Properties of Bertrand curves in dual space

Transcription

1 Vol. 9(9), pp , 16 May, 2014 DOI: /IJPS Article Number: 638CA ISSN Copyright 2014 Author(s) retain the copyright of this article International Journal of Physical Sciences Full Length Research Paper Properties of Bertr curves in dual space İlkay ARSLAN GÜVEN* İpek AĞAOĞLU Department of Mathematics, Faculty of Science Arts, University of Gaziantep, Şehitkamil, 27310, Gaziantep, Turkey. Received 8 November, 2013; Accepted 19 February, 2014 Starting from ideas results given by Özkaldi, İlarslan Yaylı in (2009), in this paper we investigate Bertr curves in three dimensional dual space D 3. We obtain the necessary characterizations of these curves in dual space D 3. As a result, we find that the distance between two Bertr curves the dual angle between their tangent vectors are constant. Also, well known characteristic property of Bertr curve in Euclid space E 3 which is the linear relation between its curvature torsion is satisfied in dual space as We show that involute curves, which are the curves whose tangent vectors are perpendicular, of a curve constitute Bertr pair curves. Key words: Bertr curves, involute-evolute curves, dual space Mathematics Subject Classification: 53A04, 53A25, 53A40. INTRODUCTİON In the study of differential geometry, the characterizations of the curves the corresponding relations between the curves are significant problems. It is well known that many important results in the theory of the curves in E 3 were given by G. Monge then G. Darboux detected the idea of moving frame. After this, Frenet defined moving frame special equations which are used in mechanics, kinematics physics. A set of orthogonal unit vectors can be built, if a curve is differentiable in an open interval, at each point. These unit vectors are called Frenet frame. The Frenet vectors along the curve, define curvature torsion of the curve. The frame vectors, curvature torsion of a curve constitute Frenet apparatus of the curve. It is certainly well known that a curve can be explained by its curvature torsion except as to its position in space. The curvature ( ) torsion ( ) of a regular curve help us to specify the shape size of the curve. Such as If 0, then the curve is a geodesic. If 0 (constant) 0, then the curve is a circle with radius 1. If 0 (constant) 0 (constant), then the curve is a helix (Guggenheimer, 1963; Struik, 1988). Bertr curves can be given as another example of that relation. Bertr curves are discovered in 1850, by J. Bertr who is known for his applications of differential equations to physics, especially thermodynamics. A Bertr curve in E 3 is a curve such that its principal normal vectors are the principal normal vectors of another curve. It is proved in most studies on the subject that the characteristic property of a Bertr *Corresponding author. iarslan@gantep.edu.tr, ilkayarslan81@hotmail.com Author(s) agree that this article remain permanently open access under the terms of the Creative Commons Attribution License 4.0 International License

2 Guven Agaoglu 209 curve is the existence of a linear relation between its curvature torsion as: 1 with constants, 0 (Kühnel, 2006). Dual numbers were defined by Clifford (1849, 1879). After him E. Study used dual numbers dual vectors to clarify a mapping from dual unit sphere to three dimensional Euclidean space E 3. This mapping is called Study mapping. Study mapping corresponds the dual points of a dual unit sphere to the oriented lines in E 3. So the set of oriented lines in Euclidean space E 3 is one to one correspondence with the points of dual space in D 3. In this paper, we study Bertr curves in dual space D 3. Also, the set commutative ring with the following operations i) ( a a ii) ( a a ) ( b b ).( b b ) ( a b) ( a ) a. b ( ab The division of two dual numbers The set b ba provided b 0 can be defined as ) ). forms a PRELİMİNARİES We now recall some basic notions about dual space apparatus of curves. The set D is called the dual number system the elements of this set are in type of. Here a a are real numbers 2 0 which is called a dual unit. The elements of the set D are called dual numbers. The set D is given by is a module on the ring D which is called D. Module the elements are dual vectors consisting of two real vectors (Çöken Görgülü, 2002; Güven et al., 2011). The inner product vector product of are given by For the dual number a R is called the real part of is called the dual part of Two inner operations equality on D are defined for as ; 1) : DD D The norm a 0. is defined by is called the addition in D. 2) If the norm of is 1, then is called a dual unit vector. Let is called the multiplication in D. 3) if only if a b a b. (Köse et al., 1988; Veldkamp, 1976) be a dual space curve with differentiable vectors. The dual arc-length parameter of is defined as

