Remove pincushion/barrel distortion from a distorted [x,y] to give tangent-plane [x,y].

palUnpcd( double disco, double $\ast$ x, double $\ast$ y );

disco = double (Given)

x = double $\ast$ (Given & Returned)

y = double $\ast$ (Given & Returned)

• The distortion is of the form RP = R$\ast$(1$+$C$\ast$R$$^2), where R is the radial distance from the tangent point, C is the DISCO argument, and RP is the radial distance in the presence of the distortion.

• For pincushion distortion, C is $+$ve; for barrel distortion, C is -ve.

• For X,Y in " radians" - units of one projection radius, which in the case of a photograph is the focal length of the camera - the following DISCO values apply:

Geometry DISCO

astrograph 0.0 Schmidt -0.3333 AAT PF doublet $+$147.069 AAT PF triplet $+$178.585 AAT f/8 $+$21.20 JKT f/8 $+$13.32

• The present routine is a rigorous inverse of the companion routine palPcd. The expression for RP in Note 1 is rewritten in the form x$$^3$+$a$\ast$x$+$b=0 and solved by standard techniques.

• Cases where the cubic has multiple real roots can sometimes occur, corresponding to extreme instances of barrel distortion where up to three different undistorted [X,Y]s all produce the same distorted [X,Y]. However, only one solution is returned, the one that produces the smallest change in [X,Y].

palPcd