Hyperbolic geometry - relations between lines, curves, and hyperbolas

[Reuel]

Hi.

I studied calculus a while back but am far from a math god. I have been reading around online about hyperbolic geometry in my spare time and had a simple question about the topic.

If a straight line in Euclidean geometry is a hyperbola in the hyperbolic plane (do I have that right?) then what is the "transformation" from one to the other? For example, the line y=x in the Cartesian coordinate system would be what in the hyperbolic plane? That is, what hyperbola corresponds to y=x? Can the two be related?

The ultimate reason I am interested in knowing specifically how to go from one to the other is because I am curious as to how the hyperbolic rational expression of the form

f(x)=\frac{ax}{b+cx}​

would be expressed in non-Euclidean terms and what straight line in Euclidean geometry would lead to such a hyperbola in non-Euclidean geometry.

If any of that is nonsense, I apologize. I don't know much about the subject but am willing to learn.

Thank you for your help.

 

[HallsofIvy]

I am not sure what you are talking about. A "straight line in Euclidean geometry" isn't in the "hyperbolic plane" and so does not correspond to any thing there. You probably are talking about a "model" for the hyperbolic plane in the Euclidean plane such as the "Klein model", the "disc model" or the "half plane" model. But what a straight line represents in such a model depends upon which model you are referring to and, depending on the model, exactly which straight line. For example, any Euclidean straight line in the Klein model represents a hyperbolic straight line while only a straight line perpendicular to the boundary in the half plane model represents a straight line in hyperbolic geometry

 

[Reuel]

Yeah. I'm sure my original post had plenty of flaws. I guess all I meant was, how would the function I mentioned be represented in non-Euclidean coordinates? You mentioned several "models" and I do not know enough about them to know which would be best suited for such. Hence seeking help from experts on such subjects.