I try to teach myself a bit of non-euclidean geometry
And I am a bit stumped by the following remark in George E. Martins "The foundations of Geometry and the Non-Euclidean Plane" page 217. extra comments are in [ and ] i added some line breaks and my poblem is the bit in bold
An isometry $\alpha $ preserves d [distance] by definition.
That $\alpha $ preserves m [angle measure] means only that $ m \angle A'B'C' = m \angle ABC $.
This last equation should not be misunderstood; it does not mean what it seems to mean. It is true that if $ \angle ABC $ then $ \angle A'B'C' $ and the two angles are congruent. However the equation does not say that $\alpha $ preserves angles.
Even though $\alpha $ preserves betweenness, it is just not clear that $\alpha $ preserves rays or lines. (the function $ f $ given by $ f(x) = e^x $ preserves betweenness on the real line but does not preserves rays on the real line.) The problem is that we do not yet know if $\alpha $ is a surjection on the points.
I can just not get my head rount the bold statement, can anybody help me?
The first part says that if $b$ is between $a$ and $c$ then $f(b)$ is between $f(a)$ and $f(c)$.
Now the reason why $f(x)$ does't preserves rays is because it takes the ray $(-\infty, 0]$ into $(0,1]$ which is not a ray.