Reeb Graphs Sweep Algorithm

Reeb graph-based modeling allow the users precisely and intuitively control the morph because the topological information of the objects, represented by the structures of the Reeb graphs, is explicit and easy to understand. Moreover, the contours of the Reeb graphs also represents the geometrical information of the objects.Specifically, we study Reeb graphs, which express the connectivity of level sets. These graphs have been used in the past to construct data structures and user-interfaces for modeling and visualization applications. In computer-aided geometric design, Reeb graphs have been used to describe surface embeddings up to isotopy. Applications of this idea include the evolution of teeth contact interfaces in the chewing process. Multi-resolution versions of the Reeb graph have lead to data-base search methods for topologically similar geometric models. In the interactive exploration of scientific data, Reeb graphs are used to efficiently compute level sets. Reeb graphs can also function as a user-interface tool aiding the selection of meaningful level sets. A more extensive discussion of Reeb graphs and their variations in geometric modeling and visualization applications can be found in Topological Methods for Visualization, A.T. FOMENKO.Y.SHINAGAWA first use Reeb graphs to describe the three-dimensional model, M.HLAGA and H.CARR developed the Algorithm from O(n~2) to O(n log n) timecomplexity, n representative the number of triangle edges. Kree Cole-McLaughlin discuss the situation of two-dimensional manifold saddle point.We are interested in the topology of smooth functions as a means to analyze and visualize intrinsic properties of geometric models and scientific data. Here, we give an algorithm that constructs the Reeb graphs in details. We focus on loops in Reeb graphs and study when they occur and how they can be constructed, when the saddle point occur and disappear.