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International J.Math. Combin. Vol.2 (2010), 30-36 


Special Smarandache Curves in the Euclidean Space 


Ahmad T. Ali 
Present address: Mathematics Department, Faculty of Science, King Abdul Aziz University 
PO Box 80203, Jeddah, 21589, Saudi Arabia 
Permanent address: Mathematics Department, Faculty of Science, 
Al-Azhar University, Nasr City, 11448, Cairo, Egypt 


Email: atali71@yahoo.com 


Abstract: In this work, we introduce some special Smarandache curves in the Euclidean 
space. We study Frenet-Serret invariants of a special case. Besides, we illustrate examples 


of our main results. 
Key Words: Smarandache Curves, Frenet-Serret Trihedra, Euclidean Space. 


AMS(2000): 53A04 


§1. Introduction 


It is safe to report that the many important results in the theory of the curves in E? were 
initiated by G. Monge; and G. Darboux pionnered the moving frame idea. Thereafter, F. Frenet 
defined his moving frame and his special equations which play important role in mechanics and 
kinematics as well as in differential geometry (for more details see [1]). 

At the beginning of the 20th century, A. Einstein’s theory opened a door to new geometries 
such as Lorentzian Geometry, which is simultaneously the geometry of special relativity, was 
established. Thereafter, researchers discovered a bridge between modern differential geometry 
and the mathematical physics of general relativity by giving an invariant treatment of Lorentzian 
geometry. They adapted the geometrical models to relativistic motion of charged particles. 
Consequently, the theory of the curves has been one of the most fascinating topic for such 
modeling process. As it stands, the Frenet-Serret formalism of a relativistic motion describes 
the dynamics of the charged particles. The mentioned works are treated in Minkowski space- 
time. 

In the light of the existing literature, in [4] authors introduced special curves by Frenet- 
Serret frame vector fields in Minkowski space-time. 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 [4]. In this work, we study special Smarandache Curve in the Euclidean 
space. We hope these results will be helpful to mathematicians who are specialized on mathe- 
matical modeling. 


1Received March 31, 2010. Accepted June 8, 2010. 


Special Smarandache Curves in the Euclidean Space 31 


§2. Preliminaries 


To meet the requirements in the next sections, here, the basic elements of the theory of curves 
in the space E? are briefly presented (A more complete elementary treatment can be found in 
[2].) 


The Euclidean 3-space E? provided with the standard flat metric given by 
(,) = dx? + dz? + dz3, 


where (21, £2, £3) is a rectangular coordinate system of E3. Recall that, the norm of an arbitrary 
vector a € E? is given by |ja|| = \/(a,a). ọ is called an unit speed curve if velocity vector v of 
y satisfies ||v|| = 1. For vectors v,w € E? it is said to be orthogonal if and only if (v, w) = 0. 
Let V = 0(s) be a regular curve in E’. If the tangent vector field of this curve forms a constant 
angle with a constant vector field U, then this curve is called a general helix or an inclined 
curve. The sphere of radius r > 0 and with center in the origin in the space E? is defined by 


s? = {p = (p1, P2, p3) E E’ : (p, p) = rek; 


Denote by {T, N, B} the moving Frenet-Serret frame along the curve ¢ in the space E3. For an 
arbitrary curve y € E3, with first and second curvature, x and T respectively, the Frenet-Serret 
formulae is given by [2] 


T’ 0 K 0 T 
N |=| -k 0 r? N |, (1) 
B' 0 -Tr 0 B 


where 


(T,T) = (N,N) = (B, B) = 1, 
(T, N) = (T, B) = (T, N) = (N, B) = 0. 


The first and the second curvatures are defined by « = «(s) = ||T’(s)|| and 7(s) = — (N, B’), 
respectively. 


§3. Special Smarandache Curves in E? 


In [4] authors introduced: 


Definition 3.1 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. 


