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

It is important to note that in this function, the values of xA and yA are assigned to the

co-ordinates of A, while xB and yB are left unassigned until we choose the location B. For

example, if A’ is at the location (124,64) then ThisFarA’ will be stored as the function:

ThisFarA'=(xB 124)2+(yB 64)2

Of course, Euclidean distance is one of the simplest spatial functions. Spatial

interpolation, on the other hand, is more complex process and more interesting aspects

of Map Calculus-based GIS can be clarified by examining its implementation. In IDW,

the function operates on a set of sampled points (L1,L2,…Ln) and calculate the value for

a new location L’ by calculating:

L'=∑di=1

n

i=1n1pii∑d1p

i

Where di is the distance from L’ to the location Li, and p is a power of the distance.

Usually, the search radius is taken as a parameter of the function. In a function-based

layer implementation, the GIS will store the template for IDW, the defined radius

distance and a linkage to the set (L1,L2,…Ln). Upon request to calculate the IDW value

for a location L’, buffer operation on the basis of the search radius extracts the points

that should be included in the calculation and constructs the sub-set. This is then

followed by the computation of the equation for L’ and the sub-set which yields the

requested output. The process can be repeated for any set of points. For example, it is

possible to analyse IDW values from the set (L1,L2,…Ln), but to calculate them only for

location (K1,K2,…Kn). This will enable the linkage of two sample sets, by calculating a

precise estimate of the field that the data set L represent for the locations of the set K.

Notably, the user interface of the GIS for entering the parameters can stay the same.

The two examples above are of global functions where the implementation of Map

Calculus-enabled GIS is rather straight forward. Piecewise functions, however, represent

a different challenge. This is the case with DEMs where it is impossible to devise a global

function and the description of the field is done through tessellation and a use of a family

of functions that are fitted to each sub-domain, as is the case with finite elements

methods (Lancaster & Salkauskas, 1986). Therefore, piecewise representations can be