Bol. Soc. Paran. Mat. (3s.) v. 32 1 (2014): 51–59.c©SPM –ISSN-2175-1188 on line ISSN-00378712 in press

SPM: www.spm.uem.br/bspm doi:10.5269/bspm.v32i1.19242

Smarandache Curves on S2

Kemal Taşköprü and Murat Tosun

abstract: In this paper, we introduce special Smarandache curves accordingto Sabban frame on S2 and we give some characterization of Smarandache curves.Besides, we illustrate examples of our results.

Key Words: Smarandache Curves, Sabban Frame, Geodesic Curvature.

Contents

1 Introduction 51

2 Preliminaries 52

3 Smarandache Curves According to Sabban Frame on S2 52

3.1 γt-Smarandache Curves . . . . . . . . . . . . . . . . . . . . . . . . . 52

3.2 td-Smarandache Curves . . . . . . . . . . . . . . . . . . . . . . . . . 54

3.3 γtd-Smarandache Curves . . . . . . . . . . . . . . . . . . . . . . . . . 55

3.4 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

1. Introduction

The differential geometry of curves is usual starting point of students in field ofdifferential geometry which is the field concerned with studying curves, surfaces,etc. with the use the concept of derivatives in calculus. Thus, implicit in thediscussion, we assume that the defining functions are sufficiently differentiable,i.e., they have no concerns of cusps, etc. Curves are usually studied as subsetsof an ambient space with a notion of equivalence. For example, one may studycurves in the plane, the usual three dimensional space, curves on a sphere, etc.There are many important consequences and properties of curves in differentialgeometry. In the light of the existing studies, authors introduced new curves.Special Smarandache curves are one of them. A regular curve in Minkowski space-time, whose position vector is composed by Frenet frame vectors on another regularcurve, is called a Smarandache curve [8]. Special Smarandache curves have beenstudied by some authors ( [1], [2], [3], [4], [8]).

In this paper, we study special Smarandache curves such as γt, td, γtd - Smaran-dache curves according to Sabban frame in Euclidean unit sphere S2. We hopethese results will be helpful to mathematicians who are specialized on mathemati-cal modeling.

2000 Mathematics Subject Classification: 53A04, 53C40

51Typeset by B

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style.c© Soc. Paran. de Mat.

52 Kemal Taşköprü and Murat Tosun

2. Preliminaries

The Euclidean 3-space E3 provided with the standard flat metric given by

〈, 〉 = dx21 + dx2

2 + dx23

where (x1, x2, x3) is a rectangular coordinate system of E3. Recall that, the normof an arbitrary vector X ∈ E3 is given by ‖X‖ =

√

〈X,X〉. The curve α is called aunit speed curve if velocity vector α′ of α satisfies ‖α′‖ = 1. For vectors v, w ∈ E3,it is said to be orthogonal if and only if 〈v, w〉 = 0. The sphere of radius r = 1 andwith center in the origin in the space E3 is defined by

S2 = {P = (P1, P2, P3)|〈P, P 〉 = 1}.

Denote by {T,N,B} the moving Frenet frame along the curve α in E3. For anarbitrary curve α ∈ E3 , with first and second curvature, κ and τ respectively, theFrenet formulae is given by [5]

T ′ = κN

N ′ = −κT + τB

B′ = −τN.

Now, we give a new frame different from Frenet frame. Let γ be a unit speedspherical curve. We denote s as the arc-length parameter of γ. Let us denotet (s) = γ′ (s), and we call t (s) a unit tangent vector of γ. We now set a vectord (s) = γ (s) ∧ t (s) along γ. This frame is called the Sabban frame of γ on S2

(Sphere of unit radius). Then we have the following spherical Frenet formulae ofγ:

γ′ = t

t′ = −γ + κgd

d′ = −κgt

where is called the geodesic curvature of κg on S2 and κg = 〈t′, d〉 [6].

3. Smarandache Curves According to Sabban Frame on S2

In this section, we investigate Smarandache curves according to the Sabbanframe on S2 .Let γ = γ(s) and β = β(s∗) be a unit speed regular spherical curves onS2, and {γ, t, d} and {γβ , tβ , dβ} be the Sabban frame of these curves, respectively.

