Geometriae Dedicata

, Volume 19, Issue 2, pp 175–206 | Cite as

Recognizing Dirichlet tessellations

  • Peter F. Ash
  • Ethan D. Bolker


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Asano, T., Edahiro, M., Imai, H., Iri, M., and Murota, K., ‘Practical use of Bucketing Techniques in Computational Geometry’ (to appear in Computational Geometry, North-Holland).Google Scholar
  2. 2.
    Besag, J., ‘Spatial Interaction and the Statistical Analysis of Lattice Systems’, J. Royal Stat. Soc. Series B 36 (1974), 192–236.Google Scholar
  3. 3.
    Brostow, W., Dussault, J. P., and Fox, B. L., ‘Construction of Voronoi Polyhedra’, J. Comp. Phys. 30 (1978), 81–92.Google Scholar
  4. 4.
    Brown, K. Q., ’Voronoi Diagrams from Convex Hulls’, Inf. Proc. Lett. 7 (1979), 223–228.Google Scholar
  5. 5.
    Crapo, H., ‘Structural Rigidity’, Structural Topology (1979), 13–45.Google Scholar
  6. 6.
    Cremona, L., Graphical Statics (translation of Le figure reciproche nelle statica graphica, 1872), Oxford University Press, London, 1890.Google Scholar
  7. 7.
    Coxeter, H. S. M., Introduction to Geometry, Wiley, 1961.Google Scholar
  8. 8.
    Descartes, R., Les Principes de la Philosophie, Elzevier, Amsterdam, 1644.Google Scholar
  9. 9.
    Dirichlet, P. G. L., ‘Über die Reduction der positiven quadratische formen mit drei unbestimmten ganzen Zahlen’, J. riene angew. Math. 40 (1850), 209–227.Google Scholar
  10. 10.
    Drysdale, R. L., ‘Generalized Voronoi Diagrams and Geometric Searching’, Ph.D. thesis, Stanford University, Computer Science Department, January, 1979.Google Scholar
  11. 11.
    Gilbert, E. N., ‘Random Subdivisions of Space into Crystals’, Ann. Math. Stat. 33 (1962), 958–972.Google Scholar
  12. 12.
    Green, P. J. and Sibson, R., ‘Computing Dirichlet Tessellations in the Plane’, Comput. J. 21 (1978), 168–173.Google Scholar
  13. 13.
    Grunbaum, B., Convex Polytopes, Wiley/Interscience, New York, 1967.Google Scholar
  14. 14.
    Grunbaum, B. and Shephard, G. C., ‘Tilings with Congruent Tiles’, Bull. (New Series) Amer. Math. Soc. 3 (1980), 951–973.Google Scholar
  15. 15.
    Honda, H., ‘Description of Cellular Patterns by Dirichlet Domains: The Two-Dimensional Case’, J. Theor. Biol. 107 (1978), 523–543.Google Scholar
  16. 16.
    Horspool, R. N., ‘Constructing the Voronoi Diagram in the Plane, Technical Report SOCS 79.12, School of Computer Science, McGill University, 1979.Google Scholar
  17. 17.
    Iri, M., Murota, K., and Ohya, T., ‘A Fast Voronoi-Diagram Algorithm with Applications to Geographical Optimization’, Lecture Notes in Control and Information Sciences 59, Springer, 1984.Google Scholar
  18. 18.
    Klee, V., ‘On the Complexity of d-Dimensional Voronoi Diagrams’, Technical Report 64, Department of Mathematics, University of Washington, 1979.Google Scholar
  19. 19.
    Linhart, J., ‘Dirichletsche Zellenkomplexe mit maximaler Eckenzahl’, Geom. Dedicata 11 (1981), 363–367.Google Scholar
  20. 20.
    Loeb, A. L., Space Structures, their Harmony and Counterpoint, Addison Wesley, Reading, Mass., 1976.Google Scholar
  21. 21.
    Maxwell, J. C., ‘On Reciprocal Figures and Diagrams of Forces’, Phil. Mag. Series4 (1864), 250–261.Google Scholar
  22. 22.
    Maxwell, J. C., ‘On Reciprocal Figures, Frames, and Diagrams of Forces’, Trans. Royal Soc. Edinburgh 26 (1869–72), 1–40.Google Scholar
  23. 23.
    Miles, R. E., ‘The Random Division of Space’, Suppl. Adv. Appl. Prob. (1972), 243–266.Google Scholar
  24. 24.
    Mollison, D., ‘Spatial Contact Models for Ecological and Epidemic Spread’, J. Royal Stat. Soc., Series B 39 (1977), 283–326.Google Scholar
  25. 25.
    Nowacki, W., ‘Über allegemeine Eigenschaften von Wirkungsbereichen’, Z. Kristal. (1976), 360–368.Google Scholar
  26. 26.
    Ohya, T., Iri, M., and Murota, K., ‘A Fast Voronoi-Diagram Algorithm with Quaternary Tree Bucketing’, Inf. Proc. Lett. 18 (1984), 227–231.Google Scholar
  27. 27.
    Ohya, T., Iri, M., and Murota, K., ‘Improvements of the Incremental Method for the Voronoi Diagram with Computational Comparison of Various Algorithms’, J. Operations Res. Soc. Japan 27 (1984), 306–336.Google Scholar
  28. 28.
    Rogers, C. A., Packing and Covering, Cambridge Mathematical Tract 54, Cambridge University Press, 1964.Google Scholar
  29. 29.
    Schoenberg, I. J., Mathematical Time Exposures, Mathematical Association of America, 1982.Google Scholar
  30. 30.
    Sibson, R., ‘A Vector Identity for the Dirichlet Tessellationrs, Math. Proc. Camb. Phil. Soc. 87 (1980), 151–155.Google Scholar
  31. 31.
    Smith, C. S., ‘Grain Shapes and Other Metallurgical Applications of Topology’, Metal Interfaces, American Society for Metals, Cleveland, Ohio (1952), pp. 65–113.Google Scholar
  32. 32.
    Toussaint, G. T., ‘The Relative Neighbourhood Graph of a Finite Planar Set’, Pattern Recognition 13 (1980), 261–268.Google Scholar
  33. 33.
    Toussaint, G. T. and Menard, R., ‘Fast Algorithms for Computing the Planar Relative Neighbourhood Graph, Methods of Operations Research, Proc. Fifth Symp. on Operations Research, University of Koln, 1980, pp. 425–428.Google Scholar
  34. 34.
    Toussaint, G. T. and Battacharya, B. K., ‘On Geometric Algorithms that use the Furthest Point Voronoi Diagram’, Technical Report No. SOCS-81.3, School of Computer Science, McGill University, Montreal, Canada, 1981.Google Scholar
  35. 35.
    Toussaint, G. T., Battacharya, B. K., and Poulsen, R. S., ‘The Application of Voronoi Diagrams to Nonparametric Decision Rules’, Proc. Computer Science and Statistics: 16th Symp. on the Interface, Atlanta, Georgia, 1984.Google Scholar
  36. 36.
    Voronoi, G., ‘Nouvelles applications des parametres continus a la theorie des formes quadratiques. Deux. Mem. Recherches sur les paralleloedres primitifs, Sec. partie, J. reine angew, Math. 136 (1909), 67–181.Google Scholar
  37. 37.
    Whiteley, W., ‘Realizability of polyhedra’, Structural Topology (1979), 46–58.Google Scholar

Copyright information

© D. Reidel Publishing Company 1985

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

Personalised recommendations