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.

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.

ND OB