computes the perimeter of polygon pg and the 
function 
CLOSED(I) = 
(XCOORD(SP())) - XCOORD(EP(I))) 
AND 
(YCOORD(SP())) - YCOORD(EP())) 
returns the logical value TRUE if and only if line [ is 
a closed line. 
4. Topological spatial relations 
Topological relations are such spatial relations that 
are invariant under topological transformations 
between two topologically equivalent spaces 
(Armstrong 1979). Adjacency, overlapping, and 
containment are typical examples. It is known that 
totally eight different topological relations exist 
between two regions (Egenhofer et al 1990). 
By disregarding orders of containment and coverage 
and taking point and line objects into account, six 
topological relations between two spatial objects can 
be named (Svensson et al 1991). Table 1 presents the 
topological relations with different combinations of 
object types. 
In some combinations of object types, one 
topological relation may be implied by different 
names. For example, MEETS, EQUALS, COVERS, 
OVERLAPS indicate the same topological relation of 
two points. In order to avoid ambiguity, only 
EQUALS is defined. 
Table 1. Topological relations between objects of different types 
  
combination of object types 
  
  
topological 
relations PP PL PPg LL LPg PgPg 
DISOINT Y Y Y Y Y Y 
MEETS - - - Y Y Y 
EQUALS Y - - Y - Y 
CONTAINS - - Y - Y Y 
COVERS - Y Y Y. Y Y 
OVERLAPS - - - Y Y Y 
  
Y/- means the topological relationship exists /undefined. 
Visualization of topological relationships between 
two polygons is given in (Egenhofer et al 1990). 
5. Topological operators 
Spatial relations vary in the same topological 
relations. For example, when two polygons meet, 
they may meet at boundaries (meets-1), or they may 
meet at a corner (meets-0). It is necessary to specify 
the cases because the merging of two primitive 
polygons which meet at boundaries produces a 
primitive polygon, whereas the merging of two 
primitive polygons which meet at a corner results in 
a non-primitive polygon. 
275 
Spatial operators are needed to detect topological 
spatial relations of two objects. In order to define 
spatial operators, specific cases of spatial relations 
should be investigated. Some operators, called 
fundamental operators, must be defined and 
implemented at the system level. Others can be 
expressed by using fundamental operators and other 
given functions. 
Below, we use some examples to argue the necessary 
fundamental operators needed for detecting some 
general and specified topological relations, and to 
show how to create new operators by combinations 
of those fundamental operators and other functions 
and operators. 
Since the disjoint relation exists in any combination 
of spatial object types (see Table 1), the operator 
DISJOINT is defined as a fundamental operator. 
Example 1: Operators for checking topological 
relations between two points p and q. 
There are only two kinds of topological relations 
between two points. They are either disjoint or not 
disjoint. We define the operator EQUALS to describe 
the not-disjoint relation. 
p EQUALS gq iff NOT(p DISJOINT gq) 
EQUALS is a non-fundamental operator defined by 
the fundamental operator DISJOINT. EQUALS can 
also be defined as 
p EQUALS 3 
iff (XCOORD(p) = XCOORD(4)) AND 
(YCOORD(p) = YCOORD(4)) 
This example shows that non-fundamental 
operators can sometimes be defined in several ways. 
Example 2: Function for checking closedness of a 
line I. 
CLOSED() iff SP() EQUALS EP(I) 
This example shows that spatial operators can be 
used to define spatial functions. 
Example 3: Operators for checking spatial relations 
between a point p and a line /. 
Two general topological relations are disjoint and 
not disjoint. We describe the not-disjoint relation by 
the operator COVERS defined as 
I COVERS p iff NOT(p DISJOINT I) 
Under the COVERS relation, two cases must be 
distinguished. The point p is either located at the 
end points, or on the other place of the line I. We 
define the operator P-AT-ENDS to describe the first 
case. 
p P-AT-ENDS | iff (I COVERS p ) AND 
((p EQUALS SP(l)) OR