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.