Definition of points and angles on the sphere involved in the geometry of the smoothing of polarization maps (adapted from Keegstra et al. 1997). J is the position of the centre of the smoothing beam, and K a neighbouring pixel, with spherical coordinates (ϕ_⋆, θ_⋆) and (ϕ_k, θ_k), respectively. The great circle passing through J and K is shown in blue. The position angles ψ_⋆ and ψ_k here are in the HEALPix convention, increasing from Galactic north toward decreasing Galactic longitude (west) on the celestial sphere as seen by the observer at O.