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The title really say's it all, but once again is a curve's curvature invariant under rotation and uniform scaling?

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up vote 9 down vote accepted

A curve's curvature is invariant under rotation. Intuitively, a curve turns just as much no matter how it is oriented. More formally, for a curve $\gamma(s)$ that is parametrized by arc length, the curvature is $\kappa(s) = ||\gamma''(s)||$. Rotation does not change the length of the $\gamma''(s)$ vector, only the direction; therefore, rotation does not affect curvature.

A curve's curvature is not invariant under uniform scaling, however. Consider the example of a circle. All circles are the same up to scaling, but they don't all have the same curvature; in general, a circle of radius r has curvature 1/r.

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D'oh, I forget about think of curvature as the radius. – Jonathan Fischoff Jul 31 '10 at 23:44

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