The inversion problem for rational B\'ezier curves

TL;DR

This paper explores a numerical approach to solving the inversion problem for rational Bézier curves using resultant matrices and singular value decomposition, focusing on finding parameter values for given points on the curve.

Contribution

It introduces a method that employs resultant matrices and SVD to determine parameters on rational Bézier curves without explicit inversion formulas.

Findings

Effective parameter estimation for points on rational Bézier curves.

Utilization of SVD enhances numerical stability in inversion.

Applicable to approximate points on the curve.

Abstract

The inversion problem for rational B\'ezier curves is addressed by using resultant matrices for polynomials expressed in the Bernstein basis. The aim of the work is not to construct an inversion formula but finding the corresponding value of the parameter for each point on the curve. Since sometimes one has only an approximation of that point the use of the singular value decomposition, a key tool in numerical linear algebra, is shown to be adequate.