Abstract: Triangular B-splines lend themselves naturally to problems such as blending and the filling of polygonal holes. Here we present an automatic method for smoothly blending piecewise polynomial surfaces using triangular B-splines. The method proceeds in two phases. In the first phase, the domain of the region to be blended or filled is triangulated and populated with basis functions. In the second phase, coefficients for these basis functions are found by minimizing a functional that measures the curvature of the blending or filling surface. Examples are provided that show the use of this method for a number of blending and filling problems.