Classification topologique des solutions du Probl\`eme d'Apollonius

TL;DR

This paper uses Lie Sphere Geometry to classify solutions of the Apollonius problem, revealing that in two dimensions, the number of solutions depends solely on the topology of the configuration, simplifying previous classifications.

Contribution

It provides a topological classification of solutions to the Apollonius problem using Lie Sphere Geometry, offering a simpler and non-redundant approach compared to earlier methods.

Findings

Number of solutions depends only on topology in 2D

Classification is simpler and non-redundant

Method leverages Lie Sphere Geometry

Abstract

We give a mathematical computation of the number of solutions of Apollonius problem, by use of Lie Sphere Geometry. Unlike in higher dimensions, the number of solutions depends only on the topology of the configuration of the 3 objects. It appears that our classification is non redundant, and far simpler than those obtained previously.