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

$$K = \kappa_1\kappa_2$$ (1)

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

This can be written as a Quadratic Equation

$$\kappa^2-2H\kappa+K=0,$$ (3)

which has solutions

$$\kappa_1 = H+\sqrt{H^2-K}$$ (4)

$$\kappa_2 = H-\sqrt{H^2-K}\,.$$ (5)

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

References

Geometry Center. ``Principal Curvatures.'' http://www.geom.umn.edu/zoo/diffgeom/surfspace/concepts/curvatures/prin-curv.html.

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.