A Riemannian manifold is said to be geodesically complete if every maximal geodesic is defined for all $t\in \mathbb R$.
Poincare ball is a unit ball(open) in $\mathbb R^n$ with metric $$g_{ij} = \frac{\delta_{ij}}{(1 - x_k x^k)^2}.$$
The geodesic equation is
$$\frac{d^2x^i}{ds^2} + \Gamma^{i}_{jk}\frac{dx^j}{ds}\frac{dx^k}{ds} = 0$$
where $\Gamma^i_{jk} = \frac{2}{(1 - x_\mu x^\mu)}[\delta^i_kx_{j} + \delta^i_jx_{k} - \delta_{jk}x^i]$(after computation).
Given $x(0)=0$ and $x'(0)=v$, I wonder how to solve this geodesic equation for the geodesic curve $x$. I know the locus of it is either a straight line or a circular arc that meets the boundary of the ball orthogonally.
In the upper half plane model, the unit speed geodesics are $$ A + i e^{it}, $$ with timer reversals and time translates, same for $$ A + B \tanh t + i B \operatorname{sech} t, $$ real constant $A$ and real $B > 0.$
These map back and forth to the unit disc with $$ \frac{iz+1}{z + i}, $$ $$ \frac{z + i}{iz+1}. $$
After that you can rotate things as needed.