---
tag: NSYMQMHPDN
kind: concept
name: Geodesic
msc: [53C22]
sources:
  - url: https://en.wikipedia.org/wiki/Geodesic
same_as:
  wikidata: Q213488
  wikipedia: Geodesic
---

## Summary

A geodesic on a [[Riemannian manifold|7CBT4SM83Y]] is a curve whose velocity is parallel along itself for the [[Levi-Civita connection|CAZYGE1T4E]].
So it has zero acceleration within the manifold.
Geodesics generalise straight lines: short pieces of a geodesic are the shortest curves between their endpoints.
They are solutions of a second-order ordinary differential equation, so a geodesic is determined by its starting point and starting velocity.
Geodesics are the basic tool for studying distance, curvature and topology of Riemannian manifolds.

## Explanations

### L1 human

A geodesic is the path of someone who walks straight ahead on a curved surface without ever turning left or right.
On a plane, geodesics are straight lines traversed at constant speed.
On a round sphere, geodesics are great circles, such as the equator or the meridians.
Airline routes follow great circles because, on the sphere, short arcs of great circles are the shortest paths.