References

  1. Aboav, D.A. (1970) The arrangement of grains in a polycrystal. Metallography, 3, 383-390.
  2. ASTM E 112-82 (1982) Standard Methods for Determining Average Grain Size.
  3. Berndt S., J. Bretschneider, H. Helm and D. Stoyan (1996) Characterization of the degree of regularity of two-dimensional random quasi-lattice structures. Materials Characterization, 36, 93-101.
  4. Besterci, M., I. Kohútek, K. Sülleiová and I. Saxl (1995) Analýza prostorového rozmístení cástic v tenké fólii kompozitu na bázi Al. Kovové materiály, 33, 251-268.
  5. Boots, B.N. (1987) Edge length properties of random Voronoi polygons. Metallography, 20, 231-236.
  6. Bowyer, A. (1981) Computing Dirichlet tessellations. The Computer Journal, 24, 162-166.
  7. Brakke, K.A. (1985a) Statistics of Random Plane Voronoi Tessellations. Techn. Report, Department of Math. Sciences, Susquehanna University, Selinsgrove (nepublikováno).
  8. Brakke, K.A. (1985b) Random Voronoi Tessellations in Arbitrary Dimension. Techn. Report, Department of Math. Sciences, Susquehanna University, Selinsgrove (nepublikováno).
  9. Brakke, K.A. (1985c) Statistics of Three Dimensional Random Voronoi Tessellations. Techn. Report, Department of Math. Sciences, Susquehanna University, Selinsgrove (nepublikováno).
  10. Brakke, K.A. (1985d) 200 000 000 Random Voronoi Polygons. Techn. Report, Department of Math. Sciences, Susquehanna University, Selinsgrove (nepublikováno).
  11. Brown, G.S. (1965) Point density in stems per acre. New Zealand Forestry Service Research Notes, 38, 1-11.
    Brown, K.Q. (1979) Voronoi diagrams from convex hulls. Information Processing Letters, 9, 223-228.
  12. Coxeter, H.M.C. (1973) Regular Polytopes. Dover Publications, New York.
    Cruz-Orive, L-M. (1979) Distortion of certain Voronoi tessellations when one particle moves. J. Appl. Prob., 16, 95-103.
  13. Cwajna, J., J. Chraponski and M. Malinski (1997) Application of 3D models on materials microstructure in stereology. In: Wojnar, L., Rozniatowski, K.,
  14. Kurzydlowski, K.J. (eds.): Proc. Int. Conf. on The Quantitative Description of Materials Microstructure. Warsaw, 27-36.
  15. Dirichlet, G.L. (1850) Über die Reduction der positiven quadratischen Formen mit drei unbestimmten ganzen Zahlen. Journal für die Reine und Angewandte Mathematik, 40, 209-227.
  16. Eaton, B.C., and R.G. Lipsey (1975) The non-uniqueness of equilibrium in the Löschan location model. The American Economic Review, 66, 77-93.
  17. Finney, J.L. (1970) Random packing and the structure of simple liquids. II. The molecular geometry of simple liquids. Proceedings of the Royal Society of London A, 319, 495-507.
  18. Fischer, R.A. and R.E. Miles (1973) The role of spatial pattern in the competition between crop plants and weeds. A theoretical analysis. Mathematical Biosciences, 18, 335-350.
  19. Gilbert, E.N. (1962) Random subdivisions of space into crystals. Annals of Mathematical Statistics, 33, 958-972.
  20. Gruber, P.M. and J.M. Wills (1993) Handbook of Convex Geometry. North-Holland, Amsterdam, London, New York, Tokyo.
  21. Guibas, L. and J. Stolfi (1985) Primitives for the manipulation of general subdivisions and the computation of Voronoi diagrams. ACM Transaction on Graphics, 4, 74-123.
  22. Hamilton, W.D. (1971) Geometry of the selfish herd. Journal for the Theoretical Biology, 31, 295-311.
  23. Hasegawa, M. and M. Tanemura (1980) Spatial patterns of territories. In: Matusita K. (ed.): Recent Developments in Statistical Inference. North-Holland, Amsterdam, 73-78.
