Constrained C1 interpolation on rectangular grids

This paper is concerned with the range restricted interpolation of data on rectangular grids. The interpolant is constrained to lie on the same side of the constraint surface as the data. Sufficient non-negativity conditions on the Bezier ordinates are derived to ensure the non-negativity of a bicubic Bezier patch. The method modifies Bezier ordinates locally to fulfill the sufficient non-negativity conditions. The C¹ interpolating surface is constructed piecewise as a convex combination of two bicubic Bezier patches with the same set of boundary Bezier ordinates. The set of admissible constraint surfaces include polynomial surfaces of the form z = C(x, y) where C(x, y) =Σ_i=0³Σ_j=0³a_i.jxⁱy^j and the a_ij are real numbers, as well as C¹ spline surfaces consisting of polynomial pieces of the form z = C(x, y) on the rectangular grid. Some graphical examples are presented.