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International J.Math. Combin. Vol.3(2015), 48-54 


Spherical Images of Special Smarandache Curves in F° 


Vahide Bulut and Ali Caliskan 


(Department of Mathematics, Ege University, Izmir, 35100, Turkey) 


E-mail: vahidebulut@mail.ege.edu.tr, ali.caliskan@ege.edu.tr 


Abstract: In this study, we introduce the spherical images of some special Smarandache 
curves according to Frenet frame and Darboux frame in Æ’. Besides, we give some differential 


geometric properties of Smarandache curves and their spherical images. 
Key Words: Smarandache curves, S.Frenet frame, Darboux frame, Spherical image. 


AMS(2010): 53A04. 


§1. Introduction 


Curves especially regular curves are used in many fields such as CAGD, mechanics, kinematics 
and differential geometry. Researchers are used various curves in these fields. Special Smaran- 
dache curves are one of them. A regular curve in Minkowski spacetime, whose position vector 
is composed by Frenet frame vectors on another regular curve, is called a Smarandache curve 
([7]). Some authors have studied on special Smarandache curves ((1, 2, 7]). 


In this paper, we give the spherical images of some special Smarandache curves according 
to Frenet frame and Darboux frame in E’. Also, we give some relations between the arc length 


parameters of Smarandache curves and their spherical images. 


§2. Preliminaries 


Let a(s) be an unit speed curve that satisfies ||a’ (s)|| = 1 in E3. S.Frenet frame of this curve 
in E’ parameterized by arc length parameter s is, 


a Ger KLOR SN; T(s) x N(s) = B(s), 


where T(s) is the unit tangent vector, N(s) is the unit principal normal vector and B(s) is the 


1Received February 9, 2015, Accepted August 10, 2015. 


44 Vahide Bulut and Ali Caliskan 


unit binormal vector of the curve a(s). The derivative formulas of S.Frenet are, 


T 
N |=|- 0 7 N |, (1) 
B 0 =r 0 B 


where x = «(s) = ||T (s)|| and T = 7(s) = ||B'(s)|| are the curvature and the torsion of the 


curve a(s) at s, respectively [4]. 


Let S be a regular surface and a curve a(s) be on the surface S. Since the curve a(s) is 
also a space curve, the curve a(s) has S.Frenet frame as mentioned above. On the other hand, 
since the curve a(s) lies on the surface S, there exists another frame which is called Darboux 
frame {T,g,n} of the curve a(s). T is the unit tangent vector of the curve a(s), n is the unit 
normal of the surface S and g is a unit vector given by g = n x T.The derivative formulas of 
Darboux frame are 


T 0 Kg Kn T 
g |= kg 0 or g |> (2) 
n —Kn Ty 0 n 


where, kK, is the geodesic curvature, kK, is the normal curvature and Ty is the geodesic torsion 
of the curve a(s). The Darboux vector and the unit Darboux vector of this curve are given, 
respectively as follows 

d = TgT + Kng + Kgn 


TE E EE RE A (3) 


Ild]| fatten 
(1) a(s) is a geodesic curve if and only if kg=0. 
(2) a(s) is an asymptotic line if and only if «,=0. 
(3) a(s) is a principal line if and only if rọ =0 ([6]). 
The sphere in Æ’ with the radius r > 0 and the center in the origin is defined by [3] 


S? = {x = (£1, £2, £3) eA a = r°}. 


Let the vectors of the moving frame of a curve a(s) with non-vanishing curvature are given. 
Assume that these vectors undergo a parallel displacement and become bound at the origin O 
of the Cartesian coordinate system in space. Then the terminal points of these vectors T (s), 
N(s) and B(s) lie on the unit sphere S which are called the tangent indicatrix, the principal 


normal indicatrix and the binormal indicatrix, respectively of the curve a(s). 


The linear elements dsr, dsy and dsp of these indicatrices or spherical images can be easily 
obtained by means of (1). Since T(s), N(s) and B(s) are the vector functions representing these 


Spherical Images of Special Smarandache Curves in E? 45 


curves we find 
ds} = R?d3?, 
dsł, = (K? +77) ds?, (4) 


ds? = r°d2?. 


