<title>New results about 3D digital lines</title>

Abstract

The current definition of 3D digital lines, which uses the 2D digital lines of closest integer points (Bresenham's lines) of two projections, has several drawbacks: the discrete topology of this 3D digital line notion is not clear; its third projection is, generally, not the closest set of points of the third euclidean projection; if we consider a family of parallel euclidean lines, we do not know how many combinatorially distinct digital structures will be built by this process; and mainly the set of voxels defined in this way is not the set of closest points of the given euclidean line. And these questions are the simplest ones; many others could be asked: dependence on the choice of the projection, intersections with digital planes, intersections between 3D digital lines,... This paper gives a new definition of 3D digital lines relying on subgroups of Z<SUP>3</SUP>, whose main advantage over the former one is its ability to convert any practical question into rigorous algebraic terms. It follows previously developed ideas but with a much simpler treatment and new results. In particular, we obtain a complete description of the topology of these lines and a condition for the third projection being a 2D digital line as well as a classification of digital lines of the same direction into classes of equivalent combinatorial structure.