Range restricted scattered data interpolation using convex combination of cubic Bézier triangles

The construction of range restricted bivariate C1 interpolants to scattered data is considered. Sufficient nonnegativity conditions on the Bézier ordinates are derived to ensure that the nonnegativity of a cubic Bézier triangular patch. The gradients at the data sites are modified if necessary to ensure that the nonnegativity conditions are fulfilled. The interpolating surface is piecewise a convex combination of three cubic Bézier triangular patches with the same set of boundary Bézier ordinates. Its construction is local and is easily extended to include as upper and lower constraints to the interpolant surfaces of the form z=C(x,y) where C(x,y) is a constant, linear, quadratic or cubic polynomial. Moreover, C1 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 numerical examples are presented graphically.