ISPRS, Vol.34, Part 2W2, “Dynamic and Multi-Dimensional GIS”, Bangkok, May 23-25, 2001 
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From the viewpoint of algebra, these set operators together 
with spatial concepts (see next sub-section) form an algebra, 
called spatial algebra, for spatial relations in this study. Let O 
denote the set of all spatial objects, then the spatial relation, 
B(a, b), between object a and object b in set O can be 
represented by the following equation: 
B(a,b)= f (a Ob) = f(a'ub,anb,a\b,a/b,aAb...)(V 
where, 6>denotes the above set of set operators, i.e, Q-{u, n, 
\, /, A...}, representing union, intersection, difference, 
difference by, symmetric difference, etc. If the desired relation 
can be sufficiently described by one operator, then other 
operators may be omitted / is a function to take a type of 
values for the results of set operations (see Equation (2)). 
Equation (1) can be regarded as a simple spatial algebra for 
the description of spatial relations. Spatial relations can be 
distinguished in a 'coarse level' with this equation. For example, 
topological relations varying from disjoint relation to equal 
relation between two solid area objects can be determined by 
this equation, see Section 5. However, this equation is not able 
to describe some more detailed spatial relations. This is 
because spatial relations are not only dependent on objects 
themselves, but also dependent on their surrounding space, as 
stated the previous sections. 
Currently, only the intersection operator is widely used for the 
determination and description of spatial relations, mainly 
topological relations and order relations. In fact, some relations 
may be easily distinguished by other operators but not by the 
intersection operator. Figure 2 shows such an example which 
illustrates the superiority of 'difference' operator over 
intersection. It is clear that 'overlap' and 'contained by' can be 
easily distinguished by their 'difference' but not their 
intersection. 
Indeed, Galton (1998) has used 'intersection', 'union', 
'difference' and 'difference by' operators and the numbers of 
connected components (see Section 3.2) of different operators 
for distinguishing different spatial configurations of overlap 
relation. But the work described in this study is much beyond 
Galton's work. The algebra developed in this study is a more 
general model, which is capable of distinguishing all possible 
types of topological relations (instead of only overlap relation) 
between multiple types of objects (instead of only areas). 
Indeed, this algebra represents an integration of scattered work 
in spatial relations. 
Figure 2: Simple 'difference' operator is able to distinguish 
"overlap" and "contain", but not intersection 
3.2 Three types of values for the results of spatial algebra 
The value of {aOb) can take three different forms, i.e. 
content, dimension and the number of connected components. 
Figure 3 shows these values in the case of intersection 
operation. 'Content' is a quality measure, i.e. either 'empty' or 
'non-empty'. 'Dimension' is a quantitative measure, i.e. either 
0-dimensional (point), 1-dimensional (line) or 2-dimensional. 
For the case of'empty', a dimensional number of (-1) is usually 
used. 'Number of connected components' is quantitative 
measure in a finer level. In the case of'empty', the number is 0. 
Otherwise, the number could be any integer larger than 0. For 
example, two objects "a" and "b" have 2-dimensional overlap 
with 2 parts connected, as shown in Figure 3(d). 
Figure 3 Content, dimension and the number of connected 
components 
Mathematically, the value of each element in the( a <$) set, say 
e , could be denoted as follows: 
I {0, - 0} if/ is a function to take content, donated as f c 
{-1, 0, 1, 2, ...} if/is a function to take dimension, donated as f D 
{0, 1, 2, 3, ...}if/is a functionto takeconnectednumbe^donatedas/v 
(2) 
As a result, a spatial relation as shown in Figure 3(a) could be 
represented as B(a,b)=/ t (a6>6) = (-0, 0,-0,-0,-0). if 
B(a,b) takes content as the type of value for the spatial algebra. 
If the result of operators consists of multiple parts, then the 
highest dimension should be used for the value of f n {a0b) 
function. In addition, a combination of dimension and 
connected number values could also be used to form a value 
set, ( /d j ). For example, such a set for Figure 3(d) could be 
represented as B(a,b)= ((2,1), (2,2), (2,1), (2,1), (2,1)) 
As the content, dimension and number represent three 
different levels from coarse to fine, it is quite possible that 
content is enough to represent a particular spatial relation. In 
such a case, it is unnecessary to consider dimension or 
connected numbers. On the other hand, it is also possible that