Simplex Spline Bases on the Powell-Sabin 12-Split: Part II

Abstract

For the space \(\mathcal{S}\) of \(C^3\) quintics on the Powell-Sabin 12-split of a triangle, we determine the simplex splines in \(\mathcal{S}\) and the six symmetric simplex spline bases that reduce to a B-spline basis on each edge, have a positive partition of unity, a (barycentric) Marsden identity, and domain points with an intuitive control net. We provide a quasi-interpolant with approximation order 6 and a Lagrange interpolant at the domain points. The latter can be used to show that each basis is stable in the \(L_\infty\) norm, which yields an \(h^2\) bound for the distance between the B\'ezier ordinates and the values of the spline at the corresponding domain points. Finally, for one of these bases we provide \(C^0\), \(C^1\), and \(C^2\) conditions on the control points of two splines on adjacent macrotriangles, and a conversion to the Hermite nodal basis.