Does there exist a relationship in terms of $r$ and $n$ to represent how large $n$ must be if $r$ of the circle is given in the hyperbolic plane? (The edges of the regular $n$-gon are tangent to the circle.)
This is a follow-up question to Circles inscribed in regular polygons in hyperbolic geometry
On
Different aproach:
we can just assume that the circle you want to circumscribe is centererd at the centre of the disk of the Beltrami Klein Model. and that the segments of the n-gon are just lines in this model.
Then from Euclidean geometry we now from Regular n-gon between two concentric circles
we learn $n \geq \pi/\cos^{-1} r/R $
with $ r$ and $R$ being the euclidean measures
Taking $R=1$ we get $n \geq \pi/\arccos( r) $
Then in the Beltrami klein model we have for the length
$ h = 1/2 \ln(\frac{ 1+r}{1-r}) $
or $r = \frac{e^{2h}-1}{e^{2h}+1} $
where h = the hyperbolic radius of the circle
so putting it all together you get:
$$n \geq \frac{\pi}{\arccos( \frac{e^{2h}-1}{e^{2h}+1})} $$
In the picture (Klein model) below it is clearly shown that the half of the central angle $\Phi(=\frac{2\pi}{n})$ equals the the angle of parallelism belonging to $\rho$, the radius of the circle. That is, $$\Pi(\rho)\ge \frac{\pi}{n}$$ or equivalently $$n\ge \frac{\pi}{\Pi(\rho)}$$is the condition that the tangent lines meet within the plane or exactly on the absolute.
One of the known formulas for the angle of parallelism is $$\tan(\Pi(\rho))=\frac{1}{\sinh(\rho)}.$$ Whit this, our condition is $$\frac{1}{\sinh(\rho)} \ge \tan \frac{\pi}{n} >\frac{\pi}{n}, $$ since $n\ge 3.$ From here $$\pi \sinh(\rho)< n.$$ Or
where $\circ \rho$ is the circumference of the hyperbolic circle of radius $\rho$. Note that the last three inequalities give sufficient conditions only. But if $n$ is large enough they are sharp lower bounds for $n$. Think of ($\tan(x)>x$).