PRINCIPAL CURVATURE

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, a result of Euler (1760), and are called 'principal directions'.
The product of the two principal curvatures $k\_1\; k\_2$ is the Gaussian curvature, $K$ and the average $(k\_1+k\_2)/2$ is the mean curvature, $H$.
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.

Contents

Classification of points on a surface

Lines of curvature

See also

External links

Classification of points on a surface

★ At 'elliptical' points, both principal curvatures have the same sign, and the surface is locally convex.

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★ At 'umbilic' points, both principal curvatures are equal and every tangent vector can be considered a principal direction.

★ At 'hyperbolic' points , the principal curvatures have opposite signs, and the surface will be locally saddle shaped.

★ At 'parabolic' points, one of the principal curvatures is zero. Parabolic points generally lie in a curve separating elliptical and hyperbolic regions.

Lines of curvature

The 'lines of curvature' are curves which are always tangent to a principal direction (they are integral curves for the principal curvature line fields). There will be two lines of curvature through each non-umbilic point and the lines will cross at right angles.
At umbilics the lines form one of three configurations ''star'', ''lemon'' and ''lemonstar'' (or ''monstar''). These points are also called Darbouxian Umbilics, in honor to
Gaston Darboux, the first to make a systematic study in Vol. 4, Note 7, of his famous Leçons (1896).

See also

★ curvature

External links

★ Historical Comments on Monge's Ellipsoid and the Configuration of Lines of Curvature on Surfaces Immersed in 'R'³