Araştırma Makalesi / Research Article

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AKÜ FEMÜBİD 16 (2016)031301 (569-575) DOI:10.5578/fmbd.32172

AKU J. Sci. Eng. 16 (2016)031301 (569-575)

Araştırma Makalesi / Research Article

On Involutes of Order k of a Space-like Curve in Minkowski 4-space IE

₁⁴

Günay Öztürk

Kocaeli University, Art and Science Faculty, Department of Mathematics, Kocaeli, TURKEY e-posta: ogunay@kocaeli.edu.tr

Geliş Tarihi:25.04.2016 ; Kabul Tarihi:07.10.2016

Keywords Involute; Space-like Curve; W-curve; Helix.

Abstract

The orthogonal trajectories of the first tangents of a curve x are called the involutes of x. In this study, we give a characterization of involutes of order k of a space-like curve x with time-like principal normal in Minkowski 4-space IE . ₁⁴

4

IE

1

Minkowski 4-uzayında bir Space-like Eğrinin k’yinci Mertebeden İnvolütleri Üzerine

Anahtar kelimeler İnvolüt; Space-like Eğri;

W-eğrisi; Helis.

Özet

Bir x eğrisinin birinci teğetlerinin dik yörüngelerine eğrinin involütleri adı verilir. Bu çalışmada, IE ⁴₁ Minkowski 4-uzayında time-like asli normalli bir space-like eğrinin k’yinci mertebeden involütlerinin bir karakterizasyonunu verdik.

1. Introduction

4

IE Minkowski space-time 1 IE is a pseudo-⁴₁ Euclidean space IE provided with the standart flat ⁴ metric given by

gdx₁² dx²₂ dx₃² dx²₄, (1) where (x₁,x₂,x₃,x₄) is a rectangular coordinate system in IE₁⁴. Since g is an indefinite metric, recall that a vector v ∈IE⁴₁ can have one of the three causal characters; it can be space-like if g(v,v)0 or v0, time-like if g(v,v)0, and null (light-like) if g(v,v)0 and v0. Similarly, an arbitrary curve

) s ( x

x in IE can be locally space-like, time-like ₁⁴ or null if all of its velocity vectors x(s) are respectively space-like, time-like or null. Also, recall the norm of a vector v is given by v  g(v,v). Therefore, v is a unit vector if g(v,v)1. Next, vectors v, w in IE₁⁴ are said to be orthogonal if

0 ) w , v (

g  . The velocity of the curve x(s) is given by x(s) . Space-like or time-like curve x(s) is said

to be parametrized by arc-length function s, if 1

)) s ( x ), s ( x (

g    (O’Neill, 1983).

Let x(s) be a space-like curve with a time- like principal normal in the space-time IE⁴₁, parametrized by arc-length function s. Then we have the following Frenet equations (Walfare, 1995):











































4 3 2 1

3 3 2

2 1

1

4 3 2 1

V V V V

0 k 0 0

k 0 k 0

0 k 0 k

0 0 k 0

V V V V

, (2)

where V , ₁ V , ₂ V₃ and V are the Frenet vectors ₄ satisfy the equations:

g(V₁,V₁)g(V₃,V₃)g(V₄,V₄)1, g(V₂,V₂)1. Here k , ₁ k , ₂ k₃ are respectively, the first, the second and the third curvatures of the curve x(s).

Definition 1. (Yılmaz and Turgut, 2008) Let )

a , a , a , a (

a ₁ ₂ ₃ ₄ , b(b₁,b₂,b₃,b₄) and )

c , c , c , c (

c ₁ ₂ ₃ ₄ be vectors in IE⁴₁. The vector

Afyon Kocatepe University Journal of Science and Engineering

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AKÜ FEMÜBİD 16 (2016) 031301 570

product in Minkowski space-time IE is defined by ⁴₁ the determinant

4 3 2 1

c c c c

b b b b

a a a a

e e e e c

b a







 , (3)

where e , ₁ e , ₂ e₃ and e are mutually ₄ orthogonal vectors satisfying the equations

4 3 2

1 e e e

e    , e₂e₃e₄ e₁,

2 1 4

3 e e e

e    , e₄e₁e₂ e₃.

