Stereographic projection
Points on the real line are mapped to the unit circle using a stereographic projection from the circle's north pole.
The circle is rotated (around its centre) and translated (anywhere on the plane).
The points are then mapped back onto the real line, using another stereographic projection from the circle's north pole.
The graph on the right hand side represents the resulting function, from the real line to itself.
Claim: any rational function can be represented in such a way.
Use the slider to rotate the circle. Drag the north pole of the circle to translate it.