Left: definition of our model geometry. The wind is launched between r_in and r_out, at an angle defined by the distance of the focus below the origin, d_f. The covering fraction f_cov defines the maximum radial extent of the wind r_max (in spherical polar coordinates), giving a spherical boundary condition. Right: example of our calculation grid (at significantly lower resolution for clarity). We split the wind into cells of cos(θ) with width dcos(θ) in the range cos(θ)∈[0, f_cov], and radius r with width dlog(r). The grid in r extends from the inner streamline to either the outer streamline (defined from r_out) or r_max, depending on whichever is smallest. Our wind cells are illustrated as the dark shaded regions. Throughout this paper we will evaluate the wind emission and density at the centre of each cell θ_i and r_j. For completeness, this also defines the length travelled along the streamline l_ij at each cell.