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 $L_\infty$