```Date: Nov 14, 2012 1:41 PM
Author: dy/dx
Subject: Re: Curvature in Cartesian Plane

On Wed, 14 Nov 2012 09:23:06 +1000, Brad Cooper wrote:> I expect that this is true...> > We have three points on a Cartesian x-y plane, and the circle that passes through these three points has a constant curvature of k.> > If we  have a doubly differentiable curve in the x-y plane that passes through these points, is there always some point on the curve which has curvature k?> > I am finding it tough to prove this. Any help appreciated.> > Cheers,> BradIf you're having difficulty proving something, it may be worth consideringthe possibility that it's false.In this case, if I imagine a V-shaped pair of line segments joining thethree points, then rounding the corner of the V so it'stwice-differentiable (but widening the V slightly so the curve still goesthrough the middle point), it's clear that the curvature goes from 0 upthrough k to a higher value at the middle point, then down through k to 0again.However, if I then imagine superimposing a high-frequency "coiling" on thiscurve, like a telephone cord projected down to 2D, arranging that it stillpass through all three points, it seems it should be possible to keep thecurvature everywhere higher than some lower bound B > k. (The curve willnow self-intersect.)
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