Projection stereographic and second fundamental form

Simone Furcas · May 9, 2015

Let r:R2 →R3 be given by the formula

$rac{2u}{u^2%20+v^2%20+1},%20\frac{2v}{u^2%20+v^2%20+1},%20\frac{-1+u^2%20+v^2}{u^2%20+v^2%20+1}).gif$

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

$Formula for the unit normal vector N equals ruXrv divided by the norm of ruXrv, where ru and rv are partial derivatives of r.$

micromass · May 10, 2015

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

Simone Furcas · May 10, 2015

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

Simone Furcas · May 12, 2015

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 · May 12, 2015

Projection stereographic and second fundamental form

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