  In the next few years, the near-infrared interferometer GRAVITY will be able
to observe the Galactic center. Astrometric data will be obtained with an
anticipated accuracy of 10 $\mu$as. To analyze these future data, we have
developed a code called GYOTO to compute orbits and images. We want to assess
the validity and accuracy of GYOTO in a variety of contexts, in particular for
stellar astrometry in the Galactic center. Furthermore, we want to tackle and
complete a study made on the astrometric displacements that are due to lensing
effects of a star of the central parsec with GYOTO. We first validate GYOTO in
the weak-deflection limit (WDL) by studying primary caustics and primary
critical curves obtained for a Kerr black hole. We compare GYOTO results to
available analytical approximations and estimate GYOTO errors using an
intrinsic estimator. In the strong-deflection limit (SDL), we choose to compare
null geodesics computed by GYOTO and the ray-tracing code named Geokerr.
Finally, we use GYOTO to estimate the apparent displacements of a star for
different angles from Sagittarius A* (Sgr A*). We have demonstrated that GYOTO
is accurate to a very high level, orders of magnitude better than the GRAVITY
requirements. GYOTO is also valid in weak- and strong-deflection regimes and
for very long integrations. At the astrometric precision that GRAVITY is aiming
for, lensing effects must always be taken into account when fitting stellar
orbits in the central parsec of the Galaxy.
