Facet connectedness of discrete hyperplanes with zero intercept: The general case

Abstract

A digital discrete hyperplane in Zd is defined by a normal vector v, a shift μ, and a thickness θ. The set of thicknesses θ for which the hyperplane is connected is a right unbounded interval of R+. Its lower bound, called the connecting thickness of v with shift μ, may be computed by means of the fully subtractive algorithm. A careful study of the behaviour of this algorithm allows us to give exhaustive results about the connectedness of the hyperplane at the connecting thickness in the case μ = 0. We show that it is connected if and only if the sequence of vectors computed by the algorithm reaches in finite time a specific set of vectors which has been shown to be Lebesgue negligible by Kraaikamp & Meester.