Gaussian Curvature Calculations

Differential Geometry · Axiom Academy

EXAMPLE Gaussian Curvature Calculations Computing the Gaussian curvature of a torus using fundamental forms Excellent work! You've completed this Gaussian curvature calculation. Here's what we learned: First Fundamental Form (E, F, G): These coefficients measure intrinsic distances on the surface. For a torus, E and G vary with position while F = 0 due to orthogonal parameterization. Second Fundamental Form (e, f, g): These coefficients measure how the surface curves in 3D space. The normal curvatures are encoded in these values. Gaussian Curvature Formula: K = (eg - f²)/(EG - F²) is an intrinsic invariant that depends only on the first fundamental form, though we compute it using both forms. Torus Curvature: The Gaussian curvature of a torus is positive on the outer equator, zero at the top/bottom, and negative on the inner equator, reflecting its saddle-like geometry. Geometric Interpretation: K > 0 indicates locally elliptic (sphere-like), K = 0 indicates parabolic (cylinder-like), and K < 0 indicates hyperbolic (saddle-like) geometry. This systematic approach works for any parametric surface. Master the computation of fundamental forms, and you can analyze the intrinsic geometry of any surface!

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