[fitswcs] Polynomial Correction Function & Alternative-WCS

[Don Wells]

Don Wells writes:
 > .. The value of n which includes all terms with (i+j)<=q is obtained by
 > setting i=0, j=q in the formula for n given above:
 > 
 >      n_q = axis*1000 + ((q + 1) * q) / 2 + q)

 > ..  The capability for 999 terms in
 > the polynomial will support terms up to q=(i+j)=21 (X^21, X^20Y,
 > X^19Y^2, .., XY^20, Y^21) for the NAXIS=2 case..

I made a mistake; the NAXIS=2 limit is not q=21 but rather q=43:

    CP1985  =   <E-format-value> / coeff for X^4*Y^39 for axis 1
    CP1986  =   <E-format-value> / coeff for X^3*Y^40 for axis 1
    CP1987  =   <E-format-value> / coeff for X^2*Y^41 for axis 1
    CP1988  =   <E-format-value> / coeff for X*Y^42 for axis 1
    CP1989  =   <E-format-value> / coeff for Y^43  for axis 1

 > .. The author has not yet derived the general formula for n and its
 > inversion formulae for NAXIS>2. Somebody should do this, and should
 > determine the q=(i+j) order limit ..

Although I do not yet have a formula for n for NAXIS=3, I do think
that I now have one for n_q:

        n_q(3) = axis*1000 + ((q + 3) * (q + 2) * (q + 1)) / (3 * 2 * 1)) - 1

I observe that n_q(2) can also be derived by setting i=(q+1),j=0:

        n_q(2) = axis*1000 + ((q + 2) * (q + 1)) / (2 * 1) - 1

(which also gives n_43(2)=989) and that the formula for NAXIS=1 can be
re-written as: 

        n_q(1) = axis*1000 + (q + 1) / (1) - 1

Comparison of these 3 n_q formulae implies that the probable general
formula for NAXIS>3 is:

        n_q(N) = axis*1000 + (q + N)! / (q! * N!) - 1

We have n_16(3)=968, so the 999-coefficient capability will support
terms X^iY^jZ^k up to (i+j+k)=16 for NAXIS=3. If the general formula
given above is correct, we have n_10(4)=1000 [sic!], i.e., we can have
terms up to 10th-order in each axis for NAXIS=4.

-Don
-- 
  Donald C. Wells         Associate Scientist         dwells at nrao.edu
		    http://www.cv.nrao.edu/~dwells
  National Radio Astronomy Observatory                +1-804-296-0277
  520 Edgemont Road,   Charlottesville, Virginia       22903-2475 USA