Representing and Understanding Non-Manifold Objects

Abstract

Solid Modeling is a well-established field. The significance of the contributions of this field is visible in the availability of abundant commercial and free modeling tools for the applications of CAD, animation, visualization etc.

There are various approaches to modeling shapes. A common problem to all of them however, is the handling of non-manifold shapes. Manifold shapes are shapes with the property of topological ``smoothness'' at the local neighbourhood of every point. Objects that contain one or more points that lack this smoothness are all considered non-manifold. Non-manifold objects form a huge catagory of shapes. In the field of solid modeling, solutions typically limit the application domain to manifold shapes. Where the occurrence of non-manifold shapes is inevitable, they are often processed at a high cost. The lack of understanding on the nature of non-manifold shapes is the main cause of it. There is a tremendous gap between the well-established mathematical theories in topology and the materialization of such knowledge in the discrete combinatorial domain of computer science and engineering. The motivation of this research is to bridge this gap between the two.

We present a characterization of non-manifoldness in 3D simplicial shapes. Based on this characterization, we propose data structures to address the applicational needs for the representation of 3D simplicial complexes with mixed dimensions and non-manifold connectivities, which is an area that is greatly lacking in the literature. The availability of a suitable data structure makes the structural analysis of non-manifold shapes feasible. We address the problem of non-manifold shape understanding through a structural analysis that is based on decomposition.