Apollonius "circle" in Hyperbolic Geometry

TL;DR

This paper explores the hyperbolic analog of Apollonius' circle, describing its algebraic curve form in the half-plane model and analyzing the probability of equal perceived segment lengths under randomness.

Contribution

It provides a simple description of the Apollonius locus in hyperbolic geometry and investigates the probability of equal segment perception in this setting.

Findings

The locus is an algebraic curve of degree four in hyperbolic geometry.

The curve can be bounded or unbounded depending on parameters.

Calculated probability of equal segment perception under random conditions.

Abstract

In Euclidean geometry the circle of Apollonious is the locus of points in the plane from which two collinear adjacent segments are perceived as having the same length. In Hyperbolic geometry, the analog of this locus is an algebraic curve of degree four which can be bounded or "unbounded". We study this locus and give a simple description of this curve using the half-plane model. In the end, we give the motivation of our investigation and calculate the probability that three collinear adjacent segments can be seen as of the same positive length under some natural assumptions about the setting of the randomness considered.