The First Fundamental Form

The First Fundamental Form is the prototype for a Riemannian metric, the central object of modern differential geometry, used throughout general relativity to describe spacetime intervals via the metric tensor $g_{\mu\nu}$. In cartography, the distortion coefficients $E, F, G$ quantify exactly how much any map projection stretches or skews distances and angles, explaining why no flat map of the Earth can be perfectly accurate (a fact provable using the Theorema Egregium built on this form). In computer graphics and computational geometry, the First Fundamental Form underlies mesh parametrization and texture-mapping algorithms, where minimizing metric distortion produces less warped textures on 3D models. In elasticity theory, the form measures strain on curved shells and membranes, since deformation is detected precisely by changes in $E, F, G$. It is also the direct ancestor of the constructions used to define the Gauss Map (which compares the First Fundamental Form to the Second Fundamental Form to extract curvature), to define Geodesics on Surfaces (curves that locally minimize the length functional built from $ds^2=E\,du^2+2F\,dudv+G\,dv^2$), and to formulate the Gauss-Bonnet Theorem relating total curvature to topology.

Ask the tutor: How does the First Fundamental Form combine with the Second Fundamental Form to define Gaussian and mean curvature?

Picture the tangent plane at a point of a surface as a private copy of $\mathbb{R}^2$, glued to the surface at that single point. The parametrization $\mathbf{x}(u,v)$ gives this plane a natural basis $\{\mathbf{x}_u, \mathbf{x}_v\}$, but this basis is typically not orthonormal: the two basis vectors may have different lengths and may not be perpendicular. The functions $E$, $F$, $G$ are precisely the data needed to correct for this: $E$ and $G$ measure how much the coordinate grid lines are stretched, and $F$ measures how much they are skewed away from perpendicularity. Think of a rectangular grid drawn in the $(u,v)$-plane, then mapped onto the surface by $\mathbf{x}$. Near a point $q_0 = (u_0,v_0)$, a small coordinate rectangle with sides $du$ and $dv$ is mapped to a small (possibly slanted, possibly stretched) parallelogram on the surface, spanned by $\mathbf{x}_u\,du$ and $\mathbf{x}_v\,dv$. The First Fundamental Form is exactly the tool that computes the side lengths, the angle, and the area of that infinitesimal parallelogram — all without needing to know how the surface curves in $\mathbb{R}^3$, only how the tangent vectors $\mathbf{x}_u, \mathbf{x}_v$ relate to each other via dot products. This is why $E$, $F$, $G$ suffice to compute the length of any curve lying in the surface, no matter how it winds through $(u,v)$-space: a curve is built from infinitesimal displacements $du\,\mathbf{x}_u + dv\,\mathbf{x}_v$, and the form tells you the squared length of each such displacement.

Example 1: A cone of revolution (asymmetric coordinate roles). Let $\alpha \in (0,\pi/2)$ be a fixed half-angle and parametrize the cone

Here $u$ is an angular coordinate and $v$ is a radial-type coordinate along the slant — deliberately asymmetric roles, unlike a symmetric doubly-periodic parametrization. Compute the partials:

Then:

So the First Fundamental Form is

Length check. Take the specific curve $\gamma(t) = \mathbf{x}(t, 2+t)$ for $t \in [0,1]$ — a "spiral" moving in both $u$ and $v$ simultaneously with unit rates, asymmetric and non-obvious. Here $u(t)=t$, $v(t) = 2+t$, so $u'=1$, $v'=1$. Evaluate $E,F,G$ along the curve: $E(t) = (2+t)^2\sin^2\alpha$, $F=0$, $G=1$. Then

For a concrete number, take $\alpha = \pi/6$, so $\sin\alpha = 1/2$:

Substitute $s = 2+t$, $ds = dt$, $s$ from $2$ to $3$:

Using $\int \sqrt{s^2+a^2}\,ds = \frac{s}{2}\sqrt{s^2+a^2} + \frac{a^2}{2}\ln\left(s+\sqrt{s^2+a^2}\right) + C$ with $a=2$:

At $s=3$: $\frac32\sqrt{13} + 2\ln(3+\sqrt{13})$. At $s=2$: $1\cdot\sqrt{8} + 2\ln(2+\sqrt8) = 2\sqrt2 + 2\ln(2+2\sqrt2)$. So

Numerically: $\sqrt{13}\approx 3.6056$, so $\frac32\sqrt{13}\approx 5.4083$; $\sqrt2\approx1.4142$, $2\sqrt2\approx 2.8284$; $3+\sqrt{13}\approx 6.6056$, $2+2\sqrt2\approx4.8284$, ratio $\approx1.3680$, $\ln(1.3680)\approx 0.3132$, times $2 \approx 0.6264$. Sum: $5.4083 - 2.8284 + 0.6264 \approx 3.2063$. Then $L = \frac12(3.2063) \approx 1.6032$. So the arc length of this spiral on the cone, computed purely from $E,F,G$ without ever using the ambient $\mathbb{R}^3$ formula for $\alpha(t)$ directly, is approximately $1.603$. Area check. Compute the area of the region $Q = \{(u,v) : 0 \le u \le \pi/3, 1 \le v \le 3\}$ (a "slice" of the cone, angularly asymmetric versus the full $2\pi$ range). With $\alpha=\pi/6$ as above, $EG-F^2 = v^2\sin^2\alpha \cdot 1 - 0 = v^2/4$, so $\sqrt{EG-F^2} = v/2$. Then

Example 2: A Monge patch (graph of a function) — angle computation. Let $f(u,v) = u^2 v$ (a genuinely asymmetric, non-quadratic-in-both-variables function) and parametrize the graph surface

Then

So

Evaluate at the specific point $q_0=(u,v) = (1,2)$: $E = 1+4(1)(4) = 17$, $F = 2(1)(2) = 4$, $G = 1+1=2$. Consider two tangent vectors at $\mathbf{x}(1,2)$: take $w_1 = 3\mathbf{x}_u + \mathbf{x}_v$ (i.e. $(a_1,b_1)=(3,1)$) and $w_2 = -\mathbf{x}_u + 2\mathbf{x}_v$ (i.e. $(a_2,b_2)=(-1,2)$), a deliberately non-orthogonal, non-unit, asymmetric pair. Compute the angle between them using the formula from the theorem. Numerator:

Compute term by term: $17\cdot(-3) = -51$. Next, $(3)(2) + (-1)(1) = 6-1=5$, times $F=4$ gives $20$. Last term: $2\cdot1\cdot2=4$. Sum: $-51+20+4 = -27$. Denominator, first factor:

Second factor:

So $|w_1| = \sqrt{179}$, $|w_2| = \sqrt{9}=3$. Then

Numerically $\sqrt{179}\approx 13.379$, so $\cos\theta \approx -0.6727$, giving $\theta \approx 132.3^\circ$. As a consistency check, recompute $w_1, w_2$ directly as vectors in $\mathbb{R}^3$ using $\mathbf{x}_u=(1,0,4)$, $\mathbf{x}_v=(0,1,1)$ (evaluated at $(1,2)$: $2uv=4$, $u^2=1$):

Direct dot product: $w_1\cdot w_2 = -3+2-26 = -27$. Matches. Direct norms: $|w_1|^2 = 9+1+169=179$; $|w_2|^2=1+4+4=9$. Both match the intrinsic computation exactly, confirming the theorem's formula reproduces the ambient Euclidean computation without needing $\mathbb{R}^3$ coordinates at all.