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B-spline-like simplex spline bases on the Powell-Sabin 12-split
| Content Provider | Semantic Scholar |
|---|---|
| Author | Muntingh, Georg |
| Copyright Year | 2018 |
| Abstract | For spaces of constant, linear, and quadratic splines of maximal smoothness on the Powell-Sabin 12-split of a triangle, Cohen, Lyche and Riesenfeld recently discovered so-called S-bases [1]. Later, S-bases of higher smoothness [2] and on the Clough-Tocher split [3] were discovered. These S-bases are simplex spline bases with B-spline-like properties on a single macrotriangle, which are tied together across macrotriangles in a Bézier-like manner. In this presentation we extend these results to the space of C cubic splines on the 12-split, as described in [4]. 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 on the 12-split 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, C, and C-smoothness are derived. Although the cubic bases on the 12-split can only be used to define smooth surfaces over specific triangulations, we envision applications for local constructions, such as hybrid meshes and extra-ordinary points. |
| File Format | PDF HTM / HTML |
| Alternate Webpage(s) | https://cs2018.sciencesconf.org/211297/document |
| Language | English |
| Access Restriction | Open |
| Content Type | Text |
| Resource Type | Article |
