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?