(adsbygoogle = window.adsbygoogle || []).push({}); Show that the stereographic projections of the points z and [itex] 1/\overline{z} [/itex] are reflections of each other in teh equatorial plane of the Reimann sphere

ok so let z = x + iy

then [tex] \frac{1}{\overline{z}} = \frac{x - iy}{x^2 + y^2} [/tex]

so the magnitude of [tex] \frac{1}{\overline{z}} [/tex] is [tex] \frac{1}{x^2 + y^2} [/tex]

the stereogrpahic projection of z is

[tex] x_{1} = \frac{2x}{x^2 + y^2 +1} [/tex]

[tex] y_{1} = \frac{2y}{x^2 + y^2 +1} [/tex]

[tex] z_{1} = \frac{x^2 + y^2 -1}{x^2 + y^2 +1} [/tex]

for 1/ z bar is

[tex] x_{2} = \frac{2x(x^2 + y^2)^2}{1 + (x^2 + y^2)^2} [/tex]

[tex] y_{2} = \frac{2y(x^2 + y^2)^2}{1 + (x^2 + y^2)^2} [/tex]

[tex] z_{2} = \frac{1-(x^2 + y^2)^2}{1 + (x^2 + y^2)^2} [/tex]

i fail to see how these are reflections of each other, then shouldnt the x1 = - x2?? and so on??

please help!

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# Fun with a sphere

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