Antipodal stereographic coordinate relation
= Antipodal stereographic coordinate relation
{title2=$q=-1/\overline p$}
For north-pole <stereographic projection> $p=(X+iY)/(1-Z)$ on the <unit sphere>, the <antipodal pair> point has coordinate $q=-(X+iY)/(1+Z)$. Since $X^2+Y^2=1-Z^2$, this gives $p\overline q=-1$ when both coordinates are finite. On the whole <Riemann sphere>, write $q=-1/\overline p$, pairing zero with infinity. The two axis endpoints fixed by a sphere <rotation> obey this relation.