Projection stereographic and second fundamental form

[Simone Furcas]

Let r:R2 →R3 be given by the formula

Compute the second fundamental form with respect to this basis (Hint: There’s a shortcut to computing the unit normal n).

I can't find thi shortcut, does anyone help me? I'm solving it with normal vector and first and second derivate, but I obtained impossibile result... The solve is too long to write down here... I use I used

 

[micromass]

Think geometrically: what vectors are orthogonal to the surface of the sphere?

 

[Simone Furcas]

I was thinking it was very difficult... Now I think to be a bit silly! :) thx

 

[Simone Furcas]

Is it the same with an ellipsoid (a*cos(x)*sen(y),b*sen(x)*sen(y),c*cos(y)) ? N is (a*cos(x)*sen(y),b*sen(x)*sen(y),c*cos(y)) ?

 

[micromass]

No.