Rational B\'ezier Curves Approximated by Bernstein-Jacobi Hybrid Polynomial Curves

TL;DR

This paper introduces a linear method using Bernstein-Jacobi hybrid polynomials for approximating rational Bézier curves with polynomial curves, enhancing accuracy and stability.

Contribution

It presents a novel approximation technique combining weighted least-squares and Bernstein-Jacobi polynomials for rational Bézier curves.

Findings

Effective degree reduction improves approximation accuracy.

Error bounds demonstrate the method's reliability.

Examples confirm simplicity and stability of the approach.

Abstract

In this paper, we propose a linear method for $C^{(r,s)}$ approximation of rational B\'{e}zier curve with arbitrary degree polynomial curve. Based on weighted least-squares, the problem be converted to an approximation between two polynomial curves. Then applying Bernstein-Jacobi hybrid polynomials, we obtain the resulting curve. In order to reduce error, degree reduction method for B\'{e}zier curve is used. A error bound between rational B\'{e}zier curve and B\'{e}zier curve is presented. Finally, some examples and figures were offered to demonstrate the efficiency, simplicity, and stability of our methods.