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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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William Elliot

Posts: 2,637
Registered: 1/8/12
Re: Curvature on a curve and circle
Posted: Mar 4, 2013 3:00 AM
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On Mon, 4 Mar 2013, 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.

As that's was quickly written, I'll state what I
think the problem you're asking is. It this correct?

Let p:[0.h] -> R^2 be a planar loop with p(0) = p(h) = (0,0)
Let s0 be the length of p and k(x) the curvature of p at x.
Shaw that there's some x in [0,k] with k(x) = k where
k is the curvature of an arc from (0,0) to (0,h) with length s0.

> I have set up a CAS program to simulate the situation and the proposition
> held up in every case.
> I have been working with the idea that, at the required point, the normal to
> f(x) is normal to the circle.
> I am not making much headway. Any ideas appreciated.

It smells like Rolle's theorem.

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