# Question about ellipses and calculus of variations

I don't know much about calculus of variation, but I think it applies to a problem I've come across. If you have a closed loop in a 2 dimensional space defined by some parametric equation r(t), is there some minimization or maximization of parameters, like arc length or enclosed area, that will yield a family of ellipses as the solution?

I'm asking because a professor showed me an iterative process that seems to take any randomly generated polygon and by averaging the lengths of the edges, somehow "untangles" the polygon and over many iterations the shape converges to an ellipse. They had no proof for the convergence to the ellipse. I was wondering if it could be approached as a discretization of some continuous minimization process.

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Two characterizations that might lead to a variational problem for which families of ellipses are the solution are (i) the pins-and-strings locus and (ii) the image of the unit circle under a linear transformation. Do you recall any more about the iterative "averaging" process? –  anon Mar 1 '12 at 2:02