New approach to Backlund transformations for a curve and its pedal curve
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(2) 210. M. T. Sarıaydın,T. Körpınar. auto-Bäcklund transformations, when the connected PDEs are the same. To generate new solutions on the integrable theories the Bäcklund transformations are highly used. These transformations help to connect difficult PDE to simpler one that has easier solution. In the case of solutions these transformations highly effective to use generating solutions of multi-solutions from the familiar solutions. By applying Bäcklund transformations to trivial solution one can generate a non-trivial case generally [7]. Classical Bäcklund transformations studies focus on the transformation of surfaces that have constant negative curvature in R3 . For a given surface, we can create a new surface in R3 through constant negative curvature. A popular approach for using this purpose is that beginning with two surfaces in Euclidean space in a line congruence. Then, obtaining a map between M1 and M2 surfaces so that for any two corresponding points through the line is tangent to both surfaces. On the other hand, Bäcklund transformations can be integrable, in the manner of a tangent line segment at the point given on a surface of constant negative curvature. Thus, one can find a new surface with a constant negative curvature which containing line segment’s the endpoint [6, 8, 14]. In recent papers, Bäcklund transformations are widely used such as Weiss found an infinite set of distinct Bäcklund transformations and these are directly related to the periodic fixed points of Bäcklund transformations in [16]. Then, Sen was used to obtain a position type solution of the nonlinear equations describing the propagation of coupled nonlinear optical pulses. This form of the position solution is then compared with that obtained by the special limiting procedure applied to two solution [1]. In [13], Leschke was examined Willmore surfaces with Bäcklund transformation. In the end, Bäck was preparing a thesis on examining the Bäcklund transformation with minimal surfaces in [2]. Additionally, there are many works related with flows as surfaces [3, 9–12, 17]. In this paper, we obtain Bäcklund transformations for a pedal curve via parallel transport frame. Firstly, it is tersely summarized the basic concepts of Bäcklund transformations. Then, we give general conditions to construct Bäcklund transformations for a pedal curve. Finally, we compare with the results obtained Bäcklund transformations in E3 .. 2 Preliminaries The Bishop frame is a special frame because it is well defined a moving frame of the curve vanishing second derivative. Bishop frame is named in parallel transport frame. Definition of Bishop frame can be summarized as: The tangent vector of a curve in space is unique. Then, it constituted basis {M1 , M2 } consisting of rotating principal normal and binormal vector of given curve by an angle θ . So, the following equations can be written as [4, 8], T M1 ∧ M2 , M1 M2 ∧ T, M2 T ∧ M1 T, M1 T, M2 M1 , M2 0. Theorem 2.1 Let ψ be a transformation between α and β in Euclidean 3-space with β ψ(α) such that in the corresponding points we have: 1. the line segment [β(s)α(s)] at the intersection of the osculating planes of the curves has constant length r ; 2. the distance vector β(s) − α(s) has the same angle γ π/2 with the tangent vectors of the curves; 3. the binormals of the curves have the same constant angle φ 0.. 123.
(3) New approach to Bäcklund transformations for a curve and…. 211. Then, these curves are congruent with natural curvatures dγ β k1 k1α − , ds tanγ sin φ β α k2 k2 r and the curves β ψ(α) is 2Ctanγ β α + 2 (Tα cos γ + Uα sin γ ). k2α + C 2 Here, the angle γ is a solution of the differential equation [8], φ dγ β k2 cos γ tan − k1α ds 2 φ β α −k2 cos γ tan − k1 , 2 and γ C k2α tan . 2 Theorem 2.2 Assume that α : I → E3 is a regular curve in 3-dimensional space. If P is a parallel curve of a given curve in Euclidean 3-space, then, [15] 1 −2tanθ + κ1 C 2κ2 tanθ − κ1 κ2 C P± α + − ± n2 , n 1 κ1 2κ1 sec2 θ 2κ12 sec2 θ where. C. 4t 2 sec2 θ −. 4 . κ12. Let M be a developable ruled surface consisting a parallel curve given by the Theorem 2.2 in E3 . Since the tangent plane is constant along ruling of M, it is clear that the pedal of M is a curve. Thus, for the pedal of M, we can write . ˆ P(s)P(s) + Q(s)T(s), T(s) P (s) 1, ˆ where Q is the distance between the points P(s) and P(s), [1].. 3 Bäcklund transformations for a pedal curve In this section, our goal is to find Bäcklund transformations for the pedal curves in Euclidean 3-space. Theorem 3.1 Let P± α +. . 1 −2tanθ + κ1 C 2κ2 tanθ − κ1 κ2 C − ± n n2 1 κ1 2κ1 sec2 θ 2κ12 sec2 θ. 123.
