The geometry of optimal degree reduction of Bézier curves Original Research Article

Optimal degree reductions, i.e., best approximations of nth degree Bézier curves by Bézier curves of degree n − 1, with respect to different norms are studied. It is shown that for any Lp-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 Bézier points of the degree reductions are found to lie on parallel lines through the Bézier points of any Taylor expansion of degree n − 1 of the original curve. This geometric situation is shown to hold also in the case of constrained degree reduction. The Bézier points of the degree reduction are explicitly given in the unconstrained case for p = 1 and p = 2 and in the constrained case for p = 2.