ε-Topological formulation of tolerant solid modeling

Abstract

Classical theory of solid modeling relies on the notion of regular sets and presupposes exactness in both geometric data and algorithms. In contrast, modeling, exchange and translation of geometric models in engineering applications usually involve data approximations and algorithms with different numerical precisions. We argue that an appropriate formulation of these geometric modeling problems require finite size neighborhoods, leading to the notion of ε-topological operations. These operations are then used to formulate the definitions of ε-regularity and ε-solid that extend and subsume the corresponding classical concepts as exact special cases. Furthermore, the proposed theory suggests how the classical solid modeling paradigm should be extended in order to deal with the outstanding problems in geometric robustness, validation, and data translation. In particular, it explains why the current methods for validating boundary representaetions are not always sufficient and demonstrates that widely adapted geometric repairs are often unnecessary for maintaining solidity in the presence of numerical inaccuracies.