Smarandache Curves According to Sabban Frame on

Transcription

1 Smarandache Curves Accordin to Sabban Frame on S Kemal Taşköprü, Murat Tosun Faculty of Arts and Sciences, Department of Mathematics Sakarya University, Sakarya 5487 TURKEY Abstract: In this paper, we introduce special Smarandache curves accordin to Sabban frame on curves S and we ive some characterization of Smarandache AMS Subject Classification (00): 5A04, 5C Keywor : Smarandache Curves, Sabban Frame, Geodesic Curvature Introduction The differential eometry of curves is usual startin point of students in field of differential eometry which is the field concerned with studyin curves, surfaces, etc with the use the concept of derivatives in calculus Thus, implicit in the discussion, we assume that the definin functions are sufficiently differentiable, ie, 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 eometry In the liht of the existin studies, authors introduced new curves Special Smarandache curves are one of them A reular curve in Minkowski space-time, whose position vector is composed by Frenet frame vectors on another reular curve, is called a Smarandache curve [5] Special Smarandache curves have been studied by some authors [,,4,5,6] In this paper, we study special Smarandache curves such as curves accordin to Sabban frame in Euclidean unit sphere t, td, td helpful to mathematicians who are specialized on mathematical modelin - Smarandache S We hope these results will be

2 Preliminaries The Euclidean -space provided with the standard flat metric iven by, dx dx dx x, x, x is a rectanular coordinate system of Recall that, the norm of an arbitrary vector X is iven by X, X X The curve is called a unit speed curve if velocity vector of satisfies For vectors vw,, it is said to be orthoonal if and only if vw, 0 The sphere of radius r and with center in the oriin in the space is defined by S P P, P, P P, P Denote by T, N, B the movin Frenet frame alon the curve in For an arbitrary curve, with first and second curvature, and respectively, the Frenet formulae is iven by [] T N N T B B N Now, we ive a new frame different from Frenet frame Let be a unit speed spherical curve We denote s as the arc-lenth parameter of Let us denote t s unit tanent vector of We now set a vector d s s t s s, and we call ts a alon This frame is called the Sabban frame of on S (Sphere of unit radius) Then we have the followin spherical Frenet formulae of : t t d d t is called the eodesic curvature of on S and, t d [7]

3 Smarandache Curves Accordin to Sabban Frame on In this section, we investiate Smarandache curves accordin to the Sabban frame on s and s be a unit speed reular spherical curves on, t, d be the Sabban frame of these curves, respectively t - Smarandache Curves S S, and S Let,, td and Definition Let s lyin fully on S be a unit sphere in and suppose that the unit speed reular curve S In this case, t -Smarandache curve can be defined by () s t Now we can compute Sabban invariants of t - Smarandache curves Differentiatin the equation () with respect to s, we have d s t and t t d () Thus, the tanent vector of curve is to be t t d () Differentiatin the equation (5) with respect to s, we et

4 t t d (4) 4 Substitutin the equation () into equation (4), we reach t t d (5) Considerin the equations () and (), it easily seen that d t t d 4 (6) From the equation (5) and (6), the eodesic curvature of s is t, d td - Smarandache Curves Definition Let s lyin fully on = S be a unit sphere in 4 and suppose that the unit speed reular curve S In this case, td - Smarandache curve can be defined by s t d (7) Now, we can compute Sabban invariants of td - Smarandache curves Differentiatin the equation (7) with respect to s, we have

5 d s t d and t d t (8) In that case, the tanent vector of curve is as follows t t d (9) Differentiatin the equation (9) with respect to s, it is obtained that t t d (0) 4 4 Substitutin the equation (8) into equation (0), we et t t d () Usin the equations () and (), we easily find 5

6 d t t d 4 () So, the eodesic curvature of s is as follows td t, d - Smarandache Curves = Definition Let s lyin fully on S be a unit sphere in Smarandache curve can be defined by S Denote the Sabban frame of and suppose that the unit speed reular curve s t d s,,, td In this case, td - () Lastly, let us calculate Sabban invariants of td - Smarandache curves Differentiatin the equation () with respect to s, we have and Thus, the tanent vector of curve is d s t d t t d t (4) 6

7 t t d (5) Differentiatin the equation (5) with respect to s, it is obtained that t t d (6) Substitutin the equation (4) into equation (6), we reach t t d 4 (7) Usin the equations () and (5), we have d t t d 6 (8) From the equation (7) and (8), the eodesic curvature of s is 4 References t, d = 4 [] Do Carmo, M P, Differential Geometry of Curves and Surfaces, Prentice Hall, Enlewood Cliffs, NJ, 976 7

8 [] Ali A T, Special Smarandache Curves in the Euclidean Space, International Journal of Mathematical Combinatorics, Vol, 0-6, 00 [] Çetin M, Tunçer Y and Karacan M K, Smarandache Curves Accordin to Bishop Frame in Euclidean - Space, arxiv: 06 0v [math DG], 6 Jun 0 [4] Bektaş Ö and Yüce, S, Special Smarandache Curves Accordin to Darboux Frame in Euclidean - Space, arxiv: 0 480v [math DG], 0 Mar 0 [5] Turut M and Yılmaz S, Smarandache Curves in Minkowski Space-time, International J Math Combin, Vol, 5-55, 008 [6] Bayrak N, Bektaş Ö and Yüce S, Special Smarandache Curves in arxiv:04566v [mathho] 5 Apr 0 Ε, [7] Koenderink J, Solid Shape, MIT Press, Cambride, MA, 990 [8] O Neill B, Elemantary Differential Geometry, Academic Press Inc New York, 966 8