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

TL;DR

This paper characterizes simplex spline bases for $C^3$ quintic splines on the Powell-Sabin 12-split, introducing stable bases with desirable properties and providing tools for spline interpolation and smoothness conditions.

Contribution

It identifies six symmetric simplex spline bases with key properties and develops interpolation, stability analysis, and smoothness conditions for these bases.

Findings

Six symmetric simplex spline bases with positive partition of unity.

A quasi-interpolant achieving approximation order 6.

Stability bounds and smoothness conditions for adjacent spline control points.

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.