The First Fundamental Form

Measure lengths, angles, and areas on surfaces

The First Fundamental Form

How would a creature living on a surface measure distance without ever leaving it? Three numbers at each point, E, F, and G, give it a ruler and a protractor.

The Metric Tensor

Click on the surface to read E, F, G at that point. The yellow ellipse is the “unit circle” of the metric: a true circle when E = G and F = 0, stretched everywhere else.

Click on the surface to see the metric at that point
Su direction
Sv direction
Metric ellipse (unit circle in metric)
Go deeper:The Metric Coefficients E, F, G

The first fundamental form is a way to measure distances on a surface using only information intrinsic to the surface itself. It encodes how the surface inherits a metric from the surrounding 3D space.

Given the tangent vectors Su and Sv, we define three coefficients E, F, G that completely determine lengths, angles, and areas on the surface.

E = ⟨Su, Su⟩
Length² in u-direction
F = ⟨Su, Sv⟩
Dot product (angle info)
G = ⟨Sv, Sv⟩
Length² in v-direction

These three numbers at each point form a 2×2 symmetric matrix called the metric tensor or first fundamental form.

Go deeper:Arc Length, Angles, and Area

Arc Length

ds² = E du² + 2F du dv + G dv²

The infinitesimal distance formula. Integrate along a curve to get arc length.

Angle Between Curves

cos θ = I(v, w) / (|v| · |w|)

Use the metric to compute dot products and angles between tangent vectors.

Area Element

dA = √(EG - F²) du dv

The area of an infinitesimal parallelogram. Integrate to get surface area.

Go deeper:Special Case: Orthogonal Coordinates

When F = 0, the parameter curves are orthogonal (perpendicular). This simplifies many calculations:

  • • Arc length: ds² = E du² + G dv² (no cross term)
  • • Area: dA = √(EG) du dv
  • • The metric ellipse becomes aligned with the coordinate axes

Example: Spherical coordinates (φ, θ) on a unit sphere — with φ the polar angle and θ the azimuth — give F = 0, with E = 1 and G = sin²φ.

Go deeper:Matrix Notation

The first fundamental form can be written as a 2×2 matrix:

I = [E F; F G]
|v|² = vT I v

This matrix is always symmetric and positive definite(as long as Su and Sv are linearly independent). Its determinant EG - F² equals |Su × Sv|².

Key Takeaways

  • E, F, G are the coefficients of the first fundamental form (the metric tensor).
  • They measure lengths: ds² = E du² + 2F du dv + G dv².
  • They give angles and area; F = 0 means orthogonal coordinates.
  • The metric is intrinsic: a 2D being can measure it without leaving the surface.

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