I am confused by this picture:
https://en.wikipedia.org/wiki/File:HyperboloidProjection.png
What is wrong with projecting from the origin, and using the disc at $t = 1$? After doing some computation to verify my gut, it doesn't seem like the projection in the drawing with give the correct asymptote cone of the hyperbloid - in this case it doesn't seem like the map is surjective onto the hyperbloid model.
So what is going on? I'm confused.
alright, made a jpeg.
First method: start with point hy. Central projection to the Beltrami-Klein model at point k. Vertical projection to the upper hemisphere at point he. Stereographic projection from the South Pole $-1$ to point p in the Poincare disk.
Second method: project hy towards the South Pole directly to the Poincare Disk at point p.
The calculations needed to confirm that these agree can all be carried out in the illustration, let us call it the $xz$ plane, the plane defined by hy, $0,$ and $-1$ of the illustration.
I took $x > 0$ and hy at $$ hy: \; \; \left( x, 0, \sqrt {1 + x^2} \right) $$ $$ k: \; \; \left( \frac{x}{\sqrt {1 + x^2}}, 0, 1 \right) $$ $$ he: \; \; \left( \frac{x}{\sqrt {1 + x^2}}, 0, \frac{1}{\sqrt {1 + x^2}} \right) $$ $$ p: \; \; \left( \frac{x}{1 +\sqrt {1 + x^2}}, 0, 0 \right) $$ You may want the identity $$ \frac{x}{1 +\sqrt {1 + x^2}} = \frac{\sqrt {1 + x^2} - 1}{x} $$