  Spectra of composite systems (e.g., spectroscopic binaries) contain spatial
information that can be retrieved by measuring the radial velocities (i.e.,
Doppler shifts) of the components in four observations with the slit rotated by
90 degrees in the sky. By using basic concepts of slit spectroscopy we show
that the geometry of composite systems can be reliably retrieved by measuring
only radial velocity differences taken with different slit angles. The spatial
resolution is determined by the precision with which differential radial
velocities can be measured. We use the UVES spectrograph at the VLT to observe
the known spectroscopic binary star HD 188088 (HIP 97944), which has a maximum
expected separation of 23 milli-arcseconds. We measure an astrometric signal in
radial velocity of 276 \ms, which corresponds to a separation between the two
components at the time of the observations of 18 $\pm2$ milli-arcseconds. The
stars were aligned east-west. We describe a simple optical device to
simultaneously record pairs of spectra rotated by 180 degrees, thus reducing
systematic effects. We compute and provide the function expressing the shift of
the centroid of a seeing-limited image in the presence of a narrow slit.The
proposed technique is simple to use and our test shows that it is amenable for
deriving astrometry with milli-arcsecond accuracy or better, beyond the
diffraction limit of the telescope. The technique can be further improved by
using simple devices to simultaneously record the spectra with 180 degrees
angles.With tachoastrometry, radial velocities and astrometric positions can be
measured simultaneously for many double line system binaries in an easy way.
The method is not limited to binary stars, but can be applied to any
astrophysical configuration in which spectral lines are generated by separate
(non-rotational symmetric) regions.
