(p EQUALS EP()))) 
Example 4: Operators for checking spatial relations 
between a point p and a polygon pg. pg is a 
primitive polygon without holes. 
There are three kinds of spatial relations. The point 
P is either disjoint from the polygon 78 , or on the 
boundary of, or inside of the polygon P$. The first 
kind is checked by the operator DISJOINT. We 
define the operator COVERS to check the second 
kind. 
pg COVERS p 
iff BOUNDARY(pg) COVERS p 
The third kind can be checked by the operator 
CONTAINS defined as 
pg CONTAINS p 
iff NOT (pg DISJOINT p ) AND 
NOT(pg COVERS p) 
Example 5: Operators for checking spatial relations 
between two lines li and Ik. 
When two lines are not disjoint, they can be 
topologically connected in different ways. Figure 2 
shows some simple, but fundamental examples. In 
group 1, two lines are connected at a point. We 
define the operator MEETS as a fundamental 
operator to detect these relations. To distinguish the 
relation 1a, the operator INTERSECTS is defined as 
li INTERSECTS Ik 
iff (li MEETS Ik) AND 
(li DISJOINT SP(Kk)) AND 
(li DISJOINT EP(Ik)) AND 
(Ik DISJOINT SP(li)) AND 
(Ik DISJOINT EP(Ii)) 
PR os cul NN 
  
1a 1b 1C 
elt uet 
» e LIA 
3 4 
Figure 2. Topological relations between two lines. 
The relation 1c can be checked by the operator 
MEETS-AT-ENDS defined as 
li MEETS-AT-ENDS /k 
iff (li MEETS Ik ) AND 
((SP(li) EQUALS SP(Ik)) AND 
NOT (SP(li) EQUALS EP(/))) OR 
((SP(li) EQUALS EP(lk)) AND 
NOT (SP(li) EQUALS SP(Ik))) OR 
276 
((EP(li) EQUALS SP(lk)) AND 
NOT (EP(li) EQUALS EP(/k))) OR 
((EP(lj) EQUALS EP(I)) AND 
NOT (EP(li) EQUALS SP(I))) 
We consider the situations in group 2 as the 
overlapping relation in which two lines partly 
intersect. The operator OVERLAPS is defined as a 
fundamental operator to detect the overlapping 
relation. Specific cases of the overlapping relation 
can be distinguished by combining the operator 
OVERLAPS with other predefined operators. The 
relation 2a is detected by the operator 
li OVERLAPS-AT-ENDS Ik 
iff (li OVERLAPS Ik ) AND 
((li COVERS SP(lk)) OR 
(li COVERS EP(I))) AND 
(Ik COVERS SP(li)) OR 
(Ik COVERS EP(li))) 
The relation 2b is detected by the operator 
li OVERLAPS-IN-MIDDLE Ik 
iff (li OVERLAPS Ik ) AND 
(li DISJOINT SP(lk)) AND 
(li DISJOINT EP(Ik)) AND 
(Ik DISJOINT SP(li)) AND 
(Ik DISJOINT EP(li)) 
We call the situations in group 3 and 4 the covering 
relation. The operator COVERS is a fundamental 
operator to check the covering relation. 
The group 4 is a special case of covering relation in 
which two lines are equal. To detect this special 
relation, the operator EQUALS is defined as 
li EQUALS Ik 
iff (li COVERS Ik ) AND 
(SP(li) EQUALS SP(IX)) AND 
(EP(li) EQUALS EP(lk))) OR 
(EP(li) EQUALS SP(lk)) AND 
(SP(li) EQUALS EP(lk))) 
Example 6: Operators for checking spatial relations 
between a line / and a polygon 7$. 
Spatial relations under the not-disjoint condition 
can be classified into four groups (Figure 3). We 
define operators MEETS, CONTAINS, COVERS and 
OVERLAPS as fundamental operators to detect the 
spatial relations of the respective groups. 
The relations in group 1 can be distinguished by the 
following operators. 
! MEETS-AT-END P$ 
iff (I MEETS pg) AND 
(I MEETS BOUNDARY(/3)) 
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