@MISC{Ciulla_classic-curvature:a, author = {Carlo Ciulla}, title = {Classic-Curvature: A Second Order Derivatives Based Approach}, year = {} }

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Abstract

Abstract: Given a signal data in either one or in multiple dimensions, and given a model function fitted to the signal data, it is possible to calculate the curvature through the computation of all of the second order derivatives of the model function using rigorous calculus methodologies. This manuscript emphasizes on the aforementioned statement bringing to the attention that convolution operators used to approximate both of first and second order derivatives of the signal data, or filters used to approximate second order derivatives, do not allow the representation of the curvature in its full geometrical meaning. The geometrical meaning of the curvature of the signal data is that one of the arctangent of the angle formed by the tangent-line to the first order derivative of the modeled signal data. The curvature calculated summing all of the second order derivatives of Hessian of the model function fitted to the data is here termed as classic-curvature. Presentation of qualitative results is given with regard to polynomials B-Spline and Lagrange in two and three dimensions; thus confirming that regardless of the dimensionality of the problem, it is possible to advance to the calculation of the curvature through the computation of all of the second order derivatives of the modeled signal, and to reveal the relationships existing between the modeled signal and the second order derivative.