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 (ϕk, θk), respectively. The great circle passing through J and K is shown in blue. The position angles ψ⋆ and ψk 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.
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