Multi-degree Reduction of Disk Bézier Curves in L2 Norm ?

A planar Bézier curve whose control points are disks is called a disk Bézier curve. It can be looked as a parametric curve with tolerance in the plane. It is an effective tool to measure or control the error. Based on minimum mean square error, this paper presents an algorithm for optimal multi-degree reduction of disk Bézier curves in L2 norm. First, applying the orthogonal property of Jacobi polynomials, we provide the optimal multi-degree reduced polynomial approximation of the center curve of the original disk Bézier curve in L2 norm, and regard it as the center curve of the degree-reduced one; Next, we transform the other task, optimally multi-degree reduced approximating the error radius curve of the disk Bézier curve, into solving a constrained quadratic programming (QP) problem. Also this paper gives error estimation for this algorithm, and shows some numerical examples to illustrate the correctness and the validity of theoretical reasoning.