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? 



Euclidean inside Hyperbolic
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 $\mathbb{H}^2.$