In the light of the above definition, we adapt it to regular curves in the Euclidean space as 
follows: 


Definition 3.2 Let y = y(s) be a unit speed regular curve in E® and {T, N, B} be its moving 
Frenet-Serret frame. Smarandache TN curves are defined by 


1 
ea EAE (2) 


32 Ahmad T. Ali 


Let us investigate Frenet-Serret invariants of Smarandache TN curves according to y = 


qy(s). Differentiating (2), we have 


d ds¢ 1 

"= > = —_(—KT N B 3 
C= de = gg THN + 7B), (3) 

and hence PEN EEB 

-K K T 
Te = —— 4 
£ V2K2 + T2 (4) 

where 


ds QK2 + 72 
ae oN oe 6) 


In order to determine the first curvature and the principal normal of the curve ¢, we formalize 


. T N B 
ma o (6) 
ds (2K2 + 72)? 


where 
ô=- [a2 (2x? +77) 4+ 7(rK! — sr’), 


u=- |K? (2k? + 377) + T(r? — TK! 4 sr’), (7) 
n= r [r2 +77) — 2(TK' — rr')| ; 





Then, we have 


T; = ae ca (oT + uN +B). (8) 


So, the first curvature and the principal normal vector field are respectively given by 








i V24/82 + pe + 2 
Jl ere o) 
(2K? + 7?) 
a ôT + uN +B 
se (10) 
yê +++ 
On other hand, we express 
T N B 
1 
XNE T —K R T y (11) 
ô u r 
where v = V2K? +7? and l = y8? + u? + n?. So, the binormal vector is 
T 4 N B 
Be len = TH) T + [en + ôT] N -r [n+ 0) B (12) 
ul 
In order to calculate the torsion of the curve ¢, we differentiate 
—(K2 + 46/)T+ 
1 
d = (kK! = K? —77)N (13) 


v2 


+(k7 +7')B 


Special Smarandache Curves in the Euclidean Space 33 





and thus 
cl" = PEPE NE (14) 
V2 
where 
w= K? +K(7? — 3K/) K", 
Q = =k? — (7? + 3K!) — 877! +K”, (15) 
o = —k?T — TÌ’ 427K! HKT +7". 
The torsion is then given by: 
vaļi +7? — K')(ka + 7w) + K(K7 + 7')(6 —w) + (k? + &')(Ko — To) 
T= or o O (16) 


[7 (2K? + 7?) + KT kT]? H (KT — KT’)? + (263 + KT?) 





Definition 3.3 Let y = y(s) be an unit speed regular curve in E® and {T, N, B} be its moving 


Frenet-Serret frame. Smarandache NB curves are defined by 


1 
E = €(s¢) Fa ( ) (17) 
Remark 3.4 The Frenet-Serret invariants of Smarandache NB curves can be easily obtained 
by the apparatus of the regular curve y = ¥(s). 


Definition 3.5 Let y = y(s) be an unit speed regular curve in E? and {T, N, B} be its moving 
Frenet-Serret frame. Smarandache TNB curves are defined by 


1 
CEU eee EARE (18) 


Remark 3.6 The Frenet-Serret invariants of Smarandache TNB curves can be easily obtained 
by the apparatus of the regular curve 7 = ¥(s). 


§4. Examples 
Let us consider the following unit speed curve: 


= 2 gj ye Dees 
Jı = zog Sin 16s — 777 sin 36s 


Y2 = — z% cos 16s + Fe cos 36s . (19) 


13 = & sin 10s 


It is rendered in Figure 1. 


34 


Ahmad T. Ali 





Figure 1: The Curve y = 7(s) 


And, this curve’s natural equations are expressed as in [2] 


k(s) = —24sin 10s (20) 
T(s) = 24 cos 10s 


In terms of definitions, we obtain special Smarandache curves, see Figures 2 — 4. 





Figure 2: Smarandache TN Curves 


Special Smarandache Curves in the Euclidean Space 





1.0 
Figure 4: Smarandache TNB Curve 


35 


36 Ahmad T. Ali 


References 


[1] Boyer, C.B., A History of Mathematics,, John Wiley and Sons Inc., New York, 1968. 

[2] Do Carmo, M.P., Differential Geometry of Curves and Surfaces, Prentice Hall, Englewood 
Cliffs, NJ, 1976. 

[3] Scofield, P.D., Curves of Constant Precession, Amer. Math. Monthly,, 102 (1995) 531-537. 

[4] Turgut, M., Yilmaz, S., Smarandache Curves in Minkowski Space-time, Int. J. Math. 
Comb., 3 (2008) 51-55.