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We study variational problems for curves approximated by B-spline curves. Weshow that, one can obtain discrete Euler-Lagrange equations, for the datadescribing the approximated curves. Our main application is to the curvecompletion problem in 2D and 3D. In this case, the aim is to find variousaesthetically pleasing solutions as opposed to a solution of a physicalproblem. The Lagrangians of interest are invariant under the special Euclideangroup action for which B-spline approximated curves are well suited. SmoothLagrangians with special Euclidean symmetries involve curvature, torsion, andarc length. Expressions in these, in the original coordinates, are highlycomplex. We show that, by contrast, relatively simple discrete Lagrangiansoffer excellent results for the curve completion problem. The methods wedevelop for the discrete curve completion problem are general and can be usedto solve other discrete variational problems for B-spline curves.

Discrete Variational Calculus for B-spline Approximated Curves