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:
Assuming the center of the projection at (RA0=0, Dec0=0) (i.e. Cartesian position of the center is (1,0,0)), from the Cartesian coordinates of the point:
the projections are:
i.e. a is the position angle of the point (v,w) .
When the center of projection is another position (RA0, DE0), a rotation is performed to bring the center of the projection to the chosen position, using the rotation matrix:
cosDec0⋅cosRA0 cosDec0⋅sinRA0 sinDec0 –sinRA0 cosRA0 0 –sinDec0⋅cosRA0–sinDec0⋅sinRA0 cosDec0
The reverse transformation (RA and Dec from x and y) are derived by the formulae, if r is the distance (r = sqrt(x2+ y2)):
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: 21 Oct 2020