Distinguishing 3D-Topological Configurations of Two Tori

Abstract

Most of the existing work regarding topology preserving hierarchies is mainly preoccupied with 2D domains. But recently attention has turned to 3D, and more generally, nD representations. Even more than in 2D, the necessity for reducing these representations exists and motivates the research in hierarchical structures i.e. pyramids. Using representations that support any dimension, like e.g. the combinatorial map, n dimensional irregular pyramids can be built, thus obtaining reduced representations of the original data, while preserving the topology. This paper presents 3D combinatorial maps and the primitive operations needed to simplify such representations. Minimal configurations of the three primitive topological configurations, simplex, hole, and tunnel, and two possible configurations for two tori are presented. Experimental results and possible applications show the potential of the approach