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Astronomical Data Analysis Software and Systems VII
ASP Conference Series, Vol. 145, 1998
R. Albrecht, R. N. Hook and H. A. Bushouse, e
Ö Copyright 1998 Astronomical Society of the Pacific. All rights reserved.
ds.
World Coordinate Systems as Objects
R.F. Warren­Smith
Starlink, Rutherford Appleton Laboratory, Chilton, DIDCOT, Oxon,
OX11 0QX, UK
D.S. Berry
Starlink, Department of Astronomy, University of Manchester, Oxford
Road, MANCHESTER, M13 9PL, UK
Abstract. We describe a new library (AST) which provides a flexible
high­level programming interface for handling world coordinate systems
in astronomy and for producing graphical output. It includes, but is not
limited to, a wide range of celestial coordinate systems and supports the
Digitised Sky Survey plate solutions and the draft FITS WCS proposals
amongst other possibilities. AST is portable and environment indepen­
dent.
1. Introduction
Writing applications which handle non­linear world coordinate systems (WCS),
such as celestial coordinates, in a general way currently presents significant dif­
ficulties. Although good algorithms exist to transform between celestial coordi­
nate systems, understanding the relationship between the many di#erent systems
in use requires considerable expertise. Storing and retrieving WCS information
in datasets also demands familiarity with complicated and changing conven­
tions, such as the many variants of FITS in use. Presenting WCS information
graphically (e.g. as coordinate grids) is also algorithmically complex, especially
if all­sky plots which include the polar regions must be accommodated.
To address these problems, we have developed a library, AST, which pro­
vides a high­level model and programming interface for manipulating WCS data
in astronomy. AST stands for `ASTrometry Library', although astrometry is, in
fact, only a small part of its function.
Our primary objective has been to insulate programmers from the prob­
lems described above by delivering `best practice' solutions in an accessible and
flexible form.
2. Design Criteria
AST is designed to be useful in a wide range of software projects and, to this
end, dependencies on other software have been minimised (only the widely avail­
able SLALIB positional astronomy library is required). AST is implemented in
41

42 Warren­Smith and Berry
ANSI C for portability. It makes extensive use of object­oriented techniques,
but conventional C and fortran77 interfaces are provided --- the latter by an
additional C layer (so that only a C compiler is required for building). Provision
has been made for new language bindings if needed in future.
Graphical output is via a small group of functions which may easily be im­
plemented over most graphics systems (a PGPLOT implementation is provided).
A similar mechanism is used for delivering error messages. This, together with
an ability to perform I/O via text and FITS headers, ensures independence of
any particular programming environment.
Currently, AST is implemented on PC Linux, Solaris and DEC Unix.
3. Inter­Relating Coordinate Systems (Mappings)
Relationships between coordinate systems are represented within AST by objects
called Mappings. A Mapping, like any AST object, is created by a constructor
function which returns a pointer (an integer in fortran) through which the
object is manipulated.
A Mapping does not describe a coordinate system, but merely the inter­
relationship between two (unspecified) coordinate systems. It is a `black box' to
which coordinate values may be given in return for a set of transformed coor­
dinates. This operation may, in principle, be performed in either direction (the
forward and inverse transformations). A Mapping may use any number of input
and output coordinates so as to match the, possibly di#erent, dimensionalities
of the coordinate systems it inter­relates.
AST provides a selection of di#erent Mappings to support a wide range
of celestial coordinate transformations and sky projections. It also provides a
range of utility Mappings, such as linear transformations, look­up tables, etc.
An important feature is that any pair of Mappings may be combined to­
gether to form a compound Mapping, or CmpMap. A CmpMap is itself a Map­
ping, so this process may be repeated. In this way, Mappings of arbitrary
complexity may be built, giving AST great flexibility in the coordinate trans­
formations it can represent.
4. Representing Coordinate Systems (Frames)
While Mappings represent the relationships between coordinate systems, the
coordinate systems themselves are represented by objects called Frames. An
AST Frame is similar in concept to the frame one might draw around a graph.
It contains information about the labels which appear on the axes, the axis units,
a title, knowledge of how to format the coordinate values on each axis, etc. A
Frame is not, however, restricted to two dimensions and may have any number
of axes.
A basic Frame may be used to represent a Cartesian coordinate system by
setting values for its attributes (all AST objects have attributes which may be
set and enquired). Usually, this would involve setting appropriate axis labels

