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The expected number and angle of intersections between random curves in a plane

Published online by Cambridge University Press:  14 July 2016

R.R.A. Morton
R.R.A. Morton*
Affiliation:
Monash University, Australia
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Extract

Consider a plane containing a set of curves. Suppose a separate set of curves is constrained to lie at random on a particular region of this plane. In this note an expression is obtained for the expected number of intersections between the two sets of curves, and it is shown that the angles between the two sets at their points of intersection are distributed as ½ sin θ.

Type
Short Communications
Information
Journal of Applied Probability , Volume 3 , Issue 2 , December 1966 , pp. 559 - 562
Copyright
Copyright © Applied Probability Trust 

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References

[1] Kendall, M. G. and Moran, P.A.P. (1963) Geometrical Probabilities. Griffin, London (especially Chapter 3).Google Scholar
[2] Miles, R.E. (1964) Random polygons determined by random lines in a plane. Proc. Nat. Acad. Sci. U.S.A. 52, 901–907.Google Scholar
[3] Wolfowitz, J. (1949) The distribution of plane angles of contact. Quart. J. Appl. Math. 7, 117–120.Google Scholar