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Int. J. Adv. Appl. Math. and Mech. 5(3) (2018) 15-26 (ISSN: 2347-2529) IJAAMM 


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International Journal of Advances in Applied Mathematics and Mechanics 





Special Smarandache curves with respect to Darboux frame in Galilean 
3-Space 


Research Article 





Tevfik Sahin® *, Merve Okur? 


a Department of Mathematics, Faculty of Arts and Sciences, Amasya University, 05000, Amasya, Turkey 
> Institute of Science, Department of Mathematics, Amasya University, 05000, Amasya, Turkey 


Received 09 December 2017; accepted (in revised version) 14 February 2018 





Abstract: In the present paper, we investigate special Smarandache curves with Darboux apparatus with respect to Frenet and 
Darboux frame of an arbitrary curve on a surface in the three-dimensional Galilean space G3. Furthermore, we give 
general position vectors of special Smarandache curves of geodesic, asymptotic and curvature line on the surface in 
G3. As a result of this, we provide some related examples of these curves. 


MSC: 53A35 ¢ 53B30 


Keywords: Special Smarandache curve e Darboux frame e Geodesic curve e Galilean space 


© 2018 The Author(s). This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nce-nd/3.0/). 





1. Introduction 


For centuries, it was thought that Euclidean geometry is the only geometric system until the discoveries of hyper- 
bolic geometry that is a non-Euclidean geometry. In 1870, it was shown by Cayley-Klein that there are 9 different 
geometries in the plane including Euclidean geometry. These geometries are determined by parabolic, elliptic, and 
hyperbolic measures of angles and lengths. The main aim of this work is to study some special curves in Galilean 
geometry which is also among foregoing geometries. The conventional view about Galilean geometry is that it is 
relatively simpler than Euclidean geometry. There are some problems that cannot be solved in Euclidean geometry, 
however they are an easy matter in Galilean geometry. For instance, the problem of determination of position vector 
of an arbitrary curve and the problem that we study in this article can be considered as good examples for the case. 
Another advantageous of Galilean geometry is that it is associated with the Galilean principle of relativity. For more 
details about Galilean geometry, we refer the interested reader to the book by Yaglom [1]. 

The theory of curves forms an important and useful class of theories in differential geometry. The curves emerge 
from the solutions of some important physical problems. Also, mathematical models are often used to describe com- 
plicated systems that arising in many different branch of science such as engineering, chemistry, biology, etc. [2, 3] 

A curve in space is studied by assigning at each point a moving frame. The method of moving frame is a central tool 
to study a curve or a surface. The fundamental theorem of curves states that curves are determined by curvatures and 
Frenet vectors [4]. Thus, curvature functions provide us with some special and important information about curves. 
For example; line, circle, helix (circular or generalized), Salkowski curve, geodesic, asymptotic and line of curvature 
etc. All of these curves are characterized by the specific conditions imposed on their curvatures. To examine the 





* Corresponding author. 
E-mail address(es): tevfik.sahin@amasya.edu.tr (Tevfik Sahin), okurmerwe869@gmail.com ( Merve Okur). 


15 





16 Special Smarandache curves with respect to Darboux frame in Galilean 3-Space 


characteristics of this curves, it is important that the position vectors of the curves are given according to the curvature 
functions. However, this is not always possible in all geometries. For example, the problem of determination of the 
position vector of a curve in Euclidean or Minkowski spaces can only be solved for some special curve such as plane 
line, helix and slant helix. However, this problem can be solved independent of type of curves in Galilean space [5, 6]. 

Curves can also be produced in many different ways, such as solution of physical problems, trajectory of a moving 
particle, etc. [4]. In addition, one can produce a new curve by using Frenet vector fields of a given curve, such as 
evolutes and involutes, spherical indicatrix, and Smarandache curves. 

If the position vector of a curve is formed by frame vectors of 6 curve, then @ is called Smarandache curve of B 
[7]. Recently, many researchers have studied special Smarandache curves with respect to different frames in different 
spaces. In [7], the authors introduced a special case of Smarandache curves in the space Bi [8] studied special 
Smarandache curve in Euclidean space E®. In [9, 10], the authors investigate the curves with respect to Bishop and 
Darboux frame in E°, respectively. Also, [11] investigated the curves with respect to Darboux frame in Minkowski 
3—space. 

Among these studies, only [12] used general position vector with respect to Frenet frame of curve to obtain Sama- 
randache curves in Galilean space. 

