Bivariate B-splines from convex configurations

Abstract

An order-k univariate spline is a function defined over a set S of at least k+2 real parameters, called knots. Such a spline can be obtained as a linear combination of B-splines, each of them being defined over a subset of k+2 consecutive knots of S, called a configuration of S. In the bivariate setting, knots are pairs of reals and B-splines are defined over configurations of k+3 knots. Using convex pseudo-circles, we define a family of configurations that gives rise to bivariate B-splines that retain the fundamental properties of univariate B-splines. We also give an algorithm to construct such configurations.