3.1. γt-Smarandache Curves

Definition 3.1. Let S2 be a unit sphere in E3 and suppose that the unit speed

regular curve γ = γ(s) lying fully on S2. In this case, γt - Smarandache curve can

be defined by

β(s∗) =1√2(γ + t). (3.1)

Smarandache Curves on S2 53

Now we can compute Sabban invariants of γt - Smarandache curves. Differen-tiating the equation 3.1 with respect to s, we have

β′ (s∗) =dβ

ds∗ds∗

ds=

1√2(γ′ + t′)

and

tβds∗

ds=

1√2(t+ κgd− γ)

where

ds∗

ds=

√

2 + κg2

2. (3.2)

Thus, the tangent vector of curve β is to be

tβ =1

√

2 + κg2(−γ + t+ κgd) . (3.3)

Differentiating the equation 3.3 with respect to s, we get

tβ′ ds

∗

ds=

1

(2 + κg2)

3

2

(λ1γ + λ2t+ λ3d) (3.4)

where

λ1 = κgκg′ − κg

2 − 2λ2 = −κgκg

′ − 2− 2κg2 − κg

4

λ3 = 2κg + 2κg′ + κg

3.

Substituting the equation 3.2 into equation 3.4, we reach

tβ′ =

√2

(2 + κg2)

2 (λ1γ + λ2t+ λ3d) . (3.5)

Considering the equations 3.1 and 3.3, it easily seen that

dβ = β ∧ tβ =1

√

4 + 2κg2(κgγ + (−1− κg) t+ 2d) . (3.6)

From the equation 3.5 and 3.6, the geodesic curvature of β(s∗) is

κgβ = 〈tβ ′, dβ〉

= 1

(2+κg2)

3

2

(λ1κg + λ2 (−1− κg) + 2λ3) .

54 Kemal Taşköprü and Murat Tosun

3.2. td-Smarandache Curves

Definition 3.2. Let S2 be a unit sphere in E3 and suppose that the unit speed

regular curve γ = γ(s) lying fully on S2. In this case, td - Smarandache curve can

be defined by

β(s∗) =1√2(t+ d). (3.7)

Now we can compute Sabban invariants of td - Smarandache curves. Differen-tiating the equation 3.7 with respect to s, we have

β′ (s∗) =dβ

ds∗ds∗

ds=

1√2(t′ + d′)

and

tβds∗

ds=

1√2(−γ + κgd− κgt)

where

ds∗

ds=

√

1 + 2κg2

2. (3.8)

In that case, the tangent vector of curve β is as follows

tβ =1

√

1 + 2κg2(−γ − κgt+ κgd) . (3.9)

Differentiating the equation 3.9 with respect to s, it is obtained that

tβ′ ds

∗

ds=

1

(1 + 2κg2)

3

2

(λ1γ + λ2t+ λ3d) (3.10)

where

λ1 = 2κgκg′ + κg + 2κg

3

λ2 = −1− κg′ − 3κg

2 − 2κg4

λ3 = −κg2 + κg

′ − 2κg4.

Substituting the equation 3.8 into equation 3.10, we get

tβ′ =

√2

(1 + 2κg2)

2 (λ1γ + λ2t+ λ3d) . (3.11)

Using the equations 3.7 and 3.9, we easily find

dβ = β ∧ tβ =1

√

2 + 4κg2(κgγ − t+ (1 + κg) d) . (3.12)

So, the geodesic curvature of β(s∗) is as follows

Smarandache Curves on S2 55

κgβ = 〈tβ ′, dβ〉

= 1

(1+2κg2)

3

2

(λ1κg − λ2 + λ3 (1 + κg)) .

