Convexity-preserving scattered data interpolation

Abstract

This study deals with constructing a convexity-preserving bivariate $C^1$ interpolants to scattered data whenever the original data are convex. Sufficient conditions on lower bound of B\'{e}zier points are derived in order to ensure that surfaces comprising cubic B\'{e}zier triangular patches are always convex 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 be implemented. Graphical examples are presented using several test functions. Keywords: Scattered data; interpolation; convexity; continuity.