Mapping a Cube to a Sphere

After the last post, I got to thinking about how to come up with a mapping from the cube



to the points in the unit sphere. In the last post I used the mapping



Taking the length of the new point gives



So you can see that if x or y are either -1 or +1, this yeilds a vector with unit length. So, beginning with this, I decided that I'd work backward from the guess that if I had a mapping from the cube to the unit sphere, the length of a generated point would need to be



Which leads to the mapping



Trying this out yields some pretty nice results. This is the cube and its mapping onto the unit sphere. The thick lines are lines where at least two of the components are -1 or +1.



And here's another view from inside the sphere



Sorry this post is a little light on justification. I might try to explain this by looking at the level curves defined by constant x, y and z, but I'll save that for another day (maybe).