  We study variational problems for curves approximated by B-spline curves. We
show that, one can obtain discrete Euler-Lagrange equations, for the data
describing the approximated curves. Our main application is to the curve
completion problem in 2D and 3D. In this case, the aim is to find various
aesthetically pleasing solutions as opposed to a solution of a physical
problem. The Lagrangians of interest are invariant under the special Euclidean
group action for which B-spline approximated curves are well suited. Smooth
Lagrangians with special Euclidean symmetries involve curvature, torsion, and
arc length. Expressions in these, in the original coordinates, are highly
complex. We show that, by contrast, relatively simple discrete Lagrangians
offer excellent results for the curve completion problem. The methods we
develop for the discrete curve completion problem are general and can be used
to solve other discrete variational problems for B-spline curves.
