Let’s take a more in-depth look at the curvature of surfaces. The word “curvature” really corresponds to our everyday understanding of what it means for something to be curved: eggshells, donuts, and spiraling telephone cables have a lot of curvature; floors, ceilings, and cardboard boxes do not. But what about something like a beer bottle? Along one direction the bottle quickly curves around in a circle; along another direction it’s completely flat and travels along a straight line:

This way of looking at curvature — in terms of curves traveling along the surface — is often how we treat curvature in general. In particular, let \(X\) be a unit tangent direction at some distinguished point on the surface, and consider a plane containing both \(df(X)\) and the corresponding normal \(N\). This plane intersects the surface in a curve, and the curvature \(\kappa_n\) of this curve is called the *normal curvature* in the direction \(X\):

Remember the Frenet-Serret formulas? They tell us that the change in the normal along a *curve* is given by \(dN = -\kappa T + \tau B\). We can therefore get the normal curvature along \(X\) by extracting the tangential part of dN, and normalizing by any “stretching out” that occurs as we go from the domain \(M\) into \(\mathbb{R}^3\):

\[ \kappa_n(X) = \frac{-df(X) \cdot dN(X)}{|df(x)|^2}. \]

Note that normal curvature is *signed*, meaning the surface can bend toward the normal or away from it.

__Principal, Mean, and Gaussian Curvature__

At any given point we can ask: along which directions does the surface bend the most? The unit vectors \(X_1\) and \(X_2\) along which we find the maximum and minimum normal curvatures \(\kappa_1\) and \(\kappa_2\) are called the *principal directions*; the curvatures \(\kappa_i\) are called the *principal curvatures*. For instance, the beer bottle above might have principal curvatures \(\kappa_1 = 1\), \(\kappa_2 = 0\) at the marked point.

One important fact about the principal directions is that they are eigenvectors of the shape operator, in the sense that

\[ dN(X_i) = \kappa_i df(X_i). \]

(Moreover, the principal directions are *orthogonal*: \(df(X_1) \cdot df(X_2) = 0\) — see the appendix below for a proof.) The principal directions therefore tell us everything there is to know about the curvature at a point, since we can express any tangent vector \(Y\) in the basis \(\{X_1,X_2\}\) and easily compute the corresponding normal curvature. As we’ll discover, however, getting your hands directly on the principal curvatures is often quite difficult — especially in the discrete setting.

On the other hand, two closely related quantities — called the *mean curvature* and the *Gaussian curvature* will show up over and over again, and have some particularly nice interpretations in the discrete world. The mean curvature \(H\) is the arithmetic mean of principal curvatures:

\[ H = \frac{\kappa_1 + \kappa_2}{2}, \]

and the Gaussian curvature is the (square of the) geometric mean:

\[ K = \kappa_1 \kappa_2. \]

What should you visualize if someone tells you a surface has nonzero Gaussian or mean curvature? Perhaps the most elementary interpretation is that Gaussian curvature is like a logical “and” (is there curvature along *both* directions?) whereas mean curvature is more like a logical “or” (is there curvature along *at least one* direction?) Of course, you have to be a little careful here since you can also get zero mean curvature when \(\kappa_1 = -\kappa_2\).

It also helps to see pictures of surfaces with zero mean and Gaussian curvature. Zero-curvature surfaces are so thoroughly studied in mathematics that they even have special names. Surfaces with zero Gaussian curvature are called *developable surfaces* because they can be “developed” or flattened out into the plane without any stretching or tearing. For instance, any piece of a cylinder is developable since one of the principal curvatures is zero:

Surfaces with zero mean curvature are called *minimal surfaces* because (as we’ll see later) they minimize surface area (with respect to certain constraints). Minimal surfaces tend to be saddle-like since principal curvatures have equal magnitude but opposite sign:

The saddle is also a good example of a surface with negative *Gaussian* curvature. What does a surface with positive Gaussian curvature look like? The hemisphere is one example:

Note that in this case \(\kappa_1 = \kappa_2\) and so principal directions are not uniquely defined — maximum (and minimum) curvature is achieved along *any* direction \(X\). Any such point on a surface is called an *umbilic point*.

There are plenty of cute theorems and relationships involving curvature, but those are the basic facts: the curvature of a surface is completely characterized by the *principal curvatures*, which are the maximum and minimum *normal curvatures*. The Gaussian and mean curvature are simply averages of the two principal curvatures, but (as we’ll see) are often easier to get your hands on in practice.