  We rigorously construct continuous curves of rotating vortex patch solutions
to the two-dimensional Euler equations. The curves are large in that, as the
parameter tends to infinity, the minimum along the interface of the angular
fluid velocity in the rotating frame becomes arbitrarily small. This is
consistent with the conjectured existence [WOZ84,Ove86] of singular limiting
patches with 90$^\circ$ corners at which the relative fluid velocity vanishes.
For solutions close to the disk, we prove that there are "Cat's eyes"-type
structures in the flow, and provide numerical evidence that these structures
persist along the entire solution curves and are related to the formation of
corners. We also show, for any rotating vortex patch, that the boundary is
analytic as soon as it is sufficiently regular.
