Timeline for Uniqueness affine curvature

4 events

when toggle format	what		by	license	comment
Mar 13, 2015 at 18:59	vote	accept	xyz abcd
Mar 13, 2015 at 14:28	comment	added	Robert Bryant		@xyzabcd: Actually, you do not have the Euclidean theorem stated correctly. What is true is that the curvature plus the element of arc determines the curve up to Euclidean motions; thus, you need two things to determine the curve geometrically. Moreover, when the common curvature of the two curves $\gamma_i:\mathbb{S}^1\to\mathbb{R}^2$ that you want to compare is not constant, you can't usually use reparametrization to 'line things up' if they have different elements of arc, because most reparametrizations that line up the elements of arc won't preserve the curvature.
Mar 13, 2015 at 0:18	comment	added	xyz abcd		If we apply your construction to curvature, instead of affine curvature, using two circles with areas a,b, then does not your construction suggest that there are two closed, convex curves, that their Euclidean curvatures are equal, although they are not related by Euclidean motions. This would contradict the fact that the curvature in the plane determines the curve up Euclidean motions.
Mar 12, 2015 at 23:15	history	answered	Anton Petrunin	CC BY-SA 3.0