The main aim of this paper is to determine position vector of Smarandache curves of arbitrary curve on a surface 
in G3 in terms of geodesic, normal curvature and geodesic torsion with respect to the standard frame. The results 
of this work include providing Smarandache curves of some special curves such as geodesic, asymptotic curve, line 
of curvature on a surface in G3; and Smarandache curves for special cases of curves such as, Smarandache curves of 
geodesics that are circular helix, genaralized helix or Salkowski, etc. Finally, we elaborate on some special curves by 
giving their graphs. 


2. Preliminaries 


The Galilean space G3 is one of the Cayley-Klein spaces associated with the projective metric of signature (0,0, +, +) 
[13]. The absolute figure of the Galilean space is the ordered triple {w, f, I}, where w is an ideal (absolute) plane, f is 
a line (absolute line) in w, and J is a fixed eliptic involution of points of f. 

In non-homogeneous coordinates the group of isometries of G3 has the following form: 


X=a\,+X, 
Y= a21 + a22X + y COS Q + Zsing, (1) 


Z= a3) + 432X — ysin p + ZCOS Q, 


where a11, 421, 422, 431, 432, and ~ are real numbers [14]. 

If the first component of a vector is zero, then the vector is called as isotropic, otherwise it is called non-isotropic 
vector [14]. 

In G3, the scalar product of two vectors v = (v1, V2, v3) and w = (w1, w2, w3) is defined by 


V1 W), if vı #0 or w1 40 


V-Gw= ; 
G i V2 W2 + V3 W3, if vı = 0 and w = 0. 


The Galilean cross product of these vectors is defined by 


0 e2? e&3 
VxXGW=|V)] V2 V3). (2) 
Wi W2 W3 


If v-gw= 0, then v and w are perpendicular. The norm of v is defined by 


lvli = Vlv-evl. 


Let I c Rand let y : I — G3 be a unit speed curve with curvature x > 0 and torsion T. 
Then the curve y is defined by 


(x) = (x, y(x), z (x), 


and that the Frenet frame fields are given by 


T(x) = &' (x), 
y" (x) 
N(x) = ———_,, 3 
ly" lle n 


B(x) = T(x) x N(x) 


Tevfik Sahin, Merve Okur / Int. J. Adv. Appl. Math. and Mech. 5(3) (2018) 15 - 26 17 


1 
sar (0,-z” (x), y" (x)), 
where 
det(y’ (x), y" (x), y” (x)) 


= hall = 
K(x) =ly @Ilg and T(x) = eG 





(4) 


The vector fields T,N and B are called the tangent vector field, the principal normal and the binormal vector field, 
respectively [14]. Therefore, the Frenet-Serret formulas can be written in matrix form as 


f 


T 0 x 0 
N| =|0 0 Tt x (5) 
B 0 0 


T 
N 
-T B 




















There is another useful frame for study curves on a surface. For an easy reference we call this surface M. This frame 
can be formed by two basic vectors. These vectors are a unit tangent vector field T of the curve y on M and the unit 
normal vector field n of M at the point y(x) of y. Therefore, the frame field {T, Q, n} is obtained and is called Darboux 
frame or the tangential-normal frame field. Here, Q = n xgT. 


Theorem 2.1. 
Lety :IcR— M c G bea unit-speed curve, and let {T,Q,n} be the Darboux frame field of y with respect to M. Then the 
Frenet formulas in matrix form is given by 


f 





T 0O Kg Kn] [T 
Q| =|O0 0 Tg} {Q], (6) 
n 0 -Tg 0 n 

















where Kg, Kn and Tg are called geodesic curvature, normal curvature and geodesic torsion, respectively. 





Proof. It follows from solving (6) componentwise [6, 16] . 











Also, (6) implies the important relations 


x? (x) = KEKA), TX) = -Tg(x) + (7) 


x3 (x) +K? (x) 


where x(x) and T(x) are the curvature and the torsion of p, respectively. We refer to [1, 14, 15, 17] for detailed treatment 
of Galilean. 


3. Special Smarandache Curves with Darboux Apparatus with respect to Frenet frame in G3 


In this section, we will give special Smarandache curves with Darboux apparatus with respect to Frenet frame of 
a curve on a surface in G3. In order to the position vector of an arbitrary curve with geodesic curvature Kg, normal 
curvature Kn and geodesic torsion Tg on the surface in G3 [6]. 

Based on the definition of Smarandache curve in [7, 12], we will state the following definition. 


