Principal curvature

In differential geometry, the two principal curvatures at a given point of a surface are the eigenvalues of the shape operator at the point. They measure how the surface bends by different amounts in different directions at that point.

At each point p of a differentiable surface in 3-dimensional Euclidean space one may choose a unit normal vector. A normal plane at p is one that contains the normal vector, and will therefore also contain a unique direction tangent to the surface and cut the surface in a plane curve, called normal section. This curve will in general have different curvatures for different normal planes at p. The principal curvatures at p, denoted k₁ and k₂, are the maximum and minimum values of this curvature.

Here the curvature of a curve is by definition the reciprocal of the radius of the osculating circle. The curvature is taken to be positive if the curve turns in the same direction as the surface's chosen normal, and otherwise negative. The directions of the normal plane where the curvature takes its maximum and minimum values are always perpendicular, if k₁ does not equal k₂, a result of Euler (1760), and are called principal directions. From a modern perspective, this theorem follows from the spectral theorem because these directions are as the principal axes of a symmetric tensor—the second fundamental form. A systematic analysis of the principal curvatures and principal directions was undertaken by Gaston Darboux, using Darboux frames.

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