Given the Poincare Disc $D$ and its ideal boundary $S^{1}$, I want to construct a homeomorphism between $S^{1}$ and the gromov boundary of $D$, $\partial D$, of equivalence classes of geodesics given some base point p.
I can construct a bijection between $S^{1}$ and $\partial D$ by identifying equivalence classes of rays with unique points on $S^{1}$ by noting that each line approaching infinity in $D$ is the unique radius for a given angle (so these lines are unique and so their intercepts in $S^{1}$ must also be unique, hence a bijection). What I am struggling with is showing that this bijection is continuous (it is not necessary to show continuous inverse in this case). I would like to show that, given a base point p and 2 geodesics from p to $x, y \ \epsilon \ \partial D$ where $x \neq y$ that: if $u$ on the line px and $v$ on the line py are close together (i.e. $d_{D}(u, v)$ is small), then the geodesics from $x , y$ to p are close together (i.e. the gromov product between the line px and py is large); and vice versa. Ideally, a relationship between $\alpha$, the angle between the 2 geodesics at p, and the length of the geodesic (I say length as this triangle xyp will be isoscles), would be most helpful.
Any help would be much appreciated!
Edit: To clear things up: if x, y both lie on the unit circle (the boundary of the poincare disc) and x$\neq$ y, and two points u and v in D, then given a base point p in D, we can define a geodesic L1 passing through p, u and x and a second geodesic L2 passing through p, v and y. Let the angle between the two geodesics at p be $\alpha$, and both L1 and L2 have length a. I also define L3, the line from u to v of length b. I would like to show that the gromov product of u with v (=$a - \frac{1}{2} b$ here) is dependent only on alpha. Hopefully that clears things up, sorry for not being more specific earlier.