If $z\bar{z}'=-1$, do $z$ and $z'$ correspond to opposite points on the Riemann sphere? [duplicate]

Solution 1:

There's only one possible complex $w$ for which $zw$ can equal $-1$, given by $w = -{1 \over z}$. So since $z\bar{z'} = -1$ for $z$ and $z'$ corresponding to antipodal points on the Riemann sphere, if $z\bar{w} = -1$ as well you'd have to have $\bar{w} = \bar{z'}$ or $w = z'$.

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