Suppose we have a 5 tuple of positive real numbers $(l\_1,l\_2,m\_1,m\_2,m\_3)$, with $m\_i \in (0,\pi)$ for all $i$. Now fix a point $v\_1$ in the hyperbolic plane. Then consider a geodesic of length $l\_1$ starting at $v\_1$. suppose that ends at $v\_2$. At $v\_2$ draw another geodesic of length $l\_2$ which makes an angle $m\_1$ at $v\_2$ with first line. Suppose the second geodesic ends at the point $v\_3$. Then draw another geodesic of length $l\_1$ at $v\_3$ making an angle $m\_2$ with the second geodesic. Suppose end point of the third geodesic is $v\_4$. Now draw another geodesic of length $l\_2$ making an angle $m\_3$ at $v\_4$ with the third geodesic. Let the end point of the last geodesic is $v\_5$.


Then can we put some condition on the tuple to get $v\_5=v\_1$? If yes what that condition should be?


Thanks in advance.



When a polygonal line become a loop in hyperbolic plane?
Your broken line closes iff the triangles $(v\_1,v\_2,v\_3)$
and $(v\_3,v\_4,v\_5)$ are equal, and $[v\_1,v\_3]$ is their common side. So there are two conditions: one is that $m\_1=m\_3$, by the hyperbolic law of cosines.


To state the second one,
Apply the hyperbolic law of cosines to find the length $x$ of the side $[v\_1,v\_3]$, then the hyperbolic law of sines to find the angles of the two triangles at $v\_3$, and write the condition that the sum of these angles at $v\_3$ is equal to your $m\_2$.


