Skip to main content
SpringerLink
Log in

Convexity in Discrete Space

  • Conference paper
Spatial Information Theory. Foundations of Geographic Information Science (COSIT 2003)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 2825))

Included in the following conference series:

Abstract

This paper looks at Coppel’s axioms for convexity, and shows how they can be applied to discrete spaces. Two structures for a discrete geometry are considered: oriented matroids, and cell complexes. Oriented matroids are shown to have a structure which naturally satisfies the axioms for being a convex geometry. Cell complexes are shown to give rise to various different notions of convexity, one of which satisfies the convexity axioms, but the others also provide valid notions of convexity in particular contexts. Finally, algorithms are investigated to validate the sets of a matroid, and to compute the convex hull of a subset of an oriented matroid.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Björner, A., Las Vergnas, M., Sturmfels, B., White, N., Ziegler, G.: Oriented Matroids. Encyclopedia of Mathematics and its Applications, vol. 46. CUP (1999)

    Google Scholar 

  2. Boissonnat, J.-D., Yvinec, M.: Algorithmic Geometry. Cambridge University Press, Cambridge (1998)

    MATH  Google Scholar 

  3. Cohn, A.G., Bennett, B., et al.: Qualitative spatial representation and reasoning with the region connection calculus. GeoInformatica 1, 275–316 (1997)

    Article  Google Scholar 

  4. Coppel, W.A.: Foundations of Convex Geometry. Cambridge University Press, Cambridge (1998)

    MATH  Google Scholar 

  5. de Berg, M., van Krevald, M., Overmars, M., Schwarzkopf, O.: Computational Geometry Algorithms and Applications. Springer, Heidelberg (1999)

    Google Scholar 

  6. Eckhardt, U.: Digital lines and digital convexity. In: Bertrand, G., Imiya, A., Klette, R. (eds.) Digital and Image Geometry. LNCS, vol. 2243, pp. 209–228. Springer, Heidelberg (2002)

    Chapter  Google Scholar 

  7. Galton, A.: The mereotopology of discrete space. In: Freksa, C., Mark, D.M. (eds.) COSIT 1999. LNCS, vol. 1661, pp. 251–266. Springer, Heidelberg (1999)

    Chapter  Google Scholar 

  8. Knuth, D.E.: Axioms and Hulls. LNCS, vol. 606. Springer, Heidelberg (1992)

    MATH  Google Scholar 

  9. Kovalevsky, V.A.: Finite topology as applied to image analysis. Computer Vision, Graphics and Image Processing 46, 141–161 (1989)

    Article  Google Scholar 

  10. Kovalevsky, V.A.: Finite topology and image analysis. Advances in Electronics and Electron Physics 84, 197–259 (1992)

    Google Scholar 

  11. Masolo, C., Vieu, L.: Atomicity vs Infinite divisibility of space. In: Freksa, C., Mark, D.M. (eds.) COSIT 1999. LNCS, vol. 1661, pp. 235–250. Springer, Heidelberg (1999)

    Chapter  Google Scholar 

  12. Oxley, J.G.: Matroid Theory. Oxford Graduate Texts in Mathematics. Oxford University Press, Oxford (1992)

    MATH  Google Scholar 

  13. Rosenfeld, A.: Digital topology. American Mathematical Monthly 86(8), 621–630 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  14. Roy, A.J.O., Stell, J.G.: A qualitative account of discrete space. In: Egenhofer, M.J., Mark, D.M. (eds.) GIScience 2002. LNCS, vol. 2478, pp. 276–290. Springer, Heidelberg (2002)

    Chapter  Google Scholar 

  15. Stell, J.G.: The representation of discrete multi-resolution spatial knowledge. In: Cohn, A.G., Giunchiglia, F., Selman, B. (eds.) Principles of Knowledge Representation and Reasoning: Proceedings of KR 2000, pp. 38–49. Morgan Kaufmann, San Francisco (2000)

    Google Scholar 

  16. von Randow, R.: Introduction to the Theory of Matroids. Lecture Notes in Economics and Mathematical Systems. Springer, Heidelberg (1975)

    MATH  Google Scholar 

  17. Roger Webster. Convexity. OUP (1994)

    Google Scholar 

  18. Webster, J.: Cell complexes and digital convexity. In: Bertrand, G., Imiya, A., Klette, R. (eds.) Digital and Image Geometry. LNCS, vol. 2243, pp. 272–284. Springer, Heidelberg (2002)

    Chapter  Google Scholar 

  19. Webster, J.: Cell complexes, oriented matroids and digital geometry. Theoretical Computer Science (to appear)

    Google Scholar 

  20. Welsh, D.J.A.: Matroid Theory. Academic Press, London (1976)

    MATH  Google Scholar 

  21. Winter, S., Frank, A.U.: Topology in raster and vector representation. Geoinformatica 4(1), 35–65 (2000)

    Article  MATH  Google Scholar 

  22. Winter, S.: Topological relations between discrete regions. LNCS, vol. 951, pp. 310–327 (1995)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Computing, University of Leeds, Leeds, LS2 9JT, UK

    Anthony J. Roy & John G. Stell

Authors
  1. Anthony J. Roy
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. John G. Stell
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Austrian Research Centers Department of Information Technology and Telematics,  

    Walter Kuhn

  2. Department of Spatial Information Science and Engineering, University of Maine, 04473, Orono, ME, USA

    Michael F. Worboys

  3. Department of Geography, University of Zürich,  

    Sabine Timpf

Rights and permissions

Reprints and permissions

Copyright information

© 2003 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Roy, A.J., Stell, J.G. (2003). Convexity in Discrete Space. In: Kuhn, W., Worboys, M.F., Timpf, S. (eds) Spatial Information Theory. Foundations of Geographic Information Science. COSIT 2003. Lecture Notes in Computer Science, vol 2825. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-39923-0_17

Download citation

  • DOI: https://doi.org/10.1007/978-3-540-39923-0_17

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-20148-9

  • Online ISBN: 978-3-540-39923-0

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation