Normal Curvature

Let ${\bf u}_{\bf p}$ be a unit Tangent Vector of a Regular Surface $M\subset\Bbb{R}^3$. Then the normal curvature of $M$ in the direction ${\bf u}_{\bf p}$ is

$$\kappa({\bf u}_{\bf p})=S({\bf u}_{\bf p})\cdot {\bf u}_{\bf p},$$ (1)

where $S$ is the Shape Operator. Let $M\subset\Bbb{R}^3$ be a Regular Surface, ${\bf p}\in M$, ${\bf x}$ be an injective Regular Patch of $M$ with ${\bf p}={\bf x}(u_0,v_0)$, and

$${\bf v}_{\bf p}=a{\bf x}_u(u_0,v_0)+b{\bf x}_v(u_0,v_0),$$ (2)

where ${\bf v}_{\bf p}\in M_{\bf p}$. Then the normal curvature in the direction ${\bf v}_{\bf p}$ is

$$\kappa(v{\bf p})={ea^2+2fab+gb^2\over Ea^2+2Fab+Gb^2},$$ (3)

where $E$, $F$, and $G$ are the coefficients of the first Fundamental Form and $e$, $f$, and $g$ 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 $\kappa_1$ and $\kappa_2$.

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.