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

the set of points that is used in the interpolation functions, thus reducing the repeated

extraction of points from the database.

Numerical analysis: finding “interesting” locations

One of the fascinating aspects of storing the functions as an element in the GIS (instead

of their outputs) is the ability to carry out analysis on the nature of the functions

themselves. One such example was noted above – the ability to calculate derivatives by

the manipulation of symbols. Furthermore, there are many other elements that are of

interest to the analyst and that Map Calculus-enabled GIS can implement better than

current raster-based GIS. By linking numerical and symbolic analysis libraries to the

functional engine of the GIS, it is possible to identify areas of transition, local and global

properties of function either globally, or within an area of interest. Such analysis should

be directed by the intrinsic behaviour of the function. For example, in a distance

function, the maximum value is neither interesting, nor informative. Thus, when

calculating distance, there is no need to store or evaluate maxima.

Computing needs and requirements

Probably the most limiting factor in the creation of Map Calculus-enabled GIS to date is

the need to compute and re-compute functions “on the fly”, while the user zooms in or

out from a specific location, or when the user asks to produce the contour lines from a

complex function. Users of modern GIS are accustomed to working with an interactive

system that responds within a few seconds. Therefore, there is a need to rapidly compute

a complex function such as Kriging, in order to make function-based layers fit into such

systems. Furthermore, the computational complexity of layers can vary dramatically, as

many users require the use of composite functions, such as the outcome of five or six

interpolations from different sets of points. Another aspect of the computing profile of

Map Calculus-enabled GIS is that it is characterised by intense peaks in computation, as

the user requests a calculation with every zoom in/out operation, followed by periods of

inactivity. Finally, Map Calculus-enabled GIS is likely to include utilise numerical-analysis

operations, and these type of calculation utilises a lot of computing power.

However, while only a few years ago it was impractical to suggest the implementation of

Map Calculus-enabled GIS, recent advances in computing open up the possibility of

implementing such a GIS today. The two most important ones are the increase in