1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: A steographic problem(did I get it right?)

  1. Mar 10, 2009 #1
    1. The problem statement, all variables and given/known data

    A given sphere [tex]S^2[/tex] is given by [tex]x^2 + y^2 + (z-1)^2 = 1[/tex]
    where stereographical projection [tex]\pi:\pi: S^2 \thilde \{N\} \rightarrow \mathbb{R}^2[/tex]
    which carries a point p = (x,y,z) of the sphere minus Northpole N = (0,0,2) onto the intersection of the xy plane which a straight line which connects N to p.

    (u,v)= pi(x,y,z)

    3. The attempt at a solution

    Show that [tex]\pi^{-1}: \mathbb{R}^2 \rightarrow S^2[/tex] is given by

    [tex]\pi^{-1}(x,y,z) = (\frac{4u}{u^2 + v^2 + 4}, \frac{4v}{u^2 + v^2 + 4}, \frac{2(u^2+v^2)}{u^2 + v^2 + 4})[/tex]

    We can construct the line from N to p. such that x = us, y = vs and z = 2-2s

    Thus the intersection of the line L from N to p.
    [tex](us)^2 + (vs)^2 + (1-2s)^2 = 1[/tex]

    which gives us an [tex]s = \frac{4}{u^2 + v^2 + 4}[/tex]

    thus L(s) = [tex](\frac{4u}{u^2+v^2 +4}),(\frac{4v}{u^2+v^2 +4}), (\frac{2(u^2 +v^2}{u^2+v^2 +4})[/tex]

    Show that the sphere can the coverd by stereographic projection can be covered by the two coordinant neighbourhoods.

    Let a mapping of of parameterization around the Northpole of the the sphere be the same to project the point (u,v) to (u,-v,0) which gives

    [tex]\pi_2(s,t) = (\frac{4s}{s^2+t^2 +4}),-(\frac{4t}{s^2+t^2 +4}), (\frac{2(s^2 +t^2}{s^2+t^2 +4})[/tex]

    which coveres the sphere from the equator to the North pole, while

    [tex]\pi_1(u,v) = (\frac{4u}{u^2+v^2 +4}),-(\frac{4v}{u^2+v^2 +4}), (\frac{2(u^2 +v^2}{u^2+v^2 +4})[/tex]

    thus they are covered by two neighbourhoods...

    How is this??
  2. jcsd
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Can you offer guidance or do you also need help?
Draft saved Draft deleted