Brisochi Formula

Author

Eric W. Weisstein
May 3, 2004

This notebook downloaded from http://mathworld.wolfram.com/notebooks/DifferentialGeometry/BrioschiFormula.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/BrioschiFormula.html.

©2005 Wolfram Research, Inc. except for portions noted otherwise

Formula

BrioschiFormula[{x_,y_,z_},{u_,v_}]:=Module[{e,f,g,ev,eu,fv,fu,gv,gu},
{e,f,g}=FirstFundamentalForm[{x,y,z},{u,v}];
{eu,fu,gu}=D[{e,f,g},u];
{ev,fv,gv}=D[{e,f,g},v];
(Det[{
{-D[ev,v]/2+D[fu,v]-D[gu,u]/2,eu/2,fu-ev/2},
{fv-gu/2,e,f},
{gv/2,f,g}}]-
Det[{{0,ev/2,gu/2},{ev/2,e,f},{gu/2,f,g}}])/(e g-f^2)^2
]

Check

General

Simplify[GaussianCurvature[{e[u,v],f[u,v],g[u,v]},{u,v}]-BrioschiFormula[{e[u,v],f[u,v],g[u,v]},{u,v}]]

Sphere

Simplify[BrioschiFormula[{Sin[u]Cos[v],Sin[u]Sin[v],Cos[u]},{u,v}]]