Franklin writes: << Actually, I would think that two degrees of freedom is the ideal: one for how long the oval is compared to its width (eccentricity), and one for relative size of big end vs. small end (ovalicity?).
I agree. In fact, here's a somewhat imprecise physical basis for determining such an oval (as I mentioned to WFL): Let kmin and kmax, 0 < kmin < kmax, be the curvatures at the two local maxima our oval will have. Now choose a radius r intermediate between rmin = 1/kmax and rmax = 1/kmin. (A natural choice is the arithmetic mean of rmin and rmax, i.e., r = 1/harmonic_mean(kmin, kmax), since 1/curvature = radius of osculating circle.) Now consider a perfect circle C of radius r made a thin strip of spring steel (an idealized such circle would be ideal). Constrain one point of C to have curvature kmin, and the antipodal point to have curvature kmax. Then let C relax into its natural shape under these constraints. That's my ideal oval. --Dan