= Connectivity of three acute-angle sphere points
{title2=$P(\text{connected})=(\pi+2)/(4\pi)$}
Join three independent uniform points on a <sphere> when their central angle is acute. Given angle $\gamma$ between the first pair, the third point's two favorable <hemispheres> overlap in a <spherical lune> of area fraction $(\pi-\gamma)/(2\pi)$. If $\gamma<\pi/2$, connection requires membership in their union, of probability $1/2+\gamma/(2\pi)$. If $\gamma>\pi/2$, it requires membership in their intersection, of probability $1/2-\gamma/(2\pi)$. Integrating these against the central-angle density $\sin\gamma/2$ proves the displayed probability. Connectivity is weaker than all three pairwise angles being acute.
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