Perturbing Bézier coefficients for best constrained degree reduction in the L2-norm

This paper first shows how the B ezier coefficients of a given degree n polynomial are perturbed so that it can be reduced to a degree m ð< nÞ polynomial with the constraint that continuity of a prescribed order is preserved at the two endpoints. The perturbation vector, which consists of the perturbation coefficients, is determined by minimizing a weighted Euclidean norm. The optimal degree n 1 approximation polynomial is explicitly given in B ezier form. Next the paper proves that the problem of finding a best L2-approximation over the interval 1⁄20; 1 for constrained degree reduction is equivalent to that of finding a minimum perturbation vector in a certain weighted Euclidean norm. The relevant weights are derived. This result is applied to computing the optimal constrained degree reduction of parametric B ezier curves in the L2-norm. 2003 Elsevier Inc. All rights reserved.