A normalized representation of super splines of arbitrary degree on Powell–Sabin triangulations

Abstract

In the paper, a family of bivariate super spline spaces of arbitrary degree defined on a triangulation with Powell–Sabin refinement is introduced. It includes known spaces of arbitrary smoothness r and degree $$3r-1$$ but provides also other choices of spline degree for the same r which, in particular, generalize a known space of $$\mathscr {C}^{1}$$ cubic super splines. Minimal determining sets of the proposed super spline spaces of arbitrary degree are presented, and the interpolation problems that uniquely specify their elements are provided. Furthermore, a normalized representation of the discussed splines is considered. It is based on the definition of basis functions that have local supports, are nonnegative, and form a partition of unity. The basis functions share numerous similarities with classical univariate B-splines.