Optimal Degree Reduction of Free Form Curves

Abstract

Optimal degree reductions, i.e. best approximations of n-th degree Bezier curves by Bezier curves of degree n - 1, with respect to different norms are studied. It is shown that for any L_p-norm the Euclidean degree reduction where the norm is applied to the Euclidean distance function of two curves is identical to component-wise degree reduction. The Bezier points of the degree reductions are found to lie on parallel lines through the Bezier points of any Taylor expansion of degree n - 1 of the original curve. The Bezier points of the degree reduction are explicitly given p = 1 and p = 2.