The First Fundamental Form

Measure lengths, angles, and areas on surfaces

The First Fundamental Form

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.

Interactive: The Metric Tensor

Click on the surface to see the metric at that point. The yellow ellipse shows the “unit circle” in the metric — directions where |v| = 1. When E = G and F = 0, this is a circle; otherwise it's stretched into an ellipse.

Click on the surface to see the metric at that point
Su direction
Sv direction
Metric ellipse (unit circle in metric)

The Metric Coefficients E, F, G

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.

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.

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 (θ, φ) on a sphere give F = 0, with E = 1 and G = sin²φ.

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 (metric tensor)
  • They measure lengths: ds² = E du² + 2F du dv + G dv²
  • They determine angles: F = 0 means orthogonal coordinates
  • They give area: dA = √(EG - F²) du dv
  • The metric is intrinsic — a 2D being can measure it without leaving the surface

Next: The Second Fundamental Form — capturing how the surface curves in 3D space.