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:

x = r ⋅ sin(a)
y = r ⋅ cos(a)

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:

(u, v, w) = (cosDec⋅cosRA, cosDec⋅sinRA, sinDec)

the projections are:

x = cos^–1(u) ⋅ v/sqrt(v²+w²)
y = cos^–1(u) ⋅ w/sqrt(v²+w²)

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

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The reverse transformation (RA and Dec from x and y) are derived by the formulae, if r is the distance (r = sqrt(x²+ y²)):

u = cos(r)
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.

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last update: 13 Feb 2024