A note on constrained degree reduction of polynomials in Bernstein–Bézier form over simplex domain

AbstractIn the paper [H.S. Kim, Y.J. Ahn, Constrained degree reduction of polynomials in Bernstein–Bézier form over simplex domain, J. Comput. Appl. Math. 216 (2008) 14–19], Kim and Ahn proved that the best constrained degree reduction of a polynomial over d-dimensional simplex domain in L2-norm equals the best approximation of weighted Euclidean norm of the Bernstein–Bézier coefficients of the given polynomial. In this paper, we presented a counterexample to show that the approximating polynomial of lower degree to a polynomial is virtually non-existent when d≥2. Furthermore, we provide an assumption to guarantee the existence of solution for the constrained degree reduction

Approximating tensor product Bézier surfaces with tangent plane continuity

AbstractWe present a simple method for degree reduction of tensor product Bézier surfaces with tangent plane continuity in L2-norm. Continuity constraints at the four corners of surfaces are considered, so that the boundary curves preserve endpoints continuity of any order α. We obtain matrix representations for the control points of the degree reduced surfaces by the least-squares method. A simple optimization scheme that minimizes the perturbations of some related control points is proposed, and the surface patches after adjustment are C∞ continuous in the interior and G1 continuous at the common boundaries. We show that this scheme is applicable to surface patches defined on chessboard-like domains