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3.5 Gaussian and mean curvatures

From (3.49), (3.50), it is readily seen that the Gaussian and mean curvatures are the product and the average of the two principal curvatures, respectively:
$\displaystyle K = \kappa_{max}\kappa_{min}\;,$     (3.61)
$\displaystyle H = \frac{\kappa_{max} + \kappa_{min}}{2}\;.$     (3.62)

The sign of the Gaussian curvature coincides with sign of $ LN-M^2$ , since $ K = \frac{LN - M^2}{EG - F^2}$ (see (3.46)) and $ EG-F^2>0$ . Consequently a point on a surface is elliptic if $ K>0$ ( $ \kappa_{max}$ and $ \kappa_{min}$ are of the same sign), hyperbolic if $ K<0$ ( $ \kappa_{max}$ and $ \kappa_{min}$ have different signs) and parabolic if $ K=0$ and $ H\neq 0$ (either $ \kappa_{max}$ or $ \kappa_{min}$ is zero), flat or planar point if $ K=H=0$ ( $ \kappa_{max} =\kappa_{min}=0$ ).

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December 2009