Definition 3.1. 
Let y(x) be a unit speed curve in G3 and T,N,B be the Frenet frame field along with y. Special Smarandache TN, TB 
and TNB curves are, respectively, defined by 


YIn=T+N (8) 
YrpB =T+B (9) 
yrs = T+N+B. (10) 


The following result which is stated as theorem is our main work in this article. 


18 Special Smarandache curves with respect to Darboux frame in Galilean 3-Space 


Theorem 3.1. 
The TN, TB and TNB special Smarandache curves with Darboux apparatus of y with respect to Frenet frame are, respec- 
tively, written as 


YTN = ( 1, f Nidx+ SSM S Ndx + =N ) 


_ 1 1 
YTB ={ 1, JS Nidx o 7 VTA ) (11) 
1 1 
YTNB = ( 1, f Ni\dx+ veraa Ma — Na); f Nodx+ gue + N2) ) 


where 
Nj = xgsin( f tgdx]+xncos{ | rgdx), 


Na =xgcos( [ redx)—Kysin{ f teaz). 


Proof. The position vector of an arbitrary curve with geodesic curvature Kg, normal curvature Kn and geodesic tor- 
sion Tg on the surface in G3 which is introduced by [6] as follows 
| x, [Uf (Kg (x) sin(f Tg (x)dx) — Kp (x) fT g(x) sin(f Tg (x)dx)dx)dx)dx, | 
y(x) = . (12) 


SU gcos([tgdx)— Kn f tg cos({tgdx)dx)dx)dx 
The derivatives of this curve are, respectively, given by; 
| 1, f(kgsin(f Tgdx)-Kn fT gsin({tgds)dx)dx, | 
I 
y x)= , 


J(kgcos([tgdx)— Ky {Tg cos({tgdx)dx)dx 


0, KgSin( f Tgdx)-Kn f Tgsin(fTgdx)dx, 
y" (x) = l 
KgCOS(f Tgdx)—Kn f Tgcos(f Tgdx)dx 
The Frenet frame vector fields with Darboux apparatus of y are determined as follows 
| 1, f (kgsin(f Tgdx)-Kn f Tgsin(fTgdx)dx)dx, | 
T= 


J(kgcos(ftgdx)—Kn [tg cos({tgdx)dx)dx 


í 0, Kg Sin(f Tgdx) +Kncos(fTgdx), 
N=-—— 
V Kg?tKn? | Kgcos(fTgdx)-Knsin(fTgdx) 

B= 


1 | 0, —Kg cCOS( f Tgdx) +Knsin(f Tgdx), | 


Kg? + Kn? 


Using the Definition 3.1, we obtain desired results. 
We now provide some applications of this theorem for some special curves. 


Kgsin( {tT gdx) +x, cos(fTgdx) 














4. Applications 


We begin with studying Smarandache curves of important special curves lying on surfaces such as geodesic, asym- 
totic and curvature (or principal) line. Also, we will provide special cases such as helix and Salkowski curve of these 
curves. 

Let y be regular curve on a surface in G3 with the curvature x, the torsion T, the geodesic curvature k g the normal 
curvature Kn and the geodesic torsion Tg. 


Tevfik Sahin, Merve Okur / Int. J. Adv. Appl. Math. and Mech. 5(3) (2018) 15 - 26 19 


Definition 4.1. 
[4] We can say that y is 


geodesiccurve => Kg =0, 
asymptoticcurve <= Kk, =0, 


lineof curvature = Tg =0. 


Also, We can say that y is called: 





K,T r 
K=0 => a straight line. 
T=0 => a plane curve. 
K = cons.>0,T=cons.>O <> a circular helix or W-curve. (13) 
+ = cons. => a generalized helix. 
K = cons.,T # Cons. <> Salkowski curve [18, 19]. 
K # CONS.,T = CONS. <> _ anti-Salkowski curve [19]. 


4.1. The position vectors of Smarandache curves of a general geodesic curve in G3 


Theorem 4.1. 
The position vectors a g(x) of Smarandache curves of a family of geodesic curve in G3 are provided by 


en = ( 1, [xncos({tgdx)dx+cos({tgdx),— f Knsin(f Tgdx)dx-sin(fTtgdx) iF 
aže = ( 1, [Kncos(f tgdx)dx+sin(ftgdx),— fxnsin({tgdx)dx+cos({tgdx) iF 


1, fxn cos(f{ tT gdx)dx+cos(f{tgdx) +sin(ftgdx), 
E = 
QTNB = 


- f xnsin(f tgdx)dx+cos(ftgdx) —sin({tgdx) 


Proof. The above equations are obtained as general position vectors for TN, TB and TNB special Smarandache curves 
with Darboux apparatus of a geodesic curve on a surface in G3 by using the Definition 4.1 and Theorem 3.1. 














