Inextensible Flows of Curves in Minkowskian Space

Transcription

1 Adv. Studies Theor. Phys., Vol. 2, 28, no. 16, Inextensible Flows of Curves in Minkowskian Space Dariush Latifi Department of Mathematics, Faculty of Science University of Mohaghegh Ardabili P.O. Box , Ardabil, Iran Asadollah Razavi Faculty of Mathematics and Computer Science Amirkabir University of Technology No. 424, Hafez Ave., Tehran, Iran Abstract In this paper we investigate inextensible flow of curves in Minkowski 3-space. Necessary and sufficient conditions for an inextensible curve flow are expressed as a partial differential equation involving the curvature and torsion. Mathematics Subject Classification: 53C44; 53A4; 53A5 Keywords: Minkowski plane curve, Heat flow, Curvature flow, Inextensible 1 Introduction It is well known that many nonlinear phenomena in physics, chemistry and biology are described by dynamics of shapes, such as curves and surfaces, and the evolution of curve and surface has significant applications in computer vision and image processing [15]. It has been known that a great number of nonlinear evolution equations are related to motion of curves in certain geometries [14, 13]. For instance, the Mullins s nonlinear diffusion model of groove development [13] describes the curve shortening problem. Hasimoto [7] showed that the Schrödinger equation arises from motion of inextensible curves in R 3.

2 762 D. Latifi and A. Razavi The flow of a curve is said to be inextensible if the arclength is preserved. Physically, inextensible curve flows give rise to motions in which no strain energy is induced. The swinging motion of a cord of fixed length, for example can be described by inextensible curve flows. Such motions arise quite naturally in a wide range of physical applications. For example, both Chirikjian and Burdick [1] and Mochiyama et al. [12] study the shape control of hyper-redundant, or snake-like, robots. Inextensible curve flows also arise in the context of many problem in computer vision [8, 11] and computer animation [2], and even structural mechanics [16]. What the above problems share in common is the need to mathematically describe the inextensible time evolution of curves. There have been numerous studies in the literature on plane curve flows, particularly on evolving curves in the direction of their curvature vector field. Particularly relevant to this paper are the methods developed by Gage and Hamilton [4] and Grayson [6] for studying the shrinking of closed plane curves to a circle via the heat equation. In [5] Gage also studies area-preserving evolutions of plane curves. In [9, 1] Kwon et al. study inextensible flows of curves and developable surface in R 3. In this paper we investigate inextensible flow of curves in Minkowski 3-space. Necessary and sufficient conditions for an inextensible curve flow are expressed as a partial differential equation involving the curvature and torsion. We use some idea from [9, 1] in this paper. 2 Inextensible curve flows in Minkowski space Let V 3 define a three-dimensional flat space with the line element ds 2 = η μν dx μ dx ν, 1 where μ, ν =1, 2, 3,, x μ =x, y, z and η μν = dig1,ɛ,ɛ. If ɛ =1, then V 3 = E 3 is a Euclidean 3 space and if ɛ = 1 then V 3 = M 3 is a Minkowskian 3 space. Hence, Eq.1 explicitly takes the form ds 2 = dx 2 + ɛdy 2 + ɛdz 2. Let C be a curve on V 3 defined by α : I V 3 and parameterized by its arc length s I. An orthonormal frame T μ,n μ,b μ at each point of C is defined by η μν T μ T ν =1, η μν N μ N ν = ɛ, η μν B μ B ν = ɛ,

3 Inextensible flows of curves in Minkowskian space 763 where T = dα, all the other products vanish. ds The Serret-Frenet equations are dt μ ds = knμ, dn μ ds = ɛkt μ τb μ, db μ ds = τnμ, where k and τ are the curvature and the torsion scalar of the curve C at any point s. The vectors T,N,and B are, respectively, the tangent, normal and bi-normal vectors to the curve at any point s [3]. If ɛ = 1 then we write <v,w>for the value ηv, w. A curve C, locally parameterized by α : I R M 3, is said to be timelike, spacelike, or null curve if < dα, dα > is positive, negative, or zero, respectively. dt dt Throughout this article, we assume that F :[,l] [,ω] M 3 is a oneparameter family of smooth timelike curves in Minkowski space, where l is the arclength of the initial curve. Let u be the curve parametrization variable, u l. The arclength of F is given by where F su = u F du = < F, F > 1 2. The operator = 1 v, is given in terms of u by where v = F. The arclength parameter is ds = vdu. Any flow of F can be represented as F = ft + gn + hb. Letting the arclength variation be su, t = u vdu, in the Euclidean space the requirement that the curve not be subject to any elongation or compression can be expressed by the condition su, t = u vdu = for all u [,l]. Definition 2.1 A curve evolution F u, t and its flow F in M 3 are said to be inextensible if F =. The necessary and sufficient conditions for inextensible flow in M 3 are then given by the following theorem.

