Euler-Savary s Formula for the Planar Curves in Two Dimensional Lightlike Cone

Transcription

1 International J.Math. Combin. Vol.1 (010), Euler-Saary s Formula for the Planar Cures in Two Dimensional Lightlike Cone Handan BALGETİR ÖZTEKİN and Mahmut ERGÜT (Department of Mathematics, Fırat Uniersity, 3119 Elazığ, TÜRKİYE) handanoztekin@gmail.com & mergut@firat.edu.tr Abstract: In this paper, we study the Euler-Saary s formula for the planar cures in the lightlike cone. We first define the associated cure of a cure in the two dimensional lightlike cone Q.Then we gie the relation between the curatures of a base cure, a rolling cure and a roulette which lie on two dimensional lightlike cone Q. Keywor: Lightlike cone, Euler Saary s formula, Smarandache geometry, Smarandachely denied-free. AMS(010): 53A04, 53A30, 53B30 1. Introduction The Euler-Saary s Theorem is well known theorem which is used in serious fiel of study in engineering and mathematics. A Smarandache geometry is a geometry which has at least one Smarandachely denied axiom(1969), i.e., an axiom behaes in at least two different ways within the same space, i.e., alidated and inalided, or only inalided but in multiple distinct ways. So the Euclidean geometry is just a Smarandachely denied-free geometry. In the Euclidean plane E,let c B and c R be two cures and P be a point relatie to c R. When c R roles without splitting along c B, the locus of the point P makes a cure c L. The cures c B, c R and c L are called the base cure, rolling cure and roulette, respectiely. For instance, if c B is a straight line, c R is a quadratic cure and P is a focus of c R, then c L is the Delaunay cure that are used to study surfaces of reolution with the constant mean curature, (see [1]).The relation between the curatures of this cures is called as the Euler-Saary s formula. Many studies on Euler-Saary s formula hae been done by many mathematicians. For example, in [4], the author gae Euler-Saary s formula in Minkowski plane. In [5], they expressed the Euler-Saary s formula for the trajectory cures of the 1-parameter Lorentzian spherical motions. On the other hand, there exists spacelike cures, timelike cures and lightlike(null) cures in semi-riemannian manifol. Geometry of null cures and its applications to general reletiity in semi-riemannian manifol has been constructed, (see []). The set of all lightlike(null) 1 Receied February 1, 010. Accepted March 30,010.

2 116 Handan BALGETİR ÖZTEKİN and Mahmut ERGÜT ectors in semi-riemannian manifold is called the lightlike cone. We know that it is important to study submanifol of the lightlike cone because of the relations between the conformal transformation group and the Lorentzian group of the n-dimensional Minkowski space E1 n and the submanifol of the n-dimensional Riemannian sphere S n and the submanifol of the (n+1)-dimensional lightlike cone Q n+1. For the studies on lightlike cone, we refer [3]. In this paper, we hae done a study on Euler-Saary s formula for the planar cures in two dimensional lightlike cone Q. Howeer, to the best of author s knowledge, Euler-Saary s formula has not been presented in two dimensional lightlike cone Q. Thus, the study is proposed to sere such a need. Thus, we get a short contribution about Smarandache geometries. This paper is organized as follows. In Section, the cures in the lightlike cone are reiewed. In Section3, we define the associated cure that is the key concept to study the roulette, since the roulette is one of associated cures of the base cure. Finally, we gie the Euler-Saary s formula in two dimensional cone Q. We hope that, these study will contribute to the study of space kinematics, mathematical physics and physical applications.. Euler-Saary s Formula in the Lightlike Cone Q Let E 3 1 be the 3 dimensional Lorentzian space with the metric where x = (x 1, x, x 3 ), y = (y 1, y, y 3 ) E 3 1. The lightlike cone Q is defined by g(x, y) = x, y = x 1 y 1 + x y x 3 y 3, Q = {x E 3 1 : g(x, x) = 0}. Let x : I Q E 3 1 be a cure, we hae the following Frenet formulas (see [3]) x (s) = α(s) α (s) = κ(s)x(s) y(s) (.1) y (s) = κ(s)α(s), where s is an arclength parameter of the cure x(s). κ(s) is cone curature function of the cure x(s),and x(s), y(s), α(s) satisfy x, x = y, y = x, α = y, α = 0, x, y = α, α = 1. For an arbitrary parameter t of the cure x(t), the cone curature function κ is gien by dx dt, d x dt d x κ(t) = dt, d x dx dt dt, dx dt dx dt, dx 5 (.) dt Using an orthonormal frame on the cure x(s) and denoting by κ, τ, β and γ the curature, the torsion, the principal normal and the binormal of the cure x(s) in E 3 1, respectiely, we

3 Euler-Saary s Formula for the Planar Cures in Two Dimensional Lightlike Cone 117 hae x = α α = κx y = κβ, where κ 0, β, β = ε = ±1, α, β = 0, α, α = 1, εκ < 0. Then we get β = ε κx y κ, ετγ = εκ εκ (x + 1 y). (.3) κ Choosing we obtain γ = κ = εκ, εκ (x + 1 y), (.4) κ τ = 1 (κ ). (.5) κ Theorem.1 The cure x : I Q is a planar cure if and only if the cone curature function κ of the cure x(s) is constant [3]. If the cure x : I Q E 3 1 is a planar cure, then we hae following Frenet formulas x = α, α = ε εκβ, (.6) β = εκα. Definition. Let x : I Q E 3 1 be a cure with constant cone curature κ (which means that x is a conic section) and arclength parameter s. Then the cure x A = x(s) + u 1 (s)α + u (s)β (.7) is called the associated cure of x(s) in the Q, where {α, β} is the Frenet frame of the cure x(s) and {u 1 (s), u (s)} is a relatie coordinate of x A (s) with respect to {x(s), α, β}. Now we put dx A = δu 1 α + δu β. (.8) Using the equation (.) and (.6), we get dx A = (1 + du 1 εκu )α + (u 1 ε εκ + du )β. (.9) Considering the (.8) and (.9), we hae δu 1 δu = (1 + du 1 εκu ) = (u 1 ε εκ + du ) (.10)

