-----Original Message----- From: math-fun-bounces+andy.latto=pobox.com@mailman.xmission.com [mailto:math-fun-bounces+andy.latto=pobox.com@mailman.xmission .com] On Behalf Of Daniel Asimov Sent: Tuesday, February 27, 2007 5:20 PM To: math-fun Subject: Re: [math-fun] Re: Simplest Ovals
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.
This doesn't work; the result is only infinitesimally different from a circle of radius R. Spring steel minds sudden changes of the first derivative, but does not resist sudden changes of the second derivative. So you end up with a circle of radius R, except for a slight distortion in the billionth of an inch near each point of constrained curvature. Andy Latto andy.latto@pobox.com