Epsilon-regular sets and intervals

Regularity of sets (both open and closed) is fundamental in the classical theory of solid modeling and is implicit in many shape modeling representations. However, strictly speaking, the notion of regularity cannot be applied to real world shapes and/or computed geometric models that usually exhibit irregularity in the forms or errors, uncertainty, and/or approximation. We propose a notion of ε-regularity that quantifies regularity of shapes in terms of set intervals and subsumes the classical notions of open and closed regular sets as special exact cases. Our formulation relies on ε-topological operations that are related to, but are distinct from, the common morphological operations. We also show that ε-regular interval is bounded by two sets, such that the Hausdorff distance between the sets, as well the Hausdorff distance between their boundaries, is at most ε. Many applications of ε-regularity include geometric data translation and solid model validation.