Malfatti's problem on the hyperbolic plane

TL;DR

This paper extends Malfatti's circle problem to the hyperbolic plane, providing a solution for constructing three cycles that touch each other and two given cycles, filling a gap in geometric problem research.

Contribution

The paper introduces the first solution to Malfatti's problem in the hyperbolic plane, expanding classical circle packing problems to non-Euclidean geometry.

Findings

Solved Malfatti's problem on the hyperbolic plane.

Established methods for constructing tangent cycles in hyperbolic geometry.

Bridged the gap between Euclidean, spherical, and hyperbolic circle packing problems.

Abstract

More than two centuries ago Malfatti (see \cite{malfatti}) raised and solved the following problem (the so-called Malfatti's construction problem):Construct three circles into a triangle so that each of them touches the two others from outside moreover touches two sides of the triangle too. It is an interesting fact that nobody investigated this problem on the hyperbolic plane, while the case of the sphere was solved simultaneously with the Euclidean case. In order to compensate this shortage we solve the following exercise: {\em Determine three cycles of the hyperbolic plane so that each of them touches the two others moreover touches two of three given cycles of the hyperbolic plane.}