Bivariate simplex B-splines: a new paradigm

A construction of bivariate splines is described, based on a new concept of higher degree Delaunay configurations. The crux of this construction is that knot-sets for simplex B-splines are selected by considering groups of "nearby" knots. The new approach gives rise to a natural generalization of univariate splines in that the linear span of this collection of B-splines forms a space which is analogous to the classical univariate splines. This new spline space depends uniquely and in a local way on the prescribed knot locations, and there is no need to use auxiliary or perturbed knots as in some earlier constructions.