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Principal Curvatures

The Maximum and Minimum of the Normal Curvature $\kappa_1$ and $\kappa_2$ at a given point on a surface are called the principal curvatures. The principal curvatures measure the Maximum and Minimum bending of a Regular Surface at each point. The Gaussian Curvature $K$ and Mean Curvature $H$ are related to $\kappa_1$ and $\kappa_2$ by

$\displaystyle K$ $\textstyle =$ $\displaystyle \kappa_1\kappa_2$ (1)
$\displaystyle H$ $\textstyle =$ $\displaystyle {\textstyle{1\over 2}}(\kappa_1+\kappa_2).$ (2)

This can be written as a Quadratic Equation
\end{displaymath} (3)

which has solutions
$\displaystyle \kappa_1$ $\textstyle =$ $\displaystyle H+\sqrt{H^2-K}$ (4)
$\displaystyle \kappa_2$ $\textstyle =$ $\displaystyle H-\sqrt{H^2-K}\,.$ (5)

See also Gaussian Curvature, Mean Curvature, Normal Curvature, Normal Section, Principal Direction, Principal Radius of Curvature, Rodrigues's Curvature Formula


Geometry Center. ``Principal Curvatures.''

Gray, A. ``Normal Curvature.'' §14.2 in Modern Differential Geometry of Curves and Surfaces. Boca Raton, FL: CRC Press, pp. 270-273, 277, and 283, 1993.

© 1996-9 Eric W. Weisstein