Hostname: page-component-74656f9884-8r8nv Total loading time: 0 Render date: 2026-11-09T14:49:17.431Z Has data issue: false hasContentIssue false
  • English
  • Français

A third note on recent research in geometrical probability

Published online by Cambridge University Press:  01 July 2016

D. V. Little
D. V. Little*
Affiliation:
University of Sheffield
Get access Rights & Permissions [Opens in a new window]

Abstract

Recent research on topics related to geometrical probability is reviewed. The survey includes articles on random points, lines, flats, and networks in Euclidean spaces, pattern recognition, random coverage and packing, random search, stereology, and probabilistic aspects of integral geometry.

Keywords

RANDOM CONVEX HULLSRANDOM SIMPLICESSYLVESTER'S PROBLEMGEOMETRICAL PROBABILITYUNIFORM CIRCULAR DISTRIBUTIONSPOISSON POINT PROCESSESRANDOM LINESINVARIANT IMBEDDINGLINE PROCESSESRANDOM NETWORKSRANDOM TESSELLATIONSRANDOM SECANTSPOISSON FLATSRANDOM POINTS AND SETS ON A SPHEREPATTERN RECOGNITIONRANDOM SEARCHRANDOM COVERAGERANDOM PACKINGRANDOM CLUSTERSSTEREOLOGYESTIMATION OF PARTICLE SIZERANDOM SETSINTEGRAL GEOMETRYAGGLOMERATION PROBABILITIES.

Information

Type
Research Article
Information
Advances in Applied Probability , Volume 6 , Issue 1 , March 1974 , pp. 103 - 130
Copyright
Copyright © Applied Probability Trust 1974 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Bibliography (A): Papers cited in previous bibliographies

Bernal, J. D. and Mason, J. (1960) Co-ordination of randomly packed spheres. Nature 188, 910–911.CrossRefGoogle Scholar
Geciauskas, E. P. (1966) Distribution of the distance between two points in an oval. Litovsk. Mat. Sb. 6, 245–248.Google Scholar
Kendall, M. G. and Moran, P. A. P. (1963) Geometrical Probability. Griffin's Statistical Monographs and Courses. No. 5. Griffin, London.Google Scholar
Moran, P. A. P. (1966) A note on recent research in geometrical probability. J. Appl. Prob. 3, 453–463.CrossRefGoogle Scholar
Palasii, I. (1960) On some random space filling problems. Publ. Math. Inst. Hungar. Acad. Sci. 5, 353–360.Google Scholar
Rényi, A. and Sulanke, R. (1963) Über die konvexe Hülle von n zufälligen gewälten Punkten. I. Z. Wahrscheinlichkeitsth. 2, 75–84.CrossRefGoogle Scholar
Rényi, A. and Sulanke, R. (1964) Über die konvexe Hülle von n zufälligen gewälten Punkten. II. Z. Wahrscheinlichkeitsth. 3, 138–147.CrossRefGoogle Scholar
Rényi, A. and Sulanke, R. (1968) Zufällige konvexe Polygone in einen Ringgebeit. Z. Wahrscheinlichkeitsth. 9, 146–157.CrossRefGoogle Scholar
Roberts, F. D. K. and Storey, S. H. (1968) A three-dimensional cluster problem. Biometrika 55, 258–260.CrossRefGoogle Scholar
Solomon, H. (1965) Random packing density. Proc. Fifth Berkeley Symp. Math. Statist. and Prob. 3, 119–134. University of California Press.Google Scholar

