Hyperboloid to Poincaré
The exact link between spacetime-style hyperbolas and the Poincaré disk: project the hyperbola t² − x² = 1 from the point (0, −1) onto the line t = 0.
Projection of the hyperbola t squared minus x squared equals one from the point (0, minus 1) onto the t equals zero line, showing the image is tanh of r over 2.
1.10
The point at hyperbolic distance r lands at tanh(r/2), so the whole infinite curve fits inside the interval from −1 to 1. In 2D the same projection turns the hyperboloid t² − x² − y² = 1 into the Poincaré disk.