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Some properties of chord length distributions

Published online by Cambridge University Press:  14 July 2016

J. Gates
J. Gates*
Affiliation:
Thames Polytechnic
*
∗Postal address: School of Mathematics, Statistics and Computing, Thames Polytechnic, Wellington St, Woolwich, London SE18 6PF, UK.
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Abstract

The chord length distributions of planar convex sets are discussed, particularly the density values at the extremes of the range; there is a qualitative distinction between polygons and sets with smooth boundaries. The distance between convex sets is related to the distance between distribution functions.

Keywords

RANDOM CHORD LENGTHINITIAL AND TERMINAL DENSITIESHAUSDORFF DISTANCE
Type
Research Papers
Information
Journal of Applied Probability , Volume 24 , Issue 4 , December 1987 , pp. 863 - 874
Copyright
Copyright © Applied Probability Trust 1987 

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References

[1] Abramowitz, M. and Stegun, I. A. (1965) Handbook of Mathematical Functions. Dover, New York.Google Scholar
[2] Bonnesen, T. and Fenchel, W. (1934) Theorie der Konvexen Korper. Ergeb. Math., Springer-Verlag, Berlin.Google Scholar
[3] Coleman, R. (1969) Random paths through convex bodies. J. Appl. Prob. 6, 430–441.CrossRefGoogle Scholar
[4] Copson, E. T. (1935) The Theory of Functions of a Complex Variable. Clarendon Press, Oxford.Google Scholar
[5] Enns, E. G. and Ehlers, P. F. (1978) Random paths through a convex region. J. Appl. Prob. 15, 144–152.Google Scholar
[6] Gates, J. (1982) Recognition of triangles and quadrilaterals by chord length distributions. J. Appl. Prob. 19, 873–879.Google Scholar
[7] Geciauskas, E. (1977) The distribution function of the distance between two points in a convex domain. Adv. Appl. Prob. 9, 472–478.CrossRefGoogle Scholar
[8] Horowitz, M. (1965) Probability of random paths across elementary geometrical shapes. J. Appl. Prob. 2, 169–177.CrossRefGoogle Scholar
[9] Mallows, C. L. and Clark, J. M. (1970) Linear intercept distributions do not characterise plane sets. J. Appl. Prob. 7, 240–244.CrossRefGoogle Scholar
[10] Matheron, G. (1975) Random Sets and Integral Geometry. Wiley, New York.Google Scholar
[11] Piefke, F. (1978) Beziehungen zwischen der Sehnenlängenverteilung und der Verteilungen des Abstandes zwei zufälliger Punkte in Eikorper. Z. Wahrscheinlichkeitsth. 43, 129–134.Google Scholar
[12] Ruben, H. (1978) On the distance between points in polygons. In Geometrical Probability and Biological Structures: Buffon's 200th Anniversary , ed. Miles, R. E. and Serra, J. Springer-Verlag, Berlin.Google Scholar
[13] Santaló, L. A. (1976) Integral Geometry and Geometric Probability. Encyclopaedia of Mathematics and its Applications, 1. Addison-Wesley, Reading, MA.Google Scholar
[14] Sulanke, R. (1965) Die Verteilung der Sehnenlängen an ebenen und räumlichen Figuren. Math. Nachr. 23, 51–74.CrossRefGoogle Scholar
[15] Voss, K. (1982) Powers of chords for convex sets. Biom. J. 5, 513–516.Google Scholar