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.
KeywordsDigital topology digital surfaces surfels curves
- 3.T.Y. Kong. A Digital Fundamental Group, volume 13. 1989.Google Scholar
- 4.A. Lenoir. Fast estimation of mean curvature on the surface of a 3d discrete object. In Proceedings of DGCI’97, Lecture Notes in Computer Science, volume 1347, pages 213–222, 1997.Google Scholar
- 5.R. Malgouyres and A. Lenoir. Topology preservation within digital surfaces. Machine Graphics and Vision, 7(1/2):417–426, 1998. Proceeding of the Computer Graphics and Image Processing.Google Scholar