Some knowledge of the recent history of multivariate interpolation helps in appreciating the problem of constructing C1 interpolants of data values that are associated with arbitrarily distributed nodes on the surface of a sphere. Recent approaches include:
(1) C0 multivariate interpolants [1],
(2) C1 interpolants with nodes on the boundary of a planar region [2],
(3) C1 interpolants with nodes arbitrarily distributed over a plane (see, e.g., [3]), and
(4) the present problem (e.g., [4]).
Here the author extends to a sphere a planar interpolation scheme based on (Thiessen) triangulation of the nodes and estimation of gradients at the nodes (e.g., [3]). Since Lawson has also done this [4], Renka here makes several claims for his variations in the details of the shared approach to the problem. For the weighted least squares fit of a quadratic to data values at nearby nodes to estimate partial derivatives, the method of [3] produces superior accuracy to that of [4]. For univariate interpolation along line segments interior to the triangle and its extension to the sphere, the use of “the planar interpolant on the underlying planar triangle with boundary data obtained by projecting values and gradients onto the plane,” is both more efficient and more accurate than that of [4]. The author implemented his algorithm and Lawson’s “6-arc method” as Method I and Method II, respectively, on the IBM 3033 at the Oak Ridge National Laboratory and compared the programs for four sets of test data. He obtained the following results: “Method I was significantly more accurate than Method II in all cases. Since the difference in accuracy increases with density of the nodal distribution, this result would likely be reversed on a very sparse distribution.” Overall, “Method I is shown to be about 1.5 times as fast as Method II.”
A description of ACM Transactions on Mathematical Software Algorithm 623, Interpolation on the Surface of a Sphere, follows the main article. Program listings are available from the ACM Algorithm Distribution Service.
