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?
Begin with the two dimensional upper half plane. One type of geodesic is, with constant $A,$ $$ x = A, \; \; y = e^t $$
The other type is a semicircle, now with constant $B > 0,$ $$ x = A + B \tanh t \; , \; \; y = B \operatorname{sech} t $$
The simplest way to map these, back and forth, to the unit disc is to regard the $y$ direction as imaginary, then use the Moebius transformations $$ \frac{z+i}{iz+1} $$ $$ \frac{iz+1}{z+i} $$