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Astronomical Data Analysis Software and Systems IV
ASP Conference Series, Vol. 77, 1995
R. A. Shaw, H. E. Payne, and J. J. E. Hayes, eds.
Representations of Celestial Coordinates in FITS
E. W. Greisen
National Radio Astronomy Observatory
M. Calabretta
Australia Telescope National Facility
Abstract. The initial descriptions of the FITS format provided a simí
plified method for describing the physical coordinate values of the image
pixels, but deliberately did not specify any of the detailed conventions
required to convey the complexities of actual image projections. Building
on conventions in wide use within astronomy, this paper proposes changes
to the simple methods for describing coordinates and proposes detailed
conventions for describing most of the methods by which spherical coorí
dinates may be projected onto a twoídimensional plane. Simple methods
for converting from the existing coordinate conventions are described.
This paper does not attempt to address the politically sensitive questions
of frequency/velocity coordinates, nor does it address various other types
of coordinates, such as time.
1. Introduction
The initial paper describing the Flexible Image Transport System, or FITS forí
mat, (Wells, Greisen, & Harten 1981) proposed keywords to describe the physií
cal coordinates of the image. They were CRPIXn for the reference pixel location
on pixel axis n, CRVALn for the coordinate value at that pixel, CDELTn for the
increment at that pixel in the coordinate value, and CTYPEn for the type of
coordinate. Coordinate rotation---of an unspecified nature---was allowed, and
a few possible values for CTYPEn 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 develí
opment of the AIPS software package of the National Radio Astronomy Obí
servatory, Greisen (1983) developed FITSílike 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 specí
tral domains and are fundamental to the present proposal. Greisen defined the
reference pixel for celestial coordinates to be the tangent point of the projecí
tion. He specified that the first four characters of CTYPEn should be used to
give the type of celestial coordinate while the next four characters specified the
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type of projection (e.g., DECííTAN). Greisen (1983) gave the mathematics for
four projections: orthographic (SIN), gnomic (TAN), zenithal equidistant (ARC),
and a special coordinate used by EastíWest radio interferometers (NCP). In a
second paper, Greisen (1986) added specifications for the stereographic (STG),
sinusoidal (GLS), HammeríAitoff (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 specií
fies a method to describe the units of the coordinates and to provide a second
coordinate description for an axis.
2. Coordinate Computation
In the current proposal, we regard the conversion from simple pixel counts to a
full coordinate description as a multiístep process containing one optional and
four required steps. These steps are indicated conceptually in Figure 1. 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 (PCiiijjj)
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 (CDELTi) 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 CTYPEn. For simple linear axes, the true coordinate
is found by adding the offset found above to the reference pixel value given by
CRVALn. Otherwise, some function of the offset(s), the CRVALn, and, perhaps,
other parameters must be established by convention and agreement. For celesí
tial 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
formulae 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 2. 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 PROJPj
are defined to provide additional parametric information needed by some of the
projections.

3
ACTUAL PIXEL
COORDS
optional correction table
IDEAL PIXEL
COORDS
linear transformation: multiply by PCnnnmmm
CRPIXn
PC matrix to rotate and skew
TRUE PLANE PIXEL
COORDS
Scale to physical units CDELTn
RELATIVE PHYSICAL
COORDS
map projection per Section 4 CTYPEn
PROJPi
NATIVE SPHERICAL
COORDS
rotation via equations 6 CRVALn
LONGPOLE
CELESTIAL
COORDS
Figure 1. Conversion of pixel to celestial spherical coordinates
0
30
60
90
120
150
180 180 210
240
270
300
330
60
30
0
í30
í60
0
30
60
90
120
150
180 180
210
240
270
300
330
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í60
Figure 2. Conversion of native (left) to standard (right) spherical
coordinates for the HammeríAitoff projection.

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3. Other Proposed Conventions
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 CTYPEn
keyword and that they would be in the basic SI units. Outside of celestial coorí
dinates, both of these assumptions have apparently failed in practice. Therefore
we propose that a new characterívalued keyword CUNITn be added to describe
the units used for coordinates on axis n. For celestial angular coordinates, folí
lowing 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.
CmVALn coordinate value at reference pixel
CmPIXn reference pixel array location
CmELTn coordinate increment at reference pixel
CmYPEn axis type (8 characters)
CmNITn units of CmVALn and CmELTn (character valued)
where m = 2; 3; : : : ; 9 for the second through ninth alternate axis coordinate and
n = 1; 2; : : : ; 999 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 equaí
tor and equinox in years. MJDíOBS gives the modified Julian date of observation
in days and RADECSYS gives the frame of reference of equatorial coordinates as
FK4, FK4íNOíE, FK5, GAPPT.
References
Greisen, E. W. 1983, AIPS Memo No. 27 (Charlottesville, National Radio Así
tronomy Observatory)
Greisen, E. W. 1986, AIPS Memo No. 46 (Charlottesville, National Radio Así
tronomy Observatory)
Greisen E. W., Calabretta, M. 1994, in preparation 1
Wells, D. C., Greisen, E. W., & Harten, R. H. 1981, A&AS, 44, 363
1 http://fits.cv.nrao.edu/documents/wcs/wcs.html