73 kB PostScript reprint

Astronomical Data Analysis Software and Systems IV

ASP Conference Series, Vol. 77, 1995

Book Editors: R. A. Shaw, H. E. Payne, and J. J. E. Hayes

Electronic Editor: H. E. Payne

**E. W. Greisen**

National Radio Astronomy Observatory

**M. Calabretta**

Australia Telescope National Facility

The initial paper describing the Flexible Image Transport System, or
FITS format, (Wells, Greisen, & Harten 1981) proposed keywords to
describe the physical coordinates of the image. They were `
CRPIX`* n* for the reference pixel location on pixel axis * n*,
` CRVAL`* n* for the coordinate value at that pixel, `
CDELT`* n* for the increment at that pixel in the coordinate value,
and ` CTYPE`* n* for the type of coordinate. Coordinate
rotation---of an unspecified nature---was allowed, and a few possible
values for ` CTYPE`* n* were proposed. The original authors
chose to defer discussion of the technical details of coordinate
specification until the basic FITS format was accepted generally and
until a deeper understanding of image coordinate specification and
computation could be obtained.

The time for that discussion is now. While participating in the
development of the AIPS software package of the National Radio
Astronomy Observatory, Greisen (1983) developed FITS-like syntax and
semantics to define both velocity and celestial coordinates. The
latter have been widely used for interchanging imagery from a number
of instruments at widely differing spectral domains and are
fundamental to the present proposal. Greisen defined the reference
pixel for celestial coordinates to be the tangent point of the
projection. He specified that the first four characters of `
CTYPE`* n* should be used to give the type of celestial coordinate
while the next four characters specified the type of projection (e.g.,
` DEC--TAN`). Greisen (1983) gave the mathematics
for four projections: orthographic (` SIN`), gnomic (` TAN`),
zenithal equidistant (` ARC`), and a special coordinate used by
East-West radio interferometers (` NCP`). In a second
paper, Greisen (1986) added specifications for the stereographic (`
STG`), sinusoidal (` GLS`), Hammer-Aitoff (` AIT`), and Mercator
(` MER`) projections. The current proposal (Greisen and
Calabretta, 1994) extends these earlier, widely tested proposals to
clarify the logical process by which celestial coordinates are
computed, and to specify a very wide range of possible projections. In
addition, the current proposal specifies a method to define skew,
offset rotations, and even rotations of axes of different physical
type into each other. It also specifies a method to describe the
units of the coordinates and to provide a second coordinate
description for an axis.

**Figure:** Conversion of pixel to celestial spherical coordinates
Original PostScript figure (3 kB)

In the current proposal, we regard the conversion from simple pixel
counts to a full coordinate description as a multi-step process
containing one optional and four required steps. These steps are
indicated conceptually in Figure . The first and
optional step is used to correct the actual image pixel numbers into
those which would have been recorded by an ideal instrument. The
corrections in this ``pixel regularization table'' are expected to be
rather small, so that they may be ignored except in high precision
computations. In the second step, for all types of coordinates, the
vector of reference pixels is subtracted from the vector of pixel
numbers and the result multiplied by a pixel conversion (` PC`*
iiijjj*) matrix to convert from pixel numbers to offsets from the
reference pixel along physical axes but still in pixel units. The
third step is a multiplication by a diagonal matrix (` CDELT`*
i*) to convert to relative coordinate in physical units.

The fourth step in the process of finding the true coordinates depends
on the type of axis given in ` CTYPE`* n*. For simple linear
axes, the true coordinate is found by adding the offset found above to
the reference pixel value given by ` CRVAL`* n*. Otherwise,
some function of the offset(s), the ` CRVAL`* n*, and, perhaps,
other parameters must be established by convention and agreement. For
celestial coordinates, the proposed fourth step involves converting
the linear offsets into longitudes and latitudes in the ``native
coordinate system'' for the specified type of projection. These are
rotated, in the fifth step, by the usual spherical formulæ to
longitudes and latitudes in the desired standard coordinates (e.g.,
Equatorial, Galactic, etc.) The native coordinate
system is, for azimuthal and conical projections, one which has its
north pole at the reference pixel. For cylindrical and conventional
projections, the native coordinate system has its origin at the
reference pixel. The rotation from native to standard coordinates is
illustrated in Figure . The keyword ` LONGPOLE` is
proposed to specify the native longitude of the north pole of the
standard system. The default value for ` LONGPOLE` is to be 180
degrees to support current usage. Extra keywords ` PROJP`* j*
are defined to provide additional parametric information needed by
some of the projections.

**Figure:** Conversion of native (left) to standard (right) spherical
coordinates for the Hammer-Aitoff projection.
Original PostScript figure (6 kB)

The original FITS paper (Wells, Greisen, & Harten 1981) naively assumed
that the units along each axis could be implied simply by the contents
of the ` CTYPE`* n* keyword and that they would be in the basic
SI units. Outside of celestial coordinates, both of these assumptions
have apparently failed in practice. Therefore we propose that a new
character-valued keyword ` CUNIT`* n* be added to describe the
units used for coordinates on axis * n*. For celestial angular
coordinates, following the proposed projection conventions, these
units will be degrees (` 'deg '`). Additional discussion and
agreements will be needed to determine how one will represent other
coordinate types.

In some cases, the axes of an image may be described as having more
than one coordinate. An example of this would be the frequency,
velocity, and wavelength along a spectral axis (only * one* of
which, of course, could be linear). To allow up to 8 additional
descriptions of each axis, we propose the addition of the follow
optional, but now reserved, keywords.

where for the second through ninth alternate axis coordinate and for axis 1 through 999.

To improve the use of these coordinates for astrometric purposes,
three new keywords are proposed. ` EQUINOX` replaces ` EPOCH`
for the epoch of the mean equator and equinox in years. ` MJD-OBS`
gives the modified Julian date of observation in days and `
RADECSYS` gives the frame of reference of equatorial coordinates as
` FK4`, ` FK4-NO-E`, ` FK5`, ` GAPPT`.

Greisen, E. W. 1986, AIPS Memo No. 46 (Charlottesville, National Radio Astronomy Observatory)

Greisen E. W., Calabretta, M. 1994, in preparation

Wells, D. C., Greisen, E. W., & Harten, R. H. 1981, A&AS, 44, 363

73 kB PostScript reprint

adass4_editors@stsci.edu