(4) 212. M. T. Sarıaydın,T. Körpınar. be parallel curve of given a space curve in Euclidean 3-space. If M is a developable ruled surface consisting this parallel curve, then the Bishop frame of the pedal of M can be written by T (1 + Q )t + Qκ1 n1 + Qκ2 n2 , N1 π1 t + π2 n1 + π3 n2 , N2 σ1 t + σ2 n1 + σ3 n2 , where. . . π1 cos φ Q − Q κ12 + κ22 + sin φ Q 2 κ1 κ2 − Q 2 κ12 κ2 ,. . π2 cos φ κ1 + 2κ1 Q + Qκ1 + sin φ(−κ2 − Q κ2 + Qκ2. × 1 + Q + Q Q κ2 ). π3 cos φ κ2 + 2Q κ2 + Qκ2 + sin φ(κ1 + Q κ1 1 + Q. + Q Q κ1 (κ1 − 1) + qκ22 ). . . σ1 cos φ Q 2 κ1 κ2 − Q 2 κ12 κ2 + sin φ Q − Q κ12 + κ22 ,. . σ2 cos φ −κ2 − Q κ2 + Qκ2 1 + Q + Q Q κ2. + sin φ κ1 + 2κ1 Q + Qκ1 ,. σ3 cos φ κ1 + Q κ1 1 + Q + Q Q κ1 (κ1 − 1) + qκ22. + sin φ κ2 + 2Q κ2 + Qκ2 n 2 ,. 2 and we choose 1 + Q + Q 2 κ12 + κ22 1. Proof Proof is clear from Bishop equations. Theorem 3.2 Assume that ω is transforms from Pˆ to B in Euclidean 3- space. That is, B ˆ Then, the Bäcklund transformation of this curve is given by ω(P).. B α + Q(1 − κ1 μ − κ2 η + r cos γ (1 + Q ) + r sin γ π1 )t 1 2κ2 tan θ − κ1 κ2 C. + Qμ + r cos γ Qκ1 + r sin γ π2 n1 + − κ1 2κ12 sec2 θ −2 tan θ + κ1 C. + + r sin γ π + r cos γ Qκ Qη 3 2 n2 . 2κ1 sec2 θ. (1).
(5) Proof Let TB , N1B , N2B be Bishop frame of the B curve in E3 . Denoting the unit vector Pˆ ˆ can write Ω1Pˆ , N2 and Ω1Pˆ , N2B to the positively orthonormal frames B − P by Ω ,then we.
(6) Ω1Pˆ , Ω2Pˆ , Ω3Pˆ and Ω1B , Ω2B , Ω3B , where Ω3Pˆ N2 , Ω3B N2B . Moreover, the angle γ. 123.
(7) New approach to Bäcklund transformations for a curve and…. 213.
(8). is between Ω1Pˆ and T. If we rotate the frames {T, N1 , N2 } and TB , N1B , N2B , then we are .
