Principal Curvatures

Differential Geometry · Axiom Academy

Understanding the eigenvalues of the shape operator and their geometric meaning 1. Definition: Eigenvalues of the Shape Operator The principal curvatures κ₁ and κ₂ at a point on a surface are defined as the eigenvalues of the shape operator S (also called the Weingarten map). 2. Maximum and Minimum Normal Curvatures The principal curvatures represent the maximum and minimum values of the normal curvature κₙ(v) as the direction v varies over all tangent directions. For any tangent vector v, the normal curvature is bounded: κ₂ ≤ κₙ(v) ≤ κ₁ (assuming κ₁ ≥ κ₂). 3. Computation from Fundamental Forms Principal curvatures are computed from the first fundamental form I (which measures distances on the surface) and the second fundamental form II (which measures how the surface bends). The eigenvalue equation becomes: det(II - κI) = 0, which yields a quadratic equation in κ. The sign of a principal curvature indicates the direction of bending: Positive curvature (κ > 0): The surface curves toward the normal vector (like the outside of a sphere). Negative curvature (κ The surface curves away from the normal vector (like a saddle). Zero curvature (κ = 0): The surface is flat in that direction (like a cylinder). Principal curvatures encode the essential geometric information about a surface: • Mean curvature: H = (κ₁ + κ₂)/2 measures average bending • Gaussian curvature: K = κ₁κ₂ determines intrinsic geometry • Principal directions: Orthogonal directions of maximum and minimum bending

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