ISPRS, Vol.34, Part 2W2, "Dynamic and Multi-Dimensional GIS”, Bangkok, May 23-25, 2001 
interaction among whole objects, Voronoi regions of spatial 
objects are utilised. 
The remainder of this paper is structured as follows: Section 2 
introduces the basic strategies of the new approach. In Section 
3, spatial components are introduced to set algebra to form a 
spatial algebra for spatial relations. In Section 4, Voronoi 
regions are employed to make this spatial algebra more 
generalised. Section 5 and Section 6 examine possible spatial 
relations described and distinguished by this spatial algebra. 
Some conclusions are made in Section 7. 
2. THE STRATEGIES USED IN THIS STUDY 
In order to develop an appropriate strategy of study, a critical 
examination of existing models seems appropriate. 
As mentioned in the previous section, in the 
decomposition-based approach, a spatial object is 
decomposed into several topological components such as the 
interior and boundary. The interaction between these 
components of objects determines the relations between 
spatial objects. Most of existing models are based on point-set 
topology. The typical examples are the 4-intersection model by 
Egenhofer and Franzona (1991) and its extension to 
9-intersection model (Egenhofer and Herring 1991). However, 
there are some imperfections associated with this model 
(Clementini et al. 1993, Molenaar et al. 1994, Chen et al. 2001). 
For example, 9-intersection model results in too many relations 
for humans to use and the list of cases resulting from this 
approach is not directly related to the users' interpretation of 
topological facts (Clementini et al. 1993, 1995). More seriously, 
the use of complement as exterior in the 9-intersection model 
makes the three components linearly dependent (Li et al. 2000, 
Chen et al. 2001). 
Now let's turn to the whole-based approaches. The 
whole-based approach directly uses spatial objects instead of 
their components for the description of spatial relations, thus 
the problems associated with decomposition-based approach 
can be avoided. However, it is not sufficient to consider only 
objects themselves for distinguishing spatial relations 
(Egenhofer et al. 1993). Indeed, many relations cannot be 
distinguished if only the objects themselves are used. 
As a consequence of these reasons, in this study, the 
whole-based approach will be used as a basis. In order to 
overcome the shortcomings with the whole-based approach, 
an additional parameter should be introduced and this 
parameter must be insensitive to the dimensionality of space. 
On the hand, it must be closely related to the object as so to 
have functions similar to those of boundary and exterior. As a 
result, the Voronoi region of an object is selected, because 
Voronoi regions have many good properties (Gold 1992, Li et al. 
1999). [A Voronoi region (or Thiessen polygon) for a point is 
the locus of points closer to that point than to any other given 
one]. 
Another observation arising from the analysis of existing 
literature is that only the ‘intersection’ operator out of the many 
Set operators are in use. This is perhaps the most expensive 
one in terms of computation. There is no reason why other 
operators cannot be used. Therefore, it is attempted to explore 
the full range of set operators to constitute a spatial algebra for 
the spatial relations. 
In summary, the basic strategies adopted here are: 
(1) 
(2) 
a spatial object is treated in a whole; 
Voronoi region of an object is employed to enhance 
its interconnection with neighbours; 
appropriate operators from set operators are 
utilised to distinguish the spatial relations between 
neighbouring spatial objects; and 
several types of values are used for the 
computational results of set operations, e.g. content, 
dimension and number of connected components 
and so on. 
3. SPATIAL ALGEBRA FOR SPATIAL RELATIONS: AN 
INTEGRATION 
In the previous section, the basic strategies employed in this 
study are outlined. From this section on, the development of 
spatial algebra using such strategies will be described. In this 
section, a number of set operators will be employed and spatial 
concepts are embedded into the algebra. 
3.1 Set operators for spatial relations 
Spatial objects are often regarded as sets in space in the 
context of GIS. This is very important as it means that objects 
can be manipulated by ordinary set operators: union, 
intersection, set difference and symmetrical difference and so 
on. At the same time, spatial relations can be considered as 
the result for handling these "sets". In fact, the theory of sets is 
the basis of the description and determination of spatial 
relations, especially, topological relations can be regarded as 
detailed relations between sets (Worboys, 1995). Figure 1 
illustrates working principles of set operators, using line and 
area objects as examples.