What are geodesics in H$^2$?

Question

Specifically, I am looking at a question that asks

What axioms for a projective plane fail in the this space?

1. Any two “points” are contained in a unique “line.”
2. Any two “lines” contain a unique “point.”
3. There exist four “points”, no three of which are in a “line.”

Points: Points of H$^2$

Lines: geodesics in H$^2$

I assume points in $H^2$ are just regular points in the upper half plane,

but what are geodesics? Semi Circles with the center on the x axis?

Answer 1

The geodesics on $H^2$ are, indeed, semi-circles centered on the real line and also vertical half-lines. This is the hyperbolic (curvature=-1) plane, so the only projective axiom which can be violated is the second one - and, indeed, there are lines which do not intersect (e.g., semi-circles with the same center and different radii).