I'm trying to understand the 'paradigm shift' from Euclidean to non-euclidean geometry. Though I can understand simple models like why the angle in a triangle on a sphere would not add to 180 degrees etc. I'm struggling to understand why certain models - such as the 'pseudosphere or the Poincare disc', are 'special.'
Can someone please
(a) describe these (and any other important models) to me using maths that doesn't require a lot of prerequisite knowledge but is still in depth and
(b) explain the significance of these models in particular over any curved space.
Thanks! And excuse my ignorance.
Let's start with the Poincaré models, disk and half-plane. You can transform one to the other using a Möbius transformation. The points inside of the disk resp. the upper half of the plane form the set of hyperbolic points. The lines and circles perpendicular to the boundary (unit circle resp. horizontal axis) form the hyperbolic lines. This is what makes up your model: a translation from terms in some formal system to terms in a system you are familiar with. This mapping might be somewhat counter-intuitive in that the term “line” is not actually modeled by a line, but by a circle, with few exceptions.
Now people have shown that this model satisfies all the axioms of Euclidean geometry except for the one about parallels, which it replaces by its own hyperbolic version. (This property of satisfying the axioms makes it a model in the sense of mathematical logic.) So in this sense, this model of hyperbolic geometry deomstrates that if Euclidean geometry is consistent (something we tend to assume but cannot prove without making assumptions) then hyperbolic geometry is consistent. Historically this is special, because it demonstrates that there do exist geometries in which the hyperbolic version of the prallel axiom holds instead of the Euclidean one. I like to phrase that axiom as “given a line and a point, there exist at least two lines through the point which do not intersect the first line”.
I would also mention the Beltrami-Klein model which lives inside a conic section and uses straight lines to model hyperbolic lines. It comes at the cost of a more involved definition of angles, which makes it harder for building intuition. But since you can again convert one of these models into the other, they are equivalent. Not so the tractricoid, which is what most people mean when they use the less well-defined term pseudosphere. Building on the axioms, one can show that hyperbolic geometry is geometry on a surface of constant negative Gaussian curvature. And the tractricoid is such a surface, but it only models a portion of the hyperbolic plane before rolling up on itself. So it's a partial model only, but it has the benefit that lines and angles all have the definitions you'd intuitively expect, with the caveat that lines are actually geodesics following the curvature, not straight lines in space.
So how are these surfaces different from arbitrary curved surfaces, whether embedded into 3d space or not? The key here is the constantness of the negative curvature. You can have various Riemannian manifolds with different curvatures in different points, but as soon as you have the same negative curvature everywhere, and don't have lines meet themselves after finite distance, then you have hyperbolic geometry.
There is no way you can embed that whole hyperbolic plane into 3d space in a way which preserves lengths and angles, which is why these distorting models are so useful. Some people also like to use the hyperboloid model as a 3d model, but keep in mind that this still does not use the same metric in 3d Euclidean space and in the hyperbolic geometry. The hyperboloid actually has positive curvature in the common sense, and it takes a weird definition of metric to make it have constant negative curvature.