Bibliography (B): Papers not previously cited

Ajne, B. (1968) A simple test for uniformity of a circular distribution. Biometrika 55, 343–354.CrossRefGoogle Scholar
Ambartzumjan, R. V. (1969) Intersections of complex curves and random straight lines. Sov. Mat. Dokl. 10, pt. 2, 865–868.Google Scholar
Ambartzumjan, R. V. (1970) Invariant imbedding in the theory of random lines. (In Russian) Izv. Akad. Nauk Armjan. SSR. Ser. Mat. 5, 167–206.Google Scholar
Ambartzumjan, R. V. (1971) On random plane mosaics. Sov. Mat. Dokl. 12, pt. 2, 1349–1353.Google Scholar
Ambartzumjan, R. V. (1971) Probability distributions in the geometry of clusters. Stud. Sci. Math. Hungar. 6, 235–241.Google Scholar
Baxter, G. (1961) A combinatorial lemma for complex numbers. Ann. Math. Statist. 32, 901–904.CrossRefGoogle Scholar
Beresford, R. H. (1969) Statistical geometry of random heaps of equal hard spheres. Nature 224, 550–553.CrossRefGoogle Scholar
Blaisdell, B. E. and Solomon, H. (1970) On random sequential packing in the plane and a conjecture of Palasti. J. Appl. Prob. 7, 667–698.CrossRefGoogle Scholar
Blaschke, W. (1935a) Integralgeometrie 1. Ermittlung der Dichten für lineare Unterräume im En . Hermann, Paris. (Act. Sci. Indust. 252).Google Scholar
Blaschke, W. (1935b) Integralgeometrie 2. Zu Ergebnissen von M. W. Crofton. Bull. Math. Soc. Roumaine des Sci. 37, 3–11.Google Scholar
Carnal, H. (1970) Die konvexe Hülle von n rotationssymmetrisch verteilen Punkten. Z. Wahrscheinlichkeitsth. 15, 168–176.CrossRefGoogle Scholar
Coleman, R. (1969) Random paths through convex bodies. J. Appl. Prob. 6, 430–441.CrossRefGoogle Scholar
Corte, H. (1969) On the distribution of mass density in paper. I. Das Papier 23, 381–393 (corrections: 495).Google Scholar
Corte, H. (1970) On the distribution of mass density in paper. II. Das Papier 24, 261–271.Google Scholar
Crain, I. K. (1972) Monte Carlo simulation of random Voronoi polygons: preliminary results. Search 3, 220–221.Google Scholar
Davidson, R. (1969) Line processes, roads and fibres. Bull. Inst. Internat. Statist. 43, pt. 2, 322–324.Google Scholar
Davidson, R. (1970) Construction of line processes; second order properties. Izv. Akad. Nauk Armjan. SSR. Ser. Mat. 5, 219–234.Google Scholar
Dodson, C. T. J. (1970) Statistical analysis of patterns of deformation in flat bonded fibrous networks. J. Phys. D 3, 269–276.CrossRefGoogle Scholar
Dodson, C. T. J. (1971) Spatial variability and the theory of sampling in random fibrous networks. J. Roy. Statist. Soc. B 33, 88–94.Google Scholar
Domb, C. (1972) A note on the series expansion method for clustering problems. Biometrika 59, 209–211.CrossRefGoogle Scholar
Dvoretzky, A. (1956) On covering a circle by randomly placed arcs. Proc. Nat. Acad. Sci. U.S.A. 42, 199–203.CrossRefGoogle ScholarPubMed
Echlin, P., Ralph, B. and Weibel, E. R. (Editors) (1972) Proceedings of the Third International Conference for Stereology. J. Microscopy 95.Google Scholar
Eckler, A. R. (1969) A survey of coverage problems associated with point and area targets. Technometrics 11, 561–589.CrossRefGoogle Scholar
Elias, H. (Editor) (1967) Proc. Second Internat. Congress for Stereology. Chicago, 1967. Springer-Verlag, Berlin.CrossRefGoogle Scholar
Finney, J. L. (1970) Random packing and the structure of simple liquids. I and II. Proc. Roy. Soc. A 319, 479–493; 495–507.Google Scholar
Fisher, L. D. (1966) The convex hull of a sample. Bull. Amer. Math. Soc. 72, 555–558.CrossRefGoogle Scholar
