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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

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

where ${\bf N}$ is the Normal Vector, $E$, $F$, and $G$ the coefficients of the first Fundamental Form
\end{displaymath} (3)

and $e$, $f$, and $g$ the coefficients of the second Fundamental Form given by
$\displaystyle e$ $\textstyle =$ $\displaystyle -{\bf N}_u\cdot{\bf x}_u={\bf N}\cdot{\bf x}_{uu}$ (4)
$\displaystyle f$ $\textstyle =$ $\displaystyle -{\bf N}_v\cdot{\bf x}_u={\bf N}\cdot{\bf x}_{uv}$  
  $\textstyle =$ $\displaystyle {\bf N}_{vu}\cdot{\bf x}_{vu}=-{\bf N}_u\cdot{\bf x}_v$ (5)
$\displaystyle g$ $\textstyle =$ $\displaystyle -{\bf N}_v\cdot{\bf x}_v={\bf N}\cdot{\bf x}_{vv}.$ (6)

See also Fundamental Forms, Shape Operator


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.

© 1996-9 Eric W. Weisstein