Topological Spatial Relations and Operators 
Zhexue Huang 
Environmental and Natural Resources Information Systems 
Department of Photogrammetry 
Royal Institute of Technology 
100 44 Stockholm, Sweden 
huang@fmi.kth.se 
Abstract: 
There is a growing interest in investigating the underlying topological relations of spatial objects, and defining 
corresponding operators to check these relations. The investigation has recently resulted in a complete set of 
eight different spatial relationships between two regions. To check these eight spatial relations between two 
polygons, six fundamental topological spatial operators are required. More operators for checking specific spatial 
relations between two objects can be created by combining the fundamental operators with other operators. These 
operators provide a sound basis for designing a spatial query language and the query language with 
implementation of such spatial operators can become a useful tool for spatial querying and analysis. 
KEY WORDS: GIS, Spatial, Theory. 
1. Introduction 
A geographical information system (GIS) can be 
considered as a spatial database system on which a set 
of application programs operates. These application 
programs provide tools for the GIS user to make 
spatial queries and analyses about objects in the 
database. The object-oriented (OO) approach has 
proved to be a powerful tool for designing spatial 
databases, especially heterogeneous spatial databases 
(Oosterom et al 1989, Kemp 1990). 
An OO spatial database consists of a collection of 
spatial objects, which, together with their (spatial) 
relations, represent spatial information about reality 
(Molenaar 1991). An important feature of an OO 
database system is that each object has an identity 
which allows the user to distinguish and address it 
(Unland et al 1990). Besides its identity a spatial 
object also carries two kinds of data: spatial data, 
which describe the location and geometry of the 
object in space, and attribute data, which represent 
non-spatial properties of the object. 
Spatial relations of objects are an important part of 
the spatial information. The complexity of spatial 
relations among objects creates difficulties to 
explicitly represent all kinds of spatial relations in a 
database. An alternative is to define spatial functions 
in the database query language to discover spatial 
relations of objects. 
There is a growing interest in investigating the 
underlying topological relations of spatial objects 
(Egenhofer et al 1990, 1991, Kainz 1989, 1990), and 
defining corresponding operators to check these 
relations (Svensson et al 1991). The investigation of 
topological spatial relations of objects in a topological 
space has recently resulted in a complete set of eight 
different spatial relationships between two regions 
.(Egenhofer et al 1990, 1991). To check these eight 
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spatial relations between two polygons, six 
fundamental topological spatial operators are 
required. The terminology used for polygons can also 
be adopted for defining spatial operators for points 
and lines and combinations of these. These operators 
provide a sound basis for designing a spatial query 
language and the query language with 
implementation of such spatial operators can 
become a useful tool for spatial querying and analysis 
(Svensson et al 1991). 
This paper is organized as follows. Section 2 
discusses types of spatial objects. Spatial data types are 
proposed for handling different types of objects in 
databases. Functions operating on the spatial data 
types are given in section 3. Section 4 discusses 
topological spatial relations between two objects. 
How to use given spatial functions and operators to 
define new operators to detect detailed spatial 
relations is demonstrated in section 5. Section 6 gives 
some examples showing the use of these operators in 
querying a spatial database. Some conclusions are 
drawn in section 7. 
2. Object types and spatial data types 
Any object that is related to a location in space is said 
to be spatial. Some spatial objects are complex, for 
example, a road network. Others are simple, for 
example, oil wells which are usually presented as 
points in maps. The complex objects are composed of 
simple objects. 
Three basic types of spatial objects exist in the two 
dimensional space. They are points, lines, areas, 
which will be called polygons from now, and each of 
these three basic types has a primitive form. Any 
point is primitive. À line is primitive if it has no 
loop between its two ends, excluding the simply 
closed line whose two ends coincide. A polygon is