BÄCKLUND TRANSFORMATIONS ACCORDING TO BISHOP FRAME IN EUCLIDEAN 3-SPACE

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1 iauliai Math. Semin., 7 15), 2012, 4149 BÄCKLUND TRANSFORMATIONS ACCORDING TO BISHOP FRAME IN EUCLIDEAN 3-SPACE Murat Kemal KARACAN, Yilmaz TUNÇER Department of Mathematics, Usak University, Usak, Turkey; s: murat.karacan@usak.edu.tr, yilmaz.tuncer@usak.edu.tr Abstract. In this paper, we dene Bäcklund transformations of curves according to Bishop frame which preserving the natural curvatures under certain assumptions in Euclidean 3-space. Key wor and phrases: Bäcklund transformations, Bishop frame Mathematics Subject Classication: 53A Preliminaries In the 1890s, Bianchi, Lie, and nally Bäcklund looked at what are now called Bäcklund transformations of surfaces. In modern parlance, they begin with two surfaces in Euclidean space in a line congruence: there is a mapping between the surfaces M 1 and M 2 such that the line through any two corresponding points is tangent to both surfaces. Bäcklund proved that if a line congruence satised two additional conditions, that the line segment joining corresponding points has constant length, and that the normals at corresponding points form a constant angle, then the two surfaces are necessarily surfaces of constant negative curvature. He was also able to show that a Bäcklund transformation is integrable, in the sense that given a point on a surface of constant negative curvature and a tangent line segment at that point, a new surface of constant negative curvature can be found, containing the endpoint of the line segment,that is a Bäcklund transform of the original surface.

2 42 M. K. Karacan, Y. Tunçer The classical Bäcklund theorem studies the transformation of surfaces of constant negative curvature in R 3 by realizing them as the focal surfaces of a pseudo-spherical line congruence. The integrability theorem says that we can construct a new surface in R 3 with constant negative curvature from a given one. In 19, Tenenblat and Terng established a high dimension generalization of Bäcklund's theorem which is very interesting both for physical and mathematical reasons. After that, Chern and Terng customized Bäcklund theorem for ane surfaces 6. By the same year, this transformation was reduced to corresponding asymptotical lines by Terng 20 and following years Tenenblat expanded the Bäcklund transformation of two surfaces in R 3 1 to space forms 18. In 1990, Palmer constructed a Bäcklund transformation between spacelike and timelike surfaces of constant negative curvature in E At that decade, some researchers gave Bäcklund transformations on Weingarten surfaces 2, 5, 7 and 21. In 1998, Calini and Ivey 3 proposed a geometric realization of the Bäcklund transformation for the sine-gordon equation in the context of curves of constant torsion. Since the asymptotic lines on a pseudospherical surface have constant torsion, the Bäcklund transformation can be restricted to get a transformation that carries constant torsion curves to constant torsion curves. Later, the converse of the idea was proved and generalized for the n dimensional case by Nemeth 12. In 13, Nemeth studied a similar concept for constant torsion curves in the 3-dimensional constant curvature spaces. Shief and Rogers used an analogue of the classical Bäcklund transformation for the generation of soliton surfaces 16. In 8, Chou, Kouhua and Yongbo obtained the Bäcklund transformation on timelike surfaces with constant mean curvature in R 2,1. Zuo, Chen, Cheng studied Bäcklund theorems in three dimensional de Sitter space and anti-de Sitter space 22. Abdel-Baky presented the Minkowski versions of the Bäcklund theorem and its application by using the method of moving frames 1. Gurbuz studied Bäcklund transformations in R1 n 9. Using the same method Ozdemir and Coken have studied Bäcklund transformations of non-lightlike constant torsion curves in Minkowski 3-space 14. Also, Cengiz and Gurbuz have studied Bäcklund transformations of curves in the Galilean and Pseudo-Galilean spaces 4. In this paper, we show that a restriction of Bäcklund theorem on space curves satisfying the given three conditions preserves the rst and second curvaturesnatural curvatures) of the curves according to Bishop frame in Euclidean 3-space. This section is taken from 4.

