Given the of the *plate centre* (the tangent point)
and the of a star within the field, the standard
coordinates can be determined by calling
sla_S2TP
(single precision) or
sla_DS2TP
(double precision). The reverse transformation, where the
is known and we wish to find the , is carried out by calling
sla_TP2S
or
sla_DTP2S.
Occasionally we know the both the and the of a
star and need to deduce the of the tangent point;
this can be done by calling
sla_TPS2C
or
sla_DTPS2C.
(All of these transformations apply not just to but to
other spherical coordinate systems, of course.)
Equivalent (and faster)
routines are provided which work directly in instead of
spherical coordinates:
sla_V2TP and
sla_DV2TP,
sla_TP2V and
sla_DTP2V,
sla_TPV2C and
sla_DTPV2C.

Even at the best of times, the tangent plane projection is merely an
approximation. Some telescopes and cameras exhibit considerable pincushion
or barrel distortion and some have a curved focal surface.
For example, neither Schmidt cameras nor (especially)
large reflecting telescopes with wide-field corrector lenses
are adequately modelled by tangent-plane geometry. In such
cases, however, it is still possible to do most of the work
using the (mathematically convenient) tangent-plane
projection by inserting an extra step which applies or
removes the distortion peculiar to the system concerned.
A simple *r _{1}*=

SLALIB includes a group of routines which can be put together to build a simple plate-reduction program. The heart of the group is sla_FITXY, which fits a linear model to relate two sets of coordinates, in the case of a plate reduction the measured positions of the images of a set of reference stars and the standard coordinates derived from their catalogue positions. The model is of the form:

*x*_{p} = *a* + *bx*_{m} + *cy*_{m}

*y*_{p} = *d* + *ex*_{m} + *fy*_{m}

where the *p* subscript indicates ``predicted'' coordinates
(the model's approximation to the ideal ``expected'' coordinates) and the
*m* subscript indicates ``measured coordinates''. The
six coefficients *a-f* can optionally be
constrained to represent a ``solid body rotation'' free of
any squash or shear distortions. Without this constraint
the model can, to some extent, accommodate effects like refraction,
allowing mean places to be used directly and
avoiding the extra complications of a
full mean-apparent-observed transformation for each star.
Having obtained the linear model,
sla_PXY
can be used to process the set of measured and expected
coordinates, giving the predicted coordinates and determining
the RMS residuals in *x* and *y*.
The routine
sla_XY2XY
transforms one into another using the linear model. A model
can be inverted by calling
sla_INVF,
and decomposed into zero points, scales, *x*/*y* nonperpendicularity
and orientation by calling
sla_DCMPF.

Starlink User Note 67

P. T. Wallace

12 October 1999

E-mail:ptw@star.rl.ac.uk