On Helices and Bertrand Curves in Euclidean 3-Space
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ON HELİCES AND BERTRAND CURVES IN EUCLİDEAN 3-SPACE Murat Babaarslan Department of Mathematics, Faculty of Science and Arts Bozok University, Yozgat, Turkey [email protected] Yusuf Yaylı Department of Mathematics, Faculty of Science Ankara University, Tandoğan, Ankara, Turkey [email protected] Abstract- In this article, we investigate Bertrand curves corresponding to spherical images of the tangent indicatrix, binormal indicatrix, principal normal indicatrix and Darboux indicatrix of a space curve in Euclidean 3-space. As a result, in case of a space curve is general helix, we show that curve corresponding to spherical images of its tangent indicatrix and binormal indicatrix are circular helices and Bertrand curves. Similarly, in case of a space curve is slant helix, we demonstrate that the curve corresponding to spherical image of its principal normal indicatrix is circular helix and Bertrand curve. Key Words- Helix, Bertrand curve, Spherical Images 1. INTRODUCTION In the differential geometry of a regular curve in Euclidean 3-space E3 , it is well known that, one of the important problems is characterization of a regular curve. The curvature and the torsion of a regular curve play an important role to determine the shape and size of the curve 1,2. Natural scientists have long held a fascination, sometimes bordering on mystical obsession for helical structures in nature. Helices arise in nanosprings, carbon nonotubes, helices, DNA double and collagen triple helix, the double helix shape is commonly associated with DNA, since the double helix is structure of DNA 3 . This fact was published for the first time by Watson and Crick in 1953 4. They constructed a molecular model of DNA in which there were two complementary, antiparallel (side-by-side in opposite directions) strands of the bases guanine, adenine, thymine and cytosine, covalently linked through phosphodiesterase bonds 5,6 . Helix is one of the most fascinating curve in science and nature. Also we can see the helix curve or helical structures in fractal geometry, for instance hyperhelices 7. From the view of differential geometry, a helix is a geometric curve with non-vanishing constant curvature and with non-vanishing constant torsion 8 . The helix may be called a circular helix or W-curve 9,10 . Circular helix is the simplest three- dimensional spirals. One of the most interesting spiral examples are k -Fibonacci 1 spirals. These curves appear naturally from studying the -Fibonacci numbers F k k ,nn0 and the related hyperbolic k -Fibonacci function. Fibonacci numbers and the related Golden Mean or Golden section appear very often is theoretical physics and physics of the high energy particles 11,12. Three-dimensional k Fibonacci spirals were studied from a geometric point of view in 13 . A classical result about helix stated by Lancret in 1802 and first proved by Saint Venant in 1845 (see 14 for details). Thereafter Bertrand curves discovered by J. Bertrand in 1850 are one of the important and interesting topic of classical special curve theory. A Bertrand curve is defined as a special curve which shares its principal normals with another special curve (called Bertrand mate). Note that Bertrand mates are particular examples of offset curves used in computer-aided design 15 . The relationships among the curves are very important in differential geometry and mathematical physics. For example, Izumiya and Takeuchi 16 have shown that cylindrical helices can be constructed from plane curves and Bertrand curves can be constructed from spherical curves. Kula and Yaylı 17 have studied spherical images of tangent indicatrix and binormal indicatrix of a slant helix and they have shown that spherical images are spherical helix. Yalınız and Hacısalihoğlu 18 find the relation of between Frenet frame, slope axis and the curvature , torsion of null generalized helix in 4 -dimensional Lorentzian space. Camcı, İlarslan, Kula and Hacısalihoğlu 3 have given some characterizations for a non-degenerate curve to be a generalized helix by using harmonic curvatures of the curve in n -dimensional Euclidean space E n . The aim of this paper is to investigate Bertrand curves corresponding to spherical images of the tangent indicatrix, binormal indicatrix, principal normal indicatrix and Darboux indicatrix of a space curve in Euclidean 3-space. In case of a space curve is general helix, we show that curve corresponding to spherical images of its tangent indicatrix and binormal indicatrix are circular helices and Bertrand curves. Similarly, in case of a space curve is slant helix, we demonstrate that the curve corresponding to spherical image of its principal normal indicatrix is circular helix and Bertrand curve. 