Sammendrag
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. These 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 talk 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 C0, C1, and C2-smoothness are derived.
Although the cubic bases 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, with the potential to be used in isogeometric analysis.
In this talk 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 C0, C1, and C2-smoothness are derived.
Although the cubic bases 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, with the potential to be used in isogeometric analysis.