Let be a unit tangent vector of a regular surface . Then the normal curvature of in the direction is
(1)
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where is the shape operator. Let be a regular surface, , be an injective regular patch of with , and
(2)
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where . Then the normal curvature in the direction is
(3)
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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 .