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I have not heard of any decent way to ascertain the curvature of a given curve (I'm sure there is one), so I have conversing the matter with some of the more clever people I know. There seem to be two sensible approaches: to use the previously determined length of the curve between two points on it (come to think of it this only works for regular planar functions) and to subtract the length of a straight line between those two points. All "additional" length will have been generated by curving, so by dividing the value by the difference between the values of x of the two points we may (or may not) have reached a value for the curvature.

Another possibly viable approach would be to derive the function for two points:

1/f'(P1)

would then be the normal for the tangent of the curve. The two tangents will meet at some point (x, y). If a circle is drawn thus that it is centered at (x, y) and that P1 and P2 are on it's radius, a larger circle will denote less curvature (draw a picture and figure it out). This only works if the points are infinitely close to each other, so some kind of limit calculation ought to be done. The curvature of a given bit of the curve would then be the Riemann integral function of the curvature function. Or something like that.

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