Are there any "simple and intuitive" models of hyperbolic geometry? 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:

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*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?
 A: You asked

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?

and the simple answer is it is not possible. The reason is that
what you want can only be done for a geometry that has greater
curvature than the space it is embedded in. Thus, in Euclidean
space of zero curvature we can have nice models of positively
curved surfaces. If we lived in a negatively curved space, then
we could have nice models of surfaces of greater curvature such
as the Euclidean plane. However, we still could not have nice
models of surfaces with more negative curvatures.
One reason, among others, for this situation is that
the perimeter of a circle expands linearly as the radius
increases in Euclidean spaces. In hyperbolic spaces,
the perimeter expands essentially exponentially. Thus,
there is no room in Euclidean space to contain all of
the perimeter in a nice way without compromises.