Now, we will give the position vectors for special Smarandache curves of some special cases of a geodesic curve in 
G3. 


Corollary 4.1. 
The position vectors of special Smarandache curves of a family of geodesic curve that is a circular helix in G3 are given 
by the equations 


1, 2sin(cx + c1) + cos(cx + c1), 
§ pe 
acn TN(X) = 
£ cos(cx + c1) — 2 sin(cx + c1) 


aË B0) =( 1, (£42) sin(cx + cy), (£2) cos(cx + ci) | 


1, ($£) sin(cx + c) + cos(cx + c1), 


§ 
a”) TNB(X) = i 
ch (=$) cos(cx + c1) — sin(cx + cy) 


where c, cı and e are integral constants. 


20 Special Smarandache curves with respect to Darboux frame in Galilean 3-Space 


Corollary 4.2. 


The position vectors of special Smarandache curves of a family of geodesic curve that is a generalized helix in G3 are 
given by the equations 


1, 4 sin(d f xndx) + cos(d f Kndx), 


ort n(x) = 
4 cos(d fk ndx) —sin(d f Kndx) 
l, 4 sin(d fxndx) +sin(d f Kkndx), 
a$ rB (x) = 
4 cos(d f Kndx) +cos(d f Kndx) 
l, #1 sin(d f xndx) +cos(d f Kndx), 
a$ rne (x) = 


dil cos(d f Kndx)-sin(d f Kkndx) 
where d is integral constant. 


Corollary 4.3. 


The position vectors of Smarandache curves of a family of geodesic that is a Salkowski curve in G3 are given by the 
equations 


1,mfcos([tgdx)dx+cos([Tgdx), 


as qn (x) = 
-m f sin(f Tgdx)dx-sin(fTgdx) 
1, m f cos( f Tgdx)dx +sin(fTgdx), 
aË r(x) = 
-m f sin(f Tgdx)dx+ cos(fTgdx) 
1, m f cos( f Tgdx)dx + cos(f Tgdx)+sin(fTgdx), 
af tng (X) = 


-m f sin(f Tẹdx)dx+ cos(f Tgdx)-sin(fTgdx) 
where m is an integral constant. 


Corollary 4.4. 


The position vectors of Smarandache curves of a family of geodesic that is a anti-Salkowski curve in G3 are given by the 
equations 


l, [Kn cos(cx + c))dx+cos(cx+ c1), 


aË ern (X) = 
— f Knsin(cx+ cı)dx-— sin(cx + cı) 
l, [Kn cos(cx+ cı)dx + sin(cx+ c1), 
a’ se = 
— f Knsin(cx+ cı)dx + cos(cx +c) 
l, [Kn cos(cx + cı)dx + cos(cx+ c1) +sin(cx+ c1), 
§ = 
Q4sTNB(X) = 


— fKnsin(cx+ cı)dx + cos(cx + c1) — sin(cx + c1) 


where c and c, are integral constants. 


We want to note that above corollaries can be proved by using the Eqs. (7), (13) and Theorem 4.1. 


Tevfik Sahin, Merve Okur / Int. J. Adv. Appl. Math. and Mech. 5(3) (2018) 15 - 26 21 


4.2. The position vectors of Smarandache curves of an general asymptotic curve in G3 


Theorem 4.2. 
The position vectors a g(x) of Smarandache curves of a family of asymptotic curve in G3 are provided by 


Bix (1, fresin tedxvax sin | rgdx, [recos f teands+cos f reas] 


bir = (i [ xesin([ tsdax-cos f rgd, [ccos({ edad sin f ras] 


1, fxgsin({tgdx)dx+sin [tgdx—cos [Tt gdx, 
Bone = 
[xgcos({tgdx)dx+cos ftgdx+sin ftgds 


Proof: By using the Definition 4.1 in Theorem 3.1, then the above equations are obtained as general position vectors 
for TN, TB and TNB special smarandache curves with Darboux apparatus of an asymptotic curve on a surface in 
G3. 














