Geometric Splines and Interpolation on S^2: Numerical Experiments

Abstract

Several different procedures are presented to produce smooth interpolating curves on the two-sphere S2. The first class of methods is a combination of the pull back/push forward technique with unrolling data from S2 into a tangent plane, solving there the interpolation problem, and then wrapping the resulting interpolation curve back to the manifold. The second method results from converting a variational problem into a finite dimensional optimisation problem by a proper discretisation process. It turns out that the resulting curves look very similar. The main difference though is that the first approach gives closed form solutions to the interpolation problem, whereas the second method results in a finite number of points. These points then require further treatment, e.g. one could connect them by geodesic arcs, i.e. by great circle segments, to get an approximate solution to the variational problem. Although the result would not be smooth, it seems to be the best that one can get if the dicretisation process is combined with a sufficiently cheap interpolation procedure.