Epi-convergence of discrete elastica

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Abstract
Curves that pass through specified locations with specified orientations and minimize an energy functional are called elastica. While physical splines readily assume minimal energy configurations, finding the numerical solutions of variational problems involving integrals of nonlinear functions of the curvature remains quite a formidable challenge.Approximate solutions of such problems yield satisfactory results and the computeraided design field relies heavily on polynomial or rational curve designs. In this paper we discuss a method for discretizing the problem of nonlinear spline design, an alternative to the more traditional approach of discretizing the differential equations that solve the variational problems involved. We show that discretizing the energy functionals (i.e, considering polygonal approximations of the curves and finding the ones that minimize their “energy” defined directly in terms of turn angles and segment length) is an approach that is simpler and leads to solutions that, in the limit of very small segment lengths, converge to the optimal continuous solutions.

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