[fitswcs] TAN+poly & astrometric discussions in WCS paper-2

[Mark Calabretta]

On Thu 2000/01/13 08:24:10 -0000, Peter Bunclark wrote
in a message to: Mark Calabretta <Mark.Calabretta at atnf.csiro.au>
and copied to: Don Wells <dwells at NRAO.EDU>, fitswcs at NRAO.EDU

>proposal already has ARC, and it already has the poly,
>just not at the same time - but the combination of these is the
>simplest, lowest power, description of a Schmidt plate.

My precepts are as summarized by Don (The radial terms in TAN+poly will
carry the difference between TAN and ARC ...) and supported by, for
example, Section 8.3.4 of Murray (Measurements on a Schmidt plate can
therefore be reduced in exactly the same way...).

The prescription looks simple enough, apparently you simply need to
adjust the coefficient which corresponds to the third order radial
distortion in an ARC+poly fit.  However, on working through the details
I found that it glosses over some unpleasantness.

Suppose we have the following ARC+poly fit for a Schmidt plate:

   X = f(x,y)
   Y = g(x,y)

where (x,y) are plate coordinates and (X,Y) are the ARC analogue of the
standard coordinates, (xi,eta), which refer to a TAN projection.  We
have effectively been saying that you can convert (X,Y) to (xi,eta) via

   xi  = X*(1 + (X^2 + Y^2)/3)
   eta = Y*(1 + (X^2 + Y^2)/3)

where I'm working in radians.  However, when expressed in terms of (x,y)
we get

   xi  = f(x,y) + [f(x,y)^3/3 + f(x,y)*g(x,y)^2/3]
   eta = g(x,y) + [g(x,y)^3/3 + g(x,y)*f(x,y)^2/3]

and in general the horrible messes in brackets may contain terms of
0th-order and above.  The translation only comes out simply if you make
a further approximation for the terms in brackets:

   xi  = f(x,y) + x*(x^2 + y^2)/3
   eta = g(x,y) + x*(x^2 + y^2)/3

This is what I had in mind, and further that this extra, static,
correction would be absorbed into terms which must already be present
to account for refraction (Koenig, 1962, Eq.20).

I guess what you and Brian McLean are saying is that the further,
simplifying approximation isn't good enough.  Is that correct?

If the CRPIXi and CDj_i are used to account for an affine transformation
as suggested in the second-last paragraph of CCS Section 4.1.2, then the
0th-order terms of f(x,y) and g(x,y) vanish and the terms in brackets
would be reduced to 3rd-order and above.  Would that make the simplifying
approximation acceptable?  Is it reasonable in practice?

>I guess I didn't explain my point - what I meant was, there are very
>few folk who think they understand the mathematics of astrometry.  There
>are vanishigley few who in fact actually do understand it. Hence, for most
>of us, we rely on a standard, meeting our needs, for which some clever
>person has written some rigorous software.  In fact, early in my career
>I have written the plate-solution stuff with nothing more than
>a spherical-astronomy book and a Fortran compiler, but more recently,
>I wouldn't dream of not using the far more trustworthy astrometric
>libraries available, or better, complete packages.

...but doesn't this argue against ARC+poly - all the texts and papers
I've come across reduce Schmidt plates using a gnomonic projection.

Cheers, Mark