Degree reduction of B\'ezier curves

Abstract

We consider the problem of approximation of Bézier curves of degree n by Bézier curves of reduced degree m(＜n) with respect to the Tchebycheff, $L^1$ and $L^2$-norm. For one-degree reduction, a simple and elegant method is proposed by the use of the Tchebycheff polynomials of the first kind, the second kind and Legendre polynomials in each norm. This method is obtained by means of `filter bank process``, which consists of the synthesis filters and analysis filters. For the best approximations with endpoint interpolation, we summerize the best degree reduction schemes in the Tchebycheff and $L^2$-norm, which were given in [3, 7]. For the $L^1$-norm, we obtain the best one-degree reduction of Bézier curves of degree ≤5 with endpoint interpolation by using perfect spline. For the general degree n, a ``good`` one-degree reduction is proposed by the use of an appropriate transform of the Tchebycheff polynomials of the second kind. Although this scheme does not give the best approximation, the subdivision algorithm suggested in this thesis is useful in implementations. For the higher degree reduction, the recursive application of one-degree reduction was suggested in [6, 21]. They are not best in general. In this thesis, the best two-degree reduction of Bézier curves of degree ≤4 is given by the use of the classical approximation theory in the Tchebycheff and $L^1$-norm. In the $L^2$-norm, the best degree reduction is easily obtained for any degree n.