3 Bäcklund transformations according to Bishop frame Introduction Let α : I R E 3 be an arbitrary curve in Euclidean 3-space E 3. Recall that the curve α is said to be of unit speed or parametrized by arc length function s) if α, α = 1, where, is the standard scalar inner) product of E 3 given by x, y = x 1 y 1 + x 2 y 2 + x 3 y 3 for each x = x 1, x 2, x 3 ), y = y 1, y 2, y 3 ) E 3. In particular, the norm of a vector x E 3 is given by x = x, x. Denote by {T s), Ns), Bs)} the moving Frenet frame along the unit speed curve α. Then the Frenet formulas are given by T N B = 0 κ 0 κ 0 τ 0 τ 0 Here T, N and B are called the tangent, the principal normal and the binormal vector el of the curves, respectively. κs) and τs) are called, curvature and torsion of the curve α, respectively 18. The ability to ride along a three-dimensional space curve and illustrate the properties of the curve, such as curvature and torsion, would be a great asset to mathematicians. The classic Serret-Frenet frame provides such ability, however, the Serret-Frenet frame does is not dened for all points along every curve. A new frame is needed for the kind of mathematical analysis that is typically done with computer graphics. The relatively parallel adapted frame or Bishop frame could provide the desired means to ride along any given space curve. The Bishop frame has many properties that make it ideal for mathematical research. Another area of interested about the Bishop frame is so-called normal development, or the graph of the twisting motion of Bishop frame. This information along with the initial position and orientation of the Bishop frame provide all of the information necessary to dene the curve. The Bishop frame may have applications in the area of Biology and Computer graphics. For example, it may be possible to compute information about the shape of sequences of DNA using a curve dened by the Bishop frame. The Bishop frame may also provide a new way to control virtual cameras in computer animations 11. The Bishop frame or parallel transport frame is an alternative approach to dening a moving frame that is well dened even when the curve has vanishing second derivative. We can parallel transport an orthonormal frame along a curve simply by parallel transporting each component of the frame. T N B.

4 44 M. K. Karacan, Y. Tunçer The parallel transport frame is based on the observation that, while T s) for a given curve model is unique, we may choose any convenient arbitrary basis Us), V s)) for the remainder of the frame, so long as it is in the normal plane perpendicular to T s) at each point. If the derivatives of Us), V s)) depend only on T s) and not each other, we can make Us) and V s) vary smoothly throughout the path regardless of the curvature. In addition, suppose the curve α is an arc length-parametrized C 2 curve. Suppose we have C 1 unit vector el U and V = T U along the curve α so that T, U = T, V = U, V = 0, i.e., T, U, V will be a smoothly varying right-handed orthonormal frame as we move along the curve to this point, the Frenet frame would work just ne if the curve were C 3 with κ 0). But now we want to impose the extra condition that U, V = 0.We say the unit rst normal vector eld U is parallel along the curve α. This means that the only change of U is in the direction of T. A Bishop frame can be dened even when a Frenet frame cannot e.g., when there are points with κ = 0) 10. Therefore, we have the alternative frame equations One can show that κs) = k1 2 + k2 2, T U V = 0 k 1 k 2 k k T U V. 2.1) ) θs) = arctan k2, k 1 0, τs) = dθs) k 1. So that k 1 and k 2 eectively correspond to a Cartesian coordinate system for the polar coordinates κ, θ with θ = τs). The orientation of the parallel transport frame includes the arbitrary choice of integration constant θ 0, which disappears from τ and hence from the Frenet frame) due to the dierentiation 1011, 17. Thus, the relation matrix may be expressed as T = T, Bishop curvatures are dened by N = U cos θs) V sin θs), B = U sin θs) + V cos θs). k 1 = κs) cos θs), k 2 = κs) sin θs).

5 Bäcklund transformations according to Bishop frame Bäcklund transformation according to Bishop frame Theorem 3.1. Suppose that ψ is a transformation between two curves α and β in Euclidean 3-space with β = ψα) such that in the corresponding points we have: a) the line segment βs)αs) at the intersection of the osculating planes of the curves has constant length r; b) the distance vector βs) αs) has the same angle γ π 2 tangent vectors of the curves; c) the binormals of the curves have the same constant angle ϕ 0. Then these curves are congruent with natural curvatures k β 1 = k α 1 = dγ, k β 2 = k α 2 = tan γ sin ϕ, r and the transformation of the curves is given by where C = k α 2 tan ϕ 2 β = α + 2C tan γ k2 α)2 + C 2 T α cos γ + U α sin γ), and γ is a solution of the dierential equation dγ = kβ 2 cos γ tan ϕ 2 kα 1 = k α 2 cos γ tan ϕ 2 kβ 1. with the Proof. Denote by T α, U α, V α ) and T β, U β, V β ) the Bishop frames of the curves α and β in the Euclidean 3-space E 3. Let V β be a unit second principal normal of β.if we denote by W1 α the unit vector of β α, then we can complete W1 α, V α and W1 α, V β to the positively oriented orthonormal frames W1 α, W 2 α, W 3 α) and W β 1, W β 2, W β 3 ), where W 3 α = V α, W β 3 = V β, and γ is the angle between W1 α and T α. The frames W1 α, W 2 α, W 3 α) and W β 1, W β 2, W β 3 ) can be obtained by rotating the framest α, U α, V α ) and T β, U β, V β ) around V α and V β with an angle γ, respectively. So, we can write W α 1 cos γ sin γ 0 T α W2 α = sin γ cos γ 0 U α W3 α V α and W α 1 W β 2 W β 3 = cos γ sin γ 0 sin γ cos γ T β U β V β.

