  Many real transportation and mobility networks have their vertices placed on
the surface of the Earth. In such embeddings, the edges laid on that surface
may cross. In his pioneering research, Moon analyzed the distribution of the
number of crossings on complete graphs and complete bipartite graphs whose
vertices are located uniformly at random on the surface of a sphere assuming
that vertex placements are independent from each other. Here we revise his
derivation of that variance in the light of recent theoretical developments on
the variance of crossings and computer simulations. We show that Moon's
formulae are inaccurate in predicting the true variance and provide exact
formulae.