Curvature and torsion appear here as quotients of linear elements; choosing the orientation 
of the spherical image by the orientation of the curve a(s) we have from (4) 


dsrT 
K = —— 


T” Ir] = de (5) 


Moreover, from (5) we obtain the Equation of Lancret ([5]) 


ds*, = ds? + ds?,. (6) 


§3. Special Smarandache Curves According to S.Frenet Frame In F’ 


3.1 TN- Smarandache Curves 


Let a(s) be a unit speed regular curve in E? and {T, N, B} be its moving S.Frenet frame. A 
Smarandache TN curve is defined by ([1]) 


wo a 
BE ay EEN): (7) 


Let moving S. Frenet frame of this curve be {T*, N*, B*}. 
3.1.1 Spherical Image of the Unit Vector T; 


We can find the relation between the arc length parameters ds* and ds as follows 


ds* 22 + 72 
ae a (8) 


From the equations (5) and (8) we have 


VIET 
AST. 
ees 


From the equation (5) we obtain the spherical image of the unit vector T3 as 


ds. ~  V2y/62 + p? + 7? (10) 
= Ko = Sor 
ds* ( [IRZ F 72)" 





ds* = 





where 
ea VIVERE (11) 
(VIRF) 


46 Vahide Bulut and Ali Caliskan 


Here ((1]), 








Then, from the equations (9) and (10) 


Jer FRR 
vË +e tN | (12) 


ds) = ST 
j k (2K? + 72)3/2 


is obtained. 


3.1.2 Spherical Image of the Unit Vector Nj 


If we use the equation (6) we have 


i 
ds, 


EN = y (P + (r. (13) 





Besides, from the equations (6), (8) and (13) 


ds = y (K*)? + isn (14) 


is obtained, where 


v2 [(«? +77 — K’) (ko + Tw) +6 (sr +7) (p-—w)+ (x + K’) (Ko — ré)| 











ai 1 172 1 1\2 2 (15) 
[T (262 + 77) + KT — KT] + (TK — KT’) + (263 + KT?) 
and 
w=K+kK G 3K) k”, 
ġ= -k-k (? + 3K) —3rr +k, 
o= —K27 -T 42rK tar HT. 
3.1.3 Spherical Image of the Unit Vector B% 
From the equations (5) and (15) we have 
d. * 
-B =r. (16) 


On the other hand, the following formula is found from the equations (5), (8) and (16). 


VILE 


Spa SB (17) 


dsp =T 


4 


Example 1 Let the curve a (s) = ($ 


sint, 2 — cost, 3 sin t) is given. T N-Smarandache curve 


Spherical Images of Special Smarandache Curves in E? 47 


of this curve is found as 


1 3 
(cost — sint), — (sin t + cost), ——= (cost — sint) | . 


s |4 
BIS) = 155 Z N 


The spherical images of T*, N* and B* for the curve 3(s*) are shown in Figures 1, 2 and 3, 
respectively. 




















Figure 3 Spherical image of B* 


3.2 NB- Smarandache Curves 


Let a(s) be a unit speed regular curve in FÆ? and {T, N, B} be its moving S. Frenet frame. 
Smarandache NB curve is defined by ([1]) 


o 1 


(N +B). (18) 


48 Vahide Bulut and Ali Caliskan 


3.2.1 Spherical Image of the Unit Vector T; 


From the equations (5) and (8) we have 

V2K2 + 7? d 

——— dsr. 
KV/2 A 


From the equation (5), we obtain the spherical image of the T3 as 


dsp, — VZR +H 4+ 


— => 
ds* (262 + 72)? 


ds* = 





9 


where 
v1 = xt (2+7) +7? (t=) ; 
y2 = — [e (2%? +37? + 2r') +7 (a = ann J] : 
J3 = 2k? (r E =) =T aa + ann’ ) í 

Then, the following formula is obtained from the equations (9) and (20). 


VEE 


ds; = ST 
- k (2K? + 72)3/2 


3.2.2 Spherical Image of the Unit Vector Nj 


The spherical image of Nj can be found by using the equation (6) as 











ds* 2 2 

BN = Vln)? + (ref, 

s 
where 

g V2 (k3 +71) (27? a k?) 
(273 — 2x2)? + (KT! — TK’)? + (—K3 HKT RT) 
and 
p3 = —-T? 3TT + «274 T, 


pı =k? +K (? + 2r’) ae Ses 
Besides, from the equations (6), (8) and (22) 


V2K2 + 72 J 
EER, S 
V2 VRZ +T? a 


ds = («*)? + (r*)? 


is obtained. 


