---
tag: H5GB19W2H3
kind: concept
name: Elliptic geometry
msc: [51M10, 51M09]
specializes: [SPCYMFN5V2]
sources:
  - url: https://en.wikipedia.org/wiki/Elliptic_geometry
same_as:
  wikidata: Q1332450
  wikipedia: Elliptic geometry
---

## Summary

Elliptic geometry is the [[non-Euclidean geometry|SPCYMFN5V2]] of constant positive curvature.
In it, any two points determine exactly one line, and any two lines meet in exactly one point.
Its standard model is the [[real projective space|1DDYB6VVSR]], seen as the sphere with antipodal points identified.
There are no parallel lines, and the angle sum of a triangle exceeds $\pi$.
Together with Euclidean and [[hyperbolic geometry|GN4N94GQ0Z]] it is one of the three classical geometries of constant curvature.

## Explanations

### L1 human

Start with [[spherical geometry|SQ9RQVANMG]] on a globe, where the lines are great circles.
There, two great circles always cross twice, at opposite points of the globe.
Now declare each pair of opposite points to be a single point.
Then two lines cross exactly once, and two points lie on exactly one line, as in the plane, yet there are still no parallels.
The resulting geometry is elliptic geometry; it is finite in size, and no two points are more than a quarter of a great circle apart.