First Fundamental Form Examples

Differential Geometry · Axiom Academy

EXAMPLE First Fundamental Form Examples Computing E, F, G coefficients and arc length on surfaces Excellent work! You've mastered computing the first fundamental form. Here's what we learned: Sphere: The coefficients are E = R², F = 0, G = R²sin²(u), giving ds² = R²(du² + sin²(u)dv²) Plane: The simplest case with E = G = 1 and F = 0, recovering the Euclidean metric ds² = du² + dv² Torus: Shows how both radii contribute: E = r², F = 0, G = (R + r cos u)² Arc Length: The first fundamental form allows us to compute intrinsic geometric quantities like curve length on any surface Orthogonality: When F = 0, the parameter curves are orthogonal, simplifying many calculations The first fundamental form encodes all intrinsic geometry of a surface. Practice computing E, F, G for different parametrizations to build geometric intuition!

This is the written version of the interactive lesson above. See the full Differential Geometry course.