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?
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? 


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 $\mathbb{H}^2.$ 


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 vectorlike 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 BeltramiKlein model the binary operation $\oplus$ is given by the formula for relativistic velocityaddition. 