Let x(s) be a space-like curve in IE . The ⁴₁ Frenet frame vectors V , ₁ V , ₂ V₃, V and Frenet ₄ curvatures k₁, k₂, k₃ are given by

) s ( x

) s ( ) x s ( V₁



  ,

) s ( x ) s ( x ) s ( x

) s ( x ) s ( x ) s ( ) x s (

V4   





 

∧

∧ ,

) s ( x ) s ( x V

) s ( x ) s ( x ) V

s ( V

4 4

3    

 

 

 , (4)

) s ( x V V

) s ( x V ) V

s ( V

4 3

2   

 

 

and

2 2

1 x(s)

)) s ( x ), s ( V ( ) g s (

k 

  _,

) s ( k ) s ( x

)) s ( x ), s ( V ( ) g s ( k

1 3 3

2 

  , (5)

) s ( k ) s ( k ) s ( x

)) s ( x ), s ( V ( ) g s ( k

2 1 4

) ıv ( 4

3   ,

respectively, where  is vector product in IE ⁴₁ (Gluck, 1966).

A curve which has constant first Frenet curvature IE is called a Salkowski curve ₁⁴ (Salkowski, 1909). ( or T.C-curve (Kılıç et al. 2008)).

An arbitrary curve is called W-curve or (circular) helix if it has constant Frenet curvatures (Klein and

Lie, 1871). Meanwhile, a curve with constant curvature ratios IE is called a ccr-curve ₁⁴ (Monterde, 2007), (Öztürk et al. 2008).

In (Öztürk et al.) (2016), the authors gave a characterization of involutes of order k of a given curve in IE . They obtain some results about the ⁿ involutes of order 1, 2, 3 of a given curve in IE , ³ IE⁴ , respectively.

In the present study, we give a characterization of involutes of order k of a space- like curve x in Minkowski space-time IE . ₁⁴

2. Involute curves of order k

Definition 2. Let x(s) be a regular space-like curve in IE given with arc-length parameter s. Then the ⁴₁ curves which are orthogonal to the system of k- dimensional osculating hyperplanes of x are called the involutes of k (or k involute) of the curve x ^th (Balazenka and Zeljka 1999). For simplicity, we call the involutes of order 1, the involute of the given curve.

In order to find the parametrization of involutes x(s) of order k of the curve x in IE , we ₁⁴ put











 ^k

1 i

i

i(s)V(s), k 3 )

s ( x ) s (

x , (6)

where _i is a differentiable function and s, which is not necessarily an arc-length parameter, is the parameter of x(s).

Furthermore, the involutes

x

of order k of the curve x in IE are detemined by ₁⁴

. 3 k i 1 , 0 )) s ( V ), s ( x (

g  _i  ≤ ≤ ≤ (7)

2.1. Involute curves of order 1

Theorem 1. Let x(s) be a space-like curve with time-like principal normal in IE given with the ⁴₁

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AKÜ FEMÜBİD 16 (2016) 031301 571

Frenet curvatures k , ₁ k , ₂ k₃. Then, the involute

x

of the curve x is a time-like curve with the Frenet frame vectors V₁, V₂, V₃, V₄ and Frenet curvatures k₁, k₂, k₃ which are given by

2 1(s) V V  ,

2 2 2 1

3 2 1 1

2 k k

V k V ) k s (

V 

  ,

























 

4 2 2 2 1

3 1 2 1

1 1 2 2

2 2 2 1 3

V ) k k ( D

V ) C k A k ( k

V ) C k A k ( k k k W ) 1 s (

V _{, (8)}



₂ ₁ ₁ ₃ ₂ ₁ ₄



4 k DV k DV (k A k C)V

W ) 1 s (

V      ,

and



 ²¹ ²²

1

k ) k

s (

k ,

) k k ( ) W s (

k ₂

2 2 1 2 2





 , (9)

_



 





 





 





 

 ₂

3 1 2 1

3 1 2 2

2 2 2 1

3 D(k C k A) k k D

) D C k )(

C k A k ( W

k ) k

s ( k

respectively, where k1

) s c ( 



 ,







k₁ 2k₁

A ,





 





k²₁ k²₂

B ,









k₂ 2k₂

C ,



k₂k₃

D ,

and

. ) k k k k ( ) k k ( k k

) C k A k ( ) k k ( D W

2 2 1 2 1 2 2 2 1 2 3 2 2

2 1 2 2 2 2 1 2

 

 











 (10)

Proof. Let x(s)be the involute of a space-like curve x with time-like principal normal in IE . Then by ₁⁴ the use of (6) with (7), we get 1₁(s)0, and furthermore ₁(s)(cs) for some constant c. We have the following parametrization

