NC− Smarandache Curves of Mannheim Curve Couple According to Frenet Frame

International J.Math. Combin. Vol.1(2015), 1-13

N C Smarandache Curves of Mannheim Curve CoupleAccording to Frenet Frame

Suleyman S ENYURT and Abdussamet C ALISKAN(Faculty of Arts and Sciences, Department of Mathematics, Ordu University, 52100, Ordu/Turkey)E-mail: [email protected]

Abstract: In this paper, when the unit Darboux vector of the partner curve of Mannheimcurve are taken as the position vectors, the curvature and the torsion of Smarandache curveare calculated. These values are expressed depending upon the Mannheim curve. Besides,we illustrate example of our main results.

Key Words: Mannheim curve, Mannheim partner curve, Smarandache Curves, Frenetinvariants.

AMS(2010): 53A04

1. IntroductionA regular curve in Minkowski space-time, whose position vector is composed by Frenet framevectors on another regular curve, is called a Smarandache curve ([12]). Special Smarandachecurves have been studied by some authors .Melih Turgut and Suha Ylmaz studied a special case of such curves and called it Smarandache T B2 curves in the space E14 ([12]). Ahmad T.Ali studied some special Smarandache curvesin the Euclidean space. He studied Frenet-Serret invariants of a special case ([1]). MuhammedC etin , Ylmaz Tuncer and Kemal Karacan investigated special Smarandache curves accordingto Bishop frame in Euclidean 3-Space and they gave some differential goematric properties ofSmarandache curves, also they found the centers of the osculating spheres and curvature spheresof Smarandache curves ([5]). Senyurt and C alskan investigated special Smarandache curves interms of Sabban frame of spherical indicatrix curves and they gave some characterization ofSmarandache curves ([4]). OzcanBektas and Salim Yuce studied some special Smarandachecurves according to Darboux Frame in E 3 ([2]). Nurten Bayrak, OzcanBektas and Salim Yuce3studied some special Smarandache curves in E1 [3]. Kemal Tas.kopru, Murat Tosun studiedspecial Smarandache curves according to Sabban frame on S 2 ([11]).In this paper, special Smarandache curve belonging to Mannheim partner curve suchas N C drawn by Frenet frame are defined and some related results are given.1 Received

September 8, 2014, Accepted February 12, 2015.

2

Suleyman S ENYURT and Abdussamet C ALIS KAN

2. Preliminaries

The Euclidean 3-space E 3 be inner product given byh, i = x21 + x32 + x23where (x1 , x2 , x3 ) E 3 . Let : I E 3 be a unit speed curve denote by {T, N, B} themoving Frenet frame . For an arbitrary curve E 3 , with first and second curvature, and respectively, the Frenet formulae is given by ([6], [9])T = N

N = T + B

(2.1)

B = N.

For any unit speed : I E3 , the vector W is called Darboux vector defined byW = (s)T (s) + (s) + B(s).If consider the normalization of the Darboux C =

cos =

1W , we havekW k

(s)(s), sin =,kW kkW k

C = sin T (s) + cos B(s)

(2.2)

where (W, B) = . Let : I E3 and : I E3 be the C 2 class differentiable unitspeed two curves and let {T (s), N (s), B(s)} and {T (s), N (s), B (s)} be the Frenet frames ofthe curves and , respectively. If the principal normal vector N of the curve is linearlydependent on the binormal vector B of the curve , then () is called a Mannheim curve and( ) a Mannheim partner curve of (). The pair (, ) is said to be Mannheim pair ([7], [8]).The relations between the Frenet frames {T (s), N (s), B(s)} and {T (s), N (s), B (s)} are asfollows:T = cos T sin B

where (T, T ) = ([8]).

N = sin T + cos B

(2.3)

B = Ncos = dsdsdssin = ds

(2.4).

Theorem 2.1([7]) The distance between corresponding points of the Mannheim partner curvesin E3 is constant.

