If you understand it geometrically, then do you see what the simplest point-to-point mapping is that will give you your homeomorphism?

What point of C should the point (3, 0, 2sqrt(2)) in H map to?

What point in H should the point (-sqrt(3)/2, 1/2, 10) in C map to? What about the point ( 0, -1, 10) in C?

The answer to your question is that it is possible to explicitly/analytically write down the homeomorphism, and its inverse.

As always, do a few concrete examples, like with the points I suggested above, to help you "see" what's happening.