Local configurations in discrete combinatorial surfaces

Abstract

Representing discrete objects by polyhedral complexes, we can define all conceivable topological characteristics of points in discrete objects, namely those of vertices of polyhedral complexes. Such a topological characteristic is determined for each point by observing a configuration of object points in the 3 × 3 × 3 local point set of its neighbors. We study a topological characteristic such that the point is in the boundary of a 3D polyhedral complex and the boundary forms a 2D combinatorial surface. By using the topological characteristic, we present an algorithm which examines whether the central point of a local point set is in a combinatorial surface, and show how many local point configurations exist in combinatorial surfaces in a 3D discrete space.