As was explained in Section 6.3, Equations (16) – (17) can be viewed as wave-map equations with
hyperbolic space as a target. As a consequence, isometries of hyperbolic space map solutions to solutions. In
the context of spikes, one isometry of , which is particularly important (as was noted in ) is
We shall refer to this isometry as an inversion; in the upper half plane, it corresponds to an inversion in the
unit circle with center at the origin.