  24. Hermann, H. (1991) Stochastic Models of Heterogeneous Materials. Mat. Sci. Forum 78. Trans. Tech. Publ., Zürich.
  25. Hermann, H., H. Wendrock and D. Stoyan (1989) Cell-area distribution of planar Voronoi mosaics. Metallography, 23, 189-200.
  26. Hinde, A.L. and R.E. Miles (1980) Monte Carlo estimates of the distribution of the random polygons of the Voronoi tessellation with respect to a Poisson process. J. Statist. Comput. Simul., 10, 205-223.
  27. Honda, H. (1983) Geometrical models for cells in tissues. International Review of Cytology, 81, 191-248.
  28. Honda, H., T. Morita and A. Tanabe (1979) Establishment of epidermal cell columns in mammalian skin: computer simulation. Journal of Theoretical Biology, 81, 745-759.
  29. Hoofd, L., Z. Turek, K. Kubat, B.E.M. Ringnalda and S. Kazda (1985) Variability of intercapillary distance estimated on histological sections of rat heart. Advances in Experimental Medicine and Biology, 191, 239-247.
  30. Horálek, V. (1990) ASTM grain-size model and related random tessellation models. Materials Characterization, 25, 263-284.
  31. Hotelling, H. (1929) Stability in competition. Economic Journal, 39, 41-57.
  32. Huang, T., T. Tsuji, A.D. Rey and M.R. Kamal (1998) The development of spherulitic domain in polymer films. Polymeric Materials Science and Engineering, 79, 132-133.
  33. Chiu, S.N. (1994) Mean-value formulae for the neighbourhood of the typical cell of a random tessellation. Adv. Appl. Prob., 26, 565-576.
  34. Chiu, S.N. (1995) Aboav-Weaire's and Lewis' laws. A Review. Materials Characterization, 34, 149-165.
  35. Chiu, S.N., R. van de Weygaert and D. Stoyan (1996) The sectional Poisson-Voronoi tessellation is not a Voronoi tessellation. Adv. Appl. Prob., 28, 356-376.
  36. Chraponski, J. and M. Malinski (1997) Estimation of grain size I, II, III. In: Wojnar, L., Rozniatowski, K., Kurzydlowski, K.J. (eds.): Proc. Int. Conf. on The Quantitative Description of Materials Microstructure. Warsaw, 215-222, 223-228, 229-234.
  37. Chraponski, J., M. Malinski and J. Cwajna (1994) Stereological parameters of model polycrystalline structures built from polyhedra of various shape and size. Acta Stereol., 13, 299-304.
  38. King, T. (1966) Random fragmentation in two and three dimensions. Zs. f. Astrophysik, 64, 433-439.
  39. Klee, V. (1980) On the complexity of d-dimensional Voronoi diagrams. Arch. Math., 34, 75-80.
  40. Kohútek, I. and I. Saxl (1993) Properties of the Voronoi tessellation corresponding to the generalized planar Gauss-Poisson process. Acta Stereol., 12, 155-162.
  41. Lewis, F.T. (1928) The correlation between cell division and the shapes and sizes of prismatic cells in the epidermis of Cucumis. Anatomical Record, 38, 341-376.
  42. Lewis, F.T. (1943) The geometry of growth and cell division in epithelial mosaics. American Journal of Botany, 30, 766-776.
  43. Lorz, U. (1990) Cell-area distributions of planar sections of spatial Voronoi mosaics. Materials Characterization, 25, 297-309.
  44. Lorz, U. (1992) Distribution of cell characteristics of the spatial Poisson-Voronoi tessellation and plane sections. In: Eckhardt, U., Hübler, A., Nagel, W.,
  45. Werner G. (eds.): Geometrical Problems of Image Processing. Research in Informatics, Vol. 4. Akademie-Verlag, Berlin, 171-178.
  46. Lorz, U. and U. Hahn (1993) Geometric Characteristics of Random Spatial Voronoi Tessellations and Planar Sections. Preprint 93-05. TU Bergakademie Freiberg, Freiberg.
