Smooth interpolation to scattered data by bivariate piecewise polynomials of odd degree (Q920542)

An algorithm for constructing bivariate spline interpolants to scattered data is developed. Given scattered data points \(x^ 1,...,x^ n\) in \({\mathbb{R}}^ 2\) and a corresponding triangulation, the authors discuss the dimension of spaces of splines of degree k and smoothness m with respect to this triangulation, and the description of the smoothness properties with the aid of Bézier ordinates. In the first step of the algorithm, a continuous spline p of degree k is constructed which interpolates given data at \(x^ 1,...,x^ n\) and satisfies certain differentiability conditions at these points. In a second step, the interpolant p is perturbed such that the resulting interpolating spline s is m-times continuously differentiable. This approach guarantees that s is as close as possible to p and that polynomials of degree m are reproduced. For a given smoothness m of the spline interpolants, the method keeps their degree k as low as possible. A similar version was already given by \textit{T. A. Grandine} [Comput. Aided Geom. Des. 4, 307-319 (1987; Zbl 0637.65008)] for the case of differentiable cubic splines. Finally, the authors describe several numerical examples for quintic splines of smoothness two.