3 210 Int. J. Phys. Sci. Now we will give dual Frenet vectors of the dual curve BERTRAND CURVES IN D3 Here, we define Bertr curves in dual space D 3 give characterizations theorems for these curves. with the dual arc-length parameter s. Then Definition 1 is called the unit tangent vector of the vector which is given by. The norm of Let D 3 be the dual space with stard inner product be the dual space curves. If there exists a corresponding relationship between the dual space curves so that the principal normal vectors of are linear dependent to each other at the corresponding points of the dual curves, then are called Bertr curves in D 3. is called curvature function of. Here is no pure-dual. Then the unit principal normal vector of is defined as The vector is called the binormal vector of. Also, we call the vectors Frenet trihedron of at the point. The derivatives of dual Frenet vectors can be written in matrix form as: Let the curves be Bertr curves in D 3, parameterized by their arc-length s Let respectively. indicate the unit Frenet frame along the unit Frenet frame along Also are the curvature torsion of Similarly,, respectively. are the curvature torsion of, respectively. In the following theorems, we obtain the characterizations of a dual Bertr curve. Theorem 1 Let be two curves in D 3. If are Bertr curves, then which are called Frenet formulas (Köse et al. (1988). The function such that is called s I D (constant). the torsion of which is no pure-dual. For a general parameter t of a dual space curve, the curvature torsion of can be calculated as: Proof: Let be Bertr curves. If are Bertr curves, then it can be written from Figure 1;

4 Guven Agaoglu 211 Theorem 2 ˆ( s) Ns ˆ () N ˆ () s ˆ( s) ˆ ˆ Let be two curves in D 3. If are Bertr curves, then the dual angle between the tangent vectors at the corresponding points of the dual Bertr curves is constant. Proof: Let be two Bertr curves in D 3. If the dual angle between the tangent vectors ˆ O Figure 1. Bertr pair curves. ˆ at the corresponding points of is D, then we write (3) for the vectors of By differentiating the Equation 1 with respect to s applying the Frenet formulas, (1) If we differentiate the last equation of the above, we obtain If we take the inner product of the last equation of the (4) is obtained. If we take the inner product of the Equation 2 with both sides, is found.thus, we notice that (2) above with both sides we use the Frenet formulas, we have d cos 0. ds So cos constant is found D. This completes the proof. (constant). If we use Conclusion 1 the Equation 1 we get If be dual Bertr curves, then the distance between the corresponding points of them the dual angle between the tangent vectors are constant. Theorem 3 s I D (constant). Let be two curves in D 3. If are Bertr curves, are the curvature

5 212 Int. J. Phys. Sci. T ˆ( s ) N ˆ () s Nˆ () s ˆ Figure 2. Involute curves of. ˆ T ˆ () s T ˆ () s ˆ torsion of, are the curvature.cot (constant). 3.9 (9) torsion of, respectively, then Finally, from the Equations 8 9, we find (5) are constant. Conclusion 2 Proof: Let are Bertr curves. If the dual angle between the tangent vectors at the corresponding points of is D, then from the previous proof we have If be dual Bertr curves, then the characterization of being Bertr curve in dual space is same with Euclid space. Theorem 4 Let be a plane curve in D 3. If are the involutes of, then are Bertr curves in D 3. From the Equation (2) we write (6) Proof: Let be the involutes of as in Figure 2. It can be written from Caliskan et al. (in press) In above equations, if we take into account (7) (10),, are the curvature then we get 1. ( s) cot.. ( s). 3.8 (8) torsion of, respectively, (constant). If is a plane curve, then We can write (11)

6 Guven Agaoglu 213 From the last two equation given above, we have So is a plane curve. Similarly, is a plane curve. Özkaldi S, İlarslan K, Yayli Y (2009). On Mannheim partner curve in dual space, An. Şt. Univ. Ovidius Constanta, 17(2): Struik DJ (1988). Lectures on Classical Differential Geometry. Dover, New York. P.221. Veldkamp GR (1976). On the use of dual numbers, vectors matrices in instantaneous, spatial kinematics, Mech. Mach. Theory, 11(2): Consequently, the principal normal vectors of are linearly dependent to each other at the corresponding points of the dual curves. In that case curves are Bertr curves in D 3. Conflict of Interest The author(s) have not declared any conflict of interests. REFERENCES Caliskan M, Senyurt S, Bilici M (in press). Some characterizations for the involute evolute curves in Dual Space. (in press) Çöken AC, Görgülü A (2002). On the dual Darboux rotation axis of the dual space curve. Demonstratio Mathematica. 35(2): Guggenheimer HW (1963). Differential Geometry. McGraw-Hill, New York. P.378. Güven İA, Kaya S, Hacısalihoğlu HH (2011). On closed ruled surfaces concerned with dual Frenet Bishop frames. J. Dynamical Syst. Geometric Theories, 9(1): Köse Ö, Nizamoğlu Ş, Sezer M (1988). An explicit characterization of dual spherical curves, Doğa TU. J. Math. 12(3): Kühnel W (2006). Differential Geometry: curves-surfaces-manifolds, second ed., Am. Math. Soc. USA. P.380.