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I have read that there is a model of Euclidean Geometry in the Hyperbolic Plane, but can't find any description on the web in a digestible form and thought I'd ask this question: If one can describe the Euclidean Model of Geometry found in the Hyperbolic Plane in plain English, with math symbols here and there, but no need for formal proofs or anything at this point. I would just like to get an intuition of how it can be found in the Hyperbolic Plane.

To me this sounds like there is some way to create a model of say a cube or a sphere or something 2D like a rectangle or triangle, from the Hyperbolic Plane alone. It would be interesting to see an example of something 2D and/or 3D to help illuminate, but if it is too complicated then just a description of how you can apply the model would be a good start.

They say Hyperbolic Geometry is a non-Euclidean Geometry, but this would seem to mean that it is both non-Euclidean and yet capable of modeling Euclidean geometry, or something like that.

In addition to providing an explanation, please provide an example so I can see how it is applied.

user10869858
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    Read Wikipedia Horosphere for the story. – Somos Feb 20 '19 at 04:38
  • Doesn't have enough information unfortunately. In addition, I think that is 3D hyperbolic geometry and I'm wondering about 2D (hyperbolic plane). – user10869858 Feb 20 '19 at 04:41
  • What kind of model are you looking for? – Somos Feb 20 '19 at 04:46
  • A model of euclidean geometry in 2d hyperbolic plane. – user10869858 Feb 20 '19 at 04:46
  • Related (duplicate?): "Hyperbolic critters studying Euclidean geometry" – Blue Feb 20 '19 at 04:46
  • You can take a model of hyperbolic plane in the Euclidean plane, and invert it to make a model of Euclidean plane in the hyperbolic plane (simply consider the hyperbolic truth "real" and the Euclidean picture the modelled thing). It will keep properties of the original model, such as conformality or mapping straight lines to straight lines. However, when you do this with the Poincaré disk model or Klein disk model, you are of course only modelling a disk rather than the full plane. – Zeno Rogue Feb 20 '19 at 16:39
  • Here is a simulation of a horosphere in 3D hyperbolic space: http://www.roguetemple.com/z/hyper/online2.php?c=-geo+1+-srx+4+-zoom+0.1+-ruggeo+0+-rugtsize+4096+-rugmodelscale+1.2+-rugon+-viz+-rugpers Click on "full screen" then use arrow keys to rotate, and PageUp/PageDown to come closer tor go away. It shows only a small part, the actual horosphere is infinite of course (it would be possible to show the whole tiled horosphere but I do not have such thing implemented at the moment). The horosphere is tiled with the hexagonal tiling, proving that the geometry of it is Euclidean. – Zeno Rogue Feb 20 '19 at 16:56
  • It would help if you could tell us where you read that "there is a model of Euclidean Geometry in the Hyperbolic Plane". As others have told you, the standard model of Euclidean geometry within hyperbolic geometry is that $n$-dimensional Euclidean geometry is modelled as a horosphere in $n+1$-dimensional hyperbolic geometry. – Lee Mosher Feb 22 '19 at 23:16

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Given the Hyperbolic plane and ond any origin point, introduce $(r,\theta)$ polar coordinates. That is, each point in the plane has a distance $0\le r$ from the origin and an angle of $\theta$ from a reference angle. Map any point with $(r,\theta)$ coordinates to the point in the Euclidean plane with the same polar coordinates. This is a one-to-one mapping between the entire Hyperbolic plane and the entire Euclidean plane giving a model of the Euclidean plane in the Hyperbolic plane. Of course, it is not an isometric or even conformal model.

If you are allowed to use Hyperbolic 3 space, then any Horosphere is an isometric embedded model of the Euclidean plane. This is exactly similar to the situation in Euclidean 3 space where the surface of any sphere (antipodal points identified) is an isometric embedded model of the elliptic plane in Elliptic geometry.

Somos
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  • Without having any more context than this it is hard to understand, though I will take this as true and try to figure out some more. Maybe you could show an example, and explain why this works. – user10869858 Feb 20 '19 at 04:57
  • You have to identify the points on the antipodes of the sphere to make it an elliptic plane. – Zeno Rogue Feb 20 '19 at 16:06
  • Also, horosphere is not only a "conformal" model of Euclidean plane in $H^3$, but even an isometric one -- both angles and distances are correct, so it is much more accurate than e.g. the Poincaré disk model which is conformal but not isometric, and thus it looks nicely but it does not give right ideas about distances. (Same with the sphere.) – Zeno Rogue Feb 20 '19 at 16:35
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    @ZenoRogue Thanks for reminding me. – Somos Feb 20 '19 at 16:49
  • Instead of a distance and an angle, you can also use two distances (to two different points). That's basically how this works. – Christopher King Feb 27 '19 at 03:10