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I would like to do some stuff with modeling geometry in hyperbolic space in software. When I look up information on hyperbolic space, however, I tend to find only information on working with

*models*of hyperbolic space. For example I find lots of information on the poincare disc model, and information on, well, if you want to have a line here's the thing which in the poincare disc model is dual to a hyperbolic line, if you want to do a translation here's the thing which in the poincare disc model is dual to a hyperbolic translation, etc.

What I cannot find is any information on how to work with hyperbolic geometry "natively"-- that is, in an actual hyperbolic geometry rather than a euclidean model. What I would like to do is work with actual points, lines, translations, rotations etc in hyperbolic space, and then only project them onto the poincare disc when I need to display them. However I cannot find any way to mathematically represent these objects in hyperbolic space itself (other than in euclidean systems, like the poincare disc, which are dual to hyperbolic space). Do any such ways exist, is there even such a thing as a "coordinate in hyperbolic space"? Are there any resources on this subject I ought to be aware of?

(Also, a somewhat tangential question-- are "hyperbolic space" and "de sitter space" the same thing or different?)