each (non-axial) point the particle interacts with just [*two*]{} photons — one approaching the axis and one already receding from it (the latter started from smaller equatorial radius than the former, so it has been bent more).

Unfortunately, equation (\[theta(r)\]) represents only an [*implicit*]{} relation between $(r,\theta)$ and $r_{\rm eq}$ and can only be solved numerically for $r_{\rm eq}$ in general. More precisely it determines the trajectory of the photon as parametrized by its starting radius $r_{\rm eq}$, which can in principle be inverted to “reconstruct" $r_{\rm eq}$ as a function of the actual photon’s position $(r,\theta)$ (this inversion is unique, at least if restricting to $r_{\rm eq}$ larger than a certain radius slightly above $3M$ in order to discard photons which make more than one full revolution in $\theta$ before reaching infinity). Being able to trace $r_{\rm eq}$ from the actual position $(r,\theta)$ within the photon flux, one then also learns the distribution of photon energy $E$ in space (and thus of their impact parameter $b\!=\!\sqrt{\cal K}/E$ as well), because $r_{\rm eq}$ is uniquely related to $E_{\rm eq}$ (at least at $r_{\rm eq}\!>\!3M$, which is relevant), namely $$E(r,\theta)=E_{\rm eq}{}_{(r,\theta)}=\sqrt{\cal K}\,(N_{\rm eq}/r_{\rm eq})_{(r,\theta)}$$ where ${\cal K}$ is an absolute constant, 