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Projection stereographic and second fundamental form

  1. May 9, 2015 #1
    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 gif.latex?N=\frac{ruXrv}{||ruXrv||}.gif
  2. jcsd
  3. May 10, 2015 #2
    Think geometrically: what vectors are orthogonal to the surface of the sphere?
  4. May 10, 2015 #3
    I was thinking it was very difficult... Now I think to be a bit silly! :) thx
  5. May 12, 2015 #4
    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)) ?
  6. May 12, 2015 #5
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