Geometry used for the single-star algorithm (top panels) and for the binary code (bottom panels). Top left panel: cylindrical coordinate system (blue) with corresponding Cartesian reference frame (red) for an observer’s direction used to obtain the synthetic spectrum of a given (single) object described in a spherical coordinate system (magenta) with corresponding Cartesian reference frame (black). Top middle panel: p − ζ (or -plane) of the cylindrical coordinate system (perpendicular to the observer’s direction). Top right panel: an arbitrary -slice, with pointing towards the observer. The grey circles indicate the radial grid of the spherical coordinate system of a considered (single) object, and the black arrows correspond to the individual rays for each impact parameter p. The radiative transfer is then performed along each ray with corresponding discretized coordinates indicated by the red dots. Bottom left panel: two local coordinate systems Σ₁ and Σ₂ describing two individual objects indicated by the cyan and magenta quarter-circles, respectively. Both coordinate systems are embedded within a global coordinate system Σ₀ (typically but not necessarily chosen to be the centre of mass of both objects). The vectors t₁ and t₂ describe the origin of the local coordinate systems in the global coordinate system. Bottom middle panel: as top middle panel, however showing the triangulation of the -plane for an arbitrary orbital configuration in a global cylindrical coordinate system within the binary algorithm. Bottom right panel: as top right panel, but now for the binary system. The individual -rays propagating through the system of object 1 (blue) and object 2 (red) potentially need to be merged.