## 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 .

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

References

Euler, L. Recherches sur la courbure des surfaces.'' Mém. de l'Acad. des Sciences, Berlin 16, 119-143, 1760.

Gray, A. Normal Curvature.'' §14.2 in Modern Differential Geometry of Curves and Surfaces. Boca Raton, FL: CRC Press, pp. 270-273 and 277, 1993.

Meusnier, J. B. Mémoire sur la courbure des surfaces.'' Mém. des savans étrangers 10 (lu 1776), 477-510, 1785.