Normal Curvature

Let u_(p) be a unit tangent vector of a regular surface M subset R^3. Then the normal curvature of M in the direction u_(p) is


where S is the shape operator. Let M subset R^3 be a regular surface, p in M, x be an injective regular patch of M with p=x(u_0,v_0), and


where v_(p) in M_(p). Then the normal curvature in the direction v_(p) is


where E, F, and G are the coefficients of the first fundamental form and e, f, and g are the coefficients of the second fundamental form.

The maximum and minimum values of the normal curvature at a point on a regular surface are called the principal curvatures kappa_1 and kappa_2.

See also

Curvature, Fundamental Forms, Gaussian Curvature, Mean Curvature, Principal Curvatures, Shape Operator, Tangent Vector

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Euler, L. "Recherches sur la courbure des surfaces." Mém. de l'Acad. des Sciences, Berlin 16, 119-143, 1760.Gray, A. "Normal Curvature." §18.2 in Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 363-367, 1997.Meusnier, J. B. "Mémoire sur la courbure des surfaces." Mém. des savans étrangers 10 (lu 1776), 477-510, 1785.

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Normal Curvature

Cite this as:

Weisstein, Eric W. "Normal Curvature." From MathWorld--A Wolfram Web Resource.

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