Error analysis for operations in solid modeling in the presence of uncertainty

Abstract

Abstract. The problem of maintaining consistent representations of solids, in computer-aided design, and of giving rigorous proofs of error bounds for operations such as regularized Boolean intersection, has been widely studied for at least two decades. One of the major difficulties is that the representations used in practice are not only in error, they are fundamentally inconsistent. Such inconsistency is one of the main bottlenecks in downstream applications. This paper provides a framework for error analysis in the context of solid modeling, in the case where the data is represented using the standard representational method, and where the data may be uncertain. Included are discussions of ill-condition, error measurement, stability of algorithms, inconsistency of defining data, and the question of when we should invoke methods outside the scope of numerical analysis. A solution to the inconsistency problem is proposed and supported by theorems: it is based on the use of Whitney extension to define sets, called QuasiNURBS sets, which are viewed as realizations of the inconsistent data provided to the numerical method. A detailed example illustrating the problem of regularized Boolean intersection is also given