The First Fundamental Form

Imagine you are an ant living on a surface, with no way to look off it into the surrounding space. You can still do geometry: you can measure how far apart two points are along the surface, the angle at which two paths cross, the area of a field. Everything you can measure this way is encoded in one object — the first fundamental form, also called the metric. It is the surface's built-in ruler and protractor.

The idea grows straight out of parametrized surfaces. Take a tiny step (du, dv) in the parameter domain. The surface responds with a tiny displacement vector

d\sigma = \sigma_u\,du + \sigma_v\,dv,

and the length of that little step is what we want. Length comes from the dot product, so we square it.

Squaring the step: E, F, G

Take the squared length of d\sigma by dotting it with itself and expanding — exactly as you would expand (a+b)^2, but with dot products:

ds^2 = d\sigma \cdot d\sigma = (\sigma_u\,du + \sigma_v\,dv)\cdot(\sigma_u\,du + \sigma_v\,dv).

Three distinct dot products fall out. Name them:

E = \sigma_u \cdot \sigma_u, \qquad F = \sigma_u \cdot \sigma_v, \qquad G = \sigma_v \cdot \sigma_v.

and the squared step length becomes the first fundamental form:

ds^2 = E\,du^2 + 2F\,du\,dv + G\,dv^2.

Three numbers E, F, G — functions of (u,v) — are all you need to turn parameter-steps into real distances on the surface. Notice E = \lVert\sigma_u\rVert^2 and G = \lVert\sigma_v\rVert^2 are the squared speeds along the coordinate curves, while F measures how skew the coordinate grid is: when F = 0 the u- and v-curves cross at right angles.

What the metric measures

Once you have E, F, G, all the intrinsic measurements follow. This is the payoff — three quantities, three geometric jobs.

Length of a curve. A curve in the parameter domain, t \mapsto (u(t), v(t)), has arc length gotten by integrating the metric along it:

L = \int \sqrt{E\,\dot u^2 + 2F\,\dot u\,\dot v + G\,\dot v^2}\;dt.

Angle between two directions. Two tangent directions \mathbf{a}, \mathbf{b} (given by their (du,dv) components) meet at an angle read straight off the metric, because it is the surface's inner product:

\cos\theta = \frac{\mathrm{I}(\mathbf{a},\mathbf{b})}{\sqrt{\mathrm{I}(\mathbf{a},\mathbf{a})}\,\sqrt{\mathrm{I}(\mathbf{b},\mathbf{b})}}.

Area. A little parameter rectangle du \times dv maps to a parallelogram spanned by \sigma_u\,du and \sigma_v\,dv. Its area is \lVert\sigma_u\times\sigma_v\rVert\,du\,dv, and the Lagrange identity turns that cross-product length into E, F, G:

dA = \sqrt{EG - F^2}\;du\,dv. On a regular surface with tangent vectors \sigma_u,\sigma_v:

Three worked metrics

The plane, \sigma=(u,v,0): here \sigma_u=(1,0,0) and \sigma_v=(0,1,0), so

E = 1,\quad F = 0,\quad G = 1 \;\Longrightarrow\; ds^2 = du^2 + dv^2.

That is just Pythagoras — exactly what a flat sheet should give.

The cylinder of radius R, \sigma=(R\cos u, R\sin u, v): with \sigma_u=(-R\sin u, R\cos u, 0) and \sigma_v=(0,0,1),

E = R^2,\quad F = 0,\quad G = 1 \;\Longrightarrow\; ds^2 = R^2\,du^2 + dv^2.

The sphere of radius R, \sigma=(R\cos u\sin v, R\sin u\sin v, R\cos v). Differentiating and dotting,

E = R^2\sin^2 v,\quad F = 0,\quad G = R^2 \;\Longrightarrow\; ds^2 = R^2\sin^2 v\,du^2 + R^2\,dv^2.

The \sin^2 v in E is the whole story of a globe: near the equator (v=\tfrac\pi2) a degree of longitude is wide, but near the poles (v\to0) it shrinks to nothing — the lines of longitude crowd together. The metric knows this without ever leaving the surface.

Feel the metric in a parallelogram

The metric is really just the geometry of the two tangent vectors. Drag the sliders to set the lengths \lVert\sigma_u\rVert, \lVert\sigma_v\rVert and the angle \theta between them. The shaded parallelogram is the image of a unit parameter square. Watch the live readouts: E=\lVert\sigma_u\rVert^2, G=\lVert\sigma_v\rVert^2, F=\lVert\sigma_u\rVert\lVert\sigma_v\rVert\cos\theta, and the area element \sqrt{EG-F^2} — which is exactly the parallelogram's area. Set \theta=90^\circ and F drops to zero.

Vignettes

The first fundamental form is a complete summary of the surface's intrinsic geometry — the geometry available to a creature confined to the surface, with no view of the ambient 3D space. Lengths, angles, areas, shortest paths, even the curvature that Gauss called remarkable — all of it is computable from E, F, G alone.

This is a genuinely startling idea. It means two surfaces that look utterly different from the outside — a flat sheet of paper and the same sheet rolled into a cylinder — can be intrinsically identical: same metric, same ant-geometry. Bend paper without stretching it and the metric never changes. The forthcoming Theorema Egregium pushes this to its stunning conclusion: curvature itself is one of these intrinsic quantities, locked inside E, F, G, and no amount of bending can fool it.

Two traps live here. First: E, F, G depend on the parametrization, not just the surface. Re-parametrize (say, measure longitude in radians vs. degrees) and the three numbers change. But the lengths, angles and areas they compute come out the same — the coordinates are scaffolding, and the geometry underneath is invariant. Do not attach physical meaning to a raw value of E without knowing the parametrization.

Second: the area element is \sqrt{EG - F^2}\,du\,dv, with the square root — not (EG - F^2)\,du\,dv. Dropping the root is one of the most common slips in a surface-area integral. A quick sanity check: for the plane E=G=1, F=0, so \sqrt{EG-F^2}=1 and dA = du\,dv, the ordinary flat area. Without the root you would still get 1 here — which is exactly why the plane is a bad test case and why the mistake survives so long. Try the cylinder (\sqrt{R^2\cdot1-0}=R vs. R^2) to catch it.