The Second Fundamental Form and Curvature

The first fundamental form measures lengths and areas within the surface — but it says nothing about how the surface bends in the surrounding space. A flat sheet of paper and that same sheet rolled into a cylinder have the very same first fundamental form, yet one is flat and one is curved. To capture bending we need a new object that watches the unit normal tilt as we move along the surface. That object is the second fundamental form.

The idea: slice the surface with a plane containing the normal \mathbf{n}. The slice is a curve — a normal section — and how sharply that curve bends tells you how the surface bends in that direction. Turn the slicing plane and the bending changes. The second fundamental form is the machine that reports it.

L, M, N: how the normal turns

Take a curve on the surface and follow its acceleration \ddot\sigma. Part of that acceleration lies in the tangent plane (that's the curve wandering sideways), and part points along the normal \mathbf{n} (that's the surface itself forcing the curve to bend). Only the normal part is about the surface's shape. Projecting the second derivatives of \sigma onto \mathbf{n} gives three coefficients:

L = \sigma_{uu}\cdot\mathbf{n}, \qquad M = \sigma_{uv}\cdot\mathbf{n}, \qquad N = \sigma_{vv}\cdot\mathbf{n}.

and the second fundamental form packages them exactly like the first:

\mathrm{II} = L\,du^2 + 2M\,du\,dv + N\,dv^2.

Whereas the first form \mathrm{I} was built from first derivatives dotted with each other, the second form \mathrm{II} is built from second derivatives dotted with the normal. First form: how fast you move. Second form: how fast the normal tips.

Normal curvature: bending in a chosen direction

Pick a tangent direction, described by its parameter step (du, dv). The normal curvature in that direction is the ratio of the two forms:

\kappa_n = \frac{\mathrm{II}}{\mathrm{I}} = \frac{L\,du^2 + 2M\,du\,dv + N\,dv^2}{E\,du^2 + 2F\,du\,dv + G\,dv^2}.

It measures how sharply the normal-section curve bends when you head off in that direction. Rotate the direction and \kappa_n changes — sometimes a lot. On a cylinder, for instance, heading around the waist bends hard, but heading straight up the axis doesn't bend at all.

As the direction sweeps a full turn, \kappa_n reaches a maximum and a minimum. These two extreme values are the principal curvatures \kappa_1 and \kappa_2, and the directions achieving them are the principal directions.

The shape operator and Euler's formula

There is a cleaner way to see the principal curvatures. Combine the two forms into a single linear map, the shape operator (or Weingarten map):

S = \mathrm{I}^{-1}\,\mathrm{II} = \begin{pmatrix} E & F \\ F & G \end{pmatrix}^{-1}\begin{pmatrix} L & M \\ M & N \end{pmatrix}.

The principal curvatures are the eigenvalues of S, and the principal directions are its eigenvectors. Because S is self-adjoint with respect to the metric, its eigenvalues are real and its eigenvectors are orthogonal — the two principal directions always cross at right angles.

Once you know \kappa_1, \kappa_2 and line up the angle \theta from the first principal direction, the normal curvature in any direction follows from a beautifully simple rule.

Let \kappa_1, \kappa_2 be the principal curvatures and \theta the angle from the first principal direction. Then:

Check the ends: at \theta = 0, \kappa_n = \kappa_1; at \theta = 90^\circ, \kappa_n = \kappa_2. Euler's formula smoothly interpolates between the two extremes.

Three shapes, three curvature stories

The plane. It never bends, so every normal section is a straight line: L = M = N = 0 and

\kappa_1 = \kappa_2 = 0.

The sphere of radius R. By symmetry every direction bends the same, and the same as a great circle of radius R:

\kappa_1 = \kappa_2 = \frac{1}{R}.

Every direction is a principal direction here — the sphere is the perfectly round case where Euler's formula gives the constant 1/R no matter what \theta is.

The cylinder of radius R. This is the interesting one. Around the waist the section is a circle of radius R; along the axis it is a straight line:

\kappa_1 = \frac{1}{R} \ \text{(around)}, \qquad \kappa_2 = 0 \ \text{(along the axis)}.

Euler's formula then says a diagonal direction on the cylinder bends by \kappa_n(\theta) = \tfrac1R\cos^2\theta — most around the waist, tapering to zero up the axis.

Watch Euler's formula sweep

The curve below is \kappa_n(\theta) = \kappa_1\cos^2\theta + \kappa_2\sin^2\theta as the direction angle \theta runs a full turn. Set the two principal curvatures with the sliders. With \kappa_1 = \kappa_2 (a sphere) the curve is flat — every direction bends equally. Make one of them zero (a cylinder) and the normal curvature dips to zero in the flat direction. Give them opposite signs and you have a saddle: the surface curves up one way and down the other, and \kappa_n crosses zero — those zero-crossings are the asymptotic directions.

Vignettes

It feels like it ought to be a special coincidence, but it is forced. The principal directions are the eigenvectors of the shape operator S, and S is self-adjoint with respect to the surface's inner product (the first fundamental form). A classic theorem of linear algebra says a self-adjoint operator has an orthonormal eigenbasis — its eigenvectors for distinct eigenvalues are always perpendicular.

So on any surface, at any point, the direction of maximum bending and the direction of minimum bending meet at a right angle. On a banana, on a Pringle, on your knee — the sharpest and gentlest curves are always perpendicular. (When the two principal curvatures are equal, as on a sphere, every direction is principal and the "right angle" is vacuously true.)

Normal curvature carries a sign, and that sign depends on which way you point the unit normal \mathbf{n}. Flip \mathbf{n} to -\mathbf{n} and every entry of the second fundamental form flips sign, so L, M, N and hence \kappa_n, \kappa_1, \kappa_2 all change sign. A sphere has \kappa_1 = \kappa_2 = +1/R with the inward normal but -1/R with the outward one.

This is genuinely different from the unsigned curvature \kappa = \lVert\ddot{\mathbf{r}}\rVert of a space curve you met in the Frenet frame, which is always \geq 0. A curve's curvature says only "how much"; a surface's normal curvature also says "which way relative to the normal" — up (toward \mathbf{n}) or down (away). Do not report a bare \kappa_n without saying which normal you chose.