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This is a followup to this question, where I learned that curvature is invariant to rotations.

I have learned of a version of curvature that is invariant under affine transformations.

I am wondering if there a is a form of curvature between the two. Invariant under uniform scaling and rotation but not all affine transformations?

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I don't know if this would suit you, but one thing you can consider (much more naive than the notion of affine curvature) is to fix a point P_0 on your curve, and then consider the function on the curve given by sending a point P to the quantity

curvature(P)/curvature(P_0) .

This is a kind of relative curvature, where you measure how much everything is curving in comparison to the curvature at P_0, and is invariant under scaling and rotation.

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    You're very welcome!2010-08-01