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B-spline-like bases for C 2 cubics on the Powell–Sabin 12-split
Tom Lyche1; Georg Muntingh2
1 University of Oslo, Department of Mathematics, P.O. Box 1053, Blindern, NO-0316, Oslo, Norway
2 SINTEF Digital, Department of Mathematics and Cybernetics, P.O. Box 124 Blindern, NO-0314, Oslo, Norway
The SMAI Journal of computational mathematics, Volume 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.

Published online:
DOI: 10.5802/smai-jcm.56
Classification: 41A15, 65D07, 65D17
Keywords: Stable bases, Powell–Sabin 12-split, Simplex splines, Marsden identity, Quasi-interpolation

Tom Lyche 1; Georg Muntingh 2

1 University of Oslo, Department of Mathematics, P.O. Box 1053, Blindern, NO-0316, Oslo, Norway
2 SINTEF Digital, Department of Mathematics and Cybernetics, P.O. Box 124 Blindern, NO-0314, Oslo, Norway
License: CC-BY-NC-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
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Tom Lyche; Georg Muntingh. B-spline-like bases for $C^2$ cubics on the Powell–Sabin 12-split. The SMAI Journal of computational mathematics, Volume S5 (2019), pp. 129-159. doi : 10.5802/smai-jcm.56. https://smai-jcm.centre-mersenne.org/articles/10.5802/smai-jcm.56/

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