Asymptotic Behavior of an Implicit Algebraic Plane Curve

TL;DR

This paper introduces the concepts of infinity branches and approaching curves to analyze the asymptotic behavior of algebraic plane curves, providing an algorithm to compare their behavior at infinity and establishing a link to the Hausdorff distance.

Contribution

It presents new notions and an algorithm for comparing the asymptotic behavior of algebraic curves at infinity, and proves a key property relating asymptotic similarity to Hausdorff distance.

Findings

Algorithm for comparing asymptotic behavior of curves

Curves with same asymptotic behavior have finite Hausdorff distance

Introduction of infinity branches and approaching curves

Abstract

In this paper, we introduce the notion of infinity branches as well as approaching curves. We present some properties which allow us to obtain an algorithm that compares the behavior of two implicitly defined algebraic plane curves at the infinity. As an important result, we prove that if two plane algebraic curves have the same asymptotic behavior, the Hausdorff distance between them is finite.