  47. Matsuda,T. and E. Shima (1984) Topology of supercluster-void structure. Progress of Theoretical Physics, 71, 855-857.
  48. Mecke, J. and L. Muche (1995) The Poisson-Voronoi tessellation I - A basic identity. Math. Nachr., 176, 199-208.
  49. Mecke, J., R.G. Schneider, D. Stoyan and W.R.R. Weil (1990) Stochastische Geometrie. Birkhäuser Verlag, Basel, Boston, Berlin.
  50. Miles, R.E. (1972) The random division of space. Adv. Appl.Prob., 4(Supplement), 243-266.
  51. Miles, R.E. (1974) A synopsis of ´Poisson flats in Euclidean spaces´. In: Harding E.F. and Kendall D.G. (eds.): Stochastic Geometry. J. Wiley & Sons, New York, 202-227.
  52. Moller, J. (1989) Random tessellations in Rd. Adv. Appl. Prob., 21, 37-73.
  53. Moller, J. (1992) Random Johnson-Mehl tessellations. Adv. Appl. Prob., 24, 814-844.
  54. Moller, J. (1994) Lectures on Random Voronoi Tessellations. Lecture Notes in Statistics 87. Springer-Verlag, New York, Berlin, Heidelberg.
  55. Moller, J. (1995) Generation of Johnson-Mehl crystals and comparative analysis of models for random nucleation. Adv. Appl. Prob., 27, 367-383.
  56. Muche, L. (1993) An incomplete Voronoi tessellation. Appl. Mathematicae, 22, 45-53.
  57. Muche, L. (1996) The Poisson-Voronoi tessellation II - Edge length distribution function. Math. Nachr., 178, 271-283.
  58. Muche, L. (1998) The Poisson-Voronoi tessellation III - Miles' formula. Math. Nachr. (in press).
  59. Niggli, R. (1927) Die topologische Strukturanalyse. Zs. f. Kristallographie, 65, 391-415.
  60. Ogawa, T. and M. Tanemura (1974) Geometrical considerations on hard core problems. Progress of Theoretical Physics, 51, 399-417.
  61. Ohya, T., M. Iri and K. Murota (1984) Improvements of the incremental method for the Voronoi diagram with computational comparisons of various algorithms. Journal of the Operations Research Society of Japan, 27, 306-336.
  62. Okabe, A. and M. Aoyagi (1991) Existence of equilibrium configurations of competitive firms on an infinite two-dimensional space. Journal of Urban Economics, 29, 349-370.
  63. Okabe, A., B. Boots and K. Sugihara (1992) Spatial Tessellations. J. Wiley & Sons, Chichester, New York, Brisbane, Toronto, Singapore.
  64. Pelikán, K., P. Ponízil and I. Saxl (1995a) Stochastic models of short fibre composites. The Polymer Processing Society, European meeting 1995, Book of abstracts. Inst. für Kunstofftechnologie, University of Stuttgart, Stuttgart, paper No. 5.31.
  65. Pelikán, K., I. Saxl and P. Ponízil (1995b) Germ-grain model of short fibre composites. Acta Stereol., 14, 75-82.
  66. Ponízil, P. and I. Saxl (1996) Booksteinuv model na kubické mrízce a Voronoiova teselace jím generovaná. In: Proc. Conf. Progress in Physical metallurgy 1996. VUT Brno, Brno, 47-50.
  67. Preparata, F.P. and M.I. Shamos (1985) Computational Geometry - An Introduction. Springer-Verlag, New York.
  68. Quine, M.P. and D.F. Watson (1984) Radial generation of n-dimensional Poisson processes. J. Appl. Prob., 21, 548-557.
  69. Rataj, J. and I. Saxl (1997) Boolean cluster models: mean cluster dilations and spherical contact distances. Math. Bohem., 122, 21-36.
  70. Rataj, J., I. Saxl and K. Pelikán (1993) Convergence of randomly oscillating point patterns to the Poisson point process. Appl. Math., 38, 221-235.
