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Topological relations between regions in ρ2 and ℤ2

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Advances in Spatial Databases (SSD 1993)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 692))

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Abstract

Users of geographic databases that integrate spatial data represented in vector and raster models, should not perceive the differences among the data models in which data are represented, nor should they be forced to apply different concepts depending on the model in which spatial data are represented. A crucial aspect of spatial query languages for such integrated systems is the need mechanisms to process queries about spatial relations in a consistent fashion. This paper compares topological relations between spatial objects represented in a continuous (vector) space of ρ2 and a discrete (raster) space of ℤ2. It applies the 9-intersection, a frequently used formalism for topological spatial relations between objects represented in a vector data model, to describe topological relations for bounded objects represented in a raster data model. We found that the set of all possible topological relations between regions in ρ2 is a subset of the topological relations that can be realized between two bounded, extended objects in ℤ2. At a theoretical level, the results contribute toward a better understanding of the differences in the topology of continuous and discrete space. The particular lesson learnt here is that topology in ρ2 is based on coincidence, whereas in ℤ2 it is based on coincidence and neighborhood. The relevant differences between the raster and the vector model are that an object's boundary in ℤ2 has an extent, while it has none in ρ2; and in the finite space of ℤ2 there are points between which one cannot insert another one, while in the infinite space of ρ2 between any two points there exists another one.

This work was partially supported by grants from Intergraph Corporation. Jayant Sharma was supported by a University Graduate Research Assistantship (UGRA) from the University of Maine. Additional support from NSF for the NCGIA under grant No. SES 88-10917 is gratefully acknowledged.

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Author information

Authors and Affiliations

  1. National Center for Geographic Information and Analysis, USA

    Max J. Egenhofer & Jayant Sharma

  2. Department of Surveying Engineering, University of Maine, 04469-5711, Orono, ME, USA

    Max J. Egenhofer & Jayant Sharma

  3. Department of Computer Science, University of Maine, 04469-5711, Orono, ME, USA

    Max J. Egenhofer & Jayant Sharma

Authors
  1. Max J. Egenhofer
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  2. Jayant Sharma
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David Abel Beng Chin Ooi

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© 1993 Springer-Verlag Berlin Heidelberg

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Egenhofer, M.J., Sharma, J. (1993). Topological relations between regions in ρ2 and ℤ2 . In: Abel, D., Chin Ooi, B. (eds) Advances in Spatial Databases. SSD 1993. Lecture Notes in Computer Science, vol 692. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-56869-7_18

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  • DOI: https://doi.org/10.1007/3-540-56869-7_18

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  • Print ISBN: 978-3-540-56869-8

  • Online ISBN: 978-3-540-47765-5

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