3.2.3 Spherical Image of the Unit Vector B3 


(19) 


(20) 


(21) 


(22) 


(23) 


(24) 


Spherical Images of Special Smarandache Curves in E? 49 


From the equations (5) and (23) we have 


ds 
=f". 25 
ds* ý (25) 





On the other hand, from the equations (5), (8) and (25) 


V2 (Kp3 + T91) (27? — r°) | V2K2 + 72 


ds, = ew y 26 
E 22)? + (Kr — TR’)” + (-K3 HRT — K'T) Ty2 z ve 





is found. 


Example 2 Let the curve a(s) = (2 sint, 2 — cost, 3 sin t) is given. NB -Smarandache curve 


of this curve is 





B (s*) ot er a ee 
== |== = —-sin =|. 
s A z sin g» cost, —= si 5 


The spherical images of T* and N* for the curve 8 (s*) are shown in Figures 4 and 5, respectively. 




















1.07 
10-05 00 05 aoio 05 00 05 10 


% 





Figure 5 Spherical image of N* 
The spherical image of B* for the curve £ (s*) is a point similar to the Figure 3. 
3.3 TB- Smarandache Curves 


Let a(s) be a unit speed regular curve in E® and {T, N, B} be its moving S.Frenet frame. 
Smarandache TB curve is defined by ([1]) 


B(s*) = — (T + B). (27) 


50 Vahide Bulut and Ali Caliskan 


3.3.1 Spherical Image of the Unit Vector T; 


We can find the spherical image of T3 from the equation (5) and obtain 


dst, a V2\/0? + 03 + 0 (28) 


ds* (26? + 72)? i 





where 
= (2%? + T?) (KT — K?) ; 


02 = (2k + 7T) (s'r — rr’) ; 


03 = (2k? + T°) (KT — T°) ; 





From the equations (5) and (8) 
VTP, 
——— ds 
V 
is obtained. Then, the formula following is acquired from equations (28) and (29). 


Joi +05 +03 (30) 


T 
k (262 + 72)3/2 


ds* = (29) 


* — 
dsp = 


3.3.2 Spherical Image of the Unit Vector N; 


If we use the equation (6) we have the spherical image of Nj as 


* 
ds _ 


ds* = (K)? + (r=). (31) 





Besides, from the equations (6), (8) and (31) 
E nE ee 
dsN = (K*)? +4 aaran (32) 


Jz V2 (T —K)* (K®3 + T®,) (33) 


is obtained, where 





®3 = ar (x -1') +r (K—7), 
®ı =K (r—K) +26 (r -%'). 
3.3.3 Spherical Image of the Unit Vector B3 
From the equations (5) and (33) we have 


dsp 
ds* 





iat (34) 


Spherical Images of Special Smarandache Curves in E? 51 


On the other hand, the following formula is found from the equations (5), (8) and (34). 


V2 (T= K)? (K®3 + 71) V 2K + ia 


7 z| as (35) 
[Fe a H [e (s a Tv2 ° 





4 


Example 3 Let the curve a (s) = (2 


sint, 2 — cost, sint) is given. TB -Smarandache curve 
of this curve is 


E A eee E, 
s E 5 COS 5 Sint, x cos 5l: 


The spherical images of T* and N* for the curve 8 (s*) are shown in Figures 6 and 7, respectively. 











-1.0-4 
T T T 05 -1.0 
10 05 00 o5 4h? °5 a 
y x 


Figure 6 Spherical image of T* 





1.0 








E) 
10 





0.5 
00 o5 


1.0 
0.5 
0.0 
0 -0.5 
y x 


-4.0 -14 


Figure 7 Spherical image of N* 
The spherical image of B* for the curve 8 (s*) is a point similar to the Figure 3. 
3.4 TNB- Smarandache Curves 


Let a(s) be a unit speed regular curve in FE? and {T, N, B} be its moving S.Frenet frame. 
Smarandache TNB curve is defined by ([1]) 


B(s*) = (T+N+B). (36) 


Remark 1 The spherical images of the curve 8 (s*) can be found in a similar way as presented 
above. 


52 Vahide Bulut and Ali Caliskan 


§4. Spherical Images of Darboux Frame {T,g,n} 


Let S be an oriented surface in Æ’. Let a(s) be a unit speed regular curves in E? and {T, g, n} 


be Darboux frame of this curve. 


