ON PARALLEL CURVES VIA PARALLEL TRANSPORT FRAME IN EUCLIDEAN 3-SPACE

AMO | Advanced Modeling and Optimization, Volume 18, Number 1, 2016

ON PARALLEL CURVES VIA PARALLEL TRANSPORT FRAME IN EUCLIDEAN 3-SPACE MUHAMMED T. SARIAYDIN AND VEDAT ASIL Abstract. In this paper, we study the parallel curve of a space curve according to parallel transport frame. Then, we obtain new results according to some cases of this curve by using parallel transport frame in Euclidean 3-space.

Additionally, we give new examples for this characterizations and we illustrate this examples in …gures.

1. Introduction To is a curve in plane, there exist two curves P+ = (s) + te2 (s) and P = (s) te2 (s) at a given distance t: But these cuves is not easy to characterize in 3-dimensional space. Then, [3] developed a new construction. This construction is carried over the three-dimensional space and as a result, two parallel curves are obtained as well. Additionally study parallel helices in three-dimensional space.

Bishop frame, which is also called alternetive or parallel frame of the curves, was introduced by L.R. Bishop in 1975 by means of parallel vector …elds. Recently, many research papers related to this concept have been treated in Euclidean space.

For example, in [13] the outhors introduced a new version of Bishop frame and an application to spherical images and in [14] the outhors studied Minkowski space in E31 .

In this paper, we obtain some characterizations about parallel curves by using Bishop frame in E3 . Additionally, we give new examples for this characterizations and we illustrate this examples in …gures.

2. Background on parallel curves In plane let a smooth curve (s) = (x(s); y(s)); where s is the arc-length and its unit tangent and unit normal vectors are e1 (s) and e2 (s) ;respectively: Then, we get P+ (S+ ) = (s) + te2 (s) and P (S ) = (s) te2 (s) ; 1991 Mathematics Subject Classi…cation. Primary 53B25; Secondary 53C40.

Key words and phrases. Bishop frame, Curves, Euclidean 3-space, Parallel curves. *AMO - Advanced Modeling and Optimization. ISSN: 1841-4311 65

66

M UHAM M ED T. SARIAYDIN AND VEDAT ASIL

where S = S (s) and S denotes the length along P ; at the distance t. Determining the length S , we can write dS =1 ds where

is the curvature of

t ;

(s) ; [3,11].

Lemma 2.1. Two curves ; : I ! E3 are parallel if their velocity vectors _ (s) and _ (s) are parallel for each s: In this case, if (s0 ) = (s0 ) for some one s0 in I then, = ; [11]: Theorem 2.2. If ; : I ! E3 are unit-speed curves such that = then, and are congruent, [11]:

=

Denote by fe1 ; e2 ; e3 g the moving Frenet–Serret frame along the curve space E3 . For an arbitrary curve with …rst and second curvature, and space E3 , the following Frenet–Serret formulae is given

and

in the in the

e_ 1 = e2 ; e_ 2 =

e 1 + e2 ;

e_ 3 =

e2 ;

where he1 ; e1 i = he2 ; e2 i = he3 ; e3 i = 1; he1 ; e2 i = he1 ; e3 i = he2 ; e3 i = 0: Here, curvature functions are de…ned by = (s) = ke_ 1 (s)k and (s) = Torsion of the curve is given by the aid of the mixed product ... ( _ ; •; ) : = 2

he2 ; e_ 3 i :

In the rest of the paper, we suppose everywhere 6= 0 and 6= 0.

The Bishop frame or parallel transport frame is an alternative approach to de…ning a moving frame that is well de…ned even when the curve has vanishing second derivative. One can express parallel transport of an orthonormal frame along a curve simply by parallel transporting each component of the frame. The tangent vector and any convenient arbitrary basis for the remainder of the frame are used.

The Bishop frame is expressed as "_1 =

1 "2

+

"_2 =

1 "1 ;

"_2 =

2 "1 ;

2 "3 ;

where (2.1) (2.2)

h"1 ; "1 i = h"2 ; "2 i = h"3 ; "3 i = 1;

h"1 ; "2 i = h"1 ; "3 i = h"2 ; "3 i = 0:

O N PARALLEL C U RVES VIA PARALLEL ...