) s ( V ) s c ( ) s ( x ) s (

x    ₁ (11) Further, differentiating the equation (11), we find

V2

) s (

x  ,

3 2 2 1

1V V k V

k ) s (

x    , (12)

4 3 2 3 2 2

2 2 2 2

1 1 1 1

V k k V ) k 2 k (

V ) k k

( V ) k 2 k ( ) s ( x





























 

where (s)₁(s)k₁(s) is a differentiable function. Substituting







k₁ 2k₁

A ,





 





k²₁ k²₂

B ,







k₂ 2k₂

C ,



k₂k₃

D .

in the last equation, we obtain

4 3 2

1 BV CV DV

AV ) s (

x      .

Furthermore, differentiating x (s) with respect to s, we get

3 3

3 3 2

2 2 1 1

1 )

ıv (

V ) D C k ( V ) D k B k C (

V ) C k A k B ( V ) B k A ( ) s ( x

 

















By the use of (12), we find

2 1(s) V V  .

While g(V₂,V₂)1, we can write 1

) V , V (

g ₁ ₁  which implies that involute

x

is a time-like curve.

Then we can compute the vector form )

s ( x ) s ( x ) s (

x     and V₄ of

x

as in the following:



 









 

 

 

 

4 1 2

3 1 1 2 2

V ) C k A k (

DV k DV ) k

s ( x ) s ( x ) s ( x and



₂ ₁ ₁ ₃ ₂ ₁ ₄



4

V ) C k A k ( DV k DV W k

1

) s ( x ) s ( x ) s ( x

) s ( x ) s ( x ) s ( ) x s ( V









 

 



 

 

 

where

W D²(k²₁k²₂)(k₂Ak₁C)² . Similarly, we can compute

























 

 

 



4 2 2 2 1

3 1 2 1

1 1 2 2 2

4

V ) k k ( D

V ) C k A k ( k

V ) C k A k ( k ) W

s ( x ) s ( x V

and

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AKÜ FEMÜBİD 16 (2016) 031301 572

























 

 

 

 

 



4 2 2 2 1

3 1 2 1

1 1 2 2

2 2 2 1 4 4 3

V ) k k ( D

V ) C k A k ( k

V ) C k A k ( k k k W

1

) s ( x ) s ( x V

) s ( x ) s ( x V V

Finally, if we calculate V3V4x(s) and substitute in (4), we get

2 2 2 1

3 2 1 1

2 k k

V k V ) k s (

V 

  .

Consequently, an easy calculation gives g(V₂(s),x(s)) k₁²k²₂ _,

2 2 2 1

3 k k

)) W s ( x ), s ( V (

g    , (13)





 





 





 

  ₂

3 1 2 1

3 1 ) 2

ıv (

4 D(k C k A) k k D

) D C k )(

C k A k ( W )) 1 s ( x ), s ( V (

g .

Hence, from equations (13) and (5), we get (9), which completes the proof.

For the case x is a W-curve, one can get the following results.

Corollary 1 Let x(s) be a space-like curve with time-like principal normal in IE given with the ₁⁴ Frenet curvatures k , ₁ k , ₂ k₃. If x is a W-curve, then the Frenet frame vectors V₁, V₂, V₃, V₄ and Frenet curvatures k₁, k₂, k₃ of the involute

x

of the curve x are given by

2 1(s) V V  ,

2 2 2 1

3 2 1 1

2 k k

V k V ) k s (

V 

  ,

4 3(s) V

V  , (14)

2 2 2 1

3 1 1 2

4 k k

V k V ) k s (

V 



 ,

and

1 2 2 2 1 1 (c s)k

k ) k

s (

k 

  ,

) k k ( k ) s c (

k ) k

s (

k ₂

2 2 1 1

3 2

2    , (15)

2 2 2 1 3

3 (c s) k k

) k s (

k    ,

respectively (Turgut et al. 2010).

Corollary 2. Let x(s) be an involute of a space-like curve x with time-like principal normal in IE given ₁⁴ with the Frenet curvatures k₁,k₂,k₃. If x is a W- curve, then

x

becomes a ccr-curve.

2.2. Involute of order 2

An involute of order 2 of a space-like curve x in IE has the parametrization ₁⁴

) s ( V ) s ( ) s ( V ) s ( ) s ( x ) s (

x  ₁ ₁ ₂ ₂ (16) where ₁, ₂ are differential functions satisfying

) s ( k ) s ( 1 ) s

( ₂ ₁

1  



) s ( k ) s ( )

s

( ₁ ₁

2 

 . (17)

From the differentiable equation system (17), we get the following result.

