Range Restricted Positivity-Preserving G Scattered Data Interpolation

Abstract : The construction of a range restricted bivariate G 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 quartic Bézier triangular patches are always positive and satisfy G continuity conditions. The gradients at the data sites are then calculated (and modified if necessary) to ensure that these conditions are satisfied. Its construction is local and easily extended to include as upper and lower constraints to the interpolating surfaces of the form z = C(x,y) where C is a polynomial of degree less or equal to 4. Moreover, G piecewise polynomial surfaces consisting of polynomial pieces of the form z = C(x,y) on the triangulation of the data sites are also admissible constraints. A number of examples are presented.