First Fundamental Form: Concepts, Formulas and Examples

The first fundamental form is the tool for measuring distances, angles and areas on a curved surface from its own coordinates. For a parametrisation X(u, v), it is the expression E du^2 + 2F du dv + G dv^2, where E, F, G come from dot products of the partial derivatives.

It is the surface's ruler and protractor: lengths of curves drawn on it, angles between them, and areas of patches all come from E, F, G alone. Remarkably this is intrinsic geometry, measurable by an ant living on the surface with no knowledge of the surrounding space.

Key Concepts

• Coefficients: E = X_u·X_u, F = X_u·X_v, G = X_v·X_v; E and G measure stretching along u and v, F measures non-perpendicularity of the grid.
• Intrinsic geometry: whatever follows from E, F, G alone, without the ambient space.
• Isometry: a map preserving the first fundamental form; rolling a flat sheet into a cylinder is one.
• Area element: dA = sqrt(EG - F^2) du dv.
• Orthogonal coordinates: F = 0 means the coordinate curves meet at right angles.

Formulas

• Form: I = E du^2 + 2F du dv + G dv^2, with EG - F^2 > 0 for a regular parametrisation.
• Curve length: integral of sqrt(E (u')^2 + 2F u'v' + G (v')^2) dt along (u(t), v(t)).
• Area: double integral of sqrt(EG - F^2) du dv over the parameter region.

Worked Example

First fundamental form of the cylinder X(u, v) = (cos u, sin u, v); area of the patch u in [0, 2 pi], v in [0, 3].

1. X_u = (-sin u, cos u, 0), X_v = (0, 0, 1).
2. E = 1, F = 0, G = 1; the grid is orthogonal.
3. Area element sqrt(EG - F^2) = 1.
4. Area = integral 0 to 2 pi, integral 0 to 3 of 1 dv du = 6 pi ≈ 18.85, i.e. circumference times height.

Common Misconceptions

• It cannot see bending. The cylinder has E = G = 1, F = 0, exactly like the plane, because rolling does not stretch. Bending needs the second fundamental form.
• E, F, G are usually functions, varying across the surface; constant values are a special convenience.
• F = 0 is not flatness: it only says the chosen grid is perpendicular, true on the sphere too.

Real-World Applications

• Cartography: projections are judged by how they distort the first fundamental form.
• Sheet metal: parts formable by pure bending are exactly those isometric to a flat sheet.
• Graphics: texture mapping is monitored through the first fundamental form to avoid distortion.

The first fundamental form E du^2 + 2F du dv + G dv^2 packages all intrinsic measurement on a surface. On the cylinder it is simply du^2 + dv^2, giving patch area 6 pi. It is the complete intrinsic geometry, blind to how the surface bends through space.