Consider the Poincare metric on the unit ball $D$ (in the complex plane) $\frac{2|dz|}{1-|z|^2}$. The distance between two points $z_0$ and $z_1$ in this metric is given by
$\rho(z_0,z_1)=\inf\{\int_0^1\frac{2|\gamma'(t)|}{1-|\gamma(t)|^2}.dt|\gamma \text{ is a curve between } {z_0} \text{ and } {z_1}\}$
Now consider the ball of radius $r$ centered at $a\in D$ in this metric, say $B_{\rho}(a,r)$. I need to prove that this $B_{\rho}(a,r)$ is a Euclidean ball as well, find its center and radius and also that its closure in the Poincare metric is compact.
Now, transferring the problem from $a$ to $0\in D$ via the map $\phi(z)=\frac{a-z}{1-\overline{a}z}$, and noting that that $\rho(0,w)=\log(\frac{1+|w|}{1-|w|})$, I find that $B_{\rho}(0,r)=\{z\in D|\log(\frac{1+|z|}{1-|z|})<r\}$ and so $B_{\rho}(a,r)=\phi(B_{\rho}(0,r))$.
Thus I have an explicit form for the ball , but still not being able determine its center or radius.
Also , the closure of $B_{\rho}(0,r)$ is the closed ball of radius $\frac{e^r-1}{e^r+1}$ and I suspect that the image of this under $\phi$ is the closure of $B_{\rho}(a,r)$, but I'm not sure though.
Please help me with these questions.