N C Smarandache Curves of Mannheim Curve Couple According to Frenet Frame

3

Theorem 2.2 Let (, ) be a Mannheim pair curves in E3 . For the curvatures and the torsionsof the Mannheim curve pair (, ) we have,ds = sin ds

= cos dsds

and

d= =ds 2 + 2 = ( sin cos ) dsds

(2.5)

(2.6)

Theorem 2.3 Let (, ) be a Mannheim pair curves in E3 . For the torsions of theMannheim partner curve we have =Theorem 2.4([10]) Let (, ) be a Mannheim pair curves in E3 . For the vector C is thedirection of the Mannheim partner curve we have1kW kC = sC+sN 2 21+1+kW kkW k

(2.7)

where the vector C is the direction of the Darboux vector W of the Mannheim curve .

3. N C Smarandache Curves of Mannheim Curve Couple According toFrenet FrameLet (, ) be a Mannheim pair curves in E 3 and {T N B } be the Frenet frame of theMannheim partner curve at (s). In this case, N C - Smarandache curve can be definedby11 (s) = (N + C ).(3.1)2Solving the above equation by substitution of N and C from (2.3) and (2.7), we obtain

1 (s) =

cos kW k + sin

qq

2 + kW k2 T + N + cos 2 + kW k2 sin kW k Bq. 2 + kW k2(3.2)

4


The derivative of this equation with respect to s is as follows,

T1 (s) =

kW k

2 +kW k2

cos

cos kW k

vu 2 2u +kW k2kW ku+u 2 +kW k2ut+vu ut

kW k

2 +kW k2

2

( 2 +kW k2 ) kW k

sin kW k

2 +kW k2

T+

+

"

2 +kW k2

2

sin B

2 +kW k22

( +kW k ) kW k

kW k

kW k

2

kW k

"

kW k

kW k

N

kW k

2 +kW k2

2

kW k

2 +kW k2

1

#

1

#

In order to determine the first curvature and the principal normal of the curve 1 (s), weformalizei h2 (r1 cos + r2 sin )T + r3 N + (r1 sin + r2 cos )BT 1 (s) =

kW k

2 +kW k2

2 +kW k2

2

+

( 2 +kW k2 ) kW k

"

kW k

2

kW k

2 +kW k2

1

#!2

where

r1

=

2 kW kq2 2 + kW k2

q! 2 + kW k2

q 2 + kW k2

q 2 + kW k2

q 2 + kW k2

kW kkW kqq22 + kW k2 + kW k2q"#

2 + kW k22kW kkW kqqq 2 + kW k2 2 + kW k2 2 + kW k2q!

2 + kW k2 kW kkW kqqq 2 + kW k2 2 + kW k2 2 + kW k2q 2 2 2 + kW k2 4 kW kkW kqq kW k 2 + kW k2 2 + kW k2

kW k

(3.3)


q 2 + kW k2 2

q 2 + kW k2 3

kW kq 2 + kW k2q3

2 + kW k2

kW kkW kqqq+222222 + kW k + kW k + kW k2q2

2 + kW k2 22kW kkW kkW kqqq kW k 2 + kW k2 2 + kW k2 2 + kW k2q

2 + kW k2 kW kqq22 kW k kW k 2 + kW k2 2 + kW k2q

2 + kW k2 kW kkW kqq 2 + kW k2 2 + kW k2q

2 + kW k2 kW kkW kqqq+ kW k 2 + kW k2 2 + kW k2 2 + kW k2qq 2 2 + kW k2 2 2 + kW k2 kW kqq kW k 2 + kW k2 2 + kW k2q 2 + kW k2 2

kW kkW kqqq+ 2 + kW k2 2 + kW k2 2 + kW k2qq! 2 + kW k2

2 + kW k2 kW kkW kqq kW k 2 + kW k2 2 + kW k2q!

