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TW 353
E. Vanraes, P. Dierckx, A. Bultheel
On the choice of the PS-triangles
Abstract
In this paper we look at different bases for the space S12(ΔPS) of C1 continuous quadratic splines on Powell-Sabin triangle splits. First, by requiring a partition of unity, we describe a general framework that leads to the notion of PS-triangles and control triangles. This is a useful property that gives insight in the shape of the surface. The problem of choosing appropriate basis functions for a certain triangulation of the domain is now equivalent to choosing the corresponding PS-triangles. Then we consider two choices more in detail: a normalised B-spline basis, and a basis based on minimal determining sets. The approach with minimal determining sets leads to bases that are stable as a function of the smallest angle in the triangulation. The basis functions are not only linearly independent but also locally linearly independent. In the normalised B-spline representation the basis functions form a convex partition of unity. Geometrically this can be interpreted as a PS-triangle containing a specific set of Bézier domain points. There is more than one triangle that satisfies this requirement. We show why the triangle with minimal area is a suitable choice and also give an alternative that is computationally more efficient.
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