World Coordinate Systems as Objects 43
Figure 1. A FrameSet containing Frames inter­connected by Mappings.
and units, for example. Like all objects, a Frame also provides methods. 1 These
perform operations such as formatting coordinate values as text, calculating
distances between points, interchanging axes, etc.
A derived class, the SkyFrame, is provided to represent celestial coordinate
systems, of which a wide range are supported and may be selected by setting
appropriate SkyFrame attributes. A SkyFrame provides the additional func­
tionality required when handling celestial coordinates --- such as sexagesimal
formatting and great circle distances. It also encapsulates knowledge of how to
convert between any pair of celestial coordinate systems, making this available
through a method.
As with Mappings (§3), it is possible to merge two Frames together into a
compound Frame, or CmpFrame, in which both sets of axes are combined. One
could, for example, have celestial coordinates on two axes and an unrelated coor­
dinate (wavelength, perhaps) on a third. Knowledge of the relationship between
the axes is preserved internally by the process of constructing the CmpFrame
which represents them.
5. Coordinate Networks (FrameSets)
Mappings and Frames may be connected together to form networks called Frame­
Sets (Figure 1). Such a network is extended by adding a new Frame and an asso­
ciated Mapping which relates the new coordinate system to one already present.
This ensures that there is always exactly one path, via Mappings, between any
pair of Frames. A method is provided for identifying this path and returning
the complete Mapping.
One of the Frames in a FrameSet is termed the base Frame. This underlies
the FrameSet's purpose, which is to calibrate datasets and other entities by
attaching coordinate systems to them. In this context, the base Frame represents
1 In AST, methods are functions which take an object pointer as their first argument.

44 Warren­Smith and Berry
Figure 2. A labelled coordinate grid for an all­sky zenithal equal area
projection in ecliptic coordinates, composed and plotted using a single
function call.
the `native' coordinate system (for example, the pixel coordinates of an image).
Similarly, one Frame is termed the current Frame and represents the `currently­
selected' coordinates. It might, typically, be a celestial coordinate system and
would be used during interactions with a user (as when plotting axes on a graph
or producing a table of results). Other Frames within the FrameSet represent
a library of alternative coordinate systems which a software user can select by
making them current.
6. Graphical Output (Plots)
Graphical output is supported by a specialised class of FrameSet called a Plot.
A Plot's base Frame corresponds with the native coordinates of the underlying
graphics system. Plotting operations are specified, using AST Plot methods, in
physical coordinates which correspond with the Plot's current Frame (typically
this might be a celestial coordinate system).
Operations, such as drawing lines, are automatically transformed from phys­
ical to graphical coordinates before plotting, using an adaptive algorithm which
ensures smooth curves (the transformation is usually non­linear). `Missing' co­
ordinates (e.g. graphical coordinates which do not project on to the celestial
sphere), discontinuities and generalised clipping are all consistently handled. It
is possible, for example, to plot in equatorial coordinates and clip in galactic
coordinates. The usual plotting operations are provided (text, markers), but
a geodesic curve replaces the primitive straight line element. There is also a
method for drawing axis lines, which are normally not geodesics.
Perhaps the most useful Plot method is for drawing fully annotated co­
ordinate grids (Figure 2). This uses a general algorithm which does not de­
pend on knowledge of the coordinates being represented, so can also handle
programmer­defined coordinate systems. Grids for all­sky projections, includ­
ing polar regions, can be drawn and most aspects of the output (colour, line
style, etc.) can be adjusted by setting appropriate Plot attributes.