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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.

This seems to be a thread construction in hyperbolic space

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: enter image description here ]

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.

helveticat
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  • I am not really sure where you are looking for, there are some books that contain some "geometrical constructions in the Poincare disk model of the hyperbolic plane" and under the tag [[hyperbolic geometry]] I have earlier asked or answered some of them. but I don't understand the construction you use completely (why specially these lines and why not others ) not the relation to the "mice problem" I do agree that "synthetic geometry in general" seem to be out of fashion these days. – Willemien Mar 06 '15 at 22:49
  • Hi @Willemien -- I'm specifically referring to thread constructions, which approximate a curve (which we can't draw directly) by constructing some of its tangent spaces (straight lines). In particular, non-trivial solutions to the mice problem are curved paths, but a repeated straight-line construction reveals them in a very clear way. I guess, really, this is the beginning of diff geom in $\mathbb{H}^2$, but I'm interested in what's known about the synthetic approach. – helveticat Mar 07 '15 at 12:24
  • things like you want i don't think are possible in synthetic geometry,(in synthetic geometry you only have circles and lines, and for hyperbolic geometry horo cycles and hypercycles) also in your muck up be aware that the distance AE is mutch larger than for example the distance MP (in the poincare Disk model the segments with the same real length get shorter in the model when the are more towards the boundary circle) – Willemien Mar 07 '15 at 18:49
  • @Willemien -- indeed, that was me being a bit lazy (and trying to heep the construction fairly clean) -- as long as the lines are tangent to the curve we're trying to approximate, it's not crucial that they're regularly-spaced as they're only a sample of the possible tangents anyway. Admittedly it would be nicer as it would give an impression of the apparent speed of motion across the disc (assuming the mice move at constant speed in hyperbolic space)... – helveticat Mar 09 '15 at 14:10
  • A pdf mentioned in http://math.stackexchange.com/q/1184542/88985 "Compass and Straightedge in the. Poincaré Disk" by Chaim Goodman-Strauss is maybe of interest : http://comp.uark.edu/~strauss/papers/hypcomp.pdf – Willemien Mar 11 '15 at 18:39

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