The game I mentioned in "Navigating though the surface of a hypersphere in a computer game" is taking shape in here. The world is a 3-sphere where everything belongs. In Euclidean coordinates, for every point $x^2 + y^2 + z^2 + w^2 = 1$. We move the objects by applying orthornormal transformations to them.
We got to a stage where we want to draw a mini-map to help the user navigate through the space, and that is the subject of this question. Like we use stereographic projection to get the 3-sphere into 3-D Euclidean space, how can we project the points of the 3-sphere into the full volume of a "common" 2-sphere? This way, all world objects could be seen in a cool sphere drawn in the corner of the screen, unlike it is with the stereographic projection, which is infinite and some points may be missing in the cropping.
Considering every point $P = (x, y, z, w) \; | \; x^2 + y^2 + z^2 + w^2 = 1$, we need a function $t$ where $t(P) = (X, Y, Z) \; | \; X^2 + Y^2 + Z^2 \leq 1$ and $t((0,0,0,-1)) = (0,0,0)$ (the center of the stereographic projection). Every point of the 3-sphere except $(0,0,0,1)$ must be defined. What is this function? I am not sure on how to look for it...