The GLS/radio projection in details

The AIPS memo 46 defines the global-sinusoidal projection (GLS) in its section 2.3, which writes as

$$x = (\alpha-\alpha_0) \cos\delta,$$ (1)

$$y = \delta-\delta_0.$$ (2)

In this case, there is no rotation of the 2D plane compared to the celestial system (e.g. ICRS). $(\alpha ,\delta )$ are the celestial coordinates, and $(\alpha_0,\delta_0)$ the coordinates of the reference point. This can not be described in the CG02 formalism, which makes this projection obsolete for writing in new FITS files.

Fortunately, the GLS projection at any declination and no rotation is equivalent to a CG02 compliant Sanson-Flamsteed (SFL) projection at zero declination (i.e. on the Equator)⁴. It is therefore possible to recast a cube in GLS projection as a cube in SFL projection just by patching the header and without any change on the data! The demonstration and the exact translation to be performed are detailed in the appendix section A.3. The Fig. 2 gives a visual representation of these transformations.

Figure: Visual representation of the GLS-to-SFL transformation. In all the plots, the grey region covers the same area on sky. Top left: the grey region symbolizes a $20 \times 12$ square degrees in the $(x,y)$ 2D AIPS-GLS projected plane. Blue lines show the abscissa and ordinate origins. There is no transformation possible to go to a native sphere in the CG02 formalism. Top right: the reference point is shifted to the Equator, i.e. by a $\delta _0$ shift ( $30^{\circ }$ in this example). The equations now match the FITS-SFL formalism at $\delta _0 = 0$. Bottom left: the area is SFL-unprojected to $(\phi ,\theta )$ spherical coordinates on the 3D native sphere. The equator and zero reference meridian are shown in green. The fiducial point $(\phi _0,\theta _0)$ of the SFL projection has coordinates $(0,0)$ by design. Bottom right: the whole sky orientation is kept the same as before, but the coordinate system represented is now the 3D celestial sphere. The equator and zero reference meridian are shown in red. $\alpha _0$ is $50^{\circ }$ in this example.