Intersection Number of Paths Lying on a Digital Surface and a New Jordan Theorem

The purpose of this paper is to define the notion of "real" intersection between paths drawn on the 3d digital boundary of a connected object. We consider two kinds of paths for different adjacencies, and define the algebraic number of oriented intersections between these two paths. We show that this intersection number is invariant under any homotopic transformation we apply on the two paths. Already, this intersection number allows us to prove a Jordan curve theorem for some surfels curves which lie on a digital surface, and appears as a good tool for proving theorems in digital topology about surfaces.