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

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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.

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