1Îą cos θ β ) N1Îą − (k2Îą cos θ β + k1Îą sin θβ ) N2Îą } , N1β = q 2 2 Îą Îą (k1 ) + (k2 )

and

1 N2β = q {− (k1Îą sin θ β + k2Îą cos θβ ) N1Îą + (k1Îą cos θβ − k2Îą sin θβ ) N2Îą } 2 2 Îą Îą (k1 ) + (k2 ) R where θβ = − Ď„ β (s)ds.

Now, we can calculate the natural curvatures of curve β, so from (2.1) we get

SMARANDACHE CURVES ACCORDING TO BISHOP FRAME IN EUCLIDEAN 3-SPACE

k1β =

q 2 2 2((k1α ) + (k2α ) ) cos θ β

k2β =

q 2 2 2((k1α ) + (k2α ) ) sin θβ

9

k1α + k2α

and

k1α + k2α

.

3.4. T N1 N2 −Smarandache Curves.

Definition 7. Let α = α(s) be a unit speed regular curve in E 3 and {Tα , N1α , N2α } be its moving Bishop frame. Tα N1α N2α −Smarandache curves can be defined by 1 β(s∗ ) = √ (Tα + N1α + N2α ) .

3

(3.17)

Now, we can investigate Bishop invariants of Tα N1α N2α −Smarandache curves according to α = α(s). Differentiating (3.17) with respect to s, we get −1 dβ ds∗ = √ ((k1α + k2α ) Tα − k1α N1α − k2α N2α ) ds∗ ds 3

·

(3.18)

β=

and Tβ

−1 ds∗ = √ ((k1α + k2α ) Tα − k1α N1α − k2α N2α ) ds 3

where ∗

ds = ds

(3.19)

v

u u 2 (k α )2 + k α k α + (k α )2 t 1 1 2 2 3

.

The tangent vector of curve β can be written as follow,

−1 α α α α α α Tβ = r n o ((k1 + k2 ) Tα − k1 N1 − k2 N2 ) .

2 (k1α )2 + k1α k2α + (k2α )2

(3.20)

Differentiating (3.20) with respect to s, we obtain

1 dTβ ds∗ α α = o 3 (λ1 Tα + λ2 N1 + λ3 N2 ) √ n α 2 ds∗ ds 2 2 α α α 2 2 (k1 ) + k1 k2 + (k2 )

(3.21) where λ1

·

2

2

·

·

4

3

= −2k1α (k2α ) − (k1α ) k2α + k1α k2α k2α − 2 (k1α ) − 2 (k1α ) k2α 2

2

3

·

4

·

−4 (k1α ) (k2α ) − 2k1α (k2α ) − 2 (k2α ) + k1α k1α k2α + k1α (k2α ) λ2

4

2

2

3

·

= −2 (k1α ) − 4 (k1α ) k2α − 4 (k1α ) (k2α ) − 2k1α (k2α ) + k1α k1α k2α ·

λ3

3

2

2

2

·

·

+2k1α (k2α ) − (k1α ) k2α − 2k1α k2α k2α

= −2 (k1α )3 k2α − 4 (k1α )2 (k2α )2 − 4k1α (k2α )3 − 2 (k2α )4 2

·

·

·

·

2

+2 (k1α ) k2α + k1α k2α k2α − 2k1α k1α k2α − k1α (k2α ) .

10

MUHAMMED C ¸ ETIN, YILMAZ TUNC ¸ ER, AND MURAT KEMAL KARACAN

Substituting (3.19) in (3.21), we get √ 3 Îą Îą ′ Tβ = n o2 (Îť1 TÎą + Îť2 N1 + Îť3 N2 ) .

2 2 Îą Îą Îą Îą 4 (k1 ) + k1 k2 + (k2 )

Then, the first curvature and the principal normal vector field of curve β are respectively, √ q 2 3 Îť1 + Îť22 + Îť23 ′

κβ = T β = n o2 2 2 4 (k1ι ) + k1ι k2ι + (k2ι ) and

1 Nβ = q (Ν1 Tι + Ν2 N1ι + Ν3 N2ι ) .

2 2 2 Îť1 + Îť2 + Îť3

(3.22)

On the other hand, we express

TÎą

1 Îą Îą

− (k Bβ = r 1 + k2 ) q

2 2 2 2 2

Îą Îą Îą Îą Îť 1 2 (k1 ) + k1 k2 + (k2 ) Îť1 + Îť2 + Îť3

N1Îą k1Îą Îť2

N2Îą k2Îą Îť3

So, the binormal vector of curve β is

.

