Alignments in two-dimensional random sets of points

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Abstract
Letnpoints in the plane be generated by some specified random mechanism and suppose thatN(∊) of theresulting triads form triangles with largest angle ≧ π – ∊. The main object of the paper is to obtain asymptotic formulae forand Var (N(∊)) when ∊ ↓ 0, and to solve the associated data-analytic problem of testing whether an empirical set ofnpoints 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 (T0,T1) (0 <T0<T1). This problem is solved by the introduction of a plot to be called thepontogramand by the introduction of simulation-based significance tests constructed by random lateral perturbations of the data.

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