``No matter how tricky you are, you cannot pave a sphere with square tiles.''

(Attributed to A. Einstein)

(Attributed to A. Einstein)

About of the difficulties encountered in astronomical mapping comes from this very profound evidence. The remaining few percents (within the errors) come from the multiplicity of coordinate systems used by the astronomers. For small field mapping, the sphere can reasonably be approximated by its tangent plane at the field center, but for fields larger than a few degrees (depending on the required positional accuracy), curvature effects become important.

As spherical plotters are not easily commercially available (and their
output hardly accepted by journal editors), the big problem is to represent
part of a sphere on a plane sheet. GREG offers several facilities to
deal with this problem, by means of commands `PROJECTION`,
`CONVERT` and `GRID` in the `GREG2` language.

The current philosophy to handle this problem is to make all plotting * in relative coordinates on the projection plane*. No drawing is done in
absolute coordinates on the sphere. This was decided because the plot page
is plane indeed. Another apparently restrictive convention in GREG
projections is that all angles are internally in the natural angular unit
Radian. Accordingly, when you give a map to be contoured, the map
coordinates should be offsets in radians from the projection center. For
user convenience, it is possible to specify limits in Degrees, Minutes or
Seconds using the command `SET ANGLE_UNIT`, but
this command has no effect on internal conversion
formulae. Again, the rationale behind this convention is that we are in
fact working in the projection plane, where angles have no meaning but only
their projections remain.

However, it is *always possible to bypass* these *apparent*
restrictions provided your understand what a conversion formula and a
projection are. As an example is always much better than lengthy
discussion, assume you want to overlay a contour map to the Equatorial and
Galactic grids relevant to the mapped area. Unfortunately, the map
coordinates are in Degrees from a point situated at (0.5,0.5) (``degrees''
in projection) from the projection center (note that this sentence is
complete nonsense, because you cannot have ``degrees'' in the projection
plane...)

SET ANGLE_UNIT/DEFAULT ! (1) LIMITS 5 -5 -5 5 ! (2) RGDATA MYMAP ! (3) LEVELS -1 1 TO 5 ! (3) RGMAP/BLANKING -2 .1 ! (3) ! SET SYSTEM EQUATORIAL ! (4) PROJECTION 6:25:30 35:40.5 0.0/TYPE GNOMONIC ! (5) SET ANGLE_UNIT DEGREES ! (6) LIMITS 4.5 -5.5 -5.5 4.5 ! (7) SHOW LIMITS ! (7) PEN 1 ! GRID 2 2 ! (8) PEN 2 ! GRID 2 2/ALTERNATE ! (9)

- Make sure to work as if you had no problem at all
- Define the map limits. As you are not worrying about the angular units, the internal limits will be just what you type.
- Read your map and draw your contours. This is the usual process for any map.
- Here start the specific astronomical problem. You know that your coordinate system is Equatorial. Specify it.
- Define the projection in this system. Note that Right Ascension is in Hours, and Declination in Degrees as usual for Equatorial coordinates. The projection type depends on your problem of course.
- Specify that your will
*now be giving limits in Degrees* - Give the limits of the map with respect to the projection center.
The system automatically converts the values typed in to internal limits
in radians, as you can check by the
`SHOW`command. Note how you handle the shift between the (0,0) of your map and the projection center which is always the (0,0) of the projection plane (compare with command (2)) - Everything is now correct to plot the grid of meridians and parallels over your map.
- You may even plot a
*Galactic*grid just by specifying the`/ALTERNATE`option in the`GRID`command.

instead of commands (6) and (7).

You can work in three different systems :

`UNKNOWN`, for which GREG makes no assumption at all about the meaning of the unprojected coordinates. In particular, there is no alternate system in this case.`EQUATORIAL`with a custom equinox (GREG assumes to be at equinox 2000.0 by default). The alternate system is`GALACTIC`of course.`GALACTIC`, with`EQUATORIAL 2000.0`system as alternate.

When you measure positions with the cursor in the last two systems, both
galactic and equatorial coordinates are given. For user convenience,
sexagesimal notations in Hours Minutes Seconds for Right Ascension and
Degrees, Minutes Seconds for Declination are used, while decimal notations
with angle in degrees are used for L and B (as well as if the system is
`UNKNOWN`).

Approximate absolute labelling in Hours and Degrees (sexagesimal notation)
can be obtained for the Box by specifying the option `/ABSOLUTE` to
command `BOX` if the system is `EQUATORIAL`.
If the system is `GALACTIC` or `UNKNOWN`, this option produces
absolute labelling in Degrees. Note that it is only meaningful for small
fields of view. Without this option, `BOX` produces relative labelling
in current angle units.

In addition, command `CONVERT` can convert absolute positions or
projected coordinates from a different projection (in any angular units),
to projected coordinates (in radians) in the current projection. This
command may be used to plot star positions on a map, and so on.