Positivity-Preserving Scattered Data Interpolating Surface using C^1 Piecewise Cubic Triangular Patches

The construction of a bivariate C^1 interpolant to scattered data is considered in which the interpolant is positive everywhere if the original data are positive. This study is motivated by earlier work in which sufficient conditions are derived on Bézier points in order to ensure that surfaces comprising cubic Bézier triangular patches are always positive and satisfy C^1 continuity conditions. Initial gradients at the data sites are estimated and then modified if necessary to ensure that these conditions are satisfied. The construction is local and easy to implement. Graphical examples are presented using two test functions.