2 + kW k2 kW kq+ kW k kW k 2 + kW k2q

2 + kW k2 kW kkW kkW kqq q, kW k 2 + kW k2 2 + kW k2 2 + kW k2

r2

=

kW kq+2 kW k 2 + kW k2

5

q3

2 + kW k2

3 kW kqq+3 kW k kW k 2 + kW k2 2 + kW k2q

2 + kW k2

kW k 2 qq+3 kW k 2 + kW k2 2 + kW k2

6


kW kq 2 + kW k2

q 2 + kW k2

2q2 kW k 2 + kW k2

kW kq 2 + kW k2

q 2 + kW k2 2

kW k

q 2 + kW k2 2

q 2 + kW k2

2 2q kW k 2 + kW k2

2 kW kq 2 + kW k2

q2 2 + kW k2

q

2 + kW k2

3 kW kkW kqq+3+ 2 + kW k2 2 + kW k2q

2 + kW k2 3

kW kkW k 2 kW kqqq2 2 + kW k2 2 + kW k2 2 + kW k2q2 2 + kW k2 2 kW kkW kqq4 kW k 2 + kW k2 2 + kW k2q

4 2 2 + kW k2 2 kW kqq2 kW k kW k 2 + kW k2 2 + kW k2q

2 + kW k2 2

2kW kkW kqq+3 kW k 2 + kW k2 2 + kW k2

r3

q

2 + kW k2 2 2

kW kq= 2+2 kW k kW k 2 + kW k2q

2 2 + kW k2

kW kqq kW k 2 + kW k2 2 + kW k2q

2 + kW k2 kW kkW kkW kqqq+ kW k 2 + kW k2 2 + kW k2 2 + kW k2qq 2 + kW k2

2 + kW k2 kW kkW kqqq 2 + kW k2 2 + kW k2 2 + kW k2qq

2 + kW k2 2 + kW k2 kW kkW kqq+ 2 + kW k2 2 + kW k2

7


q 2 + kW k2 4

kW kqq 2 + kW k2 2 + kW k2q 2 2 + kW k2 2

kW kqq kW k 2 + kW k2 2 + kW k2q

2 2 + kW k2 2 kW kkW k 2 kW kqqq 2 + kW k2 2 + kW k2 2 + kW k2qq

2 + kW k2 2 2 + kW k2 3kW kqq+2 kW k 2 + kW k2 2 + kW k2q

2 + kW k2 3 2

kW kkW kqqq+2 2 + kW k2 2 + kW k2 2 + kW k2q

2 + kW k2 kW kkW kqqq 2 + kW k2 2 + kW k2 2 + kW k2q

2 + kW k2 kW kq+ kW k kW k kW k 2 + kW k2q 2 + kW k2 kW kqq+ kW k 2 + kW k2 2 + kW k2q 2 + kW k2 kW kqq+ kW k kW k 2 + kW k2 2 + kW k2qq

2 + kW k2 2 + kW k2 2kW kkW kqq kW k 2 + kW k2 2 + kW k2

kW kqq 2 + kW k2 2 + kW k2kW k q 2 + kW k2

2

The first curvature is 1 =

kW k

2 +kW k

p2( r1 2 + r2 2 + r3 2 )"2 2

+kW k2( 2 +kW k2 )+2 kW k kW k2

kW k

2 +kW k2

1

#!2

8


The principal normal vector field and the binormal vector field are respectively given byN1 =

(r1 cos + r2 sin )T + r3 N + (r1 sin + r2 cos )Bp,r1 2 + r2 2 + r3 2B1 (s) =

123T + N + B,444

(3.4)

(3.5)

where1234

kk= r2 cos 2kW k 2 kWrcosr1 r1 2kW k 2 kW2 +kW k +kW k

r3 2kW k 2 + r3 kWsin k +kW k

= r1

kW k

2 +kW k2

kW k

kW kkW k+rr= r2 sin 211 +kW k2 2 +kW k2

r3 2kW k 2 + r3 kWk cos kW k

kW k

+kW k

vu

2 2u2 +kW k2k2 )u (r1 2 + r2 2 + r3 2 ) kW k+ (r1 2 + r2 2 + r3 2 ) (+kWukW k2 +kW k2u#!=u "

ukWk1t2 . kW k

2 +kW k2

In order to calculate the torsion of the curve 1 , we differentiate

1

=

"

h

q 2 + kW k2 i

kW kqq 2 + kW k2 2 + kW k2q2

2 + kW k2 kW kkW kq q+ kW k 2 + kW k2 2 + kW k2q

2 + kW k2 kW kqq+ sin + kW k22 + kW k2 + kW k2q!#

2 + kW k2 2kW kkW k 2qqT kW k 2 + kW k2 2 + kW k2q"

