---
tag: ZPXE5VSZ35
kind: concept
name: Conjugate points
msc: [53C22]
sources:
  - url: https://en.wikipedia.org/wiki/Conjugate_points
same_as:
  wikidata: Q5161158
  wikipedia: Conjugate points
---

## Summary

Two points on a [[geodesic|NSYMQMHPDN]] are conjugate if a nonzero [[Jacobi field|9AHMM41PJ3]] along the geodesic vanishes at both of them.
Roughly, a family of geodesics leaving the first point refocuses at the second.
They are exactly the critical values of the [[exponential map|Y7Z0AGY1A7]] along the geodesic.
A geodesic stops being length-minimizing after it passes its first conjugate point.

## Explanations

### L1 human

Send out geodesics from a point in nearly the same direction.
If, further on, they come back together to first order at some point, that point is conjugate to the starting point.
On the unit sphere, all geodesics from the north pole meet at the south pole, so the poles are conjugate along every meridian.
In the flat plane, straight lines from a point never meet again, so there are no conjugate points.