Scattered data interpolation applying rational quadratic C1 splines on refined triangulations

This paper is concerned with bivariate scattered data interpolation. It is assumed that an admissible triangulation of the data sites has been constructed which then is refined in the sense of Powell and Sabin. The aim is to show that a special class of rational quadratic C-1 splines exists which allows the Hermite interpolation problem with given functions values and gradients to be uniquely solvable. The proof is essentially based on Heindl's C-1 condition which is valid also for the present spline class. The occurring tension or rationality parameters may be used to meet further requirements; if the tension parameters increase the rational quadratic interpolants tend, at least on the interior of the triangles, to piecewise linear spline interpolating the function values.