Ytukyg

I was introduced the Poincaré Disc model of hyperbolic geometry. The concept idea was clear but I had some questions about it that I could not figure out myself.

I understand that the geometric properties come from the metric. But it seems to me that we are studying it as a subset of the extended comlex plane, and we use the natural Euclidean coordinate system to calculate our formula. The professor told me that we are using the differential structure but not the Euclidean geometry to help us with the calculations. So in some sense we are studying hyperbolic objects (by considering a different metric) on the "euclidean structure".

I don't know if this question makes any sense, but is there a natural coordinate system in which every straight line (in the Euclidean sense) is itself a hyperbolic straight line? If not, why do we still need objects from Euclidean space to study non-euclidean geometry?

Some comments in the form of an answer

1. The Poincaré disk model is only a model of the hyperbolic geometry. There are other models as well. For example the Klein disk model uses euclidean straits falling within an open disk to model the hyperbolic straights. (if this answers your question.)




2. We do not necessarily need euclidean objects if we want to study hyperbolic geometry. There is the synthetic treatment just like in the case of the euclidean geometry.




3. We use euclidean models only because some are convinced that it is easier to understand the hyperbolic concepts based on certain artificial euclidean objects.




4. The point is that there is a geometry (there are geometries...) in which the intuition is completely different from the euclidean one. To some extent -- in my opinion -- it is misleading to make the students believe that hyperbolic geometry can be understood based on strange formulas and enforced euclidean concepts.




5. Perhaps most importantly: The usual (ruler and compass based) euclidean geometry is only a model of the real euclidean geometry -- we are used to it though.

The usefulness of the Poincaré model (or of Klein's) is that it shows that if Euclidean geometry is consistent, then also hyperbolic geometry is. This is because the hyperbolic axioms are true in the model, so a contradiction in hyperbolic geometry would yield a contradiction in Euclidean geometry.

It would be wrong to prove theorems in hyperbolic geometry using the Euclidean properties of the model, because we might be using facts that only pertain to the model. Just to make a simple example: an Euclidean segment included in the Poincaré model is generally not a segment in hyperbolic geometry, so the fact that three points of the model are “Euclidean aligned” does not mean they are as “hyperbolic points”.

On the other hand, the model can give us a better insight of what's happening in hyperbolic geometry, because we can draw “hyperbolic figures” in the model; for instance, we get a grasp of “hyperbolic parallels”: two lines (seen as orthogonal circles to the boundary circle or diameters) are parallel (in hyperbolic sense) if they have a common point in the boundary circle. Similarly if we use the half-plane variant model.

Another interesting exercise is to see where Saccheri went wrong when he concluded that “the hypothesis of the acute angle is absolutely false; because it is repugnant to the nature of straight lines”

PROPOSITIO XXXIII
Hypothesis anguli acuti est absolute falsa; quia repugnans naturae lineae rectae

Full scanned version

The Euclidean metric on $mathbb{R}^2$ has Euclidean geometry. It does not follow that no continuous metrics on $mathbb{R}^2$ are hyperbolic. Some of you might be thinking Euclidean geometry is obviously true. If you say that by definition, the Euclidean metric on $mathbb{R}^2$ is how distance is defined on $mathbb{R}^2$, then you're right. According to that definition, that just means that if somebody defines a continuous hyperbolic metric of $mathbb{R}^2$, they're lying about what the distance between any two points in $mathbb{R}^2$ are. It doesn't mean that a continuous hyperbolic metric on $mathbb{R}^2$ doesn't exist.

I think at a very young age, I think it was under the age of 5, I knew quite well about the existence of 3-dimensional spheres and figured out on my own that the surface of a sphere does not have Euclidean geometry and that's how I made sense of non-Euclidean geometry from a young age. As a result, I also thought of the concept of hyperbolic geometry on my own. I think I was in Grade 1 at that time. I think that's the reason I never got confused about it when I heard about that concept later.