Alignments in two-dimensional random sets of points

Abstract

Let n points in the plane be generated by some specified random mechanism and suppose that N(∊) of the resulting triads form triangles with largest angle ≧ π – ∊. The main object of the paper is to obtain asymptotic formulae for and Var (N(∊)) when ∊ ↓ 0, and to solve the associated data-analytic problem of testing whether an empirical set of n points should be considered to contain too many such ∊-blunt triads in the situation where the generating mechanism is unknown and where all that can be said about the tolerance ∊ is that it must be allowed to take values anywhere in a given interval (T ₀, T ₁) (0 < T ₀ < T ₁). This problem is solved by the introduction of a plot to be called the pontogram and by the introduction of simulation-based significance tests constructed by random lateral perturbations of the data.