**2.1.4. Additional points**

Note that integer pixel numbers refer to the center of the pixel
in each axis, so that, for example, the first pixel runs from pixel
number 0.5 to pixel number 1.5 on every axis. Note also that the
reference point location need not be integer nor need it even occur
within the image. The original FITS paper (Wells et al.
[1981])
defined the pixel numbers to be counted from 1 to `NAXIS `*j*
( 1) on each axis in a Fortran-like order as presented in the
FITS image. ^{(1)}

A WCS representation should be invertible in the sense that a pixel coordinate, when transformed to a world coordinate, must be uniquely recoverable from that world coordinate. Note that this does not require that each pixel coordinate in an image have a valid world coordinate; as an example, pixel coordinates in the corner of a Hammer-Aitoff projection of the full sky fall outside the map boundary. Nor need each valid world coordinate correspond to a pixel coordinate; for example, the divergent poles of the Mercator projection are inaccessible. In practical terms, it means that two or more different pixel coordinates should not map to the same world coordinate, as exemplified by a cylindrical projection in which the longitude spans more than 360°. Such coordinate systems, while easy to construct, may be very difficult to interpret properly in all respects, including that of drawing a coordinate grid. Thus, while they are not explicitly prohibited, it may be expected that general WCS interpreting software may not handle them properly.

An additional convention is needed where non-linear axes must be
grouped, for example, the two axes which form a map plane. In
general, all axes in the group must have identical algorithm codes and
a scheme must be established by convention for associating members of
the group and, if necessary, their order. For example, Paper II
introduces the '*x*`LON` / *x*`LAT`' and
'*yz*`LN` / *yz*`LT`' conventions for associating
longitude/latitude
coordinate pairs. This should serve as a model for other cases. Note
that grouping is not required for linear axes which are always
separable (in the mathematical sense) by means of a rotation or skew
applied via the linear transformation matrix.

Some non-linear algorithms require parameter values, for example, conic projections require the latitudes of the two standard parallels. Where necessary, numeric parameter values will be specified via

PVi_m |
(floating-valued) |

keywords, where *i* is the intermediate world
coordinate axis number and *m* is the parameter number. Leading
zeros are not allowed and *m* may have only those values defined
for the particular non-linear algorithm in the range 0 through 99
only. There may also be auxiliary keywords which are required to
define, for example, the frames of reference used for celestial and
velocity coordinates.

A few non-linear algorithms may also require character-valued parameters, for example, table lookups require the names of the table extension and the columns to be used. Where necessary, character-valued parameters will be specified via

PS i_m |
(character-valued) |

keywords, where *i* is the intermediate world
coordinate axis number and *m* is the parameter number. Leading
zeros are not allowed and *m* may have only those values defined
for the particular non-linear algorithm in the range 0 through 99 only.

The keywords proposed above and throughout the main body of this
manuscript apply to the relatively simple images stored in the main
FITS image data and in `IMAGE` extensions Ponz et al.
[1994]).
When coordinates are used to describe image
fragments in `BINTABLE` extension tables (Cotton et al.
[1995]),
additional nomenclature conventions are required.
These are described in Sect. 3.

^{1} This convention differs from the usual
practice
in computer graphics where the pixels are counted from zero with pixel
centers as half integers (e.g. Adobe Systems, Inc.
[1999]). The
convention proposed here has been used extensively in FITS since the
format was invented and no argument has been advanced sufficiently
compelling to invalidate the many thousands of files written with that
convention. Furthermore, we regard our image samples as "voxels" in
real physical space rather than pixels in two-dimensional display
space. As such, they may be viewed from any angle via transposition
and rotation. The only point within the individual voxel that remains
invariant under those operations is its center and we argue,
therefore, that it is the center of the voxel which we should count.
Back.