Topological Relationships

Topological relationships are the qualitative relationships of geographic objects that are not affected by continuous distortion, such as translation, rotation, scale, or skew. These relationships are said to exhibit a property known as topological invariant. In other words, topological relationships describe the spatial interrelations among geographic objects, and how they are linked together, without reference to their geographic locations, dimensions, or orientations. Some examples of topological relationships are (a) touch—a point touches a line (e.g., a bridge on a river); (b) cross—a line crosses another line (e.g., a road across a river); and (c) within—a region is completely contained by the boundaries of another region (e.g., an island in a lake). In addition to topological relationships, geographic objects can also have projective relationships, such as orientation (left or right), and metric relationships, such as distance, which can be measured quantitatively. These geometric relationships are fundamental predicates for geographic representation and reasoning.

The definition of topological relationships derives from how geographic objects are understood and represented, or, more broadly, from ontology. One way to define topological relationships is Egenhofer and Herring's nine-intersection scheme, which uses discrete points, lines, and regions to represent the geometries of geographic objects. The scheme uses the interiors, boundaries, and exteriors of two geometric objects to construct a 3 × 3 matrix, in which every entry denotes whether the intersection of the two object parts is an empty set or not, tested under the topologically invariant condition. Each distinctive set of nine intersections represents a different topological relationship. Out of the 512 (29) possible combinations of the 3×3 matrix, the nine-intersection scheme identifies, for example, 8 relationships between two regions with connected boundaries (i.e., no holes or disjointed parts) and 33 relationships between two simple lines (i.e., lines without branches). Some topological relationships are illustrated in Figure 1.

Figure 1 Examples of topological relationships

Source: Adapted from Egenhofer, M. J., & Herring, J. R. (1991). Categorizing binary topological relations between regions, lines, and points in geographic databases (Technical Report No. 91–7). Santa Barbara: University of California, National Center for Geographic Information and Analysis.

The nine-intersection scheme provides a formal definition of topological relationships, and as part of the Open Geospatial Consortium specifications and ISO/TC 211 standard, some of the relationships have been implemented in commercial geographic information systems and spatial database systems to formulate qualitative queries about the connection properties of spatial objects. Its reasoning framework facilitates finding new relationships from a set of geographic objects. As the geometric representations of geographic objects become more complex and additional object properties, such as nonplanar geometry, movements of objects, or vagueness of their boundaries, are considered, the task of defining the topological relationships becomes more difficult and sometimes intractable. Many topological relationships identified in the framework have no corresponding natural-language equivalents, making them not applicable in qualitative spatial reasoning that is based on the semantics of natural languages. Given these practical purposes and concerns, topological relationships have become a topic of interdisciplinary research involving the fields of geography, geographic information science, linguistics, artificial intelligence, and cognitive science.

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