@gregeganSF Thanks for the shoutout! I have obtained that by analyzing your earlier toot https://mathstodon.xyz/@gregeganSF/110774975354528301, which could be obtained by using this technique for the horocyclic coordinates on the hyperbolic plane, covering a horodisk (I attach an animation where more of the horodisk is covered).
As I have already mentioned on 𝕏, we can also use Lobachevsky coordinates in the hyperbolic plane, covering an equidistant band.
This leaves one more natural case: azimuthal (e.g. polar) coordinates in hyperbolic or Euclidean geometry, to cover a disk.
(In spherical, we would get the same result as with longitude/latitude, although possibly not covering the whole sphere if c<1; and the Euclidean analog of longitude/latitude is boring.)