---
tag: 9KHQR16P1B
kind: concept
name: Second fundamental form
msc: [53A05]
sources:
  - url: https://en.wikipedia.org/wiki/Second_fundamental_form
same_as:
  wikidata: Q233220
  wikipedia: Second fundamental form
---

## Summary

The second fundamental form of a surface in $\R^3$ is a quadratic form on each tangent plane.
It measures how the surface bends away from that plane.
In a local parametrisation it is given by three functions, often written $e$, $f$, $g$ or $L$, $M$, $N$.
Together with the [[first fundamental form|MAJ3DH7MZS]] it determines the principal, Gaussian and mean curvatures.

## Explanations

### L1 human

Put a flat sheet of glass on a surface so that it touches at one point.
Near that point the surface pulls away from the glass, faster in some directions and slower in others.
The second fundamental form records, for each direction, how fast the surface pulls away.
On a sphere it pulls away equally in all directions, and on a saddle it rises in some directions and falls in others.