Calculating side in quadrilateral in Poincare disc

Question

If I have a quadrilateral ABCD inside the Poincaré disc such that $\angle A=\angle B=\frac{2\pi}{3}$, $AD=BC$ and we know the hyperbolic lengths of sides $AB$ and $CD$, how can I calculate the hyperbolic length of $AD$ in terms of $AB$ and $CD$?

Answer 1

As @user10354138 has pointed out, your figure is symmetrical, and $AB$, $CD$ have a common perpendicular bisector. I think that in the good situation, you can come to a solution of your problem, but I don’t know how to handle the bad situation.

The good situation occurs when $AD$ and $BC$ intersect, say at a point $U$. In the figure, I’ve drawn the common perpendicular bisector: of $AB$ at $V$ and of $CD$ at $W$. We have two right hyperbolic triangles: $\triangle AUV$ and $\triangle DUW$. The common angle at $U$ I’ve denoted $\theta$. Since we know $\angle DAV=120^\circ$, we have $\angle UAV=60^\circ$. If I’ve not mistaken my formulas, the relations among these are: $\cos\theta=\cosh a\sin60^\circ$ and $\sin\theta=\sinh a/\sinh v=\sinh b/\sinh w$, where $v$ is the length of $UA$ and $w$ is the length of $UD$. (The first equation says that this good situation occurs when $\cosh a<2/\sqrt3$, in other words, when $a<0.549$ roughly.) Of course you want $w-v$. I haven’t tried to combine all these for a possible simplification that might suggest a solution in the bad situation. I hope this has helped.