Corollary 3. Let xx(s) be a space-like Salkowski curve with time-like principal normal in IE . Then ⁴₁ the involute

x

of order 2 of the curve x has the parametrization (16) given with the coefficient functions

) s k sinh(

c ) s k cosh(

c ) s

( ₁ ₁ ₂ ₁

1  



1 1 2

1 1

2 k

) 1 s k cos h(

c ) s k s i nh(

c ) s

(   



where c and ₁ c are real constants. ₂

Theorem 2. Let xx(s) be a space-like curve with time-like principal normal in IE given with Frenet ₁⁴ curvatures k , ₁ k , ₂ k₃. Then the involute

x

of order 2 of the curve x is a space-like curve with the Frenet frame vectors V₁, V₂, V₃, V₄ and Frenet curvatures k₁, k₂, k₃ which are given by

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AKÜ FEMÜBİD 16 (2016) 031301 573

V₁(s)V₃,

2 2 2 3

4 3 2 2

2 k k

V k V ) k s (

V 

  , (18)



 







 

4 3 2 2

2 3 2 3 1 2 2 2 3 2 2 2 3

3 k (kN k L)V

V ) L k N k ( k V ) k k ( K k k W ) 1 s (

V

₄



(k₂N k₃L)V₁ k₃KV₂ k₂KV₄



W ) 1 s (

V     ,

and



 

2 2 2 3 1

k ) k

s (

k ,

) k k ( ) W s (

k ₂

2 2 3

2 2  , (19)



















 

 







 



2 3 1

2 3

1 3

2

2 2 2 2 3 3

K k k

) N k L k ( K

) L k K )(

L k N k ( W

k ) k

s (

k _,

where

2 2k





 ,



k₁k₂

K ,









k₂ 2k₂

L ,







k₃ 2k₃

N ,

and

. ) k k ( k k ) k k k k (

) k k ( K ) L k N k ( W

2 2 2 3 2 2 2 1 2 3 2 3 2

2 2 2 3 2 2 3 2



 









Proof. Let xx(s) be the involute of order 2 of a space-like curve with time-like principal normal in

4

IE . Then by the use of (16) with (7), we get 1

V3

) s (

x  , (20) where (s)₂(s)k₂(s) is a differentiable function.

3 1(s) V

V  .

While g(V3,V3)1, we can write g(V₁,V₁)1 which implies that involute

x

of order 2 is a space- like curve.

Further, the differentiation of (20) implies that x(s)k₂V₂V₃k₄V₄, (21)

. V ) k 2 k ( V ) k k

(

V ) k 2 k ( V k k ) s ( x

4 3 3 3 2 3 2

2

2 2 2 1 2 1













 

















 

Consequently, substituting



k₁k₂

K ,







k₂ 2k₂

L ,





 





k²₂ k²₃

M ,









 k₃ 2k₃

N ,

in the last vector, we obtain

. NV MV LV KV ) s (

x   ₁ ₂  ₃ ₄ (22) Furthermore, differentiating

x 

with respect to s, we get

4 3 3

3 2

2 2 1 1

1 )

ıv (

V ) M k N ( V ) N k L k M (

V ) M k K k L ( V ) L k K ( ) s ( x

















 (23)

Hence substituting (20)-(23) into (4) and (5), after making some calculations as in the previous theorem, we obtain the result.

For the case x is a W-curve, one can get the following result.

Corollary 4. Let

x

be an involute of order 2 of a space-like curve with time-like principal normal in

4

IE given with the Frenet curvatures 1 k₁,k₂,k₃. If x is a W-curve, then the Frenet frame vectors V₁, V2, V₃, V₄ and Frenet curvatures k₁, k₂, k₃ of the involute

x

of order 2 of the curve x are given by

3 1(s) V

V  ,

2 2 2 3

4 3 2 2

2 k k

V k V ) k s ( V

-

  ,

1 3(s) V

V  , (24)

2 2 2 3

4 2 2 3

4 k k

V k V ) k s (

V 

  ,

and



 

2 2 2 3 1

k ) k

s (

k ,

2 2 2 3

2 1

2 k k

k ) k

s (

k   , (25)

(6)

AKÜ FEMÜBİD 16 (2016) 031301 574

2 2 2 3

3 1

3 k k

k ) k

s (

k   ,

where (s)₂(s)k₂(s).

