Consider a simple, smooth (so that it has a well-defined tangent vector at every

label Calculus
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Consider a simple, smooth (so that it has a well-defined tangent vector at every point), closed planar curve of total length L. Prove that the total integral of curvature with respect to arclength is the same for every such curve: integral 0 to L (κdl) = 2π

Aug 3rd, 2015

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Any arc of total curvature L < π has distortion at most sec( L/2).

This can be proved using Fenchel theorem that states that the total curvature of a closed space curve X is greater than or equal to 2, i.e. k(s)ds 2.

Pick any two distinct points p, q on the curve. (We didn’t intend the theorem to apply to the constant curve!) The inscribed polygonal loop from p to q and back has total curvature 2, so the original curve has at least this much curvature.

Please let me know if you need any clarification. I'm always happy to answer your questions.
Aug 3rd, 2015

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Aug 3rd, 2015
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Aug 3rd, 2015
Sep 22nd, 2017
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