Context : 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 .

Aims : We aim at developing a framework to describe the method and to test its capabilities in a real case .

Methods : 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 .

Results : 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 { m s } ^ { -1 } , which corresponds to a separation between the two components at the time of the observations of 18 \pm 2 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 .

Conclusions : 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 .
This device together with an optimized data analysis will further reduce the measurement errors .
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 .