  Context: To quantitatively decode the information stored within an observed
spectrum, detailed modelling of the physical state and accurate radiative
transfer solution schemes are required. In the analysis of stellar spectra, the
numerical model often needs to account for binary companions and 3D structures
in the stellar envelopes. The enigmatic binary (or multiple) system LB-1
constitutes a perfect example of such a complex multi-D problem. Aims: To
improve our understanding of the LB-1 system, we directly modelled the
phase-dependent H-alpha line profiles of this system. To this end, we developed
a multi-purpose binary-object spectral-synthesis code in 3D (BOSS-3D). Methods:
BOSS-3D calculates synthetic line profiles for a given state of the
circumstellar material. The standard pz-geometry commonly used for single stars
is extended by defining individual coordinate systems for each involved object
and by accounting for the appropriate coordinate transformations. The code is
then applied to the LB-1 system, considering two main hypotheses, a binary
containing a stripped star and Be star, or a B star and a black hole with a
disc. Results: Comparing these two scenarios, neither model can reproduce the
detailed phase-dependent shape of the H-alpha line profiles. A satisfactory
match with the observations, however, is obtained by invoking a disc around the
primary object in addition to the Be-star disc or the black-hole accretion
disc. Conclusions: The developed code can be used to model synthetic line
profiles for a wide variety of binary systems, ranging from transit spectra of
planetary atmospheres, to post-asymptotic giant branch binaries including
circumstellar and circumbinary discs and massive-star binaries with stellar
winds and disc systems. For the LB-1 system, our modelling provides strong
evidence that each object in the system contains a disc-like structure.
