Distribution of three-dimensional islands from two-dimensional line segment length distribution

If an isotropic rough surface is intersected with a plane near the highest points of the surface, the resulting intersection will comprise a set of approximately circular islands whose size distribution gives some information about the nature of the contact process for the rough surface. Statistical arguments are used to predict the distribution of line segment lengths obtained when a random straight line intersects a random distribution of circular islands. The resulting expression leads to an Abel equation, which can be inverted explicitly to predict the distribution of island radii from the distribution of length segments above the given height in a two-dimensional profilometer output.The resulting expression is tested by comparing its predictions with the bearing area island distribution for an isotropic fractal rough surface defined by the random mid-point displacement (RMD) algorithm. Also, by adding extra scales in the RMD algorithm, the effect of measurement resolution on the mean island radius and the predicted distribution is explored.