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

package (such as ArcGIS) is about 800x600 pixels – due to the elements of the Graphical

User Interface which occupy the rest of the screen. This active map area can further be

divided to basic units of 2x2 pixels (usually, one pixel is too small to be visible). Thus, the

active area requires about 120,000 calculations. The speed of the calculations is the most

significant factor in making Map Calculus-enabled GIS useful and effective for its users.

This can be accomplished by the use of efficient and rapid algorithms. As the user zooms

out or in, the scale of the map changes, and new calculations for the currently displayed

area are carried out. Hence, to the system’s user, a Map Calculus-enabled GIS behaves at

the same way as the existing implementations of GIS.

Another interesting aspect of function-based layers is the ability to extend the GIS

analytical toolbox into a numerical and mathematical analysis of the functions

themselves. For example, in a complex distance function (where we use anisotropic

functions, which measure a weighted distance from several locations) there is an

advantage in calculating and locating local and global minima. This can be carried out by

analysing the behaviour of the function across the space, in a similar way to operation of

packages such as Matlab . When dealing with more complex functions, where it is

impossible to determine local minima and maxima, the visualisation of the layer can be

used as an indicator of a user’s interest in a specific area and, while calculating the values

for visualisation, the system can find and store “interesting locations” across the map.

Furthermore, using free Central Processing Unit (CPU) cycles, the system can continue

and investigate these interesting locations further and they can be visualised and stored.

A function-based layer should also store its relationship to the objects that are used as

input. Thus, in the example of the distance function used above, the function-based layer

should not just store the co-ordinates of the point that was used to calculate it, but

should be linked to the spatial objects in the database. Such an implementation will

enable the function-based layer (and all subsequent layers that use it) to be updated

instantly whenever the location of a point is being updated. This can be exemplified by a

GIS that tracks the distance amongst a set of vehicles, all of which carry Global

Positioning Satellite (GPS) receivers and transmit their locations to the GIS. The

combined layer (which might calculate the location of the “centre of gravity” amongst all

the vehicles) is updated automatically whenever the location of a vehicle is updated.

Finally, Map Calculus-enabled GIS needs to have certain management capabilities due to

the dependency between layers. In the example of the tracking function if the user