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 it determines the principal, Gaussian and mean curvatures.