Approximation and geometric modeling with simplex B-splines associated with irregular triangles

Bivariate quadratic simplicial B-splines defined by their corresponding set of knots derived from a (suboptimal) constrained Delaunay triangulation of the domain are employed to obtain a Cl-smooth surface. The generation of triangle vertices is adjusted to the area1 distribution of the data in the domain. We emphasize here that the vertices of the triangles initially define the knots of the B-splines and do generally not coincide winth the abscissae of the data. Thus, this approach is well suited to process scattered data. With each vertex of a given triangle we associate two additional points which give rise to six configurations of five knots defining six linearly independent bivariate quadratic B-splines supported on the convex hull of the corresponding five knots. If we consider the vertices of the triangulation as threefold knots, the bivariate quadratic B-splines turn into the well known bivariate quadratic Bernstein-B&ier-form polynomials on triangles. Thus we might be led 10 think of B-splines as of smoothed versions of Bernstein-Btzier polynomials with respect to the entire domain. From the degenerate Bernstein-Btzier situation we deduce rules how to locate the additional points associated with each vertex to establish knot configurations that allow the modeling of discontinuities of the function itself or any of its directional derivatives. We find that four collinear knots out of the set of five defining an individual quadratic B-spline generate a discontinuity in the surface along the tine they constitute. and that analogously three collinear knots generate a discontinuity in a first derivative. Finally, the coefficients of the linear combinations of normalized simplicial B-splines are visualized as geometric control points satisfying the convex hull property. Thus, bivariate quadratic B-splines associated with irregular triangles provide a great flexibility lo approximate and model fast changing or even functions with any given discontinuities from scattered data. An example for least squares approximation with simplex splines is presented.