# New model of non-Euclidean plane

## TL;DR

The paper introduces a novel non-Euclidean plane model based on hyperreal numbers, illustrating key geometric axioms and differences from hyperbolic geometry with accessible educational benefits.

## Contribution

It presents a new non-Euclidean plane model within hyperreal numbers, enabling representation of various geometric axioms and distinctions from hyperbolic planes.

## Key findings

- Model represents triangle angle sum as π.
- Allows visualization of parallel axiom negations.
- Highlights differences between non-Euclidean and hyperbolic planes.

## Abstract

We present a new model of a non-Euclidean plane, in which angles in a triangle sum up to $\pi$. It is a subspace of the Cartesian plane over the field of hyperreal numbers $\mathbb{R}^*$. The model enables one to represent the negation of equivalent versions of the parallel axiom, such as the existence of the circumcircle of a triangle, and Wallis' or Lagendre's axioms, as well as the difference between non-Euclidean and hyperbolic planes. The model has unique educational advantages as expounding its crucial ideas requires only the basics of Cartesian geometry and non-Archimedean fields.