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Brioschi Formula

For a curve with Metric

\end{displaymath} (1)

where $E$, $F$, and $G$ is the first Fundamental Form, the Gaussian Curvature is
K={M_1+M_2\over (EG-F^2)^2},
\end{displaymath} (2)

$\displaystyle M_1$ $\textstyle \equiv$ $\displaystyle \left\vert\begin{array}{ccccccc}
-{\textstyle{1\over 2}}E_{uv}+F_...
... F\nonumber\\
{\textstyle{1\over 2}}G_v & F & G\end{array}\right\vert\nonumber$  
$\displaystyle M_2$ $\textstyle \equiv$ $\displaystyle \left\vert\begin{array}{ccccccc}
0 & {\textstyle{1\over 2}}E_v & ...
..._v & E & F\nonumber\\
{\textstyle{1\over 2}}G_u & F & G\end{array}\right\vert,$  

which can also be written
$\displaystyle K$ $\textstyle =$ $\displaystyle -{1\over\sqrt{EG}}\left[{{\partial\over\partial u}\left({{1\over\...
...tial v}\left({{1\over\sqrt{G}}{\partial\sqrt{E}\over\partial v}}\right)}\right]$  
  $\textstyle =$ $\displaystyle -{1\over 2\sqrt{EG}}\left[{{\partial\over\partial u}\left({G_u\ov...
+{\partial\over\partial v}\left({E_v\over\sqrt{EG}}\right)}\right].$ (5)

See also Fundamental Forms, Gaussian Curvature


Gray, A. Modern Differential Geometry of Curves and Surfaces. Boca Raton, FL: CRC Press, pp. 392-393, 1993.

© 1996-9 Eric W. Weisstein