Weingarten Equations

The Weingarten equations express the derivatives of the Normal using derivatives of the position vector. Let ${\bf x}:U\to\Bbb{R}^3$ be a Regular Patch, then the Shape Operator $S$ of x is given in terms of the basis $\{{\bf x}_u, {\bf x}_v\}$ by

$$-S({\bf x}_u) = {\bf N}_u={fF-eG\over EG-F^2}{\bf x}_u+{eF-fE\over EG-F^2}{\bf x}_v$$ (1)

$$-S({\bf x}_v) = {\bf N}_v={gF-fG\over EG-F^2}{\bf x}_u+{fF-gE\over EG-F^2}{\bf x}_v,$$ (2)

where ${\bf N}$ is the Normal Vector, $E$, $F$, and $G$ the coefficients of the first Fundamental Form

$$ds^2=E\,du^2+2F\,du\,dv+G\,dv^2,$$ (3)

and $e$, $f$, and $g$ the coefficients of the second Fundamental Form given by

$$e = -{\bf N}_u\cdot{\bf x}_u={\bf N}\cdot{\bf x}_{uu}$$ (4)

$$f = -{\bf N}_v\cdot{\bf x}_u={\bf N}\cdot{\bf x}_{uv}$$

$$= {\bf N}_{vu}\cdot{\bf x}_{vu}=-{\bf N}_u\cdot{\bf x}_v$$ (5)

$$g = -{\bf N}_v\cdot{\bf x}_v={\bf N}\cdot{\bf x}_{vv}.$$ (6)

See also Fundamental Forms, Shape Operator

References

Gray, A. ``Calculation of the Shape Operator.'' §14.3 in Modern Differential Geometry of Curves and Surfaces. Boca Raton, FL: CRC Press, pp. 274-277, 1993.