4 764 D. Latifi and A. Razavi Theorem 2.1 Let F = ft + gn + hb be a smooth flow of the timelike curve F. The flow is inextensible if and only if f = gk. Proof: Since F is timelike we have v 2 =< F, F >. and commute since u and t are independent coordinates. So we have 2v v Thus we get Now let F = F, F F = 2, F F = 2, ft + gn + hb = 2v T, f g T + fvkn + f = 2v + gvk N + gkvt vτb+ h B + hvτn v = f + gvk 2 be extensible. From 2 we have su, t = = = u u v du f + gvk du for all u [,l]. This implies that f f = gvk, or = gk. The argument can be reversed to show sufficiency, completing the proof. We now restrict ourselves to arclength parameterized curves. That is, v = 1, and the local coordinate u corresponds to the curve arclength s. We require the following lemma. Lemma 2.1 T = fk + g + hτ N + gτ + h B, N = fk + g + hτ T + ψb, B h = gτ T ψn, where ψ = N,B.

5 Inextensible flows of curves in Minkowskian space 765 Proof: Using the Frenet-Serret equations and Theorem 2.1, we calculate T = F = ft + gn + hb = f g T + fkn + = fk + g + hτ N + gkt τb+ h B + hτn N + gτ + h B. Now differentiate the Frenet frame by t: = T T,N =,N + T, N = fk + g = T T,B =,B + T, B = gτ h + = N N,B =,B + N, B = ψ + N, B + hτ T, B, +, T, N, From the above we obtain N = fk + g + hτ T +ψb and B = h gτ T ψn, since N,N = B,B =. The following theorem states the conditions on the curvature and torsion for the curve flow F s, t to be inextensible. Theorem 2.2 Suppose the curve flow F = ft + gn + hb is inextensible. Then the following system of partial differential equations holds: Proof: Noting that T T k = g fk hτ gτ2 + τ h τ = k h ψ gτ kψ = τfk + g + hτ h gτ+ 2 2 = T, = [ fk + g h + hτn + gτ + g = fk hτ N + + gτ+ 2 h 2 B + gτ + h ] B fk + g τn, + hτ kt τb

6 766 D. Latifi and A. Razavi while T = kn = k N + k [ fk + g ] + hτt + ψb Hence we see that k = g fk hτ gτ2 + τ h and kψ = τ fk + g + hτ h gτ Since B while = B, we have B = = [ h 2 h 2 gτ ] gτt ψn T + ψ N ψkt τb, h gτ kn B = τn = τ N + τ [ fk + g ] + hτt + ψb. Thus τ h = k gτ ψ No other new formulas are obtained from the relation N = N. 3 Inextensible flows of Minkowski plane curves Let us investigate with more details the case of torsionless timelike inextensible flows in Minkowski space. for the sake of simplicity, let us chose our coordinate system such that the

7 Inextensible flows of curves in Minkowskian space 767 curve evolution F s, t takes place in the x, y plane. Then, the Serret-Frenet equations yields dt ds dn ds db ds = kn = kt 3 = Choosing B as the usual unit vector k in the z direction, it remains to solve the two-dimensional system of differential equation for T and N. As it can easily be verified, the general solution of 3 is given by s s F s, t = coshθsds + a, sinhθsds+, where the function θs is given in terms of the curvature k = ks byθs = ksds + φ, with a, b and φ being arbitrary constants. s References [1] G. Chirikjian, J. Burdick, A modal approach to hyper-redundant manipulator kinematics, IEEE Trans. Robot. Autom [2] M. Desbrun, M.-P. Cani-Gascuel, Active implicit surface for animation, in: Proc. Graphics Interface Canadian Inf. Process. Soc., 1998, pp [3] M. do Carmo, Differential geometry of curves and surfaces, Prentice-Hall, Englewood Cliffs, [4] M. Gage, R.S. Hamilton, The heat equation shrinking convex plane curves, J. Differential Geom [5] M. Gage, On an area-preserving evolution equation for plane curves, Contemp. Math [6] M. Grayson, The heat equation shrinks embedded plane cures to round points, J. Differential Geom [7] H. Hasimoto, A soliton on a vortex filament, J. Fluid. Mech [8] M. Kass, A. Witkin, D. Terzopoulos, Snakes: active contour models, in: Proc. 1st Int. Conference on Computer Vision, 1987, pp

8 768 D. Latifi and A. Razavi [9] D. Y. Kwon, F.C. Park, Evolution of inelastic plane curves, Appl. Math. Lett [1] D. Y. Kwon, F.C. Park, D.P. Chi, Inextensible flows of curves and developable surfaces, Appl. Math. Lett [11] H.Q. Lu, J.S. Todhunter, T.W. Sze, Congruence conditions for nonplanar developable surfaces and their application to surface recognition, CVGIP, Image Udrest [12] H. Mochiyama, E. Shimemura, H. Kobayashi, Shape control of manipulators with hyper degrees of freedom, Int. J. Robot. Res [13] W.W. Mullins, Theory of thermal grooving, J. Appl. Phys [14] Pj. Olver, Equivalence, invariants and symmetry. Cambridge: Cambridge Univ. Press; [15] G. Sapiro, Geometric partial differential equations and image analysis. Cambridge: Cambridge Univ. Press; 21. [16] D.J. Unger, Developable surfaces in elastoplastic fracture mechanics, Int. J. Fract Received: February 18, 28