= Antipodal cross-ratio and spherical distance
{title2=$[u,-1/\bar u;v,-1/\bar v]=-\tan^2(d/2)$}
Use the north-pole <stereographic projection> and the <cross-ratio> convention $[a,b;c,d]=(a-c)(b-d)/[(a-d)(b-c)]$. Antipodes have coordinates $-1/\bar u$. Rotate the first point to the south pole, whose coordinate is zero: its antipode becomes infinity, while a point at <spherical distance> $d$ has projected modulus $\tan(d/2)$. <Möbius invariance of the cross-ratio> then gives the displayed identity. Antipodal endpoints require the extended-value interpretation.
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