Fisher, L. D. (1969) Limiting sets and convex hulls of samples from product measures. Ann. Math. Statist. 40, 1824–1832.CrossRefGoogle Scholar
Fisher, L. D. (1971) Limiting convex hulls of samples: theory and function space examples. Z. Wahrscheinlichkeitsth. 18, 281–297.CrossRefGoogle Scholar
Flatto, L. and Konheim, A. G. (1962) The random division of an interval and the random covering of a circle. SIAM Rev. 4, 211–222.CrossRefGoogle Scholar
Furstenberg, H. and Tzkoni, I. (1971) Spherical functions and integral geometry. Israel J. Math. 10, 327–338.CrossRefGoogle Scholar
Geciauskas, E. P. (1967) Distribution of the distance between two points in an ovaloid. (In Russian) Litovsk. Mat. Sb. 7, 35–36.Google Scholar
Geciauskas, E. P. (1968a) The method of integral geometry for finding the distributions of chord length and distance in an oval. Litovsk. Mat. Sb. 8, 237–241.Google Scholar
Geciauskas, E. P. (1968b) Distribution of the distance from an interior point of an oval to its contour. Litovsk. Mat. Sb. 8, 735–737.Google Scholar
Geciauskas, E. P. (1969) The method of integral geometry for finding the distribution of distancei n an ovaloid. Litovsk. Mat. Sb. 9, 481–482.Google Scholar
Giger, H. (1970) Fundamental equations in stereology I. Metrika 16, 43–57.CrossRefGoogle Scholar
Giger, H. (1972) Fundamental equations in stereology II. Metrika 18, 84–93.CrossRefGoogle Scholar
Giger, H. and Hadwiger, H. (1968) Über Treffzahlwahrscheinlichkeiten in Eikörperfeld. Z. Wahrscheinlichkeitsth. 10, 329–334.CrossRefGoogle Scholar
Giger, H. and Riedwyl, H. (1970) Bestimmung der Grossenverteilung von Kugeln aus Schnittkreisradien. Biometrische Z. 12, 156–162.CrossRefGoogle Scholar
Gilbert, E. N. (1964) Randomly packed and solidly packed spheres. Canad. J. Math. 16, 286–298.CrossRefGoogle Scholar
Gilbert, E. N. (1967) Random plane networks and needle shaped crystals. Chapter 16. Applications of Undergraduate Mathematics in Engineering. (Ed. Noble, B.) MacMillan, New York.Google Scholar
Grover, N. B. and Weiss, G. H. (1968) Theoretical distribution of two-dimensional projections in electron microscopic sections. Proc. Nat. Acad. Sci. U.S.A. 60, 80–85.CrossRefGoogle ScholarPubMed
Guenther, W. C. and Terragno, P. J. (1964) A review of the literature on a class of coverage problems. Ann. Math. Statist. 35, 232–260.CrossRefGoogle Scholar
Hadwiger, H. and Streit, F. (1970) Über Wahrscheinlichkeiten räumlicher Bündelungser-scheinungen. Monatsh. Math. 74, 30–40.CrossRefGoogle Scholar
Holcomb, D. F., Iwasawa, M. and Roberts, F. D. K. (1972) Clustering of randomly placed spheres. Biometrika 59, 207–209.CrossRefGoogle Scholar
Holcomb, D. F. and Rehr, J. J. (1969) Percolation in heavily doped semi-conductors. Phys. Rev. 183, 773–776.CrossRefGoogle Scholar
Kingman, J. F. C. (1969) Random secants of a convex body. J. Appl. Prob. 6, 660–672.CrossRefGoogle Scholar
Klee, V. (1969) What is the expected volume of a simplex whose vertices are chosen at random from a given convex body? Amer. Math. Monthly 76, 286–288.CrossRefGoogle Scholar
Komenda, S., Perinova, V. and Bayer, A. (1969) Die mikroskopische Bestimmung der Grossenverteilung von Kugelelementen. Biometrische Z. 11, 73–91.CrossRefGoogle Scholar
Krickeberg, K. (1970) Invariance properties of the correlation measure of line processes. Izv. Akaa. Nauk Armjan. SSR. Ser. Mat. 5, 251–262.Google Scholar
Langford, E. (1969) Probability that a random triangle is obtuse. Biometrika 56, 689–690.CrossRefGoogle Scholar