Here, we shall call the set f"1 ; "2 ; "3 g as Bishop trihedra and curvatures. The relation matrix may be expressed as

67

1

and

2

as Bishop

e1 = " 1 ; e2 = cos (s)"2 + sin (s)"3 ; e3 = where (s) = arctan tures are de…ned by

2 1

,

sin (s)"2 + cos (s)"3 ; p 2+ (s) = _ (s) and (s) = 1 1

=

(s) cos (s) ;

2

=

(s) sin (s) :

2: 2

Here, Bishop curva-

On the other hand, " 1 = e1 ; (2.3)

"2 = cos (s)e2

sin (s)e3 ;

"3 = sin (s)e2 + cos (s)e3 :

3. Parallel Curves in 3-Dimensional Space Let a : I ! E3 be a regular curve with parametrized by arc-length. Take the derivative of (s) in accordance with s, we obtain (3.1)

( (s)

(3.2)

_ (s)( (s)

(3.3)

P)2 = t2 ; P) = 0;

P) + _ (s)2 = 0;

• (s)( (s)

for the point P. Equations (3.1), (3.2) and (3.3) have geometrical interpretation as follows: if (3.1) is regarded a spherical wave consisting of points P at the distance t from the moving point (s), then (3.2) is the enveloping surface (employing the intersection of two in…nitesimaly close waves) and (3.3) presents the double envelope (the intersection of three close waves, the multiple focus of the waves), [3].

Theorem 3.1. Let : I ! E3 be a regular curve with parametrized by arclength in 3-dimensional space. If P is a parallel curve of ; then P =

+(

1

1

2

2

tan 2 21 sec2

where C=

1 2C

)"2

s

4t2 sec2

(

2 tan + 2 1 sec2

1C

)"3 ;

4

2: 1

Proof. Considering the (2.2) and (3.2) equations, we obtain (3.4) where

P= "2 + "3 ; and

are appropriate coe¢ cients. From (3.3) and equations _ 2 = "21 = 1 and • = "_1 =

1 "2

+

2 "3 :

68

M UHAM M ED T. SARIAYDIN AND VEDAT ASIL

Then, we can write (

1 "2

+

2 "3 ) (

"2 + "3 ) + 1 = 0;

(3.5)

1

+

+ 1 = 0:

2

Also, if we write at (3.4) instead of the left side of (3.1) then, we get 2

(3.6)

2

+

= t2 :

Then from (3.5), we have (3.7)

1

=

tan :

1

Considering together (3.6) and (3.7) equations, we get (3.8)

1

2 tan + 2 1 sec2

=

where C=

1C

and

2

2 tan 2 1 sec2

=

s

1C

;

4

4t2 sec2

2: 1

In the rest of the paper, we suppose everywhere (3.9)

=

1:

And so, we get (3.10)

=

1

2

+

2

1

tan 2 21 sec2

1 2C

:

After simple computation, we get (3.11)

P =

where =(

1

2

1

Corallary 3.2. If

2

2

tan 2 21 sec2

+ "2 1 2C

"3 ;

) and

=(

2 tan 1C ): 2 1 sec2

= 0; then P =

where C=

1

+

C "3 ; 2

"2

1

s

4t2

4

2: 1

Example 3.3. Let us consider a unit speed circular helix in E3 by s s 7s = (s) = (24 cos ; 24 sin ; ): 25 25 25 One can calculate its Frenet-Serret apparatus as the following 1 s s e1 (s) = ( 24 sin ; 24 cos ; 7); 25 25 25 s s e2 (s) = ( cos ; sin ; 0); 25 25 1 s s e3 (s) = (7 sin ; 7 cos ; 24): 25 25 25

O N PARALLEL C U RVES VIA PARALLEL ...

Then, the curvatures of

69

is given by 24 ; 625 7 (s) = : 625 (s) =

Putting, (s) = where (s) =

Zs

7s ; 625

(s) ds, [8]. Then, we can write the Bishop frame from (2.3) by

0

s s 1 ( 24 sin ; 24 cos ; 7); 25 25 25 7s s 7 7s s "2 (s) = ( cos cos sin sin ; 625 25 25 625 25 7s s 7 7s s 24 7s cos sin + sin cos ; sin ); 625 25 25 625 25 25 625 s 7 7s s 7s cos + cos sin ; "3 (s) = ( sin 625 25 25 625 25 7s s 7 7s s 24 7s sin sin cos cos ; + cos ): 625 25 25 625 25 25 625 "1 (s) =

Also, the curvatures of 24 7s cos ; 625 625 7s 24 sin : 2 (s) = 625 625 1

(s) =

From (3.9), we get (3.12)

