A Desmos project which projects a selected (x,y) Euclidean point onto the hyperbolic Poincaré disk. As the distance from the origin of the initial point approaches infinity, its hyperbolic counterpart approaches the unit circle.
https://www.desmos.com/calculator/aqofk4avt3
You can move the input point around in the plane and see how the output changes.
Tags: Desmos