1 (3.23) Bβ = r (Ďƒ 1 TÎą + Ďƒ 2 N1Îą + Ďƒ 3 N2Îą ) q 2 2 2 2 2 2 (k1Îą ) + k1Îą k2Îą + (k2Îą ) Îť1 + Îť2 + Îť3

where

Ďƒ 1 = Îť3 k1Îą − Îť2 k2Îą , Ďƒ 2 = Îť3 (k1Îą + k2Îą ) + Îť1 k2Îą , Ďƒ 3 = − (Îť2 (k1Îą + k2Îą ) + Îť1 k1Îą ) .

We differentiate (3.18) with respect to s in order to calculate the torsion of curve β,

¡ ¡ ¡ ¡ ¡¡ −1 Îą Îą Îą 2 Îą 2 Îą Îą 2 Îą Îą Îą Îą Îą Îą Îą 2 Îą (k1 + k2 + (k1 ) + (k2 ) )TÎą − (k1 − (k1 ) − k1 k2 )N1 − (k2 − k1 k2 − (k2 ) )N2 , β= √ 3 and similarly 1 β = √ (Ď 1 TÎą + Ď 2 N1Îą + Ď 3 N2Îą ) , 3

¡¡¡

where Ď 1 Ď 2 Ď 3

¡¡

¡¡

¡

¡

3

2

3

2

2 (k1Îą )

3 (k2Îą )

¡ k1ι k2ι ¡ k1ι k2ι

2

= −k1Îą − k2Îą − 3k1Îą k1Îą − 3k2Îą k2Îą + (k1Îą ) + (k1Îą ) k2Îą + k1Îą (k2Îą ) + (k2Îą ) = =

¡ −3k1Îą k1Îą ¡ −2k1Îą k2Îą

− −

¡ 2k1ι k2ι ¡ 3k2ι k2ι

The torsion of curve β is

− (k1Îą ) − k1Îą (k2Îą ) − −

k2Îą −

−

+ +

¡¡ k1ι ¡¡ k2ι

3

SMARANDACHE CURVES ACCORDING TO BISHOP FRAME IN EUCLIDEAN 3-SPACE 11

τβ =   

·

 ·  2   (k1α − (k1α ) − k1α k2α )(−ρ3 k1α − ρ3 k2α − ρ1 k2α )     √ · α α α α 2 α α α 3 +(k2 − k1 k2 − (k2 ) )(ρ2 k1 + ρ2 k2 + ρ1 k1 )     · ·   2 2 −(k1α + k2α + (k1α ) + (k2α ) )(−ρ3 k1α + ρ2 k2α ) ·

·

·

(k1α k2α − k1α k2α )2 + (k1α k2α − 2 (k1α )2 k2α − 2k1α (k2α )2 − 2 (k2α )3 − k1α k2α )2 3

2

·

·

2

+(2 (k1α ) + 2 (k1α ) k2α + 2k1α (k2α ) − k1α k2α + k1α k2α )2

.  

The first and second normal field of curve β are as follow. Then, from (2.2) we obtain 

√ T 2µ cos θ λ − sin θ σ  α β 1 β 1

√ 1 + √2µ cos θβ λ2 − sin θβ σ 2 N1α N1β = √ ,  2µη  + 2µ cos θβ λ3 − sin θβ σ 3 N2α  

and



√ 2µ sin θ T λ + cos θ σ  α β 1 β 1

√ 1 , N2β = √ + √2µ sin θ β λ2 + cos θβ σ 2 N1α  2µη  + 2µ sin θ β λ3 + cos θβ σ 3 N2α q R 2 2 where θβ = − τ β (s)ds and µ = (k1α ) + k1α k2α + (k2α ) , η = λ21 + λ22 + λ23 Now, we can calculate the natural curvatures of curve β, so from (2.1) we get  

k1β =

q 3(λ21 + λ22 + λ23 ) cos θβ 2

2

4((k1α ) + k1α k2α + (k2α ) )2

and

k2β =

q 3(λ21 + λ22 + λ23 ) sin θ β 2

2

4((k1α ) + k1α k2α + (k2α ) )2

.

