Parametrization of $\epsilon$-rational curves: error analysis

TL;DR

This paper analyzes the Hausdorff distance error in the parametrization of approximately epsilon-rational curves, providing an automated approach and an upper bound for the error across a family of randomly generated curves.

Contribution

It automates the error analysis strategy for epsilon-rational curve parametrization and establishes an upper bound for the Hausdorff distance in a broad family of curves.

Findings

Hausdorff distance is small for the tested curves

An upper bound for the Hausdorff distance is established

The method is automated for broader application

Abstract

In [Computer Aided Geometric Design 27 (2010), 212-231] the authors present an algorithm to parametrize approximately $\epsilon$-rational curves, and they show in 2 examples that the Hausdorff distance, w.r.t. to the Euclidean distance, between the input and output curves is small. In this paper, we analyze this distance for a whole family of curves randomly generated and we automatize the strategy used in [Computer Aided Geometric Design 27 (2010), 212-231]. We find a reasonable upper bound of the Hausdorff distance between each input and output curve of the family.