Discrete Combinatorial Surfaces

Abstract

A new approach to the concept of discrete surfaces is proposed, It is a combinatorial approach. A surface is defined by vertices, edges, and faces satisfying the conditions of two-dimensional combinatorial manifolds. A set of voxels (points with integer coordinates) is a surface iff these points are the vertices of a two-dimensional combinatorial manifold. This approach allows introduction of several notions of discrete surfaces: The first, called a quadrangulated surface, is a combinatorial manifold whose faces are squares; the second, called a triangulated surface, is a combinatorial manifold whose faces are triangles, The last is associated with a neighborhood relation; thus, there are as many concepts of triangulated surfaces as there are neighborhood relations. As a consequence the same concepts, algorithms, and methods can be used in computer imagery and in the field of topology-based geometric modeling (so called "boundary representation").