3.3. γtd-Smarandache Curves

Definition 3.3. Let S2 be a unit sphere in E3 and suppose that the unit speed

regular curve γ = γ(s) lying fully on S2. Denote the Sabban frame of γ(s), {γ, t, d}.In this case, γtd - Smarandache curve can be defined by

β(s∗) =1√2(γ + t+ d). (3.13)

Lastly, let us calculate Sabban invariants of γtd - Smarandache curves. Differ-entiating the equation 3.13 with respect to s, we have

β′ (s∗) =dβ

ds∗ds∗

ds=

1√3(γ′ + t′ + d′)

and

tβds∗

ds=

1√3(t− γ + κgd− κgt)

where

ds∗

ds=

√

2 (1− κg + κg2)

3. (3.14)

Thus, the tangent vector of curve β is

tβ =1

√

2 (1− κg + κg2)

(−γ + (1− κg) t+ κgd) . (3.15)

Differentiating the equation 3.15 with respect to s, it is obtained that

tβ′ ds

∗

ds=

1

2√2(1− κg + κg

2)3

2

(λ1γ + λ2t+ λ3d) (3.16)

whereλ1 = −κg

′ + 2κgκg′ − 2 + 4κg − 4κg

2 + 2κg3

λ2 = −κg′ − κgκg

′ − 2− 4κg2 + 2κg + 2κg

3 − 2κg4

λ3 = −κgκg′ + 2κg − 4κg

2 + 2κg′ + 4κg

3 − 2κg4.

Substituting the equation 3.14 into equation 3.16, we reach

tβ′ =

√3

4(1− κg + κg2)

2 (λ1γ + λ2t+ λ3d) . (3.17)

Using the equations 3.13 and 3.15, we have

dβ = β ∧ tβ =1√

6√

1− κg + κg2((2κg − 1) γ + (−1− κg) t+ (2− κg) d) . (3.18)

56 Kemal Taşköprü and Murat Tosun

From the equation 3.17 and 3.18, the geodesic curvature of β(s∗) is

κgβ = 〈tβ ′, dβ〉

= 1

4√2(1−κg+κg

2)3

2

(λ1 (2κg − 1) + λ2 (−1− κg) + λ3 (2− κg)) .

3.4. Example

Let us consider the unit speed spherical curve:

γ (s) = {cos (s) tanh (s) , sin (s) tanh (s) , sech (s)} .

It is rendered in Figure 1.

Figure 1: γ = γ(s)

Considering the Sabban frame and the Sabban formulae, we have geodesic cur-vature of γ:

κg = 2sech (t) .

In terms of the definitions, we obtain Smarandache curves according to Sabbanframe on S2, see Figures 2-4.

Smarandache Curves on S2 57

Figure 2: γt - Smarandache Curve

Figure 3: td - Smarandache Curve

58 Kemal Taşköprü and Murat Tosun

Figure 4: γtd - Smarandache Curve

References

1. Ali A. T., Special Smarandache Curves in the Euclidean Space, International Journal ofMathematical Combinatorics, Vol.2, 30-36, 2010.

2. Bayrak N., Bektaş Ö. and Yüce S., Special Smarandache Curves in E3

1, arXiv: 1204.566v1

[math.HO] 25 Apr 2012.

3. Bektaş Ö. and Yüce S., Special Smarandache Curves According to Darboux Frame in Eu-clidean 3- Space, arXiv: 1203. 4830v1 [math. DG], 20 Mar 2012.

4. Çetin M., Tunçer Y. and Karacan M. K., Smarandache Curves According to Bishop Framein Euclidean 3- Space, arXiv: 1106. 3202v1 [math. DG], 16 Jun 2011.

5. Do Carmo, M. P., Differential Geometry of Curves and Surfaces, Prentice Hall, EnglewoodCliffs, NJ, 1976.

6. Koenderink J., Solid Shape, MIT Press, Cambridge, MA, 1990.

7. O’Neill B., Elemantary Differential Geometry, Academic Press Inc. New York, 1966.

8. Turgut M. and Yılmaz S., Smarandache Curves in Minkowski Space-time, International Jour-nal of Mathematical Combinatorics, Vol.3, 51-55, 2008.

Smarandache Curves on S2 59

Kemal Taşköprü

Sakarya University,

Department of Mathematics,

Faculty of Art and Science, 54187 Sakarya, Turkey

ktaskopru@gmail.com

and

Murat Tosun

Sakarya University,

Department of Mathematics,

Faculty of Art and Science, 54187 Sakarya, Turkey

tosun@sakarya.edu.tr