# Generalized Poincar\'{e} Half-Planes

## TL;DR

This paper introduces a family of generalized Poincaré upper half-plane models by replacing circular arcs with elliptical arcs, expanding the classical hyperbolic geometry framework while maintaining key axiomatic properties.

## Contribution

It generalizes the classical Poincaré upper half-plane model using elliptical arcs and proves these models satisfy hyperbolic axioms, broadening the scope of hyperbolic geometry.

## Key findings

- Generalized models are infinite in number.
- All models satisfy hyperbolic axioms.
- Models are neutral geometries, excluding parallelism.

## Abstract

In this note, we give some generalisations of the classical Poincar\'{e} upper half-plane, which is the most popular model of hyperbolic plane geometry. For this, we replace the circular arcs by elliptical arcs with center on the $x-$axis, and foci on the $x-$axis or on the lines perpendicular to the $x-$axis at the center, in the upper half-plane. Thus, we obtain a class of generalized upper half-planes with infinite number of members. Furthermore we show that every generalized Poincar\'{e} upper half-plane geometry is a neutral geometry satisfying the hyperbolic axiom. That is, it satisfies also all axioms of the Euclidean plane geometry except the parallelism.