On the shape description of general solids using Morse theory

Abstract

The automatic shape description of solids is a problem of interest in manufacturing engineering, amongst other related areas. This description can be either geometrical or topological in nature and can be applied to either surfaces or solids (embedded manifolds). Topological descriptions are specially interesting for the problem of shape comparison and retrieval, where one wants to know if a given shape resembles some other known shape. Some popular topological descriptions use Morse theory to study the topology of manifolds and encode their shape characteristics. A Morse function $f$ is defined on the manifold and the manifold’s shape is indirectly studied by studying the behavior of the critical points of $f$. This family of methods is well defined for surfaces but does not consider the case of solids. In this paper we address the topological description of solids using Morse theory. Our methodology considers three cases: solids without internal boundaries, solids with internal boundaries and thin-walled solids. We present an algorithm to identify topological changes on these solids using the principle of shape decomposition by Morse handles. The presented algorithm deals with Morse functions that produce parallel planar level sets. Future endeavors should consider other candidate functions.