We propose in this paper a new approach to three-dimensional digital lines (3DDLs) based on the study of the integer lattice generated by the projection of Z3 onto an euclidean plane which reduces the problem to dimension 2. The many properties of this lattice lead to an arithmetical definition of 3DDLs in accordance with a topological characterization. This definition is then used in an algorithm that calculates the intersection between a naive 3DDL and an arbitrary digital plane. We also show that this algorithm can be extended to calculate the intersection between a plane and a set of adjacent 3DDLs incrementally in a very efficient manner.