Smarandache Curves According to Sabban Frame S

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arXiv:1206.6229v3 [math.DG] 20 Jul 2012 Smarandache Curves According to Sabban Frame on S 2 Kemal Ta¸ sk¨opr¨ u, Murat Tosun Faculty of Arts and Sciences, Department of Mathematics, Sakarya University, Sakarya, 54187, TURKEY Abstract: In this paper, we introduce special Smarandache curves according to Sabban frame on S 2 and we give some characterization of Smarandache curves. Besides, we illustrate examples of our results. Mathematics Subject Classification (2010): 53A04, 53C40. Keywords: Smarandache Curves, Sabban Frame, Geodesic Curvature 1 Introduction The differential geometry of curves is usual starting point of students in field of differential geometry which is the field concerned with studying curves, surfaces, etc. with the use the concept of derivatives in calculus. Thus, implicit in the discussion, we assume that the defining functions are sufficiently differentiable, i.e., they have no concerns of cusps, etc. Curves are usually studied as subsets of an ambient space with a notion of equivalence. For example, one may study curves in the plane, the usual three dimensional space, curves on a sphere, etc. There are many important consequences and properties of curves in differential geometry. 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 an- other regular curve, is called a Smarandache curve [5]. Special Smarandache curves have been studied by some authors [2, 3, 4, 5, 6]. In this paper, we study special Smarandache curves such as γt, td, γtd - Smaran- dache curves according to Sabban frame in Euclidean unit sphere S 2 . We hope these results will be helpful to mathematicians who are specialized on mathe- matical modeling. 1

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

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  • arX

    iv:1

    206.

    6229

    v3 [

    math.

    DG]

    20 Ju

    l 201

    2

    Smarandache Curves According to Sabban Frame

    on S2

    Kemal Taskopru, Murat Tosun

    Faculty of Arts and Sciences, Department of Mathematics,

    Sakarya University, Sakarya, 54187, TURKEY

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

    Mathematics Subject Classification (2010): 53A04, 53C40.Keywords: Smarandache Curves, Sabban Frame, Geodesic Curvature

    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 Minkowskispace-time, whose position vector is composed by Frenet frame vectors on an-other regular curve, is called a Smarandache curve [5]. Special Smarandachecurves have been studied by some authors [2, 3, 4, 5, 6].

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

    1

  • 2 Preliminaries

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

    , = dx21 + dx22 + dx23where (x1, x2, x3) is a rectangular coordinate system of E

    3. Recall that, thenorm of an arbitrary vector X E3 is given by X =

    X,X. The curve

    is called a unit speed curve if velocity vector of satisfies = 1. Forvectors v, w E3, it is said to be orthogonal if and only if v, w = 0. Thesphere of radius r = 1 and with center in the origin in the space E3 is definedby

    S2 = {P = (P1, P2, P3)|P, P = 1}.Denote by {T,N,B} the moving Frenet frame along the curve in E3. For anarbitrary curve E3 , with first and second curvature, and respectively,the Frenet formulae is given by [1]

    T = NN = T + BB = N.

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

    = tt = + gdd = gt

    where is called the geodesic curvature of g on S2 and g = t, d [7].

    3 Smarandache Curves According to Sabban Frame

    on S2

    In this section, we investigate Smarandache curves according to the Sabbanframe on S2 .Let = (s) and = (s) be a unit speed regular sphericalcurves on S2, and {, t, d} and {, t , d} be the Sabban frame of these curves,respectively.

    3.1 t-Smarandache Curves

    Definition 3.1 Let S2 be a unit sphere in E3 and suppose that the unit speedregular curve = (s) lying fully on S2. In this case, t - Smarandache curvecan be defined by

    2

  • (s) =12( + t). (3.1)

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

    (s) =d

    dsds

    ds=

    12( + t)

    and

    tds

    ds=

    12(t+ gd )

    where

    ds

    ds=

    2 + g2

    2. (3.2)

    Thus, the tangent vector of curve is to be

    t =1

    2 + g2( + t+ gd) . (3.3)

    Differentiating the equation (3.3) with respect to s, we get

    t ds

    ds=

    1

    (2 + g2)3

    2

    (1 + 2t+ 3d) (3.4)

    where

    1 = gg g2 2

    2 = gg 2 2g2 g43 = 2g + 2g

    + g3.

    Substituting the equation (3.2) into equation (3.4), we reach

    t =

    2

    (2 + g2)2 (1 + 2t+ 3d) . (3.5)

    Considering the equations (3.1) and (3.3), it easily seen that

    d = t = 14 + 2g2

    (g + (1 g) t+ 2d) . (3.6)

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

    g = t , d

    = 1(2+g2)

    3

    2

    (1g + 2 (1 g) + 23) .