Now, we will give the position vectors for Smarandache curves of some special cases of an asymptotic curve in G3 


Corollary 4.5. 
The position vectors of Smarandache curves of a family of asymptotic curve that is a circular helix in G3 are given by the 
equations 


1,- £ cos(cx + c1) +sin(cx + c1), 
Be N(x) = 
f 


A sin(cx + c1) + cos(cx + c1) 


Be, TB (x) = ( 1, ~(“5) cos(ex+ cy), (*2) sin(ex + cy) ) 


1, — (Shy cos(cx + c1) + sin(cx + c1), 
BS TNB(X) = 
(2f) sin(cx + c1) + cos(cx + c1) 


where c, cı and f are integral constants. 


Corollary 4.6. 
The position vectors of Smarandache curves of a family of asymptotic curve that is a generalized helix in G3 are given by 
the equations 


1, — cos(k f Kgdx) + sin(k f Kgdx), 
Bent (x) = 
sin(k f Kgdx) + cos(k f Kgdx) 


PgnTB(X) = ( 1, -2cos(k f Kgdx), 2sin(k fk gdx) ) 


1, —2cos(k f Kgdx) + sin(k f Kgdx), 
Ben TNB(X) = 
2sin(k fx gdx) + cos(k fk gdx) 


where k is integral constant. 


22 Special Smarandache curves with respect to Darboux frame in Galilean 3-Space 
Corollary 4.7. 


The position vectors of Smarandache curves of a family of asymptotic curve that is a Salkowski curve in G3 are given by 
the equations 


1, f(fsin({tgdx))dx+sin(f{tgdx), 


Born (x) = 
J(f cos([ tgdx))dx+cos(f tgdx) 
1, [f sin(f tgdx))dx—cos(ftgdx), 
Bor (x) = 
J(f cos({tgdx))dx+sin([tgdx) 
1, f(fsin({tgdx))dx+sin({ tgdx) —cos(f{tgdx), 
Borns (x) = 


S(fcos({tgdx))dx + cos(ftgdx) + sin( [Tt gdx) 


where f is constant. 


Corollary 4.8. 
The position vectors of Smarandache curves of a family of asymptotic curve that is an anti-Salkowski curve in G3 are 
given by the equations 


1, f(Kgsin(cx+ c)))dx+sin(cx+ c1), 


Bastn(X) = 
I (kg cos(cx + c)))dx+cos(cx + c1) 
l, J (kg sin(cx + c1))dx-— cos(cx + c1), 
Bi 7B(X) = 
[Kg cos(cx + c1))dx + sin(cx+ c1) 
l, J (kg sin(cx + c)))dx+sin(cx+ c1) —cos(cx+c}), 
B4;TNB(X) = 


J (Kg cos(cx + c}))dx+cos(cx + c}) +sin(cx + c1) 


where c and c, are constants. 














Proof: We want to point out that above corollaries can be proved by using the Eqs. (7), (13) and Theorem 4.2. 


4.3. The position vectors of Smarandache curves of a general curvature line in G3 


Theorem 4.3. 
The position vectors y‘ (x) of Smarandache curves of a family of curvature line in G3 are provided by 





l, J(kgsinat+Ky cosa)dx+ + {kg sina+ Kn cosa), 
ye 7 VKgtkn 
a J (Kg cosa—K,ysina)dx— —1— (kg cosa- Kn sina) 
\Ketkn 
1, [(kgsina+x, cos a)dx— J 7 (kgcosa—xy, sina), 
c VV Kgtkn 
Yrs = 1 





(Kg sind +k, COS a) 


J(Kgcosa—Kysina)dx+ 
\KetKh 


Tevfik Sahin, Merve Okur / Int. J. Adv. Appl. Math. and Mech. 5(3) (2018) 15 - 26 23 


1, [(kgsina+x,cosa)dx 
1 


22 
KEtkh 





(Kg(sina— cosa) + Kn(sina + cosa)), 
Ye = 
TNB f(Kgcosa-xnsina)dx 


re aa cos a) +K,(cosa—sina)) 
K 
E n 


Proof. By using the Definition 4.1 in Theorem 3.1, then the above equations are obtained as general position vectors 
for TN, TB and TNB special smarandache curves with Darboux apparatus of a curvature line on a surface in G3. 














Now, we will give the position vectors for Smarandache curves of some special cases of a curvature line in G3 


Corollary 4.9. 


