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

Lyche, Tom; Muntingh, Georg

The SMAI Journal of computational mathematics, Tome S5 (2019), pp. 129-159.

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 Bézier-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.

Classification : 41A15, 65D07, 65D17
Mots clés : Stable bases, Powell–Sabin 12-split, Simplex splines, Marsden identity, Quasi-interpolation

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