Let $M$ be the Poincare ball model of the Hyperbolic space, and let $\zeta \in T_0M$. In my lecture notes it is claimed that $$c(t)=\tanh(\Vert \zeta \Vert t )\zeta/\Vert \zeta \Vert$$ is the geodesic that satisfies the initial conditions $c(0)=0$ and $c'(0)=\zeta$.
I know that lines through the origin are geodesics, and this is clearly a line through the origin that also satisfies the initial conditions. But my question is, where does this particular parametrization come from? How can I verify if it is correct?