TOPICS
Search

Brioschi Formula

DOWNLOAD Mathematica NotebookDownload Wolfram Notebook

For a curve with first fundamental form

ds^2=Edu^2+2Fdudv+Gdv^2,
(1)

the Gaussian curvature is

K=(M_1-M_2)/((EG-F^2)^2),
(2)

where

M_1=|-1/2E_(vv)+F_(uv)-1/2G_(uu) 1/2E_u F_u-1/2E_v; F_v-1/2G_u E F; 1/2G_v F G|
(3)
M_2=|0 1/2E_v 1/2G_u; 1/2E_v E F; 1/2G_u F G|.
(4)

For a patch where F=0, the Gaussian curvature is given by

K=-1/(sqrt(EG))[partial/(partialu)(1/(sqrt(E))(partialsqrt(G))/(partialu))+partial/(partialv)(1/(sqrt(G))(partialsqrt(E))/(partialv))]
(5)
=-1/(2sqrt(EG))[partial/(partialu)((G_u)/(sqrt(EG)))+partial/(partialv)((E_v)/(sqrt(EG)))].
(6)

See also

First Fundamental Form, Gaussian Curvature

Explore with Wolfram|Alpha

References

Gray, A. Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 504-507, 1997.

Referenced on Wolfram|Alpha

Brioschi Formula

Cite this as:

Weisstein, Eric W. "Brioschi Formula." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/BrioschiFormula.html

Subject classifications