# Curvature and the Shape Operator

## The Shape Operator

The Gaussian curvature, as it turns out, is the determinant of the special symmetrical matrix we call the shape operator:

\begin{align} S=\left[\begin{matrix} e & f \\ f & g \end{matrix}\right] = \left[\begin{matrix} \kappa_1 & 0 \\ 0 & \kappa_2 \end{matrix}\right] \\ \det\left(S\right) = eg - ff = \kappa_1\cdot\kappa_2 = \kappa \end{align}

In order for this to be true, of course, the eigenvalues of this matrix must be the principal curvatures, and if we combine this information with the knowledge that the curvature of any line travelling through this point can be calculated with just these two values (as shown by Euler), we immedialy begin to see how useful the shape operator can be!

The shape operator, in essence, tells us how the normal line changes as it travels across the surface; and in so doing, is also able to essentially describe the surface. It does so by calculating the principal curvatures at each point, which are determined by the paths upon which the normal line changes the most (the maximum curvature), or least (minimum curvature). Furthermore, the invariants of this operator (i.e. its determinant which is the Gaussian curvature) represent geometrical properties of the surface.

Having a single operator that is able to store all of this information becomes an extremely useful tool when the speed of one's computer processor is limited.