I've read many times that hyperbolic geometry is geometry on a negatively curved surface, but when I try to research it online, I usually get things like the Poincaré disk or Beltrami-Klein disk, which don't feel intuitive (especially on how distances are defined in both those models).
The current simplest model I can find is the Minkowski hyperboloid model (and both the Poincare disk and Beltrami-Klein disk models can be derived as azimuthal projections of the hyperboloid model), but there are two main difficulties I have with it:
- it relies on a "Minkowski metric" for deriving distance that is different from the Euclidean metric (and the paper doesn't seem to describe how one might discover this metric in the first place, except in the context of general relativity)
- it uses a hyperboloid of two sheets, which is a positively curved surface in 3D Euclidean space, not negatively curved
I also know of the pseudosphere model, which uses a negatively-curved surface but has complicated geodesics/lines as well as a video series called "Universal Hyperbolic Geometry" that looks promising in its simplicity but uses a completely different approach to hyperbolic geometry (and takes a very long to watch).
Because of this, is there a model of hyperbolic geometry that resolves all of these "problems" (i.e. it uses Euclidean metric, a negatively-curved surface, simple geodesics/lines, and a familiar geometric framework), or is it impossible?