Abstract
The exploration of spatial relationships is a multi-disciplinary effort involving researchers from linguistics, cognitive science, psychology, geography, cartography, semiology, computer science, surveying engineering, and mathematics. Terms like close and far or North and South are not as clearly understood as the standard relationships between integer numbers. The treatment of relationships among spatial objects is an essential task in geographic data processing and CAD/CAM. Spatial query languages, for example, must offer terms for spatial relationships; spatial database management systems need algorithms to determine relationships. Hence, a formal definition of spatial relationships is necessary to clarify the users' diverse understanding of spatial relationships and to actually deduce relationships among spatial objects. Based upon such formalisms, spatial reasoning and inference will be possible.
The topological relationships are a specific subset of the large variety of spatial relationships. They are characterized by the property to be preserved under topological transformations, such as translation, rotation, and scaling. A model of topological relations is presented which is based upon fundamental concepts of algebraic topology in combination with set theory. Binary topological relationships may be defined in terms of the boundaries and interiors of the two objects to be compared. A formalism is developed which identifies 16 potential relationships. Prototypes are shown for the eight relationships that may exist between two objects of the same dimension embedded in the corresponding space.
This research was partially funded by grants from NSF under No. IST 86-09123 (Principal Investigator: Andrew U. Frank) and Digital Equipment Corporation. The support from NSF for the NCGIA under No. SES 88-10917 is gratefully acknowledged.
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
R. Abler. The National Science Foundation National Center for Geographic Information and Analysis. International Journal of Geographical Information Systems, 1(4), 1987.
E. Carlson. Three Dimensional Conceptual Modeling of Subsurface Structures. In: ASPRS-ACSM Annual Convention, Baltimore, MD, 1987.
J.P. Corbett. Topological Principles of Cartography. Technical Report 48, Bureau of the Census, Department of Commerce, 1979.
M. Egenhofer. Appropriate Conceptual Database Schema Designs For Two-Dimensional Spatial Structures. In: ASPRS-ACSM Annual Convention, Baltimore, MD, 1987.
M. Egenhofer. Relations Between Intervals In A One-Dimensional Space. 1987. Internal Documentation, University of Maine, Orono, Department of Surveying Engineering, Orono, ME.
M. Egenhofer and A. Frank. Towards a Spatial Query Language: User Interface Considerations. In: D. DeWitt and F. Bancilhon, editors, 14th International Conference on Very Large Data Bases, Los Angeles, CA, August 1988.
M. Egenhofer et al. Computational Topology: Data Structures and Algorithms. Technical Report, Department of Surveying Engineering, University of Maine, Orono, ME, January 1989. submitted for publication.
A. Frank and W. Kuhn. Cell Graph: A Provable Correct Method for the Storage of Geometry. In: D. Marble, editor, Second International Symposium on Spatial Data Handling, Seattle, WA, 1986.
W.R. Franklin. Cartographic Errors Symptomatic of Underlying Algebra Problems. In: International Symposium on Spatial Data Handling, Zurich, Switzerland, August 1984.
P.J. Giblin. Graphs, Surfaces, and Homology. Halsted Press, John Wiley and Sons, New York, NY, 1977.
R. Güting. Geo-Relational Algebra: A Model and Query Language for Geometric Database Systems. In: J.W. Schmidt et al., editors, Advances in Database Technology—EDBT '88, International Conference on Extending Database Technology, Venice, Italy, Springer Verlag, New York, NY, 1988.
J. Herring. TIGRIS: Topologically Integrated Geographic Information Systems. In: N.R. Chrisman, editor, Eighth International Symposium on Computer-Assisted Cartography, Baltimore, MD, March 1987.
J. Jackson. Algorithms for Triangular Irregular Networks Based on Simplicial Complex Theory. In: ASPRS-ACSM Annual Convention, Baltimore, MD, March 1989.
D. Peuquet. The Use of Spatial Relationships to Aid Spatial Database Retrieval. In: D. Marble, editor, Second International Symposium on Spatial Data Handling, Seattle, WA, 1986.
D. Pullar and M. Egenhofer. Towards Formal Definitions of Topological Relations Among Spatial Objects. In: D. Marble, editor, Third International Symposium on Spatial Data Handling, Syndney, Australia, August 1988.
V.B. Robinson and R.N. Wong. Acquiring Approximate Representations of Some Spatial Relations. In: N.R. Chrisman, editor, Eighth International Symposium on Computer-Assisted Cartography, Baltimore, MD, March 1987.
H. Schubert. Topology. Allyn and Bacon, Inc., Boston, MA, 1968.
L. Talmy. How Language Structures Space. In: H. Pick and L. Acredolo, editors, Spatial Orientation: Theory, Research, and Application, Plenum Press, New York, NY, 1983.
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1989 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Egenhofer, M.J. (1989). A formal definition of binary topological relationships. In: Litwin, W., Schek, HJ. (eds) Foundations of Data Organization and Algorithms. FODO 1989. Lecture Notes in Computer Science, vol 367. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-51295-0_148
Download citation
DOI: https://doi.org/10.1007/3-540-51295-0_148
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-51295-0
Online ISBN: 978-3-540-46186-9
eBook Packages: Springer Book Archive