Notation.

A general transformation with $m$ input variables and $n$ output variables, in which both the forward and inverse mappings are defined, is denoted here by [ $m \leftrightarrow n$]. The double-ended arrow ` $\leftrightarrow$' indicates that coordinate conversion is possible in either direction. Similarly, the notation [ $m \rightarrow n$] and [ $m \leftarrow n$] is used to represent transformations where only the forward or inverse mapping (respectively) is defined, so that coordinate conversion can be performed only in the direction indicated. This notation may also be used to describe individual transformations by including a list of their input and output variables. Thus, a transformation which relates a Cartesian coordinate system to a Polar system might be denoted by [ $(x,y) \leftrightarrow (r,\theta)$].

To distinguish transformations from mappings, the notation for the latter uses braces `{...}' rather than square brackets, so that a mapping which describes how to convert a set of $m$ input values into a set of $n$ output values would be denoted by { $m \rightarrow n$}. In this case, the arrow may only point to the right because a single mapping, once separated from its parent transformation, can only perform coordinate conversion in one direction. This is always regarded as its ``forward'' direction.