This paper describes a mathematical transformation which converts a mapped circular space into a mapped square space of equal area. The advantages to such a procedure are obvious when one considers the advantages of working with a square map. One area where this technique has been used is in the creation mapped populations for statistical simulation studies (Schreuder et al. 1993). Williams and Schreuder (1995) used this technique to create a spatially realistic population which was used for testing estimators of forest condition. A similar and analogous problem is the attempt to create a two-dimensional (flat) map of the Earth's surface.
This specific problem is much simpler, as the initial mapped data is contained in a two-dimensional circle instead of the surface of a sphere. Nevertheless, the example illustrates the frequent need for data to be converted to something that is easier to manipulate: a square space where it is convenient to work with traditional Cartesian coordinates instead of two- or three- dimensional polar coordinates.
There are several different transformations which can map a circular space into a square space. However, the procedure outlined in this paper has a specific advantage: Lebesgue measure is preserved under it. This fact will be proved later in the paper.