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Concept# Principal curvature

Summary

In differential geometry, the two principal curvatures at a given point of a surface are the maximum and minimum values of the curvature as expressed by the eigenvalues of the shape operator at that 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 in 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.
The product k1k2 of the two principal curvatures is the Gaussian curvature, K, and the average (k1 + k2)/2 is the mean curvature, H.
If at least one of the principal curvatures is zero at every point, then the Gaussian curvature will be 0 and the surface is a developable surface. For a minimal surface, the mean curvature is zero at every point.
Let M be a surface in Euclidean space with second fundamental form . Fix a point p ∈ M, and an orthonormal basis X1, X2 of tangent vectors at p.

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In mathematics, the differential geometry of surfaces deals with the differential geometry of smooth surfaces with various additional structures, most often, a Riemannian metric. Surfaces have been extensively studied from various perspectives: extrinsically, relating to their embedding in Euclidean space and intrinsically, reflecting their properties determined solely by the distance within the surface as measured along curves on the surface.

The helicoid, also known as helical surface, after the plane and the catenoid, is the third minimal surface to be known. It was described by Euler in 1774 and by Jean Baptiste Meusnier in 1776. Its name derives from its similarity to the helix: for every point on the helicoid, there is a helix contained in the helicoid which passes through that point.

Gauss's Theorema Egregium (Latin for "Remarkable Theorem") is a major result of differential geometry, proved by Carl Friedrich Gauss in 1827, that concerns the curvature of surfaces. The theorem says that Gaussian curvature can be determined entirely by measuring angles, distances and their rates on a surface, without reference to the particular manner in which the surface is embedded in the ambient 3-dimensional Euclidean space. In other words, the Gaussian curvature of a surface does not change if one bends the surface without stretching it.

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