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Measurement of area by counting

Published online by Cambridge University Press:  14 July 2016

Allan M. Russell  and
Nora S. Josephson
Allan M. Russell
Affiliation:
Wesleyan University
Nora S. Josephson
Affiliation:
University of California, Riverside, and California Institute of Technology
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Summary

Through the use of a zone mapping transformation, geometrical probability is applied to the problem of measuring the area of a planar figure by counting the points covered when the figure is placed at random on a regular point lattice. An expression for the variance σ2 is obtained as a function of radius for the case of a circle. This expression is equivalent to a more general formula for σ2 derived earlier by Kendall and Rankin using analytical methods. Computer calculations of σ are given for two special cases and the results are compared with those predicted by alternative asymptotic formulae also due to Kendall and Rankin. Some examples of the use of zone mapping to treat asymmetric figures, not subject to analytical methods, are described.

Type
Research Papers
Information
Journal of Applied Probability , Volume 2 , Issue 2 , December 1965 , pp. 339 - 351
Copyright
Copyright © Applied Probability Trust 

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References

[1] Wilton, J. R. (1929) The lattice points of a circle; an historical account of the problem. Messenger of Mathematics 58, 67–80.Google Scholar
[2] Kendall, D. G. (1948) On the number of lattice points inside a random oval. Quart. J. Math. (Oxford Series) 19, 1–26.Google Scholar
[3] Kendall, D. G. and Rankin, R. A. (1953) On the number of points of a given lattice in a random hypersphere. Quart. J. Math. (2) 4, 178–189.Google Scholar
[4] Crofton, M. W. (1895) Probability: VI. On local probability. Encyclopaedia Britannica (9 ed.) 19, 784–788.Google Scholar
[5] Chalkley, H. (1943) Method for quantitative morphologic analysis of tissues. J. National Cancer Institute 4, 47–53.Google Scholar
[6] Chalkley, H., Cornfeld, J. and Park, H. (1949) A method for estimating volume-surface ratios. Science 110, 295–297.Google Scholar
[7] Cornfield, J. and Chalkley, H. W. (1951) A problem in geometric probability. J. Washington Academy of Sciences 41, 226–229.Google Scholar
[8] Chayes, Felix (1956) Petrographic Model Analysis. John Wiley and Sons, Inc., New York.Google Scholar
[9] Russell, A. M. (1956) Statistical approach to spatial measurement. Amer. J. Physiol. 24, 562–567.Google Scholar