Characterizing the finiteness of the Hausdorff distance between two algebraic curves

TL;DR

This paper provides a practical criterion to determine when the Hausdorff distance between two algebraic curves in n-dimensional space is finite, based on their asymptotic behavior at infinity.

Contribution

It offers a new characterization linking the finiteness of the Hausdorff distance to the matching of infinity branches of the curves, which can be easily checked.

Findings

Finiteness of Hausdorff distance is characterized by matching infinity branches.

The criterion applies to both parametrically and implicitly defined curves.

The approach simplifies checking the Hausdorff distance in algebraic geometry.

Abstract

In this paper, we present a characterization for the Hausdorff distance between two given algebraic curves in the $n$-dimensional space (parametrically or implicitly defined) to be finite. The characterization is related with the asymptotic behavior of the two curves and it can be easily checked. More precisely, the Hausdorff distance between two curves $\cal C$ and $\overline{\cal C}$ is finite if and only if for each infinity branch of $\cal C$ there exists an infinity branch of $\overline{\cal C}$ such that the terms with positive exponent in the corresponding series are the same, and reciprocally.