A graph-with-loop structure for a topological representation of 3D objects

Abstract

Given a cell complex K whose geometric realization |K| is embedded in R3 and a continuous function h : |K| → R (called the height function), we construct a graph Gh(K) which is an extension of the Reeb graph Rh(|K|). More concretely, the graph Gh(K) without loops is a subdivision of Rh(|K|). The most important difference between the graphs Gh(K) and Rh(|K|) is that Gh(K) preserves not only the number of connected components but also the number of "tunnels" (the homology generators of dimension 1) of K. The latter is not true in general for Rh(|K|). Moreover, we construct a map ψ : Gh(K) → K identifying representative cycles of the tunnels in K with the ones in Gh(K) in the way that if e is a loop in Gh(K), then ψ(e) is a cycle in K such that all the points in |ψ(e)| belong to the same level set in |K|. © Springer-Verlag Berlin Heidelberg 2007.