# Half space vs growing balls in the hyperbolic plane [closed]

Let $$p$$ be a point in the $$2$$-dimensional hyperbolic space $$H_2$$. Consider a normal coordinate system $$(x,y)$$ at centered at $$p$$. Let $$o_x\in H_2$$ be the point of coordinate $$(x,0)$$ and let $$C_x$$ be the hyperbolic circle centered in $$o$$ passing through $$p$$.

I have difficulties with the $$2$$ following questions:

• Given a point $$q$$ which has coordinates $$(x_q>0,y_q)$$, is there always a $$x$$ such that $$q$$ is in the interior of $$C_x$$? It is true in Euclidean geometry, I am having doubts for hyperbolic geometry...
• In case it is not true, what it the curve $$f(y)$$ such that points $$(x\leq f(y),y)$$ are not in the interior of any $$C_{x'}$$ while $$(x>f(y),y)$$ are for some $$x'$$?

Your question is actually best reformulated in terms of horospheres and horoballs. Indeed, the set of points described in normal coordinates as $$\{(x,0), x\ge 0\}$$ is precisely a geodesic ray $$\gamma$$ in the hyperbolic plane issued from the point $$p$$. Then the union of the interiors of all your $$C_x$$ is precisely the horoball in the hyperbolic plane centered at the limit point $$\gamma_\infty\in\partial \mathbf H^2$$ of the ray $$\gamma$$ and passing through the point $$p$$.
Now it is more convenient (at least for me) to use the upper hlaf space model, and take $$p=i$$ and $$\gamma$$ to be the vertical geodesic ray directed upwards, so that $$\gamma_\infty=\infty$$. Then the corresponding horosphere (or rather the horocycle, as its dimension is 1) is just the horizontal line $$y=1$$, and it is clear that even if the directing vector of a geodesic issued from $$p$$ has a positive vertical component, then the resulting endpoint still may very well end up to be below the horosphere $$y=1$$. Therefore, the answer to your first question is "no".
As for your second question, by applying some elementary hyperbolic geometry one can explicitly describe all the tangent vectors at the point $$p=i$$ (I continue talking about the upper half space model) such that the endpoints of the corresponding geodesic are within the horoball centered at $$\infty$$ and passing through $$p$$. Let $$(x,y)$$ be the coordinates of the tangent vector (since I am using the upper half space model, $$x$$ and $$y$$ are swapped here compared to your original formulation), and let $$\gamma$$ be the geodesic issued from $$p$$ in the direction $$(x,y)$$. Then the condition is that $$\gamma(d)$$ is inside the horoball (i.e., its vertical coordinate is $$>1$$), where $$d=\sqrt{x^2+y^2}$$ is the length of the tangent vector. In other words, $$d$$ should be less than the hyperbolic distance between $$p$$ and the point $$p'$$ where $$\gamma$$ intersects the horosphere $$\{y=1\}$$. The geodesic $$\gamma$$ is an arc of the circle centered at $$(\frac{y}x, 0)$$, whence $$p'=(2\frac{y}x,1)$$, so that $$d(p,p') = arcosh \bigl( 1+ 2 (\tfrac{y}x)^2 \bigr)$$ (e.g., see). Thus, the condition is $$\sqrt{x^2+y^2} > arcosh \bigl( 1+ 2 (\tfrac{y}x)^2 \bigr)$$ or $$\cosh \sqrt{x^2+y^2} > \bigl( 1+ 2 (\tfrac{y}x)^2 \;.$$
You are asking whether, for a given $$x$$ there is a threshold value $$y_0=y_0(x)$$ separating "good" and "bad" point (once again, I remind, that $$x$$ and $$y$$ are swapped compared to your original question). I am not sure this is true though. For instance, for $$x=1$$ the corresponding equation seems to have two roots.