    3

  • 3.2 td-Smarandache Curves

    Definition 3.2 Let S2 be a unit sphere in E3 and suppose that the unit speedregular curve = (s) lying fully on S2. In this case, td - Smarandache curvecan be defined by

    (s) =12(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

    (s) =d

    dsds

    ds=

    12(t + d)

    and

    tds

    ds=

    12( + gd gt)

    where

    ds

    ds=

    1 + 2g2

    2. (3.8)

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

    t =1

    1 + 2g2( gt+ gd) . (3.9)

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

    t ds

    ds=

    1

    (1 + 2g2)3

    2

    (1 + 2t+ 3d) (3.10)

    where

    1 = 2gg + g + 2g3

    2 = 1 g 3g2 2g43 = g2 + g 2g4.

    Substituting the equation (3.8) into equation (3.10), we get

    t =

    2

    (1 + 2g2)2 (1 + 2t+ 3d) (3.11)

    Using the equations (3.7) and (3.9), we easily find

    d = t = 12 + 4g2

    (g t+ (1 + g) d) . (3.12)

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

    4

  • g = t , d

    = 1(1+2g2)

    3

    2

    (1g 2 + 3 (1 + g)) .

    3.3 td-Smarandache Curves

    Definition 3.3 Let S2 be a unit sphere in E3 and suppose that the unit speedregular curve = (s) lying fully on S2. Denote the Sabban frame of (s),{, t, d}. In this case, td - Smarandache curve can be defined by

    (s) =12( + t+ d). (3.13)

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

    (s) =d

    dsds

    ds=

    13( + t + d)

    and

    tds

    ds=

    13(t + gd gt)

    where

    ds

    ds=

    2 (1 g + g2)

    3. (3.14)

    Thus, the tangent vector of curve is

    t =1

    2 (1 g + g2)( + (1 g) t+ gd) . (3.15)

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

    t ds

    ds=

    1

    22(1 g + g2)

    3

    2

    (1 + 2t+ 3d) (3.16)

    where

    1 = g + 2gg 2 + 4g 4g2 + 2g32 = g gg 2 4g2 + 2g + 2g3 2g43 = gg + 2g 4g2 + 2g + 4g3 2g4.

    Substituting the equation (3.14) into equation (3.16), we reach

    t =

    3

    4(1 g + g2)2(1 + 2t+ 3d) . (3.17)

    5

  • Using the equations (3.13) and (3.15), we have

    d = t = 161 g + g2

    ((2g 1) + (1 g) t+ (2 g) d) .(3.18)

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

    g = t , d

    = 142(1g+g2)

    3

    2

    (1 (2g 1) + 2 (1 g) + 3 (2 g)) .

    3.4 Example

    Let us consider the unit speed spherical curve:

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

    It is rendered in Figure 1.

    Figure 1: = (s)

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

    6

  • Figure 2: t - Smarandache Curve

    Figure 3: td - Smarandache Curve

    7

  • Figure 4: td - Smarandache Curve

    References

    [1] Do Carmo, M. P., Differential Geometry of Curves and Surfaces, PrenticeHall, Englewood Cliffs, NJ, 1976.

    [2] Ali A. T., Special Smarandache Curves in the Euclidean Space, InternationalJournal of Mathematical Combinatorics, Vol.2, 30-36, 2010.

    [3] Cetin M., Tuncer Y. and Karacan M. K., Smarandache Curves According toBishop Frame in Euclidean 3- Space, arXiv: 1106. 3202v1 [math. DG], 16Jun 2011.

    [4] Bektas O. and Yuce S., Special Smarandache Curves According to DarbouxFrame in Euclidean 3- Space, arXiv: 1203. 4830v1 [math. DG], 20 Mar 2012.

    [5] Turgut M. and Ylmaz S., Smarandache Curves in Minkowski Space-time,International J. Math. Combin., Vol.3, 51-55, 2008.

    [6] Bayrak N., Bektas O. and Yuce S., Special Smarandache Curves in E31 ,arXiv:1204.566v1 [math.HO] 25 Apr 2012.

    [7] Koenderink J., Solid Shape, MIT Press, Cambridge, MA, 1990.

    [8] ONeill B., Elemantary Differential Geometry, Academic Press Inc. NewYork, 1966.

    8

    1 Introduction2 Preliminaries3 Smarandache Curves According to Sabban Frame on S23.1 t-Smarandache Curves3.2 td-Smarandache Curves3.3 td-Smarandache Curves3.4 Example