What is more, projections along the short axis of the lens are more likely than projections along the long axis because the latter minimizes the ellipticity of the projected profile. For oblate halos, on the other hand, projections along the long axis of the lens are almost as likely as projections along the middle axis. This is simply because for such halos, there is little difference in the ellipticity of the projected profile between projections along the middle and long axis of the halos. Consequently, both axes result in highly effective quad lenses. Note too that for pancake-like halos, projections along the short axis are strongly avoided, since this projection minimizes both the Einstein radius and the projected axis ratio of the lens.

In short, then, prolate halos and oblate halos will have very different orientation distributions: for prolate halos, nearly all doubles will be due to projections along the long axis of the lens, while most quads will be due to projections along the middle axis of the lens, followed by projections along the short axis. For oblate halos, however, all halo orientations are almost equally likely in the case of doubly imaged systems, whereas quads strongly avoid projections 