Principal curvatures and directions

The maximum and minimum of the normal curvature $\kappa_1$ and $\kappa_2$ at a given point on a surface are called the principal curvatures. The principal curvatures measure the maximum and minimum bending of a regular surface at each point. They are the solutions to the quadratic equation $\kappa^2 -2H\kappa + K = 0$, where $H$ is the mean curvature and $K$ the Gaussian curvature.

$$\begin{array}{rcl} \kappa_1 &=& H + \sqrt{H^2 - K}\\ \kappa_2 &=& H - \sqrt{H^2 - K} \end{array}$$

The principal directions corresponding to the principal curvature are perpendicular to one another. In other words, the surface normal planes at the point and in the principal directions are perpendicular to one another, and both are perpendicular to the surface tangent plane at the point.

A direction $\vec{\tau} = du x_u + dv x_v$ is a principal direction if $\exists \kappa$ such that:

$$W_x(\vec{\tau}) = -\kappa \vec{\tau}$$

The above formula is called Rodrigues' formula. Note that it can be rewritten:

$$\frac{\delta}{\delta t}\left(\gamma(t)+\rho\vec{n}(\gamma(t)) \right) = 0$$