This paper provides a new representation for fields (continuous surfaces) in Geographical Information Systems (GIS), based on the notion of spatial functions and their combinations. Following Tomlin’s (1990) Map Algebra, the term “Map Calculus” is used

piecewise polynomial functions over triangular and quadrilateral partitions while in

computer graphic applications and, especially for applications of rendering algorithms,

there is a wide use of splines to represent fields.

Many applications of GIS utilise field models, but it is in the area of spatial analysis that

the similarity between the surface model and a well-defined mathematical function is the

clearest. This is due to the nature of spatial analysis where the user explores the formal,

quantitative structure of geographical problems (Longley & Batty, 1996). Furthermore,

within socio-economic research, it is the need to translate data from one set of areal units

to another that provides the motivation to explore different types of surfaces (Martin,

1996). As Thurstain-Goodwin (2003) demonstrated, data surfaces are valuable policy

tools when they are used to depict socio-economic data sets.

This paper describes a new way to store surface information, by using function-based

layers in a GIS. A function-based layer is defined by its mathematical and spatial function

and a number of bounded variables, while the system takes care of representing and

manipulating it in a way that is transparent to the user. Unlike current representations,

the GIS stores the function itself and not spatial objects which contain the z value in the

formula above. Such an approach existed in computer models in meteorology for many

years (Goodman, 1985) and is being used in some global climate models, but was not

adopted in GIS and spatial analysis. The main strength of the new representation is the

ability to treat each analytical layer in its symbolic form, thus making it possible to

manipulate layers (maps) through calculations of functions, and manipulates GIS layers

in a formulaic form. This can increase the range of the GIS analytical toolbox and open

up new directions in spatial analysis research. Following Tomlin’s (1990) Map Algebra,

the term Map Calculus is used here to describe the application of function-based layers in

a GIS. In essence, Map Calculus is an alternative to the current practice of using raster

layers to represent surfaces. By using Map Calculus and taking advantage of the

capabilities of existing and emerging computing environments, some of the problems

that are associated with the raster-based approach are eliminated: there is no need to

select a specific cell resolution or to consider the storage of large raster layers.

The paper is divided into four parts. First, Map Calculus is explained in detail. Second,

the computational considerations and theoretical concepts that are material to this

approach are described. Third, a simple implementation of Map Calculus in a modern

desktop GIS is presented. Fourth, the proposed approach is compared to the current