Constrained Bézier curves' best multi-degree reduction in the L 2 -norm

Abstract

Abstract In computer aided geometric design, the degree reduction of the parameter curve is a key technique in data exchange and data compression. The various existing methods of degree reduction cannot decide whether the degree reduction curve satisfying the given tolerance exists beforehand, cannot give approximation of the best multi-degree reduction, or cannot provide explicit expression and error formula of the degree reduction curve. In this paper, we propose an entirely new method, which can hurdle the above flaws completely. For a given Bexier curve of degree n, we can easily decide whether a Bezier reduction curve of degree m exists, which has equal derivatives with the given curve up to (r−1)-th and (s−1)-th orders (r, s≤m<n) respectively at the endpoints, so the approximating error is less than the given tolerance ϵ in the L 2-norm. If the curve exists, the explicit expression can be given.