I am studying Arlan Ramsay's and Robert Richtmeyer's " Introduction to hyperbolic geometry"
On page 255/6 it gives how to construct an segment with an absolute length of $ \ln (\sqrt{2} +1) $ (via the construction of a Sweikart triangle, a right triangle with two ideal points, $ \ln (\sqrt{2} +1) $ is the length of the altitude from the right angle.)
On page 278 the book mentions that all constructions are also possible with and an finite straightedge (one that can only draw a line trough any two finite points)
This made me wonder:
How to construct an segment with a known length with only a finite straight edge?
Or even:
How to construct an segment with a known length with only a limited straightedge? (a limited straightedge as a straightedge that can only draw lines beteen points that are less than some limited distance away from eachother, like a real the resulting segment will need to be shorter than the straightedge)
If $\pi/4$ is the angle of parallelism belonging to the segment to be constructed then I know what to do.
(1) Construct two lines $a$ and $b$ whose angle is $\pi/4$. Let their intersection point be denoted by $P$.
(2) Draw line $b'$, the image of $b$ by reflection in $a$.
(3) Construct the common parallel to $b'$ and $b$.
(4) This common parallel will intersect $a$ in $P'$.
(5) The segment $PP'$ is of the desired length.
Notes:
A. If you don't know how to construct the common parallel then let me know.
B. The construction described above will enable you to construct segments whose length can be associated with an angle that you can construct. (The angle of parallelism belonging to the given length.)