---
tag: SQ9RQVANMG
kind: concept
name: Spherical geometry
msc: [51M10, 51M09]
specializes: [SPCYMFN5V2]
sources:
  - url: https://en.wikipedia.org/wiki/Spherical_geometry
same_as:
  wikidata: Q326905
  wikipedia: Spherical geometry
---

## Summary

Spherical geometry is the geometry of the surface of a sphere, where the role of lines is played by great circles.
Any two great circles meet, so there are no parallel lines, and the angle sum of a triangle exceeds $\pi$.
It is the oldest [[non-Euclidean geometry|SPCYMFN5V2]] in practice, used for navigation and astronomy long before the axioms of geometry were questioned.
Identifying antipodal points turns it into [[elliptic geometry|H5GB19W2H3]].

## Explanations

### L1 human

On a globe, the shortest route between two cities follows a great circle, a circle whose centre is the centre of the Earth.
These great circles are the straight lines of spherical geometry.
The equator and any meridian are great circles; the circles of latitude other than the equator are not.
A triangle with one corner at the North Pole and two on the equator, a quarter turn apart, has three right angles.