How is the Poincare disc model or the upper half plane model a representation of hyperbolic space?

Question

I am currently working on these two models and I don't understand the connection between them and hyperbolic space. In case of spherical geometry one can imagine everything well as a 2 dimensional sphere in 3 dimensional space and all the major facts on parallel lines make sense. But in the case of hyperbolic geometry I can't really get an intuition for that. One problem is also that I have no idea what curvature really is. Could anyone please explain where this seemingly random defined metric on the upper half plane comes from and why we want half circles and vertical lines to represent the geodesics ? Thanks in advance.

Answer 1

In case of spherical geometry one can imagine everything well as a 2 dimensional sphere in 3 dimensional space and all the major facts on parallel lines make sense.

I would argue that the better analogy is elliptic geometry. That's the same as spherical geometry, except that you treat antipodal points as a unit. Let's use this example to illustrate some aspects of how a model works.

You start with a setup you understand well (or for which you have a mathematical theory). In this case it's a sphere in space. You use this to define terms of the axiomatic system you want to model. In this case we define the term “point” to mean a pair of antipodal points on the sphere. And we define the term “line” to mean a great circle on the sphere. The important part here is that the terms used in the axioms of planar geometry are just placeholders. While the people formulation these axioms had some concrete ideas of what a point or a line should be, the axioms don't depend on that meaning. So we can try to give them some new meaning and see what happens.

For this to be interesting, you want your axioms to hold with the new meaning you gave the concepts they use. If you find that all the axioms of Euclidean geometry hold, you would say that the resulting system is a model of Euclidean geometry. The system I just described only satisfies some of the axioms. “Two distinct points define a unique line” works, and depends on antipodal points being seen as a single point. “Two distinct lines have at most one point in common” again works thanks to the identification of antipodes. “Any line can be infinitely extended” is a bit of a stretch, since you will close the loop at some point so it very much depends on how exactly you formalize that vague formulation. And of course the axiom of parallels is clearly violated. So you get a model for a theory of planar geometry that either omits some axioms (leading to neutral geometry) or replaces them (leading to an axiomatisation of elliptic geometry).

But in the case of hyperbolic geometry I can't really get an intuition for that.

That's probably because you can't have a 3d model. The complete hyperbolic plane can't be embedded into 3d space in such a way that angles and path lengths would be preserved. The best you can have is some embedding of a finite part of the hyperbolic plane, e.g. using a tractricoid or Amsler's surface.

One problem is also that I have no idea what curvature really is.

There are different concepts of curvature. The one you want here is called Gaussian curvature. The general idea is that at any given point you cut the surface with a plane containing the surface normal. As you rotate the plane around the normal, you get planar curves of intersection, with different osculating circles at the point in question. Their inverse radius is the curvature of the intersection curve. That curvature will typically have a minimum for one specific cutting plane, and a maximum for a plane orthogonal to that, and the product of these two planar curve curvatures is the Gaussian curvature. It's negative if one circle touches the surface from one side and the other from the opposite side. That's why a negative curvature corresponds to a saddle-shaped surface: one direction (from head to trail of the horse) bends one way (down then up again) while the other direction (from one leg of the rider to the other) bends the other way (up then down again).

But I'm not sure that understanding curvature is very useful for hyperbolic geometry. I would suggest working mostly on the level of axioms and metrics.

Could anyone please explain where this seemingly random defined metric on the upper half plane comes from and why we want half circles and vertical lines to represent the geodesics?

That's because it works. Specifically, because these seemingly random choices satisfy all the axioms of neutral geometry and also the hyperbolic version of the parallel axiom, making this an interesting model worth studying. Keep in mind: “model” means “something which satisfies the axioms”. So by defining the term “point” to mean “a point in the upper half plane” and the term “line” to mean “semicircles and rays orthogonal to the horizontal axis” you get essential properties such as “any two distinct points define a unique line” or “two distinct lines have at most one point in common”.

The typical axioms don't talk a lot about metrics, but they do talk about congruence and the properties of the metric follow from that. This explains why the metric you have is defined in this way, it doesn't explain the historic aspect of how the researchers figured out that this specific definition of the metric and none other would have all the required properties.

It turns out that there is only a single metric that will work, except for a constant factor that can be chosen arbitrarily. That factor happens to relate to the Gaussian curvature, which can also be measured intrinsically as well, i.e. without embedding the surface into an Euclidean space. So while your can think of some form of infinite saddle surface, to me the “constant negative curvature” mostly means that there is one term in my metric and while I may or may not care about it's value, I need it to be the same in every point of my hyperbolic plane.

Note that in my write-up I've avoided the term “geodesic”. That's because the axioms of planar geometry tend to use the term “line” and I want to stress that the relationship between model and axiom is essentially just assigning meaning to terms. For further reading on this I'll suggest What's the point of the Poincaré disc model?