Brioschi Formula

For a curve with Metric

$\begin{displaymath} ds^2=E\,du^2+F\,du\,dv+G\,dv^2, \end{displaymath}$

(1)

where

, and

$\begin{displaymath} K={M_1+M_2\over (EG-F^2)^2}, \end{displaymath}$

(2)

where

$\displaystyle M_1$	$\textstyle \equiv$	$\displaystyle \left\vert\begin{array}{ccccccc} -{\textstyle{1\over 2}}E_{uv}+F_... ... F\nonumber\\ {\textstyle{1\over 2}}G_v & F & G\end{array}\right\vert\nonumber$
			(3)
$\displaystyle M_2$	$\textstyle \equiv$	$\displaystyle \left\vert\begin{array}{ccccccc} 0 & {\textstyle{1\over 2}}E_v & ... ..._v & E & F\nonumber\\ {\textstyle{1\over 2}}G_u & F & G\end{array}\right\vert,$

which can also be written

$\displaystyle K$	$\textstyle =$	$\displaystyle -{1\over\sqrt{EG}}\left[{{\partial\over\partial u}\left({{1\over\... ...tial v}\left({{1\over\sqrt{G}}{\partial\sqrt{E}\over\partial v}}\right)}\right]$
			(4)
	$\textstyle =$	$\displaystyle -{1\over 2\sqrt{EG}}\left[{{\partial\over\partial u}\left({G_u\ov... ...EG}}\right) +{\partial\over\partial v}\left({E_v\over\sqrt{EG}}\right)}\right].$	(5)

References

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