2 + kW k2

kW kkW kq+ q22 + kW k2 + kW k212

cos


"

kW k

q 2 + kW k2

h

q2 2 + kW k2

#

qN 2 + kW k2

q 2 + kW k2 i

kW kqq 2 + kW k2 2 + kW k2q2

2 + kW k2 kW kkW kq q+ kW k 2 + kW k2 2 + kW k2q

2 + kW k2 kW kqq++ cos kW k 2 + kW k2 2 + kW k2q!# !

2 2 + kW k2 kW k 2kW kqqB . kW k 2 + kW k2 2 + kW k2 sin

and thus1 =

(t1 cos + t2 sin + t3 )T + t3 N + (t2 cos t1 sin + t3 )T,2

where

t1

=

t2

q 2 + kW k2

kW kqq3 2 + kW k2 2 + kW k2qq 2 + kW k2 2 + kW k2 kW kkW kqq 2 + kW k2 2 + kW k2q

2 + kW k2 3 kW kq q kW k 2 + kW k2 2 + kW k2q 2

2 + kW k2 kW kqq kW k kW k 2 + kW k2 2 + kW k2q

3 2 + kW k2 kW kkW k2qq++kWkkWk22 + kW k2 + kW k2

=

kW k

q 2 + kW k2

kW k

q 2 + kW k2

kW kqq2 kW k 2 + kW k2 2 + kW k2q

2 + kW k2 2kW kkW kqq3 kW k kW k 2 + kW k2 2 + kW k22

9

10


+

kW k

q 2 + kW k2

kW kq 2 + kW k2q 2 + kW k2

q+ 2 + kW k2

kW kqqq+2 2 + kW k2 2 + kW k2 2 + kW k2qq

2 + kW k2 2 + kW k2 2kW kkW kqq+2 2 + kW k2 2 + kW k2kW k

kW k

q3 2 + kW k2

t3

=

q

2 + kW k2

kW kkW kqq3 kW k 2 + kW k2 2 + kW k2qq! 2 + kW k2 2 + kW k2 kW k 2qq+ 2 + kW k2 2 + kW k2qq

2 + kW k2 2 + kW k2 kW kkW kq q 2 + kW k2 2 + kW k2q!

2 + kW k2 3 2 kW kkW kq+ q kW k 2 + kW k2 2 + kW k2

3 +

The torsion is then given by1 =

det(1 , 1 , 1 ),k1 1 k21 =

122

where

2 kW k 2 kW k2 2 kW k kW kp 2p 21 = 2t1+ t1t12 kW k + kW k2 + kW k2p 2

32

kW kkW k kW k + kW k2p 2p+t+t22 kW k 2 + kW k2 + kW k2 2 + kW k2


11

2 t3

2 kW kkW k 2pp2tt33222222kWkkWk + kW k + kW k + kW kp 2

2

kW kkW k + kW kkW kp 2 p 2t2t2 p 2222 kW k kW k + kW k + kW k + kW kp 2p 22

22 + kW k + kW kkW kkW kp 2p 2p 2t2 p 2+ t2 + kW k2 + kW k2 + kW k2 + kW k2p 2

2

2

2 + kW kkW kkW k kW k kW kp 2p 2p 2t2+ t2+t3 kW k + kW k2 + kW k2 + kW k2

kW kkW kkW kkW kkW kp 2p 2p 2t3 t1 kW k kW k + kW k2 + kW k2 + kW k2p 2

2

3

22 + kW kkW kkW kkW k2p 2+t1+ t1+ t2 p 22 kW k + kW k2 + kW k2 + kW k2

3 2 3t2+t3+ t3+ t1, kW k kW k kW k+

p

kW k

2 p 2 + kW k2 2 2 kW kkW kkW kp 2p 2p+2 = kW k + kW k2 + kW k2 2 + kW k2!2p

2 + kW k2 kW k 2 kW k 2 kW k 3p 2p 22++ kW k + kW k2 + kW k2p 2p 2

kW k + kW k2 kW k + kW k2kW kp 2pp+ + kW k2 2 + kW k2 2 + kW k2p 2p

2 2 + kW k2 + kW k2kW kpp+ kW k kW k 2 + kW k2 2 + kW k2pp

2 + kW k2 2 2 + kW k2 kW kkW kkW kp 2pp 2 kW k + kW k2 2 + kW k2 + kW k2!2"

