Explicit [formula omitted] approximation of conic sections using Bézier curves of arbitrary degree

Abstract

In this paper, we propose an explicit and effective method for G1 approximation of conic sections with arbitrary degree polynomial curves. The geometric constraints can be expressed by two variables. Then we construct an objective function as the alternative approximation error function in the L2-norm, which is a quadratic function in these two variables. To minimize the objective function is equivalent to solving a system of linear equations with two variables. Since its coefficient matrix is invertible, we can explicitly obtain the unique solution and then derive the explicit G1 approximation of conic sections by Bézier curves of arbitrary degree. The proposed method has an optimal approximation in the L2-norm, i.e., the L2-distance error reaches minimum, and also can process the conic section with center angle larger than π without subdivision scheme. Finally, numerical examples demonstrate the effectiveness of our method.