4.1 Spherical Image of The Unit Vector T 


The differential geometric properties of the spherical image of the unit vector T are given as 


dT dT ds 

dsr ds dsr 
dT a ) ds 
—=(k Knn) .— 
dsr 99 dsr 


dsrT a / 2 2 
ds = Kg + KF. (37) 


On the other hand, from (4) and (37) 
K=4/K2 + R2. (38) 
can be written. 


4.2 Spherical Image of The Unit Vector g 


The differential geometric properties of the spherical image of the unit vector g are found as 


dg _ dg ds 

ds, ds ds, 
dg ds 
—? = (KT i ee 
dsg ( Kg + Tgn) dsg 


The relation between the arc length parameters are given as follows 
ds 
m (39) 


4.3 Spherical Image of The Unit Vector n 


The differential geometric properties of the spherical image of the unit vector n are given as 


dn _ dn ds 
dsn ds dS», 
dn ds 
(—KnT' — T 9g). 


dix dsn 


Spherical Images of Special Smarandache Curves in E? 53 


Also, the relation between the arc length parameters is obtained as 
ds 
Fe Vat TG: (40) 


Results: 


i) If a(s) is a geodesic curve, for kg = 0, 


dsr ds dsn / / 
= Žž T = 2 Does 2 2 
ds = kn = kK, ds = Tg, a Kn + Tg = 4/K + iT). 


Also, the unit Darboux vector is as follows 


Tol + Kng 
A Pte + K2 


ii) If a(s) is an asymptotic line, for kn=0 


dsrT ds ds 
= = Gin E DREE 2 To 
d; = kg 5K, ae Kg tT; = K + iT); T = Tg, 


and the unit Darboux vector is 


Tgl + Kgn 
J2 2 
Ta + Kg 


iii) If a(s) is a line of curvature, for tT, =0 


dsrT 
2 2 De = 
eee Ko tk, =, == = Kg; —— = kn, 


and the unit Darboux vector is 


§5. Special Smarandache Curves According To Darboux Frame In F’ 


5.1 Tg- Smarandache Curves 


Let S be an oriented surface in E’. Let a(s) be a unit speed regular curve in E’, {T, N, B} 
and {T,g,n} be its S.Frenet frame and Darboux frame, respectively. Smarandache Tg curve is 
defined by 


b (s*) = -5 (T +9). (41) 


a 


4.2 Tn- Smarandache Curves 


Let S be an oriented surface in Æ’. Let a(s) be a unit speed regular curve in E’, {T, N, B} 


54 Vahide Bulut and Ali Caliskan 


and {T,g,n} be its S.Frenet frame and Darboux frame, respectively. Smarandache Tn curve is 
defined by 


es 
P) = S(T +n). (42) 


4.3 gn- Smarandache Curves 


Let S be an oriented surface in E’. Let a(s) be a unit speed regular curve in E’, {T, N, B} 
and {T, g, n} be its S.Frenet frame and Darboux frame, respectively. Smarandache gn curve is 
defined by 


B(s") = 5 (g +n). (43) 


4.4 Tgn- Smarandache Curves 


Let S be an oriented surface in Æ’. Let a(s) be a unit speed regular curve in E’, {T, N, B} 
and {T,g,n} be its S.Frenet frame and Darboux frame, respectively. Smarandache Tgn curve 
is defined by 


A= Tgn). (44) 


(See [2].) 


Remark 2 The spherical images of these curves can be easily obtained by the similar way as 


explained in Section 4. 


§6. Conclusion 


Spherical mechanisms are very important for robotics. Spherical curves which are drawn by 
spherical mechanisms are used widely in kinematics and robotics. For this purpose, we presented 


the spherical images of special Smarandache curves and obtained some relations between them. 


References 


1] A.T.Ali, Special Smarandache curves in the Euclidean space, International Journal of 
Mathematical Combinatorics, Vol.2(2010),30-36. 

2| O.Bektas and S.Yuce, Special Smarandache curves according to Darboux frame in Eu- 
clidean 3- space, arXiv: 1203. 4830, 1, 2012. 

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

H.Guggenheimer, Diffrential Geometry, McGraw-Hill Book Company, 1963. 

E.Kreyszig, Differential Geometry, Dover Publications, 1991. 

B.O’Neill, Elemantery Differential Geometry, Academic press Inc. New York, 1966. 
M.Turgut and 8.Yilmaz, Smarandache curves in Minkowski space-time, International Jour- 
nal of Mathematical Combinatorics, Vol.3(2008), 51-55. 








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