2. PRELIMINARIES We now recall basic concept on classical differential geometry of space curves and the definitions of spherical images, general helix, circular helix, slant helix and Bertrand curve in Euclidean 3-space. After that, we describe the method to construct Bertrand curves from spherical curves. A curve : IR 3 with unit speed, is a general helix if there is some constant vector u, so that T u cos is constant along the curve, where T (s) ~(s) and we call T (s) a unit tangent vector of ~ at s. We define the curvature of ~ by (s) ~(s) . If (s) 0, then the unit principal normal vector N(s) of curve ~ at s is given by ~(s) (s)N(s) . 2 The unit vector B(s) T(s) N(s) is called the unit binormal vector of ~ at s. For the derivations of the Frenet frame the Frenet-Serret formulae hold: T(s) (s)N(s) N(s) (s)T (s) (s)B(s) (1) B(s) (s)N(s) where (s) is the torsion of the curve ~ at s. For any unit speed curve ~ : I R3 , we call W (s) (s)T (s) (s)B(s) the Darboux vector field of ~ . Spherical images(indicatrix) are well-known in classical differential geometry of curves. Let us W (s) define the curve C on S 2 by the help of vector field C(s) . This curve is W (s) called the spherical Darboux image or the Darboux indicatrix of ~ . Similarly, the unit tangent vectors along the curve ~ generate a curve (T) on S 2 . The curve (T) is called the spherical indicatrix of T or more commonly. (T) is called tangent indicatrix of the curve ~ . If ~ ~(s) is a natural representation of ~ , then (T) T(s) will be a representation of (T). One considers the principal normal indicatrix (N) N(s) and binormal indicatrix (B) B(s) 14 . For a general parameter t of a space curve ~ , we can calculate the curvature and torsion as follows: ~ ~ (t) (t) det(~ (t),~(t),~(3) (t)) (t) , (t) (2) ~ ~ 3 / 2 2 ( (t) (t)) ~ (t)~(t) It has been known that the curve ~ is a general helix if and only if (s) const. If both of (s) 0 and (s) are constant, we call it as a circular helix. A curve ~ with (s) 0 is called a slant helix if the principal normal line of ~ make a constant angle with a fixed direction. ~ is a slant helix if and only if the geodesic curvature of the spherical image of the principal normal indicatrix (N) of ~ 2 (s) (s) (3) ( 2 2 )3 / 2 is a constant function 20 . On the other hand, : IR 3 with (s) 0 is called a Bertrand curve if there exists a curve : IR 3 such that the principal normal lines of : IR 3 and at s I are equal. In this case is called a Bertrand mate of ~ . For a space curve ~(s) with (s) 0, the curve ~ is a Bertrand curve if and only if there exist nonzero real numbers A, B such that A(s) B (s) 1 for any s I 1, 19 . So a circular helix is a Bertrand curve. 3 Now we give a new frame different from Frenet frame. Let : IS 2 be unit speed spherical curve. We denote as the arc-lenght parameter of . Let us denote d t( ) ( ), and we call t( ) a unit tangent vector of at , where . d We now set a vector s( ) ( ) t( ) . By definition, we have an orthonormal frame ( ),t( ),s( ) along . This frame is called the Sabban frame of 21 . Then we have the following spherical Frenet-Serret formulae of : ( ) t( ) t( ) ( ) g ( )s( ) (4) s( ) g ( )t( ) 2 where g ( ) is the geodesic curvature of the curve in S which is given by g ( ) det( ( ),t( ),t( )). We now define a space curve ~( ) a ( )d acot s( )d c (5) 0 0 where a, are constant numbers and c is a constant vector 16 . The following theorem and its result are key in this article. Theorem 1. Under the above notation, ~ is a Bertrand curve. Moreover, all Bertrand curves can be constructed by the above method 16 . Corollary 1. The spherical curve is a circle if and only if the corresponding Bertrand curves are circular helices 16 . 3. HELİCES AND BERTRAND CURVES IN EUCLİDEAN 3-SPACE In this section, we investigate Bertrand curves corresponding to spherical images of the tangent indicatrix, binormal indicatrix, normal indicatrix and Darboux indicatrix of a space curve in Euclidean 3-space and give some results about helices and Bertrand curves. Moreover an example spherical image of tangent indicatrix of space curve and Bertrand curve is presented. Theorem 2. Let be a space curve parametrized by s arc-length parameter. Then ~( ) a T( )d acot B( )d c (6) 0 0 is a Bertrand curve, where a, are constant numbers, c is a constant vector and s T(s) ds . 0 Proof. We denote spherical image of tangent indicatrix of space curve by (T). Then by Theorem 1, Bertrand curve corresponding to (T) is ~( ) a T( )d acot T( )T( )d c .