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Stereographic projection

  1. Feb 6, 2005 #1
    So i'm trying to prove that the map
    [tex]f(x,y,z) = \frac{(x,y)}{1-z}[/tex]
    from the unit sphere S^2 to R^2 is injective by the usual means:
    [tex]f(x_1,y_1,z_1)=f(x_2,y_2,z_2) \Rightarrow (x_1,y_1,z_1)=(x_2,y_2,z_2)[/tex]
    But i can't seem to show it.... :frown:
    I end up with the result that
    [tex]\frac{x_1}{x_2}=\frac{y_1}{y_2},\frac{x_1}{x_2}=\frac{1-z_1}{1-z_2}[/tex],

    but i'm uncertain as to what this means for points on a circle......help please?
    (i have actually already found the inverse map, but i just found it a little frustrating that i couldn't prove injectiveness just straightforwardly like this..)
     
  2. jcsd
  3. Feb 7, 2005 #2

    Galileo

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    Haven't tried it, but since the domain of the function consists of points on the unit sphere, there is a restriction imposed on x,y and z, they cannot have any old values.
     
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