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Let Riemann Sphere be $X^2 + Y^2 + (Z-1/2)^2 = \frac{1}{4}$. If $z$ is the projection of $(X,Y,Z)$ and $\frac{1}{z}$ is the projection of $(X', Y', Z')$, Then we can show that $(X',Y',Z')=(X,-Y,1-Z)$.

Show that the function $\frac{1}{z}$ is represented on the Sphere by 180 degrees rotation about the diameter with endpoints $(-1/2,0,1/2)$ and $(1/2,0,1/2)$.

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See Chapter 3 on Möbius Transformations, Section IV (The Riemann Sphere) - Subsection 3 titled "Transferring Complex functions to the Sphere" of Tristan Needham's Visual Complex Analysis to tell you why $f(z) = 1/z$ induces a $180^ {\circ}$ degree rotation about of the riemann sphere about the real axis.

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