Geometric Splines and Interpolation on S2: Numerical Experiments

Several different procedures are presented to produce smooth interpolating curves on the two-sphere S
²
. The first class of methods is a combination of the pull back/push forward technique with unrolling data from S
²
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 discretisation process is combined with a sufficiently cheap interpolation procedure