# B-spline-like bases for $C^2$ cubics on the Powell-Sabin 12-split

## TL;DR

This paper introduces and analyzes B-spline-like bases, called S-bases, for $C^2$ cubic splines on the Powell-Sabin 12-split, providing formal definitions, existence results, and smoothness conditions.

## Contribution

It formally defines S-bases for spline spaces on the Powell-Sabin 12-split and investigates their existence, especially for cubic cases, along with deriving differentiation, recurrence formulas, and smoothness conditions.

## Key findings

- Existence of S-bases for certain spline spaces established.
- Derived simple formulas for differentiation and recurrence of S-bases.
- Established a Marsden identity leading to quasi-interpolants and control nets.

## 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 $C^0$-, $C^1$-, and $C^2$-smoothness are derived.