4. Curvature Spheres and Osculating Spheres of Smarandache Curves Definition 8. Let α : (a, b) → Rn be a regular curve and let F : Rn → R be a differentiable function. We say that a parametrically defined curve α and an implicity defined hypersurface F −1 (0) have contact of order k at α(t0 ) provided we have (F oα)(t0 ) = (F oα)′ (t0 ) = ... = (F oα)(k) (t0 ) = 0 but (F oα)(k+1) (t0 ) 6= 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 1. Let β(s∗ ) be a unit speed Smarandache curve of α(s) with first curvature k1β and second curvature k2β , then the centres of curvature spheres of β(s∗ )

12

MUHAMMED C ¸ ETIN, YILMAZ TUNC ¸ ER, AND MURAT KEMAL KARACAN

according to Bishop frame are k1β c(s∗ )

= β(s∗ ) +

+

∓

k2β

r

r2 (k1β )2 − (k2β )2 − 1 4(k1β )2

3k1β ± k2β

q r2 ((k1β )2 − (k2β )2 ) − 1 4k1β k2β

N1β

N2β .

Proof. Assume that c(s∗ ) is the centre of curvature sphere of β(s∗ ), then radius vector of curvature sphere is c − β = δ 1 Tβ + δ 2 N1β + δ 3 N2β .

Let we define the function F such that F = hc(s∗ ) − β(s∗ ), c(s∗ ) − β(s∗ )i − r2 .

First and second derivatives of F are as follow.

F ′ = −2δ1

F ′′ = −2(k1β δ 2 + k2β δ 3 − 1).

The condition of contact of second-order requires F = 0, F ′ = 0 and F ′′ = 0. Then, coefficients δ 1 , δ 2 , δ 3 are obtained δ1 δ2

δ3

= 0 =

=

k1β ∓ k2β

r

r2 (k1β )2 − (k2β )2 − 1 4(k1β )2

3k1β ± k2β

q r2 ((k1β )2 − (k2β )2 ) − 1 4k1β k2β

.

Thus, centres of the curvature spheres of Smarandache curve β are r

k1β ∓ k2β r2 (k1β )2 − (k2β )2 − 1 c(s∗ ) = β(s∗ ) + N1β 4(k1β )2 q 3k1β ± k2β r2 ((k1β )2 − (k2β )2 ) − 1 β + N2 .

4k1β k2β

δ 2 N1β

λN2β

is a line passing the point + Let δ 3 = λ, then c(s∗ ) = β(s∗ ) + β β ∗ β(s ) + δ 2 N1 and parallel to line N2 . Thus we have the following corollary.

Corollary 1. Let β(s∗ ) be a unit speed Smarandache curve of α(s) then each centres of the curvature spheres of β are on a line.

Theorem 2. Let β(s∗ ) be a unit speed Smarandache curve of α(s) with first curvature k1β and second curvature k2β , then the centres of osculating spheres of β(s∗ ) according to Bishop frame are c(s∗ ) = β(s∗ ) +

(k2β )′ (k1β )′ β β β ′ N1 − β ′ N2 . β 2 k2 β 2 k2 (k1 ) (k1 ) β β k1

k1

SMARANDACHE CURVES ACCORDING TO BISHOP FRAME IN EUCLIDEAN 3-SPACE 13

Proof. Assume that c(s∗ ) is the centre of osculating sphere of β(s∗ ), then radius vector of osculating sphere is c(s∗ ) − β(s∗ ) = δ 1 Tβ + δ 2 N1β + δ 3 N2β .

Let we define the function F such that F = hc(s∗ ) − β(s∗ ), c(s∗ ) − β(s∗ )i − r2 .

First, second and third derivatives of F are as follow.

F ′ = −2δ1 F ′′ = −2(k1β δ 2 + k2β δ 3 − 1)

F ′′′ = −2 (k1β )′ δ 2 − (k1β )2 δ 1 + (k2β )′ δ 3 − (k2β )2 δ 1 .

The condition of contact of third-order requires F = 0, F ′ = 0, F ′′ = 0 and F ′′′ = 0.

