Transformation variables.

The inter-conversion of coordinates in this way between two different 2-dimensional systems (e.g. between Equatorial and Galactic sky coordinates) is a familiar concept. In general, however, there is no need for the two coordinate systems to be 2-dimensional, nor even for both of them to have the same dimensionality. To cater for this general case, each transformation has a set of input and output variables associated with it. They are represented here by $(x_{1},\ldots x_{m})$ and $(y_{1},\ldots y_{n})$ respectively, where the numbers of input and output variables ($m$ and $n$) may be any positive integer - not necessarily equal. These variables are akin to the ``dummy arguments'' used in Fortran subroutines and are the means by which coordinate values are passed to and from the transformation; they may be visualised as a set of input and output ports attached to it, as follows:

$$\begin{array}{c} \mbox{Forward } \longrightarrow \\ \par m ... ...tput values} \par \\ \longleftarrow \mbox{ Inverse} \end{array}$$

Any coordinate values supplied to the ports (i.e. variables) at either end of the transformation may be converted into a corresponding set of ``transformed'' coordinates (using the information within the transformation), to be delivered through the ports (i.e. variables) at the opposite end.

As with subroutine arguments, the names assigned to the input/output variables have no absolute significance outside the transformation itself, and coordinate values must be supplied in the correct order (typically within a data array, for instance) in order to ``match up'' with the appropriate transformation variables. In practice, of course, when describing a transformation, it is convenient to retain meaningful names for its variables; these may then be referred to using the usual notation as $(\alpha,\delta)$ or $(x,y,z)$, etc., so that the expected order is clear.