Mallows, C. L. and Clark, J. M. C. (1970) Linear intercept distributions do not characterize plane sets. J. Appl. Prob. 7, 240–244.CrossRefGoogle Scholar
Mallows, C. L. and Clark, J. M. C. (1971) Corrections to ‘Linear intercept distributions …’. J. Appl. Prob. 8, 208–209.CrossRefGoogle Scholar
Mandelbrot, B. B. (1972a) Renewal sets and random cutouts. Z. Wahrscheinlichkeitsth. 22, 145–157.CrossRefGoogle Scholar
Mandelbrot, B. B. (1972b) On Dvoretzky coverings for the circle. Z. Wahrscheinlichkeitsth. 22, 158–160.CrossRefGoogle Scholar
Mantel, N. (1969) More light bulb statistics. Amer. Statist. 23, no. 2, 21–23.Google Scholar
Marriot, F. H. (1971) Buffon's problem for non-random directions. Biometrics 27, 233–235.CrossRefGoogle Scholar
Matern, B. (1960) Spatial variation. Meddelanden fran Statens Skogsforskninsinstitut 49, no. 5.Google Scholar
Matheron, G. (1967) Eléments pour une théorie des milieux poreux. Masson, Paris.Google Scholar
Matheron, G. (1972a) Ensembles fermés aléatoires, ensembles semi-markoviens et polyèdres poissoniens. Adv. Appl. Prob. 4, 508–541.CrossRefGoogle Scholar
Matheron, G. (1972b) Random sets theory and its applications to stereology. J. Microscopy 95, 15–25; 393.CrossRefGoogle Scholar
Matschinski, M. (1954) Considération statistique sur les polygones et les polyèdres. Publ. Inst. Statist. Univ. Paris 3, 179–201.Google Scholar
Matschinski, M. (1969) Détermination expérimental des probabilités du nombre des côtes des polygones. Publ. Inst. Statist. Univ. Paris 18, 246–265.Google Scholar
Meluish, F. M. and Lang, A. R. (1968) Quantitative studies of roots in soil. Soil Science 106, 16–22.Google Scholar
Meluish, F. M. and Lang, A. R. (1970) Lengths and diameters of plant roots in non-random populations Biometrics 26, 421–431.Google Scholar
Miles, R. E. (1969a) Probability distribution of a network of triangles. SIAM Rev. 11, 399–402.CrossRefGoogle Scholar
Miles, R. E. (1969b) Poisson flats in Euclidean spaces. I. Adv. Appl. Prob. 1, 211–237.CrossRefGoogle Scholar
Miles, R. E. (1969c) Asymptotic values of certain coverage probabilities. Biometrika 56, 661–680.CrossRefGoogle Scholar
Miles, R. E. (1970a) On the homogeneous planar Poisson point process. Math. Biosci. 6, 85–127.CrossRefGoogle Scholar
Miles, R. E. (1970b) Synopsis of Poisson flats in Euclidean spaces. Izv Akad. Nauk. Armjan. SSR. Ser. Mat. 5, 263–285.Google Scholar
Miles, R. E. (1971a) Isotropic random simplices. Adv. Appl. Prob. 3, 353–382.CrossRefGoogle Scholar
Miles, R. E. (1971b) Poisson flats in Euclidean spaces. II. Adv. Appl. Prob. 3, 1–43.CrossRefGoogle Scholar
Miles, R. E. (1971c) Random points, sets and tessellations on the surface of a sphere. Sankhyā A 33, 145–174.Google Scholar
Miles, R. E. (1972) Multidimensional perspectives on stereology. J. Microscopy 95, 181–195.CrossRefGoogle Scholar
Moran, P. A. P. (1969) A second note on recent research in geometrical probability. Adv. Appl. Prob. 1, 73–89.CrossRefGoogle Scholar
Mullooly, J. P. (1968) A one-dimensional random space filling problem. J. Appl. Prob. 5, 427–435.CrossRefGoogle Scholar
Nicholson, W. L. (1970) Estimation of linear properties of particle size distributions. Biometrika 57, 273–297.CrossRefGoogle Scholar
Nicholson, W. L. and Merckx, R. K. (1969) Unfolding particle size distributions. Technometrics 11, 207–223.CrossRefGoogle Scholar
Paloheimo, J. E. (1971) On a theory of search. Biometrika 58, 61–75.CrossRefGoogle Scholar
Petkantschin, B. (1936) Integralgeometrie 6. Zusammenhänge zwischen den Dichten der linearen Unterräume im n-dimensionalen Räum. Abh. Math. Seminar Hamburg 11, 249–310.CrossRefGoogle Scholar