4 118 Handan BALGETİR ÖZTEKİN and Mahmut ERGÜT Let s A be the arclength parameter of x A. Then we write dx A = dx A A. A = 1α + β (.11) and using (.8) and (.10), we get 1 = 1 + du 1 εκu = u 1 ε εκ + du. (.1) The Frenet formulas of the cure x A can be written as follows: dα A A = ε A εa κ A β A dβ A A = ε A κ A α A, (.13) where κ A is the cone curature function of x A and ε A = β A, β A = ±1 and α A, α A = 1. Let θ and ω be the slope angles of x and x A respectiely. Then where φ = ω θ. If β is spacelike ector, then we can write κ A = dω = (κ + dφ A ) 1 (.14) 1 + ε, cos φ = and sin φ =. 1 + Thus, we get and (.14) reduces to dφ = d 1 (cos 1 ) 1 + κ A = (κ If β is timelike ector, then we can write 1 ). 1 + cosh φ = 1 1 and sinh φ = 1 and we get Thus, we hae dφ = d 1 (cosh 1 ). 1 κ A = (κ ). 1 Let x B and x R be the base cure and rolling cure with constant cone curature κ B and κ R in Q, respectiely. Let P be a point relatie to x R and x L be the roulette of the locus of P.

5 Euler-Saary s Formula for the Planar Cures in Two Dimensional Lightlike Cone 119 We can consider that x L is an associated cure of x B such that x L is a planar cure in Q, then the relatie coordinate {w 1, w } of x L with respect to x B satisfies δw 1 = 1 + dw 1 ε B κ B w δw = w 1 ε B εb κ B + dw (.15) by irtue of (.10). Since x R roles without splitting along x B at each point of contact, we can consider that {w 1, w } is a relatie coordinate of x L with respect to x R for a suitable parameter s R. In this case, the associated cure is reduced to a point P. Hence it follows that Substituting these equations into (.15), we get If we choose ε B = ε R = 1, then Hence, we can put Differentiating this equations, we get δw 1 = 1 + dw 1 ε R κ R w = 0 δw = w 1 ε R εr κ R + dw = 0. (.16) δw 1 = ( ε R κ R ε B κ B )w δw = (ε B εb κ B ε R εr κ R )w 1. (.17) 0 < ( δw 1 ) ( δw ) = ( κ R κ B ) (w w 1). (.18) Proiding that we use (.16), then we hae If we consider (.19) and (.0), then we get w 1 = r sinh φ, w = r cosh φ. dw 1 = dr sinh φ + r cosh φ dφ dw = dr cosh φ + r sinh φ dφ (.19) Therefore, substituting this equation into (.14), we hae dw 1 = r κ R cosh φ 1 dw = r sinh φ κ R (.0) r dφ = r κ R + cosh φ (.1) rκ L = ±1 + cosh φ r κ R κ B (.)

6 10 Handan BALGETİR ÖZTEKİN and Mahmut ERGÜT If we choose ε B = ε R = +1, then from (.17) Hence we can put 0 < ( δw 1 ) + ( δw ) = ( κ R κ B ) (w 1 + w ) (.3) Differentiating this equations, we get and w 1 = r sin φ, w = r cos φ. dw 1 = dr sin φ + r cos φ dφ = r κ R cos φ 1 dw = dr cos φ r sin φ dφ = r sin φ κ R (.4) r dφ = r κ R cos φ (.5) Therefore, substituting this equation into (.14), we hae κb + κ R rκ L = κ R cos φ κ B r κ R κ B, (.6) where κ L = ε L κ L. Thus we hae the following Euler-Saary s Theorem for the planar cures in two dimensional lightlike cone Q. Theorem.3 Let x R be a planar cure on the lightlike cone Q such that it rolles without splitting along a cure x B. Let x L be a locus of a point P that is relatie to x R. Let Q be a point on x L and R a point of contact of x B and x R correspon to Q relatie to the rolling relation. By (r, φ), we denote a polar coordinate of Q with respect to the origin R and the base line x B R. Then curatures κ B, κ R and κ L of x B, x R and x L respectiely, satisfies cosh φ rκ L = ±1 + r κ R κ B, if ε B = ε R = 1, κb + κ R rκ L = κ R cos φ κ B r κ R κ B if ε B = ε R = +1. References [1] Hano, J. and Nomizu, K., Surfaces of Reolution with Constant Mean Curature in Lorentz-Minkowski Space, Tohoku Math. J., 36 (1984), [] Duggal, K.L. and Bejancu, A., Lightlike Submanifol of Semi-Riemannian Manifol and Applications, Kluwer Academic Publishers,1996. [3] Liu, H., Cures in the Lightlike Cone, Contributions to Algebra and Geometry, 45(004), No.1, [4] Ikawa, T., Euler-Saary s Formula on Minkowski Geometry, Balkan Journal of Geometry and Its Applications, 8(003), No.,

7 Euler-Saary s Formula for the Planar Cures in Two Dimensional Lightlike Cone 11 [5] Tosun, M., Güngör, M.A., Okur, I., On the 1-Parameter Lorentzian Spherical Motions and Euler-Saary formula, American Society of Mechanical Engineering, Journal of Applied Mechanic, ol.75, No.4, , September 007.