In Stillwells' "Sources of Hyperbolic Geometry " page 66 figure 3.3 shows an ((incomplete?) construction of hyperbolically equidistant points on a line.
I tried to reconstruct the figure but did not manage it can anybody tell me how this figure is constructed?
Or another way to construct hyperbolically equidistant points on a line.
I'm looking at the original source: Felix Klein „Vorlesungen über nicht-euklidische Geometrie“, page 172. The general idea is that the cross ratio $\operatorname{CR}(P_1,P_2;A,B)$ must be equal to the cross ratio $\operatorname{CR}(P_2,P_3;A,B)$ since length can be computed from that cross ratio. Now following Kleins description, the line $BA'$ is chosen arbitrarily through $B$, and the point $S$ is arbitrary as well. Then $A',P_1',P_2'$ follow by projection with center $S$ from line $AB$ to line $A'B$. Furthermore, $P_2P_1'$ intersects $AS$ in a point $M$, which can be used as the center of a second projection back onto $AB$. Since these projections preserve cross ratios, we have
$$ \operatorname{CR}(P_1,P_2;A,B) \overset S= \operatorname{CR}(P_1',P_2';A',B) \overset M= \operatorname{CR}(P_2,P_3;A,B) $$
These steps can be repeated to form an arbitrary number of equidistant points. As the dashed lines indicate, you can use the same points $S$ and $M$ for all of them, only the intersection points on the line $A'B$ are different for every new point you transform. In essence, projection onto $A'B$ via $S$ then down to $AB$ via $M$ is a unit step of your sequence.