Generalized Apollonius Circles As Equioptic Curves Of Circles In Constants Curvature Geometries

TL;DR

This paper generalizes Apollonius circles to constant curvature geometries, showing they coincide with equioptic curves of circle centers in hyperbolic, spherical, and Euclidean spaces.

Contribution

It extends the classical definition of Apollonius circles to hyperbolic and spherical geometries, linking them to equioptic curves of circle centers.

Findings

Apollonius circles coincide with equioptic curves in all constant curvature geometries.

The generalized definition simplifies analysis across different geometries.

The approach unifies Euclidean, hyperbolic, and spherical circle properties.

Abstract

We extend the old definition of the Apollonius circle in such a way that it results in the same curve in Euclidean geometry but will be more convenient in hyperbolic and spherical geometries. We show that there exists an Apollonius circle of the centers of two circles that coincides with their equioptic curves, as in Euclidean geometry.