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The 3rd ISPRS Workshop on Dynamic and Multi-Dimensional GIS & the 10th Annual Conference of CPGIS on Geoinformatics
Chen, Jun

ISPRS, Vol.34, Part 2W2, “Dynamic and Multi-Dimensional GIS”, Bangkok, May 23-25, 2001
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
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
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
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