From: foltinek@math.utexas.edu (Kevin Foltinek) Subject: Re: shape of bent uniform spring-coil Date: 01 Feb 2000 12:03:02 -0600 Newsgroups: sci.math Summary: [missing] In article <876hbp$9pp$1@nnrp1.deja.com> David Bernier writes: > Suppose a muscle-man (AS) bends a flexible spring-coil or rod until > the angle of rotation of the tangent to the curve while going > from one end to the other is \theta => 0 radians. What is the shape > of the curved coil? I'm not sure I'm interpreting the question as you intended it to be interpreted, so I'll rephrase and then say something about the answer. Problem: Find the curves gamma of fixed length L, which minimize total bending energy, and subject to the condition that the two tangents, at the beginning and end of the curve, are parallel. If this is indeed the problem, then the usual approximate solution is to express the total bending energy in terms of the curvature k of the curve: E[gamma] = (1/2) \int_{0}^{L} k^2(s) ds where s is arclength. The solutions to this problem are called elastica. There are variations on this problem, including: drop the constraint on the total length; use different boundary conditions; look at curves in R^2, R^3, S^2, H^2, etc.; and probably some others. If I recall correctly, the solutions involve elliptic integrals. (You'll note that, as I translated your problem, the problem is underdetermined: you need to specify more boundary data.) There are some special cases which are interesting; for example, the closed elastica are torus knots (that is, if your boundary data is that the curve be smooth closed, and total length is constrained, then the elastica lie on the surface of a symmetric torus in R^3). Some people who have recently (within 20 years) worked on this topic include: Joel Langer; David Singer; Phillip Griffiths; Robert Bryant; myself; and lots more whom I have forgotten (my excuse is that I haven't looked much at the problem since around five years ago). Langer and Singer wrote some papers in the 80s which are quite readable (assuming you're very familiar with the Frenet-Seret apparatus of curvature, torsion, Tangent, Normal, Binormal, and also very familiar with the calculus of variations). But there's a complication which needs to be addressed. (If you're looking at curves in three or more dimensions.) That is the question of just how this elastic rod is being held at the ends. Strictly speaking, in order for elastica to be a close approximation to the solution, the ends must be constrained so that they can't move back and forth (along the length of the rod), but free to rotate around the axis of the rod. Physically, you'd build this by welding the ends of the rod into a pair of ball bearing joints, and then holding on to the bearings: || || <- rod || *||* <- ball bearings; clamp here (* represents a weld) || Anyway, if you just want to clamp the rod itself, then things become more complicated, because you're now talking about "framed" elastica; there will be not only bending energy, but also a twisting energy. Someone has worked on this but I've forgotten who. Of course the problem in two dimensions is somewhat easier, because curvature is the change of angle of the tangent vector, and total curvature == 0 mod 2pi is the boundary condition which I think you want. The fact that you can explicitly write this down (which you can't really do in three dimensions) probably helps a bit. :-) Kevin.