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One can make a model of the hyperbolic plane inside the Euclidean plane, either using the conformal model or projective model.

How does one make a model of the Euclidean plane inside the hyperbolic plane?

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math.SE duplicate! math.stackexchange.com/questions/1347/… –  Qiaochu Yuan Aug 31 '12 at 16:20
    
There are infinitely many diffeomorphisms from the Euclidean plane to an open subset of the hyperbolic plane. You'd want to find a diffeomorphism with some nice property, to distinguish it from this enormous family. An obvious choice is to find a conformal mapping, but by Liouville's theorem there is none. Why do you want such a map? Is there any property you would like it to have? –  Will Sawin Aug 31 '12 at 16:25
    
I want to know what are the lines in the geometry, how to compute distance and angle.. –  i. m. soloveichik Aug 31 '12 at 17:12
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What algebraic structure would you place on $\mathbb H^2$? –  Will Sawin Aug 31 '12 at 20:03
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There's the tangent space to a point in hyperbolic space. That's Euclidean. –  Ryan Budney Aug 31 '12 at 20:42

2 Answers 2

up vote 5 down vote accepted

I believe there is no good model of $\mathbb{E}^2$ in $\mathbb{H}^2.$ However, there is an excellent model in $\mathbb{H}^3:$ any horosphere will work.

Also This is not particularly interesting, but if you use the hyperboloid model of $\mathbb{H}^2,$ you can project it (from, e.g., the point $(2, 0, 0)$ onto the $(x, y)$ plane. This will give an algebraic model of $\mathbb{E}^2$ in $\mathbb{H}^2.$

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Yes I know that. That's why I want the smaller dimension. –  i. m. soloveichik Aug 31 '12 at 17:11
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Look at the Also. –  Igor Rivin Aug 31 '12 at 17:47
    
@Igor How do I see triangles have 180 degrees? –  i. m. soloveichik Aug 31 '12 at 18:09
    
The $(x, y)$ plane is a Euclidean plane? –  Igor Rivin Aug 31 '12 at 19:21
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Hyperboloid is a graph over the xy plane. –  Deane Yang Aug 31 '12 at 22:40

This is a response to Will Sawin's comment "What algebraic structure would you place on $\mathbb{H}^2$?"

On $\mathbb{R}^n$ there is a vector space structure where

1) The metric is given by $d(\mathbf{u},\mathbf{v})=\|\mathbf{u}-\mathbf{v}\|$

and

2) given three points $U,V,W$ the angle $U\hat WV$ satisfies $\cos \theta = \frac{(-W+U)\cdot(-W+V)}{\|-W+U\|\|-W+V\|}$

Analogously, for some models of $\mathbb{H}^n$ with points identified with a subset of $\mathbb{R}^n$ there is a vector-like structure but with a noncommutative, nonassociative binary operation $\oplus$ where

1) The metric is given by $d(\mathbf{u},\mathbf{v})=\|\mathbf{u} \ominus \mathbf{v} \| $

and

2) given three points $U,V,W$ the angle $U\hat WV$ satisfies $\cos \theta = \frac{(\ominus W\oplus U)\cdot(\ominus W\oplus V)}{\|\ominus W\oplus U\|\|\ominus W\oplus V\|}$

$\| \|$ and $\cdot $ are the vector norm and dot product inherited from $\mathbb{R}^n$.

$\ominus a$ denotes the left inverse of a.

$a\ominus b$ denotes $a\oplus (\ominus b)$.

Note the use of the trig function "cos" even though this hyperbolic geometry.

For the Beltrami-Klein model the binary operation $\oplus$ is given by the formula for relativistic velocity-addition.

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