Degree reduction of the Bézier curve with $C^1$-continuity using Jacobi polynomial

Abstract

In case of one-degree reduction with $C^1$-continuity constraint for the uniform error norm, Lachance proposed the method using constrained Chebyshev polynomials which are obtained numerically by modified Remes algorithm. It is the best one-degree reduction method. In this thesis, we propose another method in the same case as above. We introduce constrained Jacobi polynomials of which B$\acute{e}$zier coefficients are represented explicitly. Furthemore if we want to approximate even degree B$\acute{e}$zier curves, we also find explicitly the error form. So we have error bounds and subdivision algorithm for one-degree reduction of even degree B$\acute{e}$zier curves. Even if our method does not give the best approximation, it is more useful and easily applicable than Lachance``s method.