Corollary 5. Let

x

be an involute of order 2 of a space-like curve x with time-like principal normal in

4

IE given with the Frenet curvatures 1 k₁,k₂,k₃. If x is a W-curve, then

x

becomes a ccr-curve.

2.3. Involute of order 3

An involute of order 3 of a space-like curve x in IE has the parametrization ₁⁴

) s ( V ) s ( ) s ( V ) s ( ) s ( V ) s ( ) s ( x ) s (

x  1 1 2 2 3 3 (26)

where ₁, ₂, ₃ are differentiable functions satisfying

) s ( k ) s ( 1 ) s

( ₂ ₁

1  

 ,

) s ( k ) s ( ) s ( k ) s ( )

s

( ₁ ₁ ₃ ₂

2  

 , (27)

) s ( k ) s ( )

s

( ₂ ₂

3 

 .

By solving the differential equation system (27), we get the following result.

Corollary 6. Let xx(s) be a space-like W-curve with time-like principal normal in IE . Then the ₁⁴ involute

x

of order 3 of the curve x has the parametrization (26) given with the coefficient functions

2 1 2 2 2

3 1

1 c

k s k k

)) ks sin(

c ) ks cos(

c ( ) k s

(   



 ,

2 1 3

2

2 k

) k ks sin(

c ) ks cos(

c ) s

(   

 ,

2 1 1 2

2 1 2

3 2

3 k

k c k

s k k k

)) ks sin(

c ) ks cos(

c ( ) k s

(   



 ,

where k k₁k₂ _, _{c ,}₁ _{c and}₂ _c₃ are real constants.

Theorem 3. Let xx(s) be a space-like curve with time-like principal normal in IE given with Frenet ₁⁴ curvatures k , ₁ k and ₂ k₃. Then the involute

x

of order 3 of the curve x is a space-like curve with the

Frenet frame vectors V₁, V₂, V₃, V₄ and Frenet curvatures k₁, k₂, k₃ which are given by

4 1(s) V V  ,

3 2(s) V

V  , (28)

2 3(s) V

V  ,

1 4(s) V V  and

 ³

1

) k s (

k ,



 ²

2

) k s (

k , (29)



 ¹

3

) k s (

k , where (s)₃(s)k₃(s).

Proof. Let xx(s) be the involute of order 3 of a space-like curve with time-like principal normal in

4

IE . Then by the use of (26) with (7), we get 1

V4

) s (

x  (30) where (s)₃(s)k₃(s) is a differentiable function.

4 1(s) V V  .

While g(V₄,V₄)1, we can write g(V₁,V₁)1 which implies that involute

x

of order 3 is a space- like curve.

Further, the differentiation of (29) implies that

4 3

3 V V

k ) s (

x    ,

. V ) k (

V ) k 2 k ( V k k ) s ( x

4 2 3

3 3 3 2 3 2





 















 

(31) Consequently, substituting





 k₂k₃ E

) k 2 k (

F 3 3





 

 k²₃ G

in the last vector, we obtain . GV FV EV ) s (

x   ₂ ₃ ₄ (32) Furthermore, differentiating

x 

with respect to s, we get

4 3 3

3 2

2 2 1

1 ) ıv (

V ) F k G ( V ) G k E k F (

V ) F k E ( EV k ) s ( x















 (33)

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AKÜ FEMÜBİD 16 (2016) 031301 575

Hence substituting (30)-(33) into (4) and (5), after some calculations as in the previous theorem, we obtain the result.

Corollary 7. The ivolute

x

of order 3 of a space-like ccr-curve x with time-like principal normal in IE is ₁⁴ also a ccr-curve in IE . ₁⁴

3. Conclusion

In recent years, many authors have studied with the involute-evolute curve couples in many paper. Turgut et al. (2010) gave the characterization of the involute of order 1 (involute) of a W-curve in IE ₁⁴

In this paper, we study involute curves of order k of a space-like curve x with time-like principal normal in Minkowski 4-space IE . First, ⁴₁ we investigate an involute curve of order 1 of a given curve. Furthermore, we give the characterizations of the involutes of order 2 and 3.

We obtain the Frenet Frame and Frenet curvatures of the involutes of order k of the curve with respect to the Frenet Frame and Frenet curvatures of the given curve.

Nowadays, as known W-curve (or helix) is very important topic in curve theory, we characterize the involutes of order k of a W-curve in IE . ₁⁴

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