Normalized trivariate B-splines on Worsey-Piper split and quasi-interpolants
In this paper, the problem of producing a trivariate piecewise quadratic \(C^1\) interpolant to positional and gradient information at the vertices of a tetrahedral partition, considered by \textit{A. J. Worsey} and \textit{B. Piper} [Comput. Aided Geom. Des. 5, No. 3, 177--186 (1988; Zbl 0654.65008)] is recalled. The authors consider a regular tetrahedral partition and give an algorithm to construct a normalized B-splines basis of the space of Worsey-Piper splines (these B-splines are all positive, have local support and form a partition of unity), based on a cubic programming problem. In the geometric framework, the problem is equivalent with founding a WP tetrahedron with minimal volume, containing all Bezier points which surround a vertex. The WP-B-spline representation of \(C^1\) trivariate quadratic polynomials or WP splines in terms of their polar forms is also introduced in order to construct discrete quasi-interpolants which have an optimal approximation order.
scientific article; zbMATH DE number 6018293