Constrained polynomial approximation of rational Bézier curves using reparameterization

Abstract

This paper proposes a novel method for polynomial approximation of rational Bézier curves with constraints. Different from the previous techniques, for a given rational Bézier curve r(t), a polynomial curve q(s) with a parameter transformation s=ϕ(t), such that q(ϕ(t)) is the closest point to the point r(t), is considered to approximate it. To minimize the distance between these two curves in the L2 norm produces a similar effect as that of the Hausdorff distance. We use a rational function s(t) of a Möbius parameter transformation to approximate the function ϕ(t). The method can preserve parametric continuity or geometric continuity of any u,v(u,v≥0) orders at two endpoints, respectively. And applying the least squares method, we deduce a matrix-based representation of the control points of the approximation curve. Finally, numerical examples show that the reparameterization-based method is feasible and effective, and has a smaller approximation error under the Hausdorff distance than the previous methods.