Decomposition of Rational Discrete Planes

This paper is a contribution to the study of rational discrete planes, i.e., sets of points with integer coordinates lying between two parallel planes. Up to translation and symmetry, they are completely determined by a normal vector \(\boldsymbol{a}\in \mathbb {N}^3\) . Excepted for a few well-identified cases, it is shown that there are two approximations \(\boldsymbol{b},\boldsymbol{c}\in \mathbb {N}^3\) of \(\boldsymbol{a}\) , satisfying \(\boldsymbol{a}=\boldsymbol{b}+\boldsymbol{c}\) , such that the discrete plane of normal \(\boldsymbol{a}\) can be partitioned into two sets having respectively the combinatorial structure of discrete planes of normal \(\boldsymbol{b}\) and \(\boldsymbol{c}\) . Christoffel graphs are used to compactly encode the structure of discretes planes. This result may have practical interest in discrete geometry for the analysis of planar features.