First Fundamental Form

First Fundamental Form: Intrinsic Geometry of a Surface

Intrinsic vs Extrinsic

Imagine a flat sheet of paper. It can be rolled into a cylinder or a cone. For a creature living on the sheet (an ant), the flat sheet and the cylinder are indistinguishable — distances, angles, and areas remain unchanged. This is intrinsic geometry — the geometry “seen” by a surface dweller.

The first fundamental form encodes precisely the intrinsic geometry: how to measure distances and areas while living on the surface, without reference to the external 3D space.

Surface and Its Tangent Space

A surface is given by a smooth mapping $r(u, v) = (x(u,v), y(u,v), z(u,v))$ from the parameter domain $(u, v)$ into $\mathbb{R}^3$.

Tangent vectors: $r_u = \partial r/\partial u = (\partial x/\partial u, \partial y/\partial u, \partial z/\partial u),\ r_v = \partial r/\partial v$. At each point they generate the tangent plane $T_pM$.

Normal: $n = (r_u \times r_v)/|r_u \times r_v|$. Perpendicular to the surface.

Regular point: $r_u \times r_v \ne 0$ — condition for the surface to be “smooth” (no self-intersections at this point).

First Quadratic Form

The first fundamental form (metric tensor):

$I = ds^2 = E, du^2 + 2F, du, dv + G, dv^2$

Coefficients: $E = r_u \cdot r_u = |r_u|^2$, $F = r_u \cdot r_v$, $G = r_v \cdot r_v = |r_v|^2$.

Physical meaning: If a curve on the surface is given by $u = u(t)$, $v = v(t)$, then its length: $L = \int \sqrt{E(u')^2 + 2F u'v' + G(v')^2},dt$. The first form is the “ruler” for measuring distances on the surface.

Area: $dS = |r_u \times r_v|, du, dv = \sqrt{EG - F^2}, du, dv$. The area of a region $D$: $S = \iint_D \sqrt{EG - F^2}, du, dv$.

Angle between curves: For curves with tangents $(u_1', v_1')$ and $(u_2', v_2')$: $ \cos \theta = \frac{E u_1'u_2' + F(u_1'v_2' + u_2'v_1') + G v_1'v_2'}{\sqrt{I_1 \cdot I_2}} $

Numerical Examples

Plane: $r(u,v) = (u, v, 0)$. $r_u = (1,0,0)$, $r_v = (0,1,0)$. $E = 1$, $F = 0$, $G = 1$. $ds^2 = du^2 + dv^2$ — Euclidean metric. $\sqrt{EG - F^2} = 1 \to$ standard area.

Unit sphere: $r(\varphi,\theta) = (\sin\varphi \cos\theta, \sin\varphi \sin\theta, \cos\varphi)$. $r_\varphi = (\cos\varphi \cos\theta, \cos\varphi \sin\theta, -\sin\varphi)$, $r_\theta = (-\sin\varphi \sin\theta, \sin\varphi \cos\theta, 0)$.

$E = r_\varphi \cdot r_\varphi = 1$, $F = r_\varphi \cdot r_\theta = 0$, $G = r_\theta \cdot r_\theta = \sin^2\varphi$.

$ds^2 = d\varphi^2 + \sin^2\varphi, d\theta^2$ — standard metric of the sphere.

$dS = \sqrt{EG-F^2} d\varphi d\theta = \sin\varphi d\varphi d\theta$. Area of the unit sphere: $\int_0^\pi \int_0^{2\pi} \sin\varphi, d\theta, d\varphi = 4\pi\ \checkmark$.

Torus: $r(u,v) = ((R + r\cos v)\cos u,, (R + r\cos v)\sin u,, r\sin v)$. $E = (R + r\cos v)^2$, $F = 0$, $G = r^2$. Area: $4\pi^2Rr$ (depends on both radii $R$ and $r$ — the product of the lengths of the circles $2\pi R$ and $2\pi r$).

Conformal Mappings: Preservation of Angles

A map $f: M \to N$ between surfaces is conformal if it preserves angles. This is equivalent to: the metric $f^*(g_N) = \lambda^2 \cdot g_M$ (scaling by a varying factor $\lambda$).

The most important example: the Mercator projection — a conformal mapping of the sphere onto the plane (1569). Angles are preserved $\to$ navigational courses are straight lines. But areas are distorted: Greenland appears as large as Africa, though in reality it is 14 times smaller.

For a conformal mapping of the sphere onto the plane: $ds^2_{\text{sphere}} = \cos^2(u/R)(du^2 + dv^2)$ (after appropriate substitutions) $\to F = 0,\ E = G$ $\to$ orthogonal isothermal coordinate system.

Intrinsic Geometry and Gauss's Theorem

A crucial fact: Gauss curvature $K = \det([L_{ij}]) / \det([g_{ij}])$ — it is computed solely using $E, F, G$ and their derivatives (Gauss's theorem). Thus, $K$ is an intrinsic property!

Corollary: it is impossible to isometrically (without distorting distances) map a sphere onto a plane. The sphere has $K = 1/R^2 > 0$, the plane $K = 0$. Different $K \to$ no isometry. Thus, the inevitable distortions of all geographical maps.

First Fundamental Form in Navigation and GPS

Practical applications of the first fundamental form cover a broad range of tasks. In navigation, computing the shortest route along the Earth's surface requires minimizing the curve length in the metric of the sphere — a problem solved directly through the coefficients $E, F, G$. Air routes follow great circle arcs precisely because these are geodesics of the spherical metric. The GPS system computes distances using an ellipsoidal model of the Earth — the WGS-84 metric with $E, F, G$ adapted to the spheroid. The difference between spherical and ellipsoidal metrics can reach several kilometers over long distances, which is critical for precise navigation. In geodesy, the concept of “normal height” relies on the gravitational potential as a function on a manifold — planning construction over large areas requires accounting for the curvature of the Earth's surface. In computer animation, surfaces of characters are given parametrically, and realistic deformation (stretching) of fabric, skin, muscles requires computing $E, F, G$ and the change of the metric under deformation. Angle preservation (conformality) and area preservation (equiareal property) are two extreme cases between which all real-world cartographic projections must balance.

The first fundamental form is indispensable in computer graphics when working with parametric meshes: the metric coefficients $E, F, G$ determine texture stretching and are used in texture mapping algorithms and the calculation of UV unwrappings of 3D objects. In numerical simulation of surface flows (liquid on a surface, heat diffusion across a shell), the Laplace–Beltrami operator, expressed via $E, F, G$, is the principal differential operator, discretized by the finite element method.

Exercise: (a) For the paraboloid $r(u,v) = (u, v, u^2+v^2)$: find $E, F, G$ and the area above the domain $u^2+v^2 \le 1$. (b) For the unit sphere: the length of the parallel $\varphi = \varphi_0$. Compare with a Euclidean circle of the same radius. (c) Prove: for the surface of revolution $r(u,v) = (f(u)\cos v, f(u)\sin v, g(u))$, $F = 0$ always (meridians and parallels are orthogonal).