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What is the difference between norm of second derivative and curvature?


Good question. In Schoenberg's case (if I remember correctly), he approximates a curve by its height over a given tangent line, because he only wants to consider small displacements. What's the difference between its second derivative of a height function and the curvature of its graph? Quite a bit, actually; take a look at the Wikipedia article on curvature for some explicit calculations.