The position vectors y° (x) of Smarandache curves of a family of curvature line with Kg = const. and Kn = const. is a 
circular helix in G3 are provided by 


1 


fg 2 
aitaz 





1, (a, sina + az cos a)x + (a sina + acosa), 














c 
YenTN(X) = 
(a, COS a— az sin a)x-— l (a cosa- asina) 
a +a? 
1 2 
1, (a, sina + az cos a)x — 1 (a cosa- asina), 
[+45 
c 
Y cnTB(X) = 
(a, COSa— az sina)x+ L (a, sina + a cosa) 
1+a3 
1, (a, sina + az cos a)x 
4 5 (a (sin a— cos a) + do(cosa+sina)), 
aitas 
c = 
Y cn TNB(X) = 


(a, cosa- a2 sin a)x 

1 
2 2 
aitas 





(a (sina + cosa) — az (cosa — sin a)) 





Proof. By using the Eqs. (7) and (13) in Theorem 4.3, we obtain the above equation. 











We will now provide some illustrative examples for arbitrary curve on a surface. 


Example 4.1. 
In (12), if we let K g(x) = sin x, Kn (x) = cos x and T g(x) = x, we obtain the following curve: 


x, 


va (xcos (1/2) — cos (1/2)) FresnelC (+4) 


Vit 
+ yn (xsin (1/2) — sin 1/2) Fresnels { +) 
y(x) =| —cos(1/2)sin (1/2 (x— 1)?) + sin (1/2) cos (1/2 (x- 1°), (14) 


-ynm (sin (1/2) — xsin (1/2)) FresnelC (#4) 
— y7 (xcos (1/2) — cos (1/2)) FresnelS 


-1 
x-1 
Vi 
-cos (1/2 (x-1)?) cos (1/2) — sin (1/2 (x- 1)?)s 


in (1/2) 


where 
2 


_ (mx mx? 
Fresnels(x) = f sin{ ==} ax, FresneiC(x)= f cos(=). 


24 Special Smarandache curves with respect to Darboux frame in Galilean 3-Space 


The special Smaradache curves of y can be obtained directly from Definition 3.1, or replacing K g(x), K,(x) and T g(x) 
by sin x, cos x and x in Theorem 3.1, respectively. In this case, the graphs of y curve and its TN, TB, TNB special Smaran- 
dache curves are given as follows Fig. 1. 














05 -0.6-0.7-0.8-0.9 -1 -1.1 


Fig. 1. y curve and the right figure is printed from outside to inside Y TNB, YTB, Y TN SMarandache curves of y 


We now consider another example for geodesic curve on surface along with their graphs. 


Example 4.2. 
Let the surface M be defined by 





Plu, v) =|uU+t v, i 


4 4 
and define the curve y which lies on the surface M as follows 


u—sin(u+v)cos(ut+v) sin(u+ v)? -— “| 








x—sin(x)cos(x) sin(x)* — x? 


y(x) = (x A ; Fl 


Thus, y is a geodesic curve with x(x) = sinx and T(x) = 1 on M in G3. Also, T,Q,n vector fields and Kp, (x), T g(x) 
curvatures are obtained by using Eq. (6), (7). Using these curvatures in Theorem 4.1, we derive special Smarandache 
curves of y [6]. 








-060.4 0.2 


a 
"2 04g6 3 


Fig. 2. @(u, v) surface and y(x) curve 








5. 


Tevfik Sahin, Merve Okur / Int. J. Adv. Appl. Math. and Mech. 5(3) (2018) 15 - 26 25 





Fig. 3. YtNB, YTB) YTN Special Smarandache curves of y, respectively. 





Dg pg. oe" 





Fig. 4. yTn, YTQ, YTQn Special Smarandache curves with respect to Darboux frame of y, respectively. 


Conclusion 


In this work, we studied general position vectors of special Smarandache curves with Darboux apparatus of an 
arbitrary curve on a surface in the three-dimensional Galilean space G3. As a result of this, we also provided special 
Smarandache curves of geodesic, asymptotic and curvature line on the surface in G3 and provided some related ex- 
amples of special Smarandache curves with respect to Frenet and Darboux frame of an arbitrary curve on a surface. 
Finally, we emphasize that one can investigate position vectors of elastic curves on a surface using the general position 
vectors of curves on a surface in Galilean space. Last but not least, we want to point out that the results of this study 
can be easily generalized to families of surfaces that have common Smarandache curves. 


Acknowledgements 


This study was supported financially by the Research Centre of Amasya University (Project No: FMB-BAP16-0213). 


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