---
tag: MAJ3DH7MZS
kind: concept
name: First fundamental form
msc: [53A05]
sources:
  - url: https://en.wikipedia.org/wiki/First_fundamental_form
same_as:
  wikidata: Q939725
  wikipedia: First fundamental form
---

## Summary

The first fundamental form of a surface in $\R^3$ is the inner product of $\R^3$ restricted to the tangent planes of the surface.
In a local parametrisation it is given by three functions $E$, $F$, $G$.
It lets one measure lengths of curves, angles and areas on the surface without leaving it.
Properties that depend only on it form the intrinsic geometry of the surface.

## Explanations

### L1 human

An ant living on a surface can measure distances and angles with a tiny ruler and protractor, but it cannot see the surrounding space.
The first fundamental form is exactly the information that the ant's ruler gives.
On a map of the Earth with longitude and latitude, it tells how many kilometres one degree of longitude is at each latitude.
Near the poles one degree of longitude is short, and the first fundamental form records this.