Then, coefficients δ 1 , δ 2 , δ 3 are obtained δ 1 = 0,

δ2 =

(k2β )′ β ′ , k (k1β )2 k2β

δ3 = −

1

(k1β )′ β ′ . k (k1β )2 k2β 1

Thus, centres of the osculating spheres of Smarandache curve β can be written as follow (k1β )′ (k2β )′ β β c(s∗ ) = β(s∗ ) + β ′ N1 − β ′ N2 . β 2 k2 β 2 k2 (k1 ) kβ (k1 ) kβ 1

1

Corollary 2. Let β(s∗ ) be a unit speed Smarandache curve of α(s) with first curvature k1β and second curvature k2β , then the radius of the osculating sphere of β(s∗ ) is r 2 2 (k1β )′ + (k2β )′ r(s∗ ) = . β ′ k (k1β )2 k2β 1

Example 1. Let α (t) be the Salkowski curve given by (4.1)

α (t) = (α1 (t) , α2 (t) , α3 (t))

where α1

=

α2

=

α3

=

and

1−n 1 1+n 1 √ − sin ((1 + 2n) t) − sin ((1 − 2n) t) − sin t 4 (1 + 2n) 4 (1 − 2n) 2 1 + m2

1−n 1 1+n 1 √ cos ((1 + 2n) t) + cos ((1 − 2n) t) + cos t 4 (1 − 2n) 2 1 + m2 4 (1 + 2n) cos (2nt) √ 4m 1 + m2 m n= √ .

1 + m2

14

MUHAMMED C ¸ ETIN, YILMAZ TUNC ¸ ER, AND MURAT KEMAL KARACAN

α (t) can be expressed with the arc lenght parameter.

We get unit speed regular curve √ by using the parametrization t = n1 arcsin n 1 + m2 s . Furthermore, graphics of √ special Smarandache curves are as follow. Here m = 3.

Figure 1 : Smarandache Curve Tα N1α

Figure 2 : Smarandache Curve Tα N2α

Figure 3 : Smarandache Curve N1α N2α

SMARANDACHE CURVES ACCORDING TO BISHOP FRAME IN EUCLIDEAN 3-SPACE 15

Figure 4 : Smarandache Curve Tα N1α N2α References [1] Ali, A. T., ”Special Smarandache Curves in the Euclidean Space”, International Journal of Mathematical Combinatorics, Vol.2, pp.30-36, 2010.

[2] Bishop, L.R., “There is more than one way to frame a curve”, Amer. Math. Monthly, Vol.82, Issue.3, pp.246-251, 1975.

[3] Bukcu, B. and Karacan, M.K., ”On The Slant Helices According to Bishop Frame”, International Journal of Computational and Mathematical Sciences, Vol 3, Number 2, Spring 2009.

[4] Bukcu, B. and Karacan, M.K., ”Special Bishop Motion and Bishop Darboux Rotation Axis of The Space Curve”, Journal of Dynamical Systems and Geometric Theories,Vol.6, 2008.

[5] Gray, A., ”Modern Differential Geometry of Curves and Surfaces with Mathematica”, 2nd edition, CRC Press, 1998.

[6] Hanson, A. J., and Ma, H., “Parallel Transport Approach to Curve Framing”, Indiana University, Techreports-TR425, January 11, 1995.

[7] Hanson, A. J., and Ma, H., “Quaternion Frame Approach to Streamline Visualization”, Ieee Transactions On Visualization And Computer Graphics, Vol.I, No.2, June 1995.

[8] Turgut, M. and Yilmaz, S., ”Smarandache Curves in Minkowski Space-time”, International Journal of Mathematical Combinatorics, Vol.3, pp.51-55, 2008.

[9] Turhan, E. and K¨ orpınar, T., ”Biharmonic Slant Helices According to Bishop Frame in E 3 ”, International Journal of Mathematical Combinatorics, Vol.3, pp.64-68, 2010.

[10] Yilmaz, S. and Turgut, M., ”On the Differential Geometry of the Curves in Minkowski Spacetime I”, Int. J. Contemp. Math. Sciences, Vol.3, No.27, pp.1343-1349, 2008.

Usak University, Instutition of Science and Technology, MSc Student, 1 Eylul Campus, 64200, Usak-TURKEY E-mail address: mat.mcetin@hotmail.com Usak University, Faculty of Sciences and Arts, Department of Mathematics, 1 Eylul Campus, 64200, Usak-TURKEY E-mail address: yilmaz.tuncer@usak.edu.tr Usak University, Faculty of Sciences and Arts, Department of Mathematics, 1 Eylul Campus, 64200 Usak-TURKEY E-mail address: murat.karacan@usak.edu.tr

This figure "fig_11.png" is available in "png" format from: http://arxiv.org/ps/1106.3202v1

This figure "fig22.png" is available in "png" format from: http://arxiv.org/ps/1106.3202v1

This figure "fig33.png" is available in "png" format from: http://arxiv.org/ps/1106.3202v1

This figure "fig44.png" is available in "png" format from: http://arxiv.org/ps/1106.3202v1