What does the primitive relations "betweenness" defined in Cayley–Klein model? What's the difference to the Euclidean geometry?

Question

What does the primitive relations "betweenness" defined in Cayley–Klein model? What's the difference to the Euclidean geometry?

In the link below, is $n$ between $k$ and $m$ in the Klein-Beltrami model example or not?

Thanks very much!

https://www.britannica.com/science/Klein-Beltrami-model

Answer 1

Assuming that one has defined the notion of betweenness for points on lines as is done in Hilbert's axiomatization (see the comment of @Blue), one could define betweenness of lines as follows.

Given three lines $L,M,N$ which are pairwise disjoint, one would first want to prove a lemma of neutral geometry: If $P,P'$ are two lines each intersecting $L,M,N$ in single points, say $l,m,n$ on $P$ and $l',m',n'$ on $P'$ respectively, then the point $m$ is between $l$ and $n$ on $P$ if and only if the point $m'$ is between $l'$ and $n'$ on $P'$.

Then one could define $M$ to between $L$ and $N$ if for some (equivalently any) $P$ as above, the point $m$ is between $l$ and $n$ on $P$.

The difference between Euclidean and hyperbolic geometry would then be the following theorem: in Euclidean geometry, given three lines which are pairwise disjoint, exactly one of those three lines is between the other two; on the other hand, in hyperbolic geometry, there exist three lines which are pairwise disjoint such that none of them is between the other two.

Here's an example of this behavior in the Cayley Klein model (aka the Klein Beltrami model, akd the Klein model). This model is an open disc in which the geodesics are chords with endpoints on the boundary circle. Let $a,b,c,d,e,f$ be six distinct points listed in counterclockwise order (say) on the boundary circle. Then let $$l = \overline{ab}, \quad m = \overline{cd}, \quad n = \overline{ef} $$ No two of the lines $l,m,n$ intersect, and none of those three lines is between the other two.

An example can be seen in this article, with the relevant lines are called $k,m,n$ (instead of $l,m,n$).