Using Jacobi polynomials for degree reduction of Bézier curves with Ck-constraints

Abstract

We propose the constrained Jacobi polynomial as an error function of good degree reduction of Bézier curve with C k -constraints at the boundaries, k=2,3. The result is a natural extension of the method proposed by Kim and Ahn (2000). The best C k -constrained degree reduction in L ∞-norm, k>0, cannot be obtained in explicit form and requires higher computational complexity such as Remes algorithm. The method of C k -constrained degree reduction using the constrained Jacobi polynomials is represented in explicit form, and its L ∞-norm error is obtainable using Newton method and is slightly larger than that of the best C k -constrained degree reduction. We also present the subdivision scheme for the C k -constrained degree reduction within given tolerance. As an illustration, our method is applied to C k -constrained degree reduction of planar Bézier curve, and compare its result to that of the best C k -constrained degree reduction.