Principal curvature

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In differential geometry, the two principal curvatures at a given point of a differentiable surface in Euclidean space are the minimum and maximum of the curvatures at that point of all the curves on the surface passing through the point.

Here the curvature of a curve is taken to be 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, otherwise negative. The directions of minimum and maximum curvature are always perpendicular and are called principal directions. At umbilic points both principal curvatures are equal and every tangent vector can be considered a principal directions.

The product of the two principal curvatures k₁k₂ is the Gaussian curvature, K and the average (k₁ + k₂) / 2 is the mean curvature, H. When both principal curvatures have the same sign the Gaussian curvature is positive and the surface is locally convex. Such points are called elliptical. When the curvatures have opposite signs the surface will be locally saddle shaped, such points are called hyperbolic. If one of the principal curvatures is zero then the Gaussian curvature is zero and the surface is parabolic. Parabolic points generally lie in a line separating elliptical and hyperbolic regions.

The lines of curvature are curves which are always tangent to a principal direction. There will be two lines of curvature through each non-umbilic point. At umbilics the lines form one of three configurations star, lemon and lemonstar or monstar.

For a developable surface, at least one of the principal curvatures is zero at every point. For a minimal surface, the two principal curvatures are equal and opposite at every point. See also the discussion in the article on curvature (section "Curvature of surfaces in 3-space").

[edit] See also

• curvature