O. Davydov and L. L. Schumaker, Interpolation and scattered data fitting on manifolds using projected Powell-Sabin splines, IMA J. Numer. Anal., 28 (2008), 785--805. DOI 10.1093/imanum/drm033

Abstract: We present methods for either interpolating data or for fitting scattered data on a two-dimensional smooth manifold $\Omega$. The methods are based on a local bivariate Powell-Sabin interpolation scheme, and make use of a family of charts $\{(U_\xi,\phi_\xi)\}_{\xi\in\Omega}$ satisfying certain conditions of smooth dependence on $\xi$. If $\Omega$ is a $C^2$-manifold embedded into $\RR^3$, then projections into tangent planes can be employed. The data fitting method is a two-stage method. We prove that the resulting function on the manifold is continuously differentiable, and establish error bounds for both methods for the case when the data are generated by a smooth function.

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