  71. Rathie, P.N. (1992) On the volume distribution of the typical Poisson-Delaunay cell. J. Appl. Prob., 29, 740-744.
  72. Rhynsburger, D. (1973) Analytic delineation of Thiessen polygons. Geographical Analysis, 5, 133-144.
  73. Saxl, I. (1996) Charakteristiky Booksteinova modelu na kubické mrízce softwarem Mathematica. Sborník semináre "Programy a algoritmy numerické matematiky. Janov n.N., 10.6.-14.6.1996. Matematický ústav AVCR, Praha, 177-183.
  74. Saxl, I. and I. Kohútek (1997) Voronoi tessellations generated by Boolean cluster fields. In: Wojnar L, Rozniatowski, K., Kurzydlowski K.J. (eds.): Proc. Int. Conf. on The Quantitative Description of Materials Microstructure. Warsaw, 481-488.
  75. Saxl, I., I. Kohútek and M. Besterci (1996) Particle cluster analysis in heterogeneous systems. In: Parilák, L., Danninger, H., Dusza, J., Weiss, B. (eds.): Deformation and Fracture in Structural PM Materials. Inst. Mater. Res. SAV, Košice, 221-230.
  76. Saxl, I., K. Pelikán, J. Rataj and M. Besterci (1995) Quantification and Modelling of Heterogeneous Systems. Cambridge Int. Science Publishing, Cambridge.
  77. Saxl, I. and J. Rataj (1991) Sférická kontaktní vzdálenost - charakteristika prostorového usporádání cástic. Pokroky práškové metalurgie (VÚPM Šumperk), No. 1-2, 88-160.
  78. Serra, J.P. (1982) Image Analysis and Mathematical Morphology. Academic Press, London.
  79. Shamos, M. I. and D. Hoey (1975) Closest-point problems. Proc. of the 16th Annual IEEE Symposium on Foundations of Computer Science. 151-162.
  80. Schückher, F. (1968) In: DeHoff, R.T., Rhines, F.N. (eds.): Quantitative Microscopy. McGraw-Hill Book Comp., New York, 201-265.
  81. Schwertel, J., and H. Stamm (1997) Analysis and modelling of tessellations by means of image analysis methods. J. Microsc., 186, 198-209.
  82. Stoyan, D. (1990) Estimation of distances and variances in Bookstein's landmark model. Biom. J., 32, 843-849.
  83. Stoyan, D., W.S. Kendall and J. Mecke (1995) Stochastic Geometry and its Applications. J. Wiley & Sons, New York.
  84. Stoyan, D. and H. Stoyan (1980) Gedanken zur Entstehung der Säulenformen bei Basalten. Zs. f. Geolog. Wissen., 8, 1529-1537.
  85. Stoyan, D. and H. Stoyan (1992) Fraktale, Formen, Punktfelder. Akademie-Verlag, Berlin.
  86. Thiessen, A.H. (1911) Precipitation averages for large areas. Monthly Weather Review, 39, 1082-1084.
  87. Underwood, E.E. (1970) Quantitative Stereology. Adison Wesley Publ. Comp., Reading.
  88. van de Weygaert, R. (1994) Fragmenting the universe III. Astron. Astrophys., 283, 361-406.
  89. Vander Voort, G.F. (1982) Grain size measurement. In: Practical Applications of Quantitative Metallography. ASTM Special technical publication 839. Philadelphia, 85-131.
  90. Voronoi, G. (1908) Nouvelles applications des parametres continus a la théorie des formes quadratiques, deuxieme memoire, recherches sur les parallelloedres primitifs. Journal für die Reine und Angewandte Mathematik, 134, 198-287.
  91. Weaire, D. (1974) Some remarks on the arrangement of grains in a polycrystal. Metallography, 7, 157-160.
  92. Wigner, E. and F. Seitz (1933) On the constitution of metallic sodium. Phys. Rev., 43, 804-810.
  93. Williams, R. (1979) Geometrical foundation of natural structures. Dover Publications Inc., New York.
  94. Zuyev, S.A. (1992) Estimates of the Voronoi polygon's geometric characteristics. Random Structures and Algorithms, 3, 149-162.