### Motivation

In the Klein disk model of the hyperbolic plane, the points are the interior of the disk, and the lines in $H^2$ correspond to lines intersecting the interior.

Similarly, the Euclidean plane can be modeled by the interior of a hemisphere of $S^2$ (or $\mathbb {RP}^2$ minus a line) so that lines in $\mathbb R^2$ are the intersections of geodesics of the sphere with the hemisphere.

In both cases, the angles aren't preserved, but the orderings of points on lines are preserved.

### Definitions

For any open convex set in $\mathbb R^2$, consider the nonempty intersections of lines with the set as lines in an ordered geometry with the induced ordering from $\mathbb R^2$. Two such geometries are equivalent if there is a bijection between them preserving lines and the ordering on each line.

Which convex open sets produce geometries equivalent to $H^2$?

Which pairs of convex open sets produce equivalent geometries?

Either line-segment preserving maps are quite flexible, or else there should be ways to recover much of the information about convex sets from their incidence geometries.

### Some weak results

You can distinguish the interior of a triangle from $H^2$ (or any other bounded convex open set) through the incidence relations. In the triangle, there are three lines such that every other line intersects at least one of the three. Any three lines through the vertices work. In $H^2$, you can always find a line disjoint from any finite collection of lines.

Similarly, if a line segment makes up part of the boundary of a set in the plane, then the incidence geometry is not $H^2$.

The incidence relation plus ordering is enough to construct ideal points of the boundary of the set. These correspond to maximal sets of rays so that for any two disjoint rays $R_1$ and $R_2$ in the set, the set of points $p$ so that for some $x_1 \in R_1$ and $x_2 \in R_2$, $p$ is between $x_1$ and $x_2$, is a triangle subgeometry.