Raynaud, H. (1970) Sur l'enveloppe convexe des nuages de points aléatoires dans Rn. I. J. Appl. Prob. 7, 35–48.Google Scholar
Rényi, A. (1958) On a one-dimensional problem concerning space filling. Publ. Math. Inst. Hungar. Acad. Sci. 3, 109–127.Google Scholar
Roach, S. A. (1968) The Theory of Random Clumping. Methuen Monograph on Applied Probability and Statistics. Methuen, London.Google Scholar
Roberts, F. D. K. (1969) Nearest neighbours in a Poisson ensemble. Biometrika 56, 401–406.CrossRefGoogle Scholar
Santaló, L. A. (1953) Introduction to Integral Geometry. Hermann, Paris. (Act. Sci. Indust. 1198).Google Scholar
Shepp, L. A. (1972a) Covering the circle with random arcs. Israel J. Math. 11, 328–345.CrossRefGoogle Scholar
Shepp, L. A. (1972b) Covering the line with random intervals. Z. Wahrscheinlichkeitsth. 23, 163–170.CrossRefGoogle Scholar
Sidak, Z. (1968) On the mean number and size of opaque particles in transparent bodies. Studies in Math. Statist. Theory and Applications. (Ed. Sarkadi, K. and Vincze, I.) Akademiai Kiado, Budapest.Google Scholar
Smalley, I. J. (1966) Contraction crack networks in basalt flows. Geol. Mag. 103, 110–114.CrossRefGoogle Scholar
Smith, W. L. (1958) Renewal theory and its ramifications. J. Roy. Statist. Soc. B 20, 273–277.Google Scholar
Spitzer, F. and Widom, H. (1961) The circumference of a convex polygon. Proc. Amer. Math. Soc. 12, 506–509.CrossRefGoogle Scholar
Stearns, M. B. (1967) Probability distribution of a network of triangles. SIAM Rev. 9, 591.Google Scholar
Streit, F. (1970) On multiple integral geometric integrals and their applications to probability theory. Canad. J. Math. 22, 151–163.CrossRefGoogle Scholar
Sulanke, R. (1966) Integralgeometrie ebener Kurvennetze. Acta. Math. Acad. Sci. Hungar. 17, 233–261.CrossRefGoogle Scholar
Tallis, G. M. (1970) Estimating the distributions of spherical and elliptical bodies in conglomerates from plane sections. Biometrics 26, 87–103.CrossRefGoogle ScholarPubMed
Tate, C. (1969) Aggregates in a random distribution of points in a cylinder. SIAM J. Appl. Math. 17, 1177–1189.CrossRefGoogle Scholar
Vinograd, O. P. and Zaragradski, I. P. (1970) Geometric probabilities in the theory of optimal search. (In Russian) Izv. Akad. Nauk Armjan. SSR. Ser. Mat. 5, 207–218.Google Scholar
Watson, G. S. (1971) Estimating functionals of particle size distributions. Biometrika 58, 483–490.CrossRefGoogle Scholar
Wicksell, S. D. (1925) The corpuscle problem. Part I. Biometrische Z. 17, 84–99.Google Scholar
Wicksell, S. D. (1926) The corpuscle problem. Part II. Biometrische Z. 18, 151–172.Google Scholar
Wong, E. and Steppe, J. A. (1969) Invariant recognition of geometric shapes. Methodologies of pattern recognition. (Ed. Watanabe, S.,) Academic Press, New York.Google Scholar
Ziezold, H. (1970) Über die Eckenanzahl zufällige konvexe Polygone. Izv. Akad. Nauk Armjan. SSR. Ser. Mat. 5, 296–312.Google Scholar

Note added in proof

Three recently published volumes contain many articles of interest is geometrical probability.

Kendall, D. G. and Hardinb, E. F. (Editors) (1973) Stochastic Analysis. Wiley, New York.Google Scholar
Kendall, D. G. and Harding, E. F. (Editors) (1973) Stochastic Geometry. Wiley, New York.Google Scholar
Nicholson, W. L. (Editor) (1972) Proceedings of the symposium on statistical and probabilistic problems in metallurgy, Seattle, 1971. Adv. Appl. Prob. Special Supplement.Google Scholar