B-spline-like bases for $C^2$ cubics on the Powell-Sabin 12-split
Abstract
For spaces of constant, linear, and quadratic splines of maximal smoothness on the Powell-Sabin 12-split of a triangle, the so-called S-bases were recently introduced. These are simplex spline bases with B-spline-like properties on the 12-split of a single triangle, which are tied together across triangles in a B\'ezier-like manner. In this paper we give a formal definition of an S-basis in terms of certain basic properties. We proceed to investigate the existence of S-bases for the aforementioned spaces and additionally the cubic case, resulting in an exhaustive list. From their nature as simplex splines, we derive simple differentiation and recurrence formulas to other S-bases. We establish a Marsden identity that gives rise to various quasi-interpolants and domain points forming an intuitive control net, in terms of which conditions for -, -, and -smoothness are derived.
Cite
@article{arxiv.1901.06885,
title = {B-spline-like bases for $C^2$ cubics on the Powell-Sabin 12-split},
author = {Tom Lyche and Georg Muntingh},
journal= {arXiv preprint arXiv:1901.06885},
year = {2019}
}