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
<bernier@my-deja.com> 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
     ||
 <o>*||*<o> <-  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.