(9) obtain Ω1Pˆ , Ω2Pˆ , Ω3Pˆ and Ω1B , Ω2B , Ω3B . Thus, ⎡ ⎤ ⎡ ⎤⎡ ⎤ Ω1Pˆ cos γ sin γ 0 T ⎢ Pˆ ⎥ ⎣ (2) ⎣ Ω2 ⎦ − sin γ cos γ 0 ⎦⎣ N1 ⎦, ˆ P N 0 0 1 2 Ω3 and. ⎡. ⎤ ⎡ ⎤⎡ B ⎤ cos γ sin γ 0 T Ω1Pˆ ⎣ Ω B ⎦ ⎣ − sin γ cos γ 0 ⎦⎣ NB ⎦. 1 2 0 0 1 N2B Ω3B. (3). Similar to Eqs. (2) and (3), if we rotate the vector Ω1Pˆ by the angle ζ , ⎤⎡ Pˆ ⎤ ⎡ ˆ⎤ ⎡ Ω1 1 0 0 Ω1P ⎥ ⎣ Ω B ⎦ ⎣ 0 cos ζ − sin ζ ⎦⎢ Ω2Pˆ ⎦. ⎣ 2 B ˆ 0 sin ζ cos ζ Ω3 Ω3P. (4). Now firstly, we can be express as follows from (2)–(4):. TB [(cos2 γ + sin2 γ cos ζ )(1 + Q ) + (cos γ sin γ ) × (1 − cos ζ )π1 + (sin γ sin ζ )σ1 ]t + [(cos2 γ + sin2 γ cos ζ )Qκ1 + (cos γ sin γ )(1 − cos ζ )π2 + (sin γ sin ζ )σ2 ]n1 + [(cos2 γ + sin2 γ cos ζ )Qκ2 + (cos γ sin γ )(1 − cos ζ )π3 + (sin γ sin ζ )σ3 ]n2 , N1B. [(cos γ sin γ )(1 − cos ζ )(1 + Q ) + (sin2 γ + cos2 γ cos ζ )π1 − (cos γ sin ζ )σ1 ]t + [(cos γ sin γ )(1 − cos ζ )Qκ1 + (sin2 γ + cos2 γ cos ζ )π2 − (cos γ sin ζ )σ2 ]n1 + [(cos γ sin γ )(1 − cos ζ )Qκ2 + (sin2 γ + cos2 γ cos ζ )π3 − (cos γ sin ζ )σ3 ]n2 ,. N2B. [−(sin γ sin ζ )(1 + Q ) + (cos γ sin ζ )π1 + cos ζ σ1 ]t + [−(sin γ sin ζ )Qκ1 + (cos γ sin ζ )π2 + cos ζ σ2 ]n1 + [−(sin γ sin ζ )Qκ2 + (cos γ sin ζ )π3 + cos ζ σ3 ]n2 .. (5). Taking derivative equations (5) and using Bishop equations, we obtain dTB. [(k1B cos γ sin γ (1 − cos ζ ) − k2B sin γ sin ζ )(1 + Q ) ds + (k1B (sin2 γ + cos2 γ cos ζ ) + k2B sin γ sin ζ )π1 + (k2B cos ζ − k1B cos γ sin ζ )σ1 ]t + [(k1B cos γ sin γ (1 − cos ζ ) − k2B sin γ sin ζ )Qκ1 + (k1B (sin2 γ + cos2 γ cos ζ ) + k2B sin γ sin ζ )π2 + (k2B cos ζ − k1B cos γ sin ζ )σ2 ]n1 + [(k1B cos γ sin γ (1 − cos ζ ) − k2B sin γ sin ζ )Qκ2 + (k1B (sin2 γ + cos2 γ cos ζ ) + k2B sin γ sin ζ )π3 + (k2B cos ζ − k1B cos γ sin ζ )σ3 ]n2 ,. 123.
