Degree reduction of Bézier curves

Abstract

We provide an simple algorithm for constructing an polynomial Bézier approximation of degree n—1 to an nth degree Bézier curve. This algorithm makes previous work of Lachance more transparent as formulas are given which express the geometric relationship between the control points. The two curves agree at the two endpoints up to a preselected differentiation order since the method is based on constrained Chebyshev polynomials in order to obtain best constrained approximations. These polynomials then allow a detailed error analysis providing apriori bounds of the pointwise approximation error. The extension to tensor product surfaces is also briefly discussed.