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 Computations of x,y in the arc projection  ← back⋅menu⋅previous⋅next

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

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.

last update: 13 Feb 2024 

 Computations of x,y in the arc projection  ← back⋅menu⋅previous⋅next