, Volume 4, Issue 4, pp 419–433 | Cite as

Basic Topological Models for Spatial Entities in 3-Dimensional Space

  • Zhilin Li
  • Yongli Li
  • Yong-qi Chen


In recent years, models of spatial relations, especially topological relations, have attracted much attention from the GIS community. In this paper, some basic topologic models for spatial entities in both vector and raster spaces are discussed.

It has been suggested that, in vector space, an open set in 1-D space may not be an open set any more in 2-D and 3-D spaces. Similarly, an open set in 2-D vector space may also not be an open set any more in 3-D vector spaces. As a result, fundamental topological concepts such as boundary and interior are not valid any more when a lower dimensional spatial entity is embedded in higher dimensional space. For example, in 2-D, a line has no interior and the line itself (not its two end-points) forms a boundary. Failure to recognize this fundamental topological property will lead to topological paradox. It has also been stated that the topological models for raster entities are different in Z2 and R2. There are different types of possible boundaries depending on the definition of adjacency or connectedness. If connectedness is not carefully defined, topological paradox may also occur. In raster space, the basic topological concept in vector space—connectedness—is implicitly inherited. This is why the topological properties of spatial entities can also be studied in raster space. Study of entities in raster (discrete) space could be a more efficient method than in vector space, as the expression of spatial entities in discrete space is more explicit than that in connected space.

topological models topological properties vector space raster space spatial entities 


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  1. 1.
    Max J. Egenhofer and Robert D. Franzosa. “Point-set topological spatial relations,” International Journal of Geographical Information Systems, Vol. 5(2):161-176, 1991.Google Scholar
  2. 2.
    Max J. Egenhofer and J. Sharma. “Topological relations between regions in ℝ2 and ℤ2,” in D. Abel and B.C. Ooi (Eds.), Advances in Spatial Databases, SSD'93, Lecture Notes in Computer Science 692, Springer-Verlag: 316-226, 1993.Google Scholar
  3. 3.
    Max J. Egenhofer, J. Sharma, and David M. Mark. “A critical comparison of the 4-intersection and 9-intersection models for spatial relations: formal analysis,” Auto-Carto, Vol. 11:1-11, 1993.Google Scholar
  4. 4.
    M. Henle. A Combinatorial Introduction to Topology. Dover Publications, Inc.: NewYork, 1979.Google Scholar
  5. 5.
    K. Jänich. Topology (translated by Silvio Levy). Springer-Verlag: 1984.Google Scholar
  6. 6.
    T.Y. Kong and A. Rosenfeld. “Digital topology: An introduction and a survey,” Computer Vision, Graphics and Image Processing, Vol. 48:357-393, 1989.Google Scholar
  7. 7.
    Y.C. Lee and Zhilin Li. “A taxonomy of 2D space tessellation,” International Archives for Photogrammetry and Remote Sensing, Vol. 32(4):344-346, 1998.Google Scholar
  8. 8.
    G. Maffini. “Raster versus vector data encoding and handling: A commentary,” Photogrammetric Engineering and Remote Sensing, Vol. 53(10):1397-1398, 1987.Google Scholar
  9. 9.
    D.J. Maguire, B. Kimber, and J. Chick. “Integrated GIS: The importance of raster,” Technical Papers, ACSM-ASPRS Annual Convention, Vol. 4:107-116, 1991.Google Scholar
  10. 10.
    D.J. Peuquet. “A conceptual framework and comparison of spatial data models,” Cartographica, Vol. 21:66-113. Reprinted in: D.J. Peuquet and D.F. Marble (Eds.), 1990. Introduction Readings in Geographic Information Systems. Taylor & Francis: 251-285, 1984.Google Scholar
  11. 11.
    A. Rosenfield and J.L. Pfalez. “Sequential operations in digital picture processing,” Journal of the Association of Computer and Machines, Vol. 13:471-497, 1966.Google Scholar

Copyright information

© Kluwer Academic Publishers 2000

Authors and Affiliations

  • Zhilin Li
    • 1
  • Yongli Li
    • 2
  • Yong-qi Chen
    • 1
  1. 1.Department of Land Surveying and Geo-InformaticsThe Hong Kong Polytechnic UniversityKowloonHong Kong, China
  2. 2.Department of Land Surveying and Geo-InformaticsThe Hong Kong Polytechnic UniversityKowloonHong Kong, China

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