[fitswcs] Polynomial corrections to WCS projections

[Doug Mink]

There are at least three ways to apply polynomial corrections to FITS
WCS pixel transformations:

1: To raw pixel coordinates before scaling, rotating, and projecting

2: To scaled and rotated pixel coordinates before projection

3: Instead of scaling and rotating pixel coordinates before projection

Right now there are two implementations of polynomial corrections to
FITS WCS:

1. TNX projection by Lindsay Davis in IRAF (type 2 above)
   The polynomial is applied after a CD matrix rotates and scales
   the image pixel coordinates.  The TAN projection is applied to
   the corrected coordinates.  If the WCS software cannot deal with
   the polynomial, the CD matrix can be used as a first order fit.
   Software must, however, recognize the RA---TNX/DEC--TNX CTYPEn
   keyword values.  The coefficients are stored in IRAF multiline
   compound keywords.

2. Traditional plate solution by Doug Mink in WCSTools (type 3 above)
   COi_nnn coefficients are used instead of CD matrix, then a TAN
   projection is applied to the resulting corrected pixels.
   A CD matrix is provided for a first-order WCS in case software
   cannot deal with the polynomial, and RA---TAN/DEC--TAN are used
   as CTYPEn values.  If WCS software was required to recognize the
   polynomial coefficients, this would be the same as the TNX with
   an identity CD matrix.

In the current Calabretta/Greisen process, where lin() scales and
rotates and cel() projects, the alternatives are:

   [(1)polynomial]        lin()             [(3)polynomial]
           |                |                       |
           |                |                       |
          lin()       [(2)polynomial]              cel()   
           |                |
           |                |
          cel()            cel()

I like type 3 because it is easiest to invert and most resembles the
traditional plate solution, but type 1 best models a detector-dependent
variation and type 2 keeps the polynomial correction relatively small.

-Doug Mink
 Telescope Data Center
 Harvard-Smithsonian Center for Astrophysics
 Cambridge, Massachusetts