# 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:

• 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?

• Are you familiar with the difference between intrinsic and extrinsic curvature, or for surfaces, the difference between the first and the second fundamental form? I am asking because the requirements from the last paragraph (Euclidean metric, negatively-curved surface, ...) seem to indicate some confusion about this... Apr 10 at 17:13
• In the last paragraph, I mean an extrinsically negatively-curved surface being used to described an intrinsically negatively-curved geometry. Apr 10 at 17:39
• Alright. Then see Somos' answer below. On a side note: you might also want to take a look at the half-plane model. I think it is quite intuitive and simple. Apr 10 at 17:56