(10) 214. M. T. Sarıaydın,T. Körpınar. dN1B. [(−k1B (cos2 γ + sin2 γ cos ζ ))(1 + Q ) + (−k1B (1 − cos ζ ) ds × (cos γ sin γ ))π1 + (−k1B (sin γ sin ζ ))σ1 ]t + [(−k1B (cos2 γ + sin2 γ cos ζ ))Qκ1 + (−k1B (1 − cos ζ )(cos γ sin γ ))π2 + (−k1B (sin γ sin ζ ))σ2 ]n1 + [(−k1B (cos2 γ + sin2 γ cos ζ ))Qκ2 + (−k1B (1 − cos ζ )(cos γ sin γ ))π3 + (−k1B (sin γ sin ζ ))σ3 ]n2 , dN2B. [(−k2B (cos γ + sin2 γ cos ζ ))(1 + Q ) + (−k2B (1 − cos ζ ) ds × (cos γ sin ζ ))π1 + (−k2B (sin γ sin ζ )σ1 ]t + [(−k2B (cos γ + sin2 γ cos ζ ))Qκ1 + (−k2B (1 − cos ζ )(cos γ sin ζ ))π2 + (−k2B (sin γ sin ζ )σ2 ]n1 + [(−k2B (cos γ + sin2 γ cos ζ ))Qκ2 + (−k2B (1 − cos ζ )(cos γ sin ζ ))π3 + (−k2B (sin γ sin ζ )σ3 ]n2 . Rearranging these equations and taking derivative equations (5): dγ dTB (cos ζ − 1) 2 + k1Pˆ (cos γ sin γ ) − (k2Pˆ sin γ sin ζ ) ds ds . Pˆ × (1 + Q ) + (k1 (cos2 γ + sin2 γ cos ζ ) + cos2 γ − sin2 γ . dγ π1 + k2Pˆ (cos2 γ + sin2 γ cos ζ ) × (1 − cos ζ ) ds dγ dγ + cos γ sin ζ σ1 t + (cos ζ − 1) 2 + k1Pˆ ds ds. Pˆ × (cos γ sin γ ) − (k2 sin γ sin ζ ) Qκ1 + k1Pˆ (cos2 γ + sin2 γ cos ζ ) . dγ + cos2 γ − sin2 γ (1 − cos ζ ) )π2 + k2Pˆ (cos2 γ + sin2 γ cos ζ ) ds dγ dγ Pˆ + cos γ sin ζ σ2 n1 + (cos ζ − 1) 2 + k1 ds ds. × (cos γ sin γ ) − (k2Pˆ sin γ sin ζ ) Qκ2 + k1Pˆ (cos2 γ + sin2 γ cos ζ ) . 2 dγ 2 π3 + k2Pˆ (cos2 γ + sin2 γ cos ζ ) + cos γ − sin γ (1 − cos ζ ) ds dγ + cos γ sin ζ σ n2 , ds 3 2 dN1B dγ cos γ − sin2 γ (1 − cos ζ ) − k1Pˆ (sin2 γ + cos2 γ cos ζ ) ds ds. + (k2Pˆ cos γ sin ζ ) (1 + Q ) + ((1 − cos ζ ) dγ Pˆ π1 × (cos γ sin γ ) 2 + k1 ds dγ σ1 t + k2Pˆ (1 − cos ζ )(cos γ sin γ ) + (sin γ sin ζ ) ds. 123.
(11) New approach to Bäcklund transformations for a curve and…. . 215. dγ − k1Pˆ (sin2 γ + cos2 γ cos ζ ) cos2 γ − sin2 γ (1 − cos ζ ) ds . dγ + (k2Pˆ cos γ sin ζ ) Qκ1 + (1 − cos ζ )(cos γ sin γ ) 2 π2 + k1Pˆ ds. + k2Pˆ (1 − cos ζ )(cos γ sin γ ) 2 dγ dγ σ2 n1 + cos γ − sin2 γ (1 − cos ζ ) + (sin γ sin ζ ) ds ds. Pˆ Pˆ 2 2 − k1 (sin γ + cos γ cos ζ ) + (k2 cos γ sin ζ ) Qκ2 dγ + (1 − cos ζ )(cos γ sin γ ) 2 + k1Pˆ π3 ds dγ σ 3 n2 , + k2Pˆ (1 − cos ζ )(cos γ sin γ ) + (sin γ sin ζ ) ds dN2B dγ. (cos γ sin ζ ) − − k1Pˆ − (k2Pˆ cos ζ ) (1 + Q ) ds ds dγ + −(sin γ sin ζ ) k1Pˆ + π1 + (−k2Pˆ (sin γ sin ζ ))σ1 t ds dγ − k1Pˆ − (k2Pˆ cos ζ ) Qκ1 + (−(sin γ sin ζ ) + (cos γ sin ζ ) − ds dγ Pˆ Pˆ π2 + (−k2 (sin γ sin ζ ))σ2 n1 + ((cos γ sin ζ ) × k1 + ds dγ dγ × − − k1Pˆ − (k2Pˆ cos ζ ) Qκ2 + −(sin γ sin ζ ) k1Pˆ + π3 ds ds + (−k2Pˆ (sin γ sin ζ ))σ3 n2 . +. . If we consider the above equations together, then it is easily seen that k2B k2Pˆ , dγ k2B cot γ (1 − cos ζ ) − k1Pˆ . ds Similarly from above equations, we can obtain dγ k1Pˆ k1B − . ds On the other hand, Pˆ is a unit speed curve. Thus, we can write. (6). ˆ 2 r 2. (B − P). (7). B − Pˆ r (T cos γ + N1 sin γ ).. (8). By these Eqs. (2) and (7), we get. From (8) and Eq. (5), it is easily seen that. cos2 γ + 2 cos γ sin γ ( 1 + Q π1 + Qκ(1 π2 + κ2 π3 ) r 1 . sin2 γ π12 + π22 + π32 ). 123.
