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I'm trying to work out the point of the Poincaré disc model (excuse the pun). As far as I can tell, it's a disc, on which the only permitted lines are a line straight across the middle, and arcs of circles that meet the edge of the disc at right angles (and segments of such arcs, so if I draw three of them together I can get a curvy-edged triangle), and hyperbolic circles (which is "just" the set of points that are a common distance from another point, but where distance is a newly defined kind of distance).

We define new meanings for distance and area and such on the disc. So now I've got a disc on which I can draw some curves and say they are of length such-and-such (using my new definition of distance), and I can say a curvy triangle is of area such-and-such, and I can prove that various allowed lines will and will not touch/cross under various circumstances and so on. I've calculated various distances and mid-points and areas and reflections and so on.

But I could come up with an infinite number of geometries by allowing lines of various types and defining length and area in my own special way (or can I?). What's so special about this disc that it's worth examining? I'm not averse to playing with new ideas just for fun, but this Poincaré disc seems like a big deal, so it must be for more than the fun of it.

Perhaps the clue is in the name; it's a Poincaré disc model, so what's it a model of?

I think if I have a hyperbolic space with some geodesics on it, I can project those geodesics onto a disc and the projections are the right shape to be allowed (i.e. arcs of circles meeting the edge of the disc at right-angles), but I don't see what that gets me that I didn't already know. Is there some link between the length of the line on the disc (as defined using my new special definition of length) that relates to the length of the geodesic in the original hyperbolic space? Can I look at the curves on the disc and from them know something about the curves in the hyperbolic space that I didn't know before? Why have I gone to the trouble of creating this disc with a new definition of length etc?

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  • $\begingroup$ So it's "just" an example of a geometry of the type described by Lobachevsky (which I have read, along with an associated commentary)? One example among many, presumably. If this is a "perfect" model, what's an "imperfect" model of such a geometry? Anyway, is that all? It's just a convenient (by being entirely contained in a disc) example of a geometry as described by Lobachevsky; essentially just a demonstration that we can construct such a geometry, and it's internally self-consistent and all that? $\endgroup$ – Moschops Jan 20 '15 at 19:30
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    $\begingroup$ In answer to the now vanished comment: I have studied it. Did you not read all of my original post? I've been drawing it, calculating distances, calculating areas, reflecting and inverting and transforming. I've read Lobachevsky, I've read commentaries, I know what Euclidean geometry is, I know what elliptic geometry is, I know what hyperbolic geometry is. What I didn't know was what the Poincaré disc model was for, and from your answers so far I think it's just a convenient way to play with a Lobachevskyian geometry.Thank you for your help. $\endgroup$ – Moschops Jan 20 '15 at 19:39
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It's a model of hyperbolic geometry in the plane, the same geometry described by Lobachevsky. There are several models of planar hyperbolic geometry. The most common ones include the Poincaré disk (or more generally ball), the Poincaré upper hald-plane (or more generally half-space), the Beltrami-Klein model which is somtimes also called the projective model, and the hyperboloid model which uses a three-dimensional Minkowsky space to embed the plane.

Comparing these models, the Poincaré disk model has the benefit that it doesn't extend to infinity (as the upper half-plane does), and preserves angles (contrary to the non-Poincaré models), and lives in the plane without need for a third dimension (as needed by the hyperboloid). But these benefits are balanced by drawbacks: the upper half-plane can use real $2\times2$ matrices to describe isometric transformations, the Beltrami-Klein model uses straight lines to model geodesics, and the hyperboloid model is very close in formulation to the geometry on a sphere.

An imperfect model of this geometry would – at least in my interpretation of the word “imperfect” – be the geometry on the tractricoid. Since the tractricoid has constant negative curvature, it closely resembles the hyperbolic plane, uisng “real” angles and “real” geodesics, as they are in the ambient three-space. But the tractricoid only models a part of the hyperbolic plane; it contains closed curves which would not close in the hyperbolic plane. So it is only a local model, which doesn't represent the global structure well. There can be no embedding of the hyperbolic plane into real three-space which uses normal Euclidean angles and geodesics.

You could define all kinds of stuff about what you call a “line”, what you call “distance” and so on. The special thing about hyperbolic geometry is the fact that it still satisfies the first four axioms of Euclid, even though it violates the fifth. In this regard it is pretty much unique.

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  • $\begingroup$ This is superbly put! $\endgroup$ – Travis Jan 21 '15 at 6:53
  • $\begingroup$ That is very helpful; thank you very much. As you'll have seen from my comments to my own question, I'd worked out that it was "just" a convenient Lobachevsky geometry; you've added very much to that by explaining what those particular conveniences are (and what inconveniences come with it that other Lobachevskyian geometries don't suffer). To no small extent, it being called a "model" didn't help me - as far as I'm concerned, it's not modelling anything. It IS a Lobachevskyian geometry! Of course, that's just me and other people probably find calling it a "model" IS helpful. Many thanks. $\endgroup$ – Moschops Jan 21 '15 at 21:39
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    $\begingroup$ @Moschops: I'm using slightly different terms. I'd say there is only one Lobachevsky geometry, but there are various models of it. Since one can translate between models, they don't describe different things, just highlight different aspects of the same thing. But the term “model” also carries strong connections to relative consistency: modeling hyperbolic geometry using Euclidean ensures that the former is consistent if the latter is. You already mentioned this consistency aspect, but for me that's the main point behind the word “model”. See e.g. mathworld.wolfram.com/Model.html $\endgroup$ – MvG Jan 21 '15 at 22:06
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    $\begingroup$ @MvG Makes sense. I was a physicist by original education, and for me a "model" is a simplified picture of something else that I actually want to know about, so I spent ages with this model trying to work out what the "something else" was, and when I couldn't piece together a suitable "something else", I couldn't see the point of it. $\endgroup$ – Moschops Jan 22 '15 at 11:26
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    $\begingroup$ @ZenoRogue: Personally I agree that the different models of the hyperbolic plane are quite akin to the different projections of a sphere. In particular, they are all just different representations of the same underlying structure. However, for many people the term projection will trigger a connotation of a 3d object that is being projected. This would lead them astray, since even though the hyperboloid is using 3 dimensions, hyperbolic geometry is not just geometry on a hyperboloid; you need the Minkowski metric to make that work. So there is some caveat here. $\endgroup$ – MvG Apr 19 '18 at 0:12

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