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Encyclopedia of Database Systems pp 3500–3506Cite as

Simplicial Complex

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Synonyms

Cell complex; CW complex; Polyhedron

Definition

A simplicial complex is a topological space constructed by gluing together dimensional simplices (points, line segments, triangles, tetrahedrons, etc.).

A simplicial complex K is a set of simplices k, which satisfies the two conditions:

  1. 1.

    Any face of a simplex in K is also in K

  2. 2.

    The intersection of any two simplices in K is a face of both simplices (or empty)

Historical Background

Raster (field) or vector (object) are the two dominant conceptualizations of space. Applications focusing on object with 2 or 3 dimensional geometry structure the storage of geometry as points, lines, surfaces, and volumes and the relations between them; a classical survey paper discussed the possible approaches mostly from the perspective of Computer Aided Design (CAD) where individual physical objects are constructed [1].

The representation of geographic information, e.g., maps, introduces consistency constraints between the objects; consider the...

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Recommended Reading

  1. Requicha A. Representation for rigid solids: theory, methods and systems. ACM Comp Surv. 1980;12(4):437–64.

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  2. Corbett JP. Topological principles in cartography. Washington, DC: Bureau of the Census, US Department of Commerce; 1979.

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  3. Härder T. New approaches to object processing in engineering databases. In: Proceedings of the 1986 International Workshop on Object-Oriented Database Systems; 1986.

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  4. Frank AU, Kuhn W. Cell graph: a provable correct method for the storage of geometry. In: Proceedings of the 2nd International Symposium on Spatial Data Handling; 1986.

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  5. Alexandrov PS. Combinatorial topology volumes 1, 2 and 3. Mineola/New York: Dover Publications, Inc.; 1960.

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  6. Egenhofer M, Herring JR. A mathematical framework for the definition of topological relationships. In: Proceedings of the 4th International Symposium on Spatial Data Handling; 1990.

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  7. Egenhofer MJ, Franzosa RD. On the equivalence of topological relations. Int J Geogr Inf Syst. 1995;9(2):133–52.

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  8. Guibas LJ, Stolfi J. A language for bitmap manipulation. ACM Trans Grap. 1982;1(3):191–214.

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  9. Lienhardt P. Extensions of the notion of map and subdivision of a three-dimensional space. In: Proceedings of the 5th Annual Symposium on Theoretical Aspects of Computer Science; 1988.

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  10. Thompson RJ. Towards a rigorous logic for spatial data representation. Doctoral thesis, Delft, NCG; 2007.

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  11. Sellis T, et al., editors. Spatiotemporal databases: the chorochronos approach. LNCS, vol. 2520. Berlin: Springer; 2003.

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  12. Timpf S. Hierarchical structures in map series. Technical University Vienna, Vienna, Ph.D thesis;1998.

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  13. Levin B. Objecthood: an event structure perspective. In: Proceedings Chicago Linguistic Society, vol. 35; 1999. p. 223–247.

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Author information

Authors and Affiliations

  1. Vienna University of Technology, Vienna, Austria

    Andrew U. Frank

Authors
  1. Andrew U. Frank
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Corresponding author

Correspondence to Andrew U. Frank .

Editor information

Editors and Affiliations

  1. Georgia Institute of Technology College of Computing, Atlanta, GA, USA

    Ling Liu

  2. University of Waterloo School of Computer Science, Waterloo, ON, Canada

    M. Tamer Özsu

Section Editor information

  1. Computer Science, University of Hagen, Hagen, Germany

    Ralf Hartmut Güting

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Frank, A.U. (2018). Simplicial Complex. In: Liu, L., Özsu, M.T. (eds) Encyclopedia of Database Systems. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-8265-9_345

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