p 2 + kW k2 3 kW kp 2++)+ kW k kW k kW k + kW k2p 2p 2

22

kW k + kW k2 kW k + kW k2kW kp 2ppp 2+22222 + kW k + kW k + kW k + kW k2#pp

2 2 + kW k2 2 + kW k2kW k 2 kW kp 2p 22 kW k + kW k2 + kW k2!2 3 kW k2p 2p 2++2 kW k kW k + kW k2 + kW k2

Example 3.1 Let us consider the unit speed Mannheim curve and Mannheim partner curve:

11(s) = cos s, sin s, s , (s) = (2 cos s, 2 sin s, s)22

The Frenet invariants of the partner curve, (s) are given as followingT (s)

=

1 2 sin s, 2 cos s, 1 ,5

12


N (s) =B (s) =C (s) = (s) = (s) =

1 sin s, cos s, 25

cos s, sin s, 022221sin s + cos s, cos s + sin s,555552 2525

In terms of definitions, we obtain special Smarandache curve, see Figure 1.

2 ........1 ...................-0.8-0.04 .......... -0.6-0.02 ....... -0.4...... -0.20.0 ....... 0.0....0.2...0.40.02 ......0.6.....0.04 ....... 0.8 1Figure 1 1 = (5 + 2 5) sin s + 10 cos s, (5 2 5) cos s + 10 sin s, 9 55 5References[1] Ali A.T., Special Smarandache curves in the Euclidean space, International Journal ofMathematical Combinatorics, Vol.2, 2010, 30-36. and Y[2] Bektas O.uce S., Special Smarandache curves according to Dardoux frame in Euclidean 3-space, Romanian Journal of Mathematics and Computer science, Vol.3, 1(2013),48-59. and Y[3] Bayrak N., Bektas O.uce S., Special Smarandache curves in E31 , International Conference on Applied Analysis and Algebra, 20-24 June 2012, Yldz Techinical University, pp.209, Istanbul.[4] C alskan A., Senyurt S., Smarandache curves in terms of Sabban frame of spherical indicatrix curves, XI, Geometry Symposium, 01-05 July 2013, Ordu University, Ordu.


13

[5] C etin M., Tuncer Y. and Karacan M.K.,Smarandache curves according to bishop frame inEuclidean 3-space, arxiv:1106.3202, vl [math.DG], 2011. on[6] Hacsalihoglu H.H., Differential Geometry, Inu University, Malatya, Mat. no.7, 1983.[7] Liu H. and Wang F.,Mannheim partner curves in 3-space, Journal of Geometry, Vol.88, No1-2(2008), 120-126(7).[8] Orbay K. and Kasap E., On mannheim partner curves, International Journal of PhysicalSciences, Vol. 4 (5)(2009), 261-264.[9] Sabuncuoglu A., Differential Geometry, Nobel Publications, Ankara, 2006.[10] Senyurt S. Natural lifts and the geodesic sprays for the spherical indicatrices of the mannheimpartner curves in E 3 , International Journal of the Physical Sciences, vol.7, No.16, 2012,2414-2421.[11] Taskopru K. and Tosun M., Smarandache curves according to Sabban frame on S 2 , Boletimda Sociedade parananse de Mathemtica, 3 srie, Vol.32, No.1(2014), 51-59 ssn-0037-8712.[12] Turgut M., Ylmaz S., Smarandache curves in Minkowski space-time, International Journalof Mathematical Combinatorics, Vol.3(2008), pp.51-55.[13] Wang, F. and Liu, H., Mannheim partner curves in 3-space, Proceedings of The EleventhInternational Workshop on Diff. Geom., 2007, 25-31.

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NC− Smarandache Curves of Mannheim Curve Couple According to Frenet Frame