Concatenating Transformations

It is common to find that the relationship between two coordinate systems is most easily expressed in terms of several transformations applied in succession. For instance, the transformation between positions on the sky and the pixel coordinates of a CCD image might be divided into two stages; the first representing the effect of imaging the sky into the focal plane of the telescope, and the second taking account of the position, size and orientation of the detector in the focal plane. The TRANSFORM facility makes explicit provision for cases such as this by allowing transformations to be concatenated.

Figure: Concatenating two transformations.

A rather general case of concatenation is illustrated in Figure . In this example, Transformation 1 relates two input variables $(x_1,x_2)$ to three ``intermediate'' variables $(y_1,y_2,y_3)$, which are also the input variables $(X_1,X_2,X_3)$ of Transformation 2. This transformation, in turn, has a single final output variable $(Y_1)$. The concatenation process involves eliminating the three intermediate variables and storing the two transformation definitions together in a single new transformation. This new transformation then has two input variables $(x_1,x_2)$ and a single output variable $(Y_1)$. Using the `.' symbol to represent concatenation, this entire process may be summarised as:

$[2 \leftrightarrow 3].[3 \leftrightarrow 1] = [2 \leftrightarrow 1]$

or, in general:

$[i \leftrightarrow j].[j \leftrightarrow k] = [i \leftrightarrow k]$

Note that the number of output variables from the first transformation must be equal to the number of input variables to the second transformation. For obvious reasons, also, it is not permitted to concatenate a transformation in which only the forward mapping is defined with another in which only the inverse mapping is defined (e.g. [ $i \rightarrow j$] could not be concatenated with [ $j \leftarrow k$]).

The result of concatenating two transformations is itself a transformation, so the process may be repeated indefinitely, making it possible for a whole sequence of transformations to be joined together and processed as a single unit. Once transformations have been combined in this way, however, they cannot later be separated.