The expected number of clumps when convex laminae are placed at random and with random orientation on a plane area

Abstract

As Armitage(1) has noted, this is a problem of some importance in dust particle counting where overlapping particles are usually counted as single particles owing to their small size and the difficulty of detecting overlapping. In his paper Armitage(1) obtained approximate formulae and also expansions for the expected number of clumps (as he called a self-overlapping group) (i) when the particles were all circular laminae (though of different sizes) and (ii) when the particles were similar rectangular laminae randomly oriented. His expansions were accurate to the first two terms, but he only gave approximate values for the third terms. However, by using a method similar to that of the author (5,6) in treating aggregates in random placings of points and to that of Hammersley (4) in treating one-dimensional problems analogous to those dealt with here, this paper gives exact formulae for any mixture of laminae of different shapes and sizes, the only restrictions being (i) that they are all convex (though they need not be symmetrical), (ii) that they are randomly oriented with no preferred direction, and (iii) that there are a large number of each kind. With these restrictions we shall show in addition that, surprisingly, the only relevant geometrical properties of the laminae are (i) their areas and (ii) their perimeter lengths, their shapes apart from this being immaterial.