Super-smooth cubic Powell-Sabin splines on three-directional triangulations: B-spline representation and subdivision

Starting from a general B-spline representation for C1 cubic Powell-Sabin splines on arbitrary triangulations, we focus on the construction of a B-spline representation for a particular subspace defined on three-directional triangulations with C2 super-smoothness over each of the macro-triangles. We analyze the properties of the basis and point out the relation with simplex splines. Furthermore, we provide explicit expressions for the B-spline coefficients of any element of the spline space, and derive subdivision rules under dyadic refinement. Finally, we show simple conditions ensuring global C2 smoothness on the domain