Example 1. Drawing graphs.

A simple illustration of mappings, transformation functions and the notation used to describe them may be taken from the [ $2 \leftrightarrow 2$] transformation commonly used to relate ``data'' coordinates $(x_d,y_d)$ to ``graph paper'' coordinates $(x_p,y_p)$ when drawing a graph. If the graph is linear, then the transformation's two mappings might typically be defined by the following transformation functions:

$$\begin{array}{cc} \mbox{Forward mapping } \left\{ \begin{arr... ...x_0 \\ y_d & = & (y_p/S_y)+y_0 \end{array} \right. \end{array}$$ (2)

where $x_0$ and $y_0$ are zero points on the two axes, and $S_x$ and $S_y$ are scale factors. In this example, ($x_d,y_d$) have been treated as input variables, while ($x_p,y_p$) are output variables. This particular choice is arbitrary, although a convention would have to be adopted before writing software which used such a transformation.

Using the notation outlined above, this simple transformation would be denoted by:

[ $(x_d,y_d) \leftrightarrow (x_p,y_p)$]

i.e. with the input variables appearing on the left. The forward mapping would then be denoted by:

{ $(x_d,y_d) \rightarrow (x_p,y_p)$}

and the inverse mapping by:

{ $(x_p,y_p) \rightarrow (x_d,y_d)$}