Simplicity surfaces: a new definition of surfaces in Z3

TLDR: It is shown that a simplicity surface may be seen as a combinatorial manifold, that is, a set of faces which are linked by an adjacency relation, and that the main existing notions of surfaces, for the 6- and the 26- adjacencies, are also simplicity surfaces.

Abstract: In this paper, we introduce a new motion of surfaces in Z3, called simplicity surfaces. In the continuous space, a surface is characterized by the fast that the neighborhood of each point of the surface is homomorphic to an Euclidean disc. The chosen approach consists in characterizing a surface in Z3 by the condition that the neighborhood of any point constitutes a simple closed curve. The major problem is than, if we consider only the usual adjacency relations, this condition is not satisfied even for the simplest surfaces, e.g. digital planes. We thus have to consider another relation. We use a relation for points in Z3 which is based on the notion of homotopy. This allows to define a surface as a connected set of points in which the neighborhood of each point constitutes a simples closed curve for this relation; such a surface is called a simplicity surface. We give two different local characterizations of simplicity surfaces. We then show that a simplicity surface may also be seen as a combinatorial manifold, that is, a set of faces which are linked by an adjacency relation. Furthermore, we show that the main existing notions of surfaces, for the 6- and the 26- adjacency, are also simplicity surfaces.© (1998) COPYRIGHT SPIE--The International Society for Optical Engineering. Downloading of the abstract is permitted for personal use only.