I just came across the following image (source) and realised something that should have been obvious to me a while ago: it should be possible to construct the envelopes of curves in the Poincaré disc just as we can on the plane.
For example, it seems evident that one could solve the "mice problem" pretty easily with this approach, just as one does in Euclidean space.
[EDIT: I mocked up one by hand using NonEuclid. Here are three mice at A, B and C in the Poincaré disc; each pursues its nearest clockwise neighbour. The constructed lines are tangents to the actual paths they follow (e.g. A-E-G-J-M-P for one mouse), which are curved:
]
My searches thus far have turned up nothing whatsoever on this topic. Thread constructions (and synthetic methods in general) seem to be very much out of fashion these days and I wonder whether this is just an area that's fallen into neglect.
So I'm looking for:
(a) Anything at all that mentions / describes this type of construction; and
(b) Any references on advanced synthetic hyperbolic geometry. By "advanced" I guess I just mean beyond this paper (which is great, incidentally), and in particular that might deal with non-straight lines.
(c) Your own advice, ideas and insights, as always.
Sadly I don't have academic journals access so online resources or books are preferred.