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.
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