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Simplicial Complex

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Encyclopedia of Database Systems

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Synonyms

CW complex; Polyhedron; Cell complex

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 [10].

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

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Recommended Reading

  1. Alexandrov P.S. Combinatorial Topology Volumes 1, 2 and 3. Dover Publications, Inc., Mineola, New York, 1960.

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  2. Corbett J.P. Topological Principles in Cartography, Bureau of the Census, US Department of Commerce, 1979.

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  3. Egenhofer M. and Herring J.R. A mathematical framework for the definition of topological relationships. In Proc. Fourth Int. Symp. on Spatial Data Handling, 1990.

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  4. Egenhofer M.J. and Franzosa R.D. On the equivalence of topological relations. Int. J. Geogr. Inf. Syst., 9(2):133–152, 1995.

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  5. Frank A.U. and Kuhn W. Cell graph: a provable correct method for the storage of geometry. In Proc. Second Int. Symp. on Spatial Data Handling, 1986.

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  6. Guibas L.J. and Stolfi J. A language for bitmap manipulation. ACM Trans. Grap., 1(3):191–214, 1982.

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  7. Härder T. New approaches to object processing in engineering databases. In Proc. 1986 Int. Workshop on Object-Oriented Database Systems, 1986.

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  8. Levin B. Objecthood: an event structure perspective. In Proc. Chicago Linguistic Society 35, 1999, pp. 223–247.

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  9. Lienhardt P. Extensions of the notion of map and subdivision of a three-dimensional space. In Proc. 5th Annual Symp. on Theoretical Aspects of Computer Science, 1988.vol. 294,

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  10. Requicha A. Representation for rigid solids: theory, methods and Systems. ACM Comp. Surv., 12(4):437–464, 1980.

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  11. T, et al. (eds.) Sellis (2003). Spatiotemporal Databases: The Chorochronos Approach. LNCS, Springer, Berlin, .vol. 2520.

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  12. Thompson R.J. Towards a Rigorous Logic for Spatial Data Representation. Doctoral thesis, Delft, NCG, 2007.

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  13. Timpf S. 1998.Hierarchical Structures in Map Series. Technical University Vienna, Vienna, Ph.D thesis,

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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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Editor information

Editors and Affiliations

  1. College of Computing, Georgia Institute of Technology, 266 Ferst Drive, 30332-0765, Atlanta, GA, USA

    LING LIU (Professor) (Professor)

  2. Database Research Group David R. Cheriton School of Computer Science, University of Waterloo, 200 University Avenue West, N2L 3G1, Waterloo, ON, Canada

    M. TAMER ÖZSU (Professor and Director, University Research Chair) (Professor and Director, University Research Chair)

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Frank, A.U. (2009). Simplicial Complex. In: LIU, L., ÖZSU, M.T. (eds) Encyclopedia of Database Systems. Springer, Boston, MA. https://doi.org/10.1007/978-0-387-39940-9_345

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