= Solution
In the ambient space of signature $(-,-,+,\ldots,+)$, AdS is $X\cdot X=-L^2$. Let an arbitrary event have embedding vector $X_P$, and let $V_P$ be any future unit timelike tangent, so
$$
X_P\cdot V_P=0,\qquad V_P\cdot V_P=-1.
$$
The complete family through that event is
$$
\boxed{X(s)=X_P\cos(s/L)+LV_P\sin(s/L).}
$$
Direct substitution gives $X^2=-L^2$ and $\dot X^2=-1$. Its ambient acceleration is $\ddot X=-X/L^2$, normal to the hyperboloid, so its tangential covariant acceleration vanishes. This proves that $s$ is <proper time> and that each curve is a <timelike geodesic>, rather than just guessing sinusoidal tracks from the diagram.
An AdS isometry can place any chosen $P$ at $T=\rho=0$. Rotate the spatial direction into the displayed signed diameter. Write the initial local velocity as $v=\tanh\chi$, with $|v|<1$. Then $X_{-1}=L\cos(s/L)$, $X_0=L\cosh\chi\sin(s/L)$ and $X_1=L\sinh\chi\sin(s/L)$. Eliminating $s$ using the global coordinates gives
$$
\boxed{\sin\rho=v\sin T,\qquad |v|<1.}
$$
Varying the speed gives the drawn family; suppressed spatial directions can be restored by varying the tangent direction. It represents all timelike initial directions after the isometry and spatial rotations.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2002/iii/paper-75-ads-geodesics.png]
{title=Timelike geodesics through an arbitrary AdS event, placed at the center by an isometry, refocusing at later antipodal events}
Every finite-speed curve turns before the conformal boundary. At proper time $\pi L$, all reach $-X_P$; at $2\pi L$ their ambient coordinates return to $X_P$. This is <Anti-de Sitter timelike geodesic refocusing>. On the <universal cover> the global time has advanced by $\pi$ and $2\pi$, respectively, so these are later events. They are not closed timelike curves. The limits $|v|\to1$ approach the null directions and are not timelike members of the family.
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