=

625 7s 1 7s sin + cos2 24 625 2 625

r

(4t2

390625 7s ) sec2 : 144 625

Therefore, by using (3.7) equation, we have (3.13)

=

7s 625 cos 24 625

1 7s 7s cos sin 2 625 625

s

4t2

390625 144

sec2

After simple computation, we get P = (24 cos

s s 7s ; 24 sin ; ) + "2 25 25 25

where 625 7s 1 7s 7s = cos + cos sin 24 625 2 625 625 and =

625 7s 1 7s sin + cos2 24 625 2 625

r

r

(4t2

(4t2

"3 ;

390625 7s ) sec2 144 625 390625 7s ) sec2 : 144 625

7s : 625

70

M UHAM M ED T. SARIAYDIN AND VEDAT ASIL

F igure 3:1; 0

s

=6

Example 3.4. Let us consider a unit speed curve in E3 by, [10], 9 sin 16s 208 The transformation matrix 2 3 2 T 1 4 N 5=4 0 B 0 =

where

(s) = (

1 sin 36s; 117 for the curve

9 1 6 cos 16s + cos 36s; sin 10s): 208 117 65 = (s) has the form 32 3 0 0 T cos(24 sin 10s) sin(24 sin 10s) 5 4 M1 5 ; sin(24 sin 10s) cos(24 sin 10s) M2

(s) =

Zs

24 cos(10s) =

24 sin 10s: 10

0

Then, we can give ‌gure of this curve as

F igure 3:2; 0

s

1

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71

References [1] V. Asil, Velocities of Dual Homothetic Exponential Motions in D3 , Iranian Journal of Science & Tecnology Transaction A: Science 31 (4) (2007), 265-271.

[2] L.R. Bishop, There is More Than One Way to Frame a Curve, Amer. Math. Monthly, 82 (3) (1975), 246-251.

[3] V. Chrastinova, Parallel Curves in Three-Dimensional Space, Sbornik 5. Konference o matematice a fyzice 2007, UNOB. [4] T. Körp¬nar, New characterization of b-m2 developable surfaces, Acta Scientiarum. Technology 37(2) (2015), 245–250 [5] T. Körp¬nar, E. Turhan, On Characterization of B-Canal Surfaces in Terms of Biharmonic B-Slant Helices According to Bishop Frame in Heisenberg Group Heis 3 , Journal of Mathematical Analysis and Applications 382 (1) (2011), 57-65.

[6] T. Körp¬nar, Faraday Tensor for Time-Smarandache TN Particles Around Biharmonic Particles and its Lorentz Transformations in Heisenberg Spacetime, Int J Theor Phys 53 (2014), 4153–4159 [7] T. Körp¬nar, New type surfaces in terms of B-Smarandache Curves in Sol 3 , Acta Scientiarum. Technology 37(2) (2015), 245–250 [8] T. Körp¬nar, A New Method for Inextensible Flows of Timelike Curves in Minkowski SpaceTime, International Journal of Partial Di¤erential Equations (2014), Article ID 517070.

[9] T. Körp¬nar, B-tubular surfaces in Lorentzian Heisenberg Group, Acta Scientiarum. Technology 37(1) (2015), 63–69 [10] T. Körp¬nar, New Characterizations for Minimizing Energy of Biharmonic Particles in Heisenberg Spacetime, Int. J. Theor. Phys., 53 (9) (2014), 3208–3218.

[11] B. O’Neil, Elementary Di¤ erential Geometry, Academic Press, New York, 1967.

[12] I.M. Yaglom, A. Shenitzer, A Simple Non-Euclidean Geometry and Its Pysical Basis, Springer-Verlag, New York, 1979.

[13] S. Y¬lmaz, M. Turgut, A New Version of Bishop Frame and An Aplication to Spherical Images , J. Math. Anal. Appl., 371 (2010), 764-776.

[14] S. Y¬lmaz, Poition Vectors of Some Special Space-like Curves According to Bishop Frame in Minkowski Space E31 , Sci Magna, 5 (1) (2010), 48-50.

[15] S. Y¬lmaz, E. Özy¬lmaz, M. Turgut, New Spherical Indicatrices and Their Characterizations, An. S ¸ t. Univ. Ovidius Constanta, 18(2) (2010), 337-354.

MuS¸ Alparslan University, Department of Mathematics,49250, MuS¸, Turkey, F¬rat University, Department of Mathematics,23119, Elaz¬G¼ , Turkey E-mail address : talatsar¬ayd¬n@gmail.com, vedatasil@gmail.com