Geometriae Dedicata

, Volume 20, Issue 2, pp 209–243 | Cite as

Generalized Dirichlet tessellations

  • Peter F. Ash
  • Ethan D. Bolker
Article

Abstract

In this paper we study how to recognize when a dissection of the plane has been constructed in one of several natural ways each of which models some phenomena in the natural or social sciences. The prototypical case is the nearest-neighbor or Dirichlet tessellation.

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References

  1. 1.
    Ash, P. and Bolker, E., ‘Recognizing Dirichlet Tessellations’, Geom. Dedicata 19 (1985) 175–206.MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Aurenhammer, F., ‘The One-Dimensional Weighted Voronoi Diagram’, Technical Report F110, Institut fur Informationsverarbeitung, Technische Institut Graz und Osterreichische Computergesellschaft, Graz, Austria, January, 1983.MATHGoogle Scholar
  3. 3.
    Aurenhammer, F., ‘Power Diagrams: Properties, Algorithms and Applications’, Technical Report F120, Institut fur Informationsverarbeitung, Technische Institut Graz und Osterreichische Computergesellschaft, Graz, Austria, June 1983.MATHGoogle Scholar
  4. 4.
    Aurenhammer, F., ‘On the Generality of Power Diagrams’, Technical Report F126, Institut fur Informationsverarbeitung, Technische Institut Graz and Osterreichische Computergesellschaft, Graz, Austria, December, 1983.Google Scholar
  5. 5.
    Bowyer, A., ‘Computing Dirichlet Tessellations’, The Computer Journal 24 (1981), 162–166.MathSciNetCrossRefGoogle Scholar
  6. 6.
    Conway, J. H. and Sloane, N. J. A., ‘Voronoi Regions of Lattices, Second Moments of Polytopes, and Quantization’, IEEE Trans. on Information Theory IT-28 (1982), 211–226.MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Coxeter, H. S. M., Introduction to Geometry, Wiley, 1961.Google Scholar
  8. 8.
    Crapo, H., ‘Structural Rigidity’, Structural Topology 1 (1979), 13–45.MathSciNetMATHGoogle Scholar
  9. 9.
    Ehrlich, P. E. and Im Hof, H. C., ‘Dirichlet Regions in Manifolds without Conjugate Points’, Comment. Math. Helv. 54 (1979), 642–658.MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Guillemin, V. and Pollack, A., Differential Topology, Prentice Hall, Englewood Cliffs, N. J., 1974.MATHGoogle Scholar
  11. 11.
    Hyson, C. D. and Hyson, W. P., ‘The Economic Law of Market Areas’, in Spatial Economic Theory (eds R. D. Dean et. al.), The Free Press, New York, 1970, pp. 165–170.Google Scholar
  12. 12.
    Imai, H., Iri, M., and Murota, K., ‘Voronoi Diagrams in the Laguerre Geometry and its Applications’, Research Memorandum RMI 83-02, Department of Mathematical Engineering and Instrumentation Physics, University of Tokyo, March, 1983 (to appear in SIAM Journal on Computing 14 (1985)).Google Scholar
  13. 13.
    Maxwell, J. C., ‘On Reciprocal Figures and Diagrams of Forces’, Phil. Mag., Series 4, 27 (1864), 250–261.Google Scholar
  14. 14.
    Maxwell, J. C., ‘On Reciprocal Figures, Frames, and Diagrams of Forces’, Trans. Roy. Soc. Edinburgh 26 (1869–72), 1–40.MATHCrossRefGoogle Scholar
  15. 15.
    Sibson, R., ‘A Vector Identity for the Dirichlet Tessellation’, Math. Proc. Camb. Phil. Soc. 87 (1980), 151–155.MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    Whiteley, W., ‘Realizability of Polyhedra’, Structural Topology 1 (1979), 46–58.MathSciNetMATHGoogle Scholar

Copyright information

© D. Reidel Publishing Company 1986

Authors and Affiliations

  • Peter F. Ash
    • 1
  • Ethan D. Bolker
    • 2
  1. 1.Department of Mathematics and Computer ScienceSt. Joseph's UniversityPhiladelphiaUSA
  2. 2.Department of Mathematics and Computer ScienceUniversity of Massachusetts/BostonBostonUSA

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