
Re: Curvature on a curve and circle
Posted:
Mar 4, 2013 2:09 AM


On Mon, 04 Mar 2013 13:16:22 +1000, Brad Cooper wrote: > The "well behaved, smooth" function f(x) has endpoints f(0) = f(h) = 0. The > curve of the function has length s1. > > An arc of a circle passing through (0, 0) and (0, h) has fixed curvature k > and its arc length is also s1. > > It is required to show that a point must exist on f(x) where curvature is > also k. ... > I am not making much headway. Any ideas appreciated. ...
Presumably "well behaved, smooth" f has a continuous curvature function, say K(f,x) for the curvature of f at x. By continuity, unless a point x exists where K(f,x) = k, either K(f,x) < k for all x, or K(f,x) > k for all x.
From <http://en.wikipedia.org/wiki/Total_curvature> which says "the total curvature of an immersed plane curve is the integral of curvature along a curve taken with respect to arclength" and that "The total curvature of a closed curve is always an integer multiple of 2?" (that last symbol is pi), and with a bit of additional reasoning about the integral of curvature across a couple of discontinuities in curvature, you can conclude that the integral from 0 to h of the curvature of f equals the integral from 0 to h of the curvature of the arc. Etc.
 jiw

