---
tag: 2K9VZ8M8AG
kind: definition
name: Non-Euclidean geometry
msc: [51M10, 51M05]
defines: SPCYMFN5V2
uses: [KRZD96GSVC, 2PC3XY0PYZ, 0MP9BCF6DP]
sources:
  - url: https://en.wikipedia.org/wiki/Non-Euclidean_geometry
---

## Definition

A plane geometry consists of points, [[lines|0MP9BCF6DP]], and relations of incidence, betweenness and congruence that satisfy a system of axioms.
A plane geometry is *non-Euclidean* if the [[parallel postulate|2PC3XY0PYZ]] fails in it.
It must satisfy the other axioms of [[Euclidean geometry|KRZD96GSVC]], with only the changes forced by the new parallel property.
The classical cases are the following.

1. Hyperbolic: for each line $\ell$ and point $P$ not on $\ell$, at least two lines through $P$ do not meet $\ell$.
2. Elliptic: any two lines meet.

## Remarks

In the hyperbolic case, all axioms of absolute geometry hold, and only the parallel axiom is replaced.
In the elliptic case, lines are closed curves, so the axioms of order must also be changed.
Both geometries have Riemannian models of constant [[Gaussian curvature|SQVQ5DQT6W]]: $-1$ for the hyperbolic plane and $+1$ for the elliptic plane.

Example: the [[Poincaré half-plane model|RHKSMPBSQ1]] is a hyperbolic plane.
Through $(1, 1)$, both the vertical line $x = 1$ and the semicircle centred at $(2, 0)$ of radius $\sqrt 2$ miss the line $x = 0$.

Non-example: in the Euclidean plane $\R^2$, the only line through $(1, 1)$ that misses $x = 0$ is $x = 1$.
So the parallel postulate holds there, and the Euclidean plane is not non-Euclidean.