As part of doing something else, I worked out an isometry from the hyperboloid model to the upper half-space model for hyperbolic 3-space. I did this by composing the isometry that goes from the hyperboloid to the ball model with one that goes from the ball to the upper half-space model, then computing a very long simplification of the coordinates.
I'm at a phase where I'm writing up my results for an article. I don't want to include (nor do I expect anyone to want to read) my long tedious calculation, and in hindsight I'm feeling like ... I can't possibly be the first one to have computed this.
So my question is: can anyone give me a reference where such a formula is given explicitly in coordinates? When you write it a certain way (that I'm writing it) it's actually quite elegant and simple, but this may not come out in more conventional approaches. Anyway, formula reference anyone?
Added January 28, 2017: link to my paper where I derive the formula. It's Theorem 5.2, on page 9 of https://arxiv.org/abs/1701.06709
Recall that the transition from the hyperboloid model to the Poincare ball model (to distinguish it from the projective Klein ball model :) ) is obtained by a technique equivalent to stereographic projection performed with pole coinciding with the tip of "the other" sheet of the hyperboloid onto the horisontal 3D hyper-plane tangent to the tip of the sheet of hyperboloid which represents the model of hyperbolic space. Now do the same, but instead of projecting onto the horizontal 3D hyper-plane, project onto a "tilted" 3D hyper-plane (i) passing through the tip of the sheet of the hyperboloid that modes the hyperbolic space and (ii) parallel to a 3D hyper-plane tangent to the light cone along some light-like line on the light-cone (generator line of the light-cone). Kind of analogous to the way a parabola is constructed as a conic section, by taking a plane tangent to the cone along a generator of the cone and then translated to intersect the cone.
Just to be clear, when I say "sheet" I mean a 3D sheet, i.e. a 3D hyper-surface, a 3D two-sheeted hyperboloid in the three plus one Minkowski space (four dimensional).
Isn't this construction featured in Benedetti-Petronio's book on hyperbolic geometry? Or maybe in Bill Thurston's notes on 3D manifolds, geometry and topology (alternatively, his book version of his notes)?
I am sorry I am answering too late maybe, I just came across your question.