Uncertainty Management in Feature-Based Geometric Modelling and Data Exchange
Sonia Abdoli, Malika Boumedien-Zidani, Neil F. Stewart · 2014
If it is desired to obtain answers with correct topological form, then the problem of computing regularized Boolean operations in solid modelling is ill-conditioned. To say that the answer has correct topological form here means that not only is the computed model topologically well-formed, but also that it has the same topological form as the true (exact) result (i.e., the exact and computed objects are linked by a homeomorphism). That the problem is ill-conditioned means that small changes in the problem data (smaller than the uncertainty in this data) can cause the true result of the operation to change topological form, so that the true result corresponding to the new data is not just erroneous, but different in kind. The ill condition implies that the resulting computational difficulty cannot be resolved by designing better numerical algorithms. It is shown in this paper, however, that in at least one situation, there is enough information available to resolve the problem of correct topological form even though the available information does not eliminate the uncertainty involved. The situation referred to is that of feature-based design, where information concerning attachment of features is available. It is shown here how to produce a posteriori guarantees on topological form in the case when the faces of the solids and features are defined by logically-locally-planar Bézier patches. This case is general enough to cover, by means of representation conversion, many practical representation methods. Our approach is based on existing methods for the production of consistent trimmed surfaces, and existing methods permitting a posteriori verification that the patches forming the faces do not have self-intersections or extraneous intersections. The main contribution of the paper is to observe that uncertainty can affect the results of computing Boolean intersections at different levels of severity and that, in certain important practical situations, although the uncertainty cannot be removed, its effects can be rendered almost completely harmless.