O. Davydov and L. L. Schumaker, Scattered data fitting on surfaces using projected Powell-Sabin splines, in  "Mathematics of Surfaces 2007," (R. Martin, M. Sabin, and J. Winkler, Eds.), pp.~138--153, LNCS 4647, Springer-Verlag, 2007.

Abstract: We present C1 methods for either interpolating data or for tting scattered data associated with a smooth function on a two-dimensional smooth manifold embedded into R3. The methods are based on a local bivariate Powell-Sabin interpolation scheme and make use of local projections on the tangent planes. The data tting method is a two-stage method. We illustrate the performance of the algorithms with some numerical examples, which, in particular, conrm the O(h3) order of convergence as the data becomes dense.

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