  A trajectory isomorphism between the two Newtonian fixed center problem in
the sphere and two associated planar two fixed center problems is constructed
by performing two simultaneous gnomonic projections in $S^2$. This isomorphism
converts the original quadratures into elliptic integrals and allows the
bifurcation diagram of the spherical problem to be analyzed in terms of the
corresponding ones of the planar systems. The dynamics along the orbits in the
different regimes for the problem in $S^2$ is expressed in terms of Jacobi
elliptic functions.
