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# Normal Curvature

Let be a unit tangent vector of a regular surface . Then the normal curvature of in the direction is

 (1)

where is the shape operator. Let be a regular surface, , be an injective regular patch of with , and

 (2)

where . Then the normal curvature in the direction is

 (3)

where , , and are the coefficients of the first fundamental form and , , and 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 and .

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

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## References

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.

Normal Curvature

## Cite this as:

Weisstein, Eric W. "Normal Curvature." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/NormalCurvature.html