The *x,y* (relative coordinates from the center) represent the *arc*
projection.
It means that, if *r* is the relative angular distance between the point and
the center of the projection, and *a* the position angle (North through East)
of the point relative to the center, the relative coordinates are:

y = r ⋅ cos(a)

Assuming the center of the projection at
*(RA _{0}=0, Dec_{0}=0)* (i.e. Cartesian position of the center is

the projections are:

y = cos

i.e. *a* is the position angle of the point *(v,w)* .

When the center of projection is another position *(RA _{0}, DE_{0})*,
a rotation is performed to bring the center of the projection
to the chosen position, using the rotation matrix:

cosDec_{0}⋅cosRA_{0}cosDec_{0}⋅sinRA_{0}sinDec_{0}–sinRA_{0}cosRA_{0}0–sinDec_{0}⋅cosRA_{0}–sinDec_{0}⋅sinRA_{0}cosDec_{0}

The reverse transformation (RA and Dec from *x* and *y*) are derived by
the formulae, if *r* is the distance (*r = sqrt(x ^{2}+ y^{2})*):

v = x sin(r)/r

w = y sin(r)/r

the reverse rotation (with the transposed rotation matrix) is performed,
and the *(u, v, w)* vector is transformed into the exact position.

Note that the computations of

Dec=Dec_0 + y RA=RA_0 + x / cos(Dec)

are only asymptotically correct, at very small distances from the projection center.

*last update: 13 Feb 2024
*