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ON HELİCES AND BERTRAND IN EUCLİDEAN 3-SPACE

Murat Babaarslan Department of , 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 indicatrix, binormal indicatrix, principal normal indicatrix and Darboux indicatrix of a space in Euclidean 3-space. As a result, in case of a space curve is general , 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 of its principal normal indicatrix is circular helix and Bertrand curve.

Key Words- Helix, Bertrand curve, Spherical Images

1. INTRODUCTION

In the differential 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  and the torsion  of a regular curve play an important role to determine the 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 , 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 , 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 . One of the most interesting examples are k -Fibonacci

1 spirals. These curves appear naturally from studying the -Fibonacci numbers F  k  k ,nn0 and the related hyperbolic k -Fibonacci . 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 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 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, 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 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 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 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 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 of ~ make a constant with a fixed direction. ~ is a slant helix if and only if the 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 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 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 .   0 0 Applying the Frenet frame and the Frenet-Serret formulae, we obtain

4   ~( )  a T ( )d  a cot B( )d  c .   0 0 This completes the proof. Then we have the following corollary of Theorem 2.

Corollary 2. Let  be a general helix parametrized by s arc-length parameter. Then   ~( )  a T ( )d  a cot B( )d  c (7)   0 0 is a circular helix and a Bertrand curve. Proof. Because of Theorem 2, Bertrand curve corresponding to spherical image of tangent indicatrix ()T of  space curve is   ~( )  a T ( )d  a cot B( )d  c .   0 0 It is not difficult to show that  be a general helix if and only if the spherical image of the tangent ():TIS 2 is a part of a circle in S 2 . Thus by Corollary 1, ~ is a circular helix.

Theorem 3. Let  be a space curve parametrized by s arc-length parameter. Then   ~( )  a B( )d  acot T( )d  c (8)   0 0 is a Bertrand curve, where a,  are constant numbers, c is a constant vector and s   B(s) ds . 0 Proof. We denote spherical image of binormal indicatrix of  space curve by (B). Hence by Theorem 1, Bertrand curve corresponding to (B) is   ~( )  a B( )d  a cot B( ) B( )d  c .   0 0 And using the Frenet frame and the Frenet-Serret formulae, we have   ~( )  a B( )d  acot T( )d  c .   0 0 The proof is completed. Thus we can give the following corollary of Theorem 3.

Corollary 3. Let  be a general helix parametrized by s arc-length parameter. Then   ~( )  a B( )d  acot T( )d  c (9)   0 0 is a circular helix and a Bertrand curve. Proof. Because of Theorem 3, Bertrand curve corresponding to spherical image of tangent indicatrix ()B of  space curve is   ~( )  a B( )d  a cot T( )d  c .   0 0 We can show that  be a general helix if and only if the spherical image of the binormal ():BIS 2 is a part of a circle in S 2 . Then by Corollary 1, ~ is a circular helix.

5 Theorem 4. Let  be a space curve parametrized by s arc-length parameter. Then   ~( )  a N( )d  a cot C( )d  c (10)   0 0 is a Bertrand curve, where a,  are constant numbers, c is a constant vector and s   N(s) ds . 0 Proof. We denote spherical image of principal normal indicatrix of  space curve by (N). Then by Theorem 1, Bertrand curve corresponding to (N) is   ~( )  a N( )d  acot N( ) N( )d  c .   0 0 Because of N( )  N( )  C( ), we get   ~( )  a N( )d  a cot C( )d  c .   0 0 This completes the proof. We can give the following results.

Corollary 4. Let  be a slant helix parametrized by s arc-length parameter. Then   ~( )  a N( )d  a cot C( )d  c (11)   0 0 is a circular helix and a Bertrand curve. Proof. Because of Theorem 4, Bertrand curve corresponding to spherical image of tangent indicatrix ()N of  space curve is   ~( )  a N( )d  a cot C( )d  c .   0 0 We can show that  be a slant helix if and only if the spherical image of the principal normal ():NIS 2 is a part of a circle in S 2 . Then by Corollary 1, ~ is a circular helix. Considering the above theorem, we also give:

Corollary 5. Let  be a general helix parametrized by s arc-length parameter. Then  ~( )  a N ( )d  acotC( )  c (12)  0 is a Bertrand curve. Proof. By Theorem 4,   (  )a N (  ) d   a cot  C (  )  c    0 0 is a Bertrand curve. Since  is a general helix, we can easily show that the Darboux indicatrix C is constant. Thus we immediately have  ~( )  a N ( )d  acotC( )  c .  0 This completes the proof.

