---
tag: 3K774WEHHY
kind: concept
name: Moving frame
msc: [53A55, 53C05]
sources:
  - url: https://en.wikipedia.org/wiki/Moving_frame
same_as:
  wikidata: Q6927157
  wikipedia: Moving frame
---

## Summary

A moving frame is a basis of the tangent space that varies smoothly from point to point.
It may live on a manifold or along a curve or surface.
Élie Cartan developed the method of moving frames.
One expresses the derivatives of the frame in terms of the frame itself, and the coefficients become the invariants of the geometry.
The Frenet frame of a space curve, whose derivative gives curvature and torsion, is the classical example.
In modern terms a moving frame is a local section of the [[frame bundle|F4HWTRNXAP]].

## Explanations

### L1 human

Instead of using one fixed set of axes for the whole space, attach a small set of axes to each point and let it move with the point.
For a curve in space, one axis can point forward, one toward the centre of bending, and one perpendicular to both.
How these axes turn as one moves along the curve tells everything about its shape.
In the plane without the origin, the polar frame, with one arrow pointing away from the origin and one turning around it, is a moving frame.