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On the number of clumps resulting from the overlap of randomly placed figures in a plane

Published online by Cambridge University Press:  14 July 2016

A. M. Kellerer
A. M. Kellerer*
Affiliation:
University of Würzburg
*
∗ Postal address: University of Würzburg, Versbacher Str. 5, D-8700 Würzburg, W. Germany. Research supported by EURATOM Contract BIO–286–81 D.
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Abstract

When two-dimensional figures, called laminae, are randomly placed on a plane domains result that can either be aggregates or individual laminae. The intersection of the union, U, of these domains with a specified field of view, F, in the plane is considered. The separate elements of the intersection are called clumps; they may be laminae, aggregates or partial laminae and aggregates. A formula is derived for the expected number of clumps minus enclosed voids. For bounded laminae homeomorphic to a closed disc with isotropic random direction the formula contains only their mean area and mean perimeter, the area and perimeter of F, and the intensity of the Poisson process.

Keywords

GEOMETRIC PROBABILITYPARTICLE COUNTINGOVERLAP CORRECTIONEDGE CORRECTIONRANDOM COVERAGE
Type
Research Papers
Information
Journal of Applied Probability , Volume 20 , Issue 1 , March 1983 , pp. 126 - 135
Copyright
Copyright © Applied Probability Trust 1983 

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References

Armitage, P. (1949) An overlap problem arising in particle counting. Biometrika 36, 257–266.CrossRefGoogle ScholarPubMed
François, H., Kurtz, N., Massue, J. P., Monnin, M., Schmitt, R. and Durrani, S. A. (eds.) (1980) Solid State Nuclear Track Detectors. Pergamon Press, Oxford.Google Scholar
Gilbert, E. N. (1965) The probability of covering a sphere with N circular caps. Biometrika 52, 323–330.Google Scholar
Kendall, M. G. and Moran, P. A. P. (1963) Geometrical Probability. Griffin, London.Google Scholar
Mack, C. (1948) An exact formula for Qk (n), the probable number of k-aggregates in a random distribution of n points. Phil. Mag. (7) 39, 778–790.Google Scholar
Mack, C. (1949) The expected number of aggregates in a random distribution of points. Proc. Camb. Phil. Soc. 46, 285–292.Google Scholar
Mack, C. (1953) The effect of overlapping in bacterial counts of incubated colonies. Biometrika 40, 220–222.Google Scholar
Mack, C. (1954) The expected number of clumps when convex laminae are placed at random and with random orientation on a plane area. Proc. Camb. Phil. Soc. 50, 581–585.Google Scholar
Mack, C. (1956) On clumps formed when convex laminae or bodies are placed at random in two or three dimensions. Proc. Camb. Phil. Soc. 52, 246–256.CrossRefGoogle Scholar
Moran, P. A. P. and Fazekas de St Groth, S. (1962) Random circles on a sphere. Biometrika 49, 389–396.CrossRefGoogle Scholar
Morton, R. R. A. (1966) The expected number and angle of intersections between random curves in a plane. J. Appl. Prob. 3, 559–562.Google Scholar
Roach, S. A. (1968) The Theory of Random Clumping. Methuen, London.Google Scholar
Santaló, L. A. (1976) Integral Geometry and Geometric Probability. Addison-Wesley, London.Google Scholar
Underwood, E. E. (1970) Quantitative Stereology. Addison-Wesley, London.Google Scholar