6 Theorem 5. Let  be a space curve parametrized by s arc-length parameter. Then   ~( )  a C( )d  acot N( )d  c (13)   0 0 is a Bertrand curve, where a,  are constant numbers,    1, c is a constant vector s and   C(s) ds . 0 Proof. We denote the Darboux indicatrix of  space curve by (C). Then by Theorem 1, Bertrand curve corresponding to (C) is   () a C ()  d    a cot  C ()()   C  d   c .    0 0 T B TB   Since C  and taking the , we have C    2  2 2  2 so we can easily obtain that C  C  N using the above , we can form   ~( )  a C( )d  acot N( )d  c .   0 0 This completes the proof.

Now we will give an example of the spherical image of the tangent inticatrix of space curve and Bertrand curve and draw their pictures by using Mathematica.

sin2 s1 s Example 1. We consider a space curve ()s (cos,  s2 ,sin2 4 s  2 ) . The spherical image of the tangent inticatrix of space curve  is  (s)  (T)  (sin s,sin scos s,cos2 s). Then by Theorem 2, we have the following Bertrand curve (a 1 and cot 1) :

      1 1sd 1 s 1 s d      coss     4 1  sin2  d   3  , 0  0  0  2 212 2 2 1 2  1 sin 1  sin      2 2       2 sins  1 3 cos(2s ) sin s  (s )  2arctan   cos(2 s )  , . 3 cos(2s )  4 2 2     s coss 3 cos(2 s ) 3 1   log 2 coss  3  cos(2 s )  sin s  22 2 2 4       

We can draw pictures of ,  and ~ , respectively (Fig. 1, Fig. 2 and Fig. 3).

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Figure 1:  space curve

Figure 2:  spherical image of the tangent indicatrix of space curve 

Figure 3: ~ Bertrand curve corresponding to spherical image of the tangent indicatrix of space curve 

8 5. REFERENCES 1. M. do Carmo, Differential Geometry of Curves and , Prentice-Hall, New Jersey, 1976 2. W. Kuhnel, Differential Geometry: curves-surfaces-, Braquschweig, Wiesbaden, 1999. 3. Ç. Camcı, K. İlarslan, L. Kula and H. H. Hacısalihoğlu, Harmonic curvatures and generalized helices in E n , Chaos, Solutions & Fractals 40, 2590-2596, 2009. 4. J. D. Watson, F. Crick, Molecular structures of nucleic acids, Nature, 171, 737- 738, 1953. 5. N. Chouaieb, A. Goriely, J. H. Maddocks, Helices, PNAS, 103, 398-403, 2006. 6. T. A. Cook, The curves of life, Constable, London, 1914; Reprinted (Dover, London, 1979). 7. C. D. Toledo-Suarez, On the arithmetic of fractal using hyperhelices, Chaos, Solutions & Fractals, 39, 342-349, 2009. 8. M. Baros, General helices and a theorem of Lancret, Proc. Amer. Math. Soc. 125, 1503-1509, 1997. 9. K. İlarslan, O. Boyacıoğlu, vectors of a spacelike W-curve in 3 Minkowski space E1 , Bull. Korean Math. Soc. 44, 429-438, 2007. 10. J. Monterde, Salkowski curves revisted: A family of curves with constant curvature and non-constant torsion, Comput. Aided Geomet. Design, 26, 271- 278, 2009. 11. M.S. El Naschie, Notes on superstings and the infinite sums of Fibonacci and Lucas numbers, Chaos, Solitions & Fractals, 12, 1937-1940, 2001. 12. M.S. El Naschie, Experimental and theoretical arguments for the number and mass of the Higgs particle, Chaos, Solitions & Fractals, 23, 1901-1908, 2005. 13. S. Falson, A. Plaza, On the 3-dimensional k-Fibonacci spirals, Chaos, Solitions & Fractals, 38, 993-1003, 2008. 14. DJ. Struik, Lectures on Classical Differential Geometry, New-York, Dover, 1988. 15. A. W. Nutbourne and R. R. Martin, Differential Geometry Applied to Design of Curves and Surfaces, Ellis Horwood, Chichester, UK, 1988. 16. S. Izumiya and N. Takeuchi, Generic properties of helices and Bertrand curves, J.geom. 74, 97-109, 2002. 17. L. Kula and Y. Yayli, On slant helix and its spherical indicatrix, Applied Mathematics and Computation 169, 600-607, 2005. 18. A. F. Yalınız and H. H. Hacısalihoğlu, Null Generalized Helices in L3 and L4 , 3 and 4-Dimensional Lorentzian Space, Mathematical and Computational Applications, 10, 105-111, 2005. 19. S. Izumiya and N. Takeuchi, Special Curves and Ruled Surfaces, Beitrage zur Algebra und Geometrie Contributions to Algebra and Geometry 44, 203-212, 2003. 20. S. Izumiya and N. Takeuchi, New special curves and developable surfaces, Turk J. Math. 28, 2004. 21. J. Koenderink, Solid shape, MIT Press, Cambridge, MA, 1990.

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