Consider you have a sphere centered at the origin.The sphere has a diameter of $\frac{1}{2} \sqrt{\frac{3}{2}}$. This means that the inscribed cube has an edge of 1.
Take any point from the plane (1,1,0)-perpendicular to Z such that x in [-0.5,0.5], y in [-0.5,0.5] z in [-0.5,0.5]. What is the coordinate of the point on the sphere for which the orthogonal(perpendicular) projection on the plane has the coordinate (x,Y,Z)? It will be (x,Y,what value does Z have?), as the projection is along Z in this case.
How would the equations look if the plane was(1,0,1)-perpendicular to Y axis?
What I am trying is to take 6 projections of the sphere, which will constitute the mapping of the sphere to cube faces.