DENSITY FUNCTION APPROXIMATION WITH PARTICULAR REFERENCE TO THE DISTRIBUTION OF OVERLAP OF CIRCLES

Given that two circles overlap, the area in common is a function of the distance between their centres. This paper adopts a suitable random distribution for the intercentre distance and then derives the distribution of the area of overlap. An approximation is sought for the density function using a criterion which enables bounds to be placed on the difference between the moments of the density function and those of the approximation. This is an approach of general applicability. The importance of matching the end-point behaviour of the density and the approximation is stressed. It is shown that the distribution of the area of overlap may be well approximated by a mixture of beta distributions in which the parameters change smoothly with the ratio of radii.