(12) 216. M. T. Sarıaydın,T. Körpınar. Finally, we obtain B Pˆ + r (T cos γ + N1 sin γ ). Corollary 3.3 Assume that ω is transforms from Pˆ to B in Euclidean 3- space. Then, dγ k1Pˆ k1B − . ds Corollary 3.4 Assume that ω is transforms from Pˆ to B in Euclidean 3-space. Then, dγ k2B cot γ (1 − cos ζ ) − k1Pˆ . ds. References 1. Asil, V., Altay, G., Sarıaydin, M.T.: The pedal cone surface consisting parallel curves according to bishop frame in E 3 . Adv. Model. Optim. 18(2), 307–313 (2016) 2. Bäck, P.: Bäcklund transformations for minimal surfaces, Linkoping University, Department of Mathematics, MSc thesis (2015) 3. Bas, S., Körpinar, T.: A new characterization of one parameter family of surfaces by inextensible flows in de-Sitter 3-space. J. Adv. Phys. 7(2), 251–256 (2018) 4. Bishop, R.L.: There is more than one way to frame a curve. Am. Math. Mon. 82(3), 246–251 (1975) 5. Clelland, J.N., Ivey, T.A.: Bäcklund transformations and Darboux integrability for nonlinear wave equations. Asian. J. Math. 13(1), 15–64 (2009) 6. Do Carmo, M.P.: Differential Geometry of Curves and Surfaces. Prentice Hall, Englewood Cliffs (1976) 7. http://nptel.kmeacollage.ac.in/courses/112105165/lec38.pdf 8. Karacan, M.K., Tuncer, Y.: Bäcklund transformations according to bishop frame in Euclidean 3-space. Siavliai Math. Semin. 7(15), 41–49 (2012) 9. Körpınar, T.: On velocity magnetic curves in terms of inextensible flows in space. J. Adv. Phys. 7(2), 257–260 (2018) 10. Körpınar, T., Demirkol, R.C.: A new characterization on the energy of elastica with the energy of Bishop vector fields in Minkowski space. J. Adv. Phys. 6(4), 562–569 (2017) 11. Körpınar, Z., ˙Inç, M.: On the Biswas–Milovic model with power law nonlinearity. J. Adv. Phys. 7(2), 239–246 (2018) 12. Körpınar, T., Asil, V., Sarıaydın, M.T., ˙Incesu, M.: A characterization for Bishop equations of parallel curves according to Bishop frame in E 3 . Bol. Soc. Paran. 33(1), 33–39 (2015) 13. Leschke, K.: Transformations on Willmore surfaces. PhD Thesis. Habilitationsschrift, Universität Augsburg (2006) 14. O’Neill, B.: Elementary differential geometry. Academic Press Inc., New York (1966) 15. Sarıaydın, M.T., Asil, V.: On parallel curves via parallel transport frame in Euclidean 3-space. Adv. Mod. Op. 18(1), 65–67 (2016) 16. Sen, D.C., Chowdhury, A.R.: On Darboux–Bäcklund transformation and position type solution of coupled nonlinear optical waves. Acta Phys. Pol. 98(1–2), 3–10 (2000) 17. Weiss, J.: Bäcklund transformations, focal surfaces and the two-dimensional Toda lattice. Phys. Lett. A. 137(7–8), 365–368 (1989). 123.
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