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Topic: Curvature on a curve and circle
Replies: 6   Last Post: Mar 11, 2013 3:54 PM

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James Waldby

Posts: 545
Registered: 1/27/11
Re: Curvature on a curve and circle
Posted: Mar 4, 2013 2:09 AM
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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 <> 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.


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