6 46 M. K. Karacan, Y. Tunçer Similarly, for a rotation around W α 1 by the angle ϕ, From the above equations, we write W β 2 = W α 2 cos ϕ W α 3 sin ϕ, W β 3 = W α 2 sin ϕ + W α 3 cos ϕ. T β = cos 2 γ + sin 2 γ cos ϕ ) T α + cos γ sin γ) 1 cos ϕ) U α + sin γ sin ϕ) V α, 3.1) U β = cos γ sin γ) 1 cos ϕ) T α + sin 2 γ + cos 2 γ cos ϕ ) U α cos γ sin ϕ) V α, 3.2) V β = sin γ sin ϕ) T α + cos γ sin ϕ) U α + cos ϕ) V α. 3.3) Using 2.1) and 3.1), 3.2), 3.3), for T β, U β and V β, we have dt β du β dv β = k β 1 U β + k β 2 V β = k β 1 cos γ sin γ1 cos ϕ) kβ 2 sin γ sin ϕ T α + k β 1 sin 2 γ + cos 2 γ cos ϕ ) + k β 2 ) cos γ sin ϕ) + k β 2 cos ϕ kβ 1 cos γ sin ϕ V α, = k β 1 T β = k β 1 cos 2 γ + sin 2 γ cos ϕ ) T α k β 1 1 cos ϕ) cos γ sin γ) U α k β 1 sin γ sin ϕ) V α, = k β 2 T β = k β 2 cos 2 γ + sin 2 γ cos ϕ ) T α k β 2 1 cos ϕ) cos γ sin γ) U α k β 2 sin γ sin ϕ) V α, U α and taking derivative of T β, U β and V β in 3.1), 3.2), 3.3) with respect to s, we get dt β = 1 cos ϕ) + k1 α 2 dγ ) cos γ sin γ) k2 α sin γ sin ϕ) kα 1 cos 2 γ + sin 2 γ cos ϕ ) + cos 2 γ1 cos ϕ) dγ U α T α

7 Bäcklund transformations according to Bishop frame du β dv β = = + k2 α cos 2 γ + sin 2 γ cos ϕ ) + dγ cos 2 γ1 cos ϕ) dγ cos γ sin ϕ V α, kα 1 sin 2 γ + cos 2 γ cos ϕ ) + k2 α cos γ sin ϕ) T α + 1 cos ϕ) cos γ sin γ) dγ + kα 1 1 cos ϕ) cos γ sin γ) + k2 α 1 cos ϕ) cos γ sin γ) + sin γ sin ϕ) dγ cos γ sin ϕ) dγ sin γ sin ϕ) kα 1 ) k2 α cos ϕ) k α 1 + dγ T α V α, ) U α k α 2 sin γ sin ϕ) V α. Then, equating the two statements above, we obtain k β 2 = k2 α, dγ = k β 2 cos γ tan ϕ 2 kα 1. Similarly, using 2.1) and 3.1), 3.2), 3.3), we have k α 1 + k β 1 = 2 dγ, k α 1 = k β 1 = dγ. Now α is a unit speed curve. Dierentiating and substituting the distance vector β α) 2 = r 2, U α β α = r T α cos γ U α sin γ), 3.4) we nd that β is also a unit speed curve. Next, taking the derivative of 3.4), we obtain T β = r sin γ) k1 α + dγ ) T α + r sin γ) k1 α + dγ ) U α + rk2 α cos γ) V α. From this equation and the Bishop frames 3.1), 3.2) and 3.3), we get k β 2 = kα 2 = tan γ sin ϕ. r

8 48 M. K. Karacan, Y. Tunçer Then, rearranging this equality, we get r = tan γ sin ϕ k α 2 ) 2k2 α tan γ tan ϕ 2 = )). k2 α)2 1 + tan 2 ϕ 2 Finally, with the aid of 3.4), the Bäcklund transformation of the curves is β = α + 2C tan γ k2 α)2 + C 2 T α cos γ + U α sin γ). References 1 R. A. Abdel Baky, The Bäcklund's theorem in Minkowski 3-space R 3 1, Appl. Math. Comput., 160, ). 2 S. G. Buyske, Geometric aspects of Bäcklund transformations of weingarten submanifol, Pacic J. Math., 166, ). 3 A. Calini, T. Ivey, Bäcklund transformations and knots of constant torsion, J. Knot Theor. Ramif., 7, ). 4 S. Cengiz, N. Gurbuz, Bäcklund transformations of curves in the Galilean and pseudo-galilean spaces, v1.pdf 5 W. Chen, H. Li, Weingarten surfaces and Sine-Gordon equation, Sci. China Ser. A, 40, ). 6 S. S. Chern, C. L. Terng, An analogue of Bäcklund's theorem in ane geometry, Rocky Mt. J. Math., 10, ). 7 T. Chou, C. Xifang, Bäcklund transformation on surfaces with ak + bh = c, Chin. J. Contemp. Math., 18, ). 8 T. Chou, Z. Kouhua, T. Yongbo, Bäcklund transformation on surfaces with constant mean curvature in R 2 1, Acta Math. Sci., Ser. B, Engl. Ed., 233), ). 9 N. Gurbuz, Bäcklund transformations of constant torsion curves in R n 1, Hadronic J., 29, ). 10 A. J. Hanson, H. Ma, Parallel transport approach to curve framing, Indiana University, Techreports, TR425, January ).

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