Degree reduction of Bézier curves

Abstract We provide an simple algorithm for constructing an polynomial Bezier approximation of degree n —1 to an n th degree Bezier 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.

[1]  Wendelin L. F. Degen,et al.  Best Approximations of Parametric Curves by Splines , 1992, Geometric Modelling.

[2]  A. J. Worsey,et al.  Degree reduction of Be´zier curves , 1988 .

[3]  Matthias Eck,et al.  Degree Reduction of Bézier Surfaces , 1992, IMA Conference on the Mathematics of Surfaces.

[4]  Josef Hoschek.  Approximate conversion of spline curves , 1987, Comput. Aided Geom. Des..

[5]  Horst Nowacki,et al.  Approximate conversion of surface representations with polynomial bases , 1985, Comput. Aided Geom. Des..

[6]  A. Robin Forrest,et al.  Interactive interpolation and approximation by Bezier polynomials , 1972, Comput. J..

[7]  Michael A. Lachance,et al.  Chebyshev economization for parametric surfaces , 1988, Comput. Aided Geom. Des..