Imperfect Form Tolerancing on Manifold Objects: A Metric Approach

Abstract

In an earlier article we proposed a modeling space for objects defined in terms of tolerances, based on Requicha's ideas on variational classes. To do this, we introduced a certain number of requirements, including the requirements that objects within tolerance should be tamely homeomorphic to the nominal solid and that variational classes should be regular, in the relative topology defined by a certain metric dw and the set of objects tamely homeomorphic to the nominal solid. The practical con sequence of d w-regularity is to exclude variational classes that require all or part of the boundary of an admissible set to be in a fixed or exact position. The subscript in dw is intended to suggest weakness, as the metric in effect introduces a weak condition to be satisfied by the boundary of acceptable r-sets. These ideas led us to the definition of the R-classes, which we claim provides the appropriate definition for a permissible variational class in the context of object tolerancing. In par ticular, we proved that the position, size, and form tolerances ( S, Mp), ( S, Ms), and ( S, M f) are R-classes, at least in the simple case when S is an n-dimensional ball. Generalized ver sions of the regularized Boolean operations, operating not on r-sets but on R -classes, were also introduced; the R-classes are closed under these generalized versions of the regularized Boolean operations. In this article we will extend these results to quite a wide class of position tolerances and present three possibilities for the introduction of a "slow variation constraint. " The latter will be done by means of new metrics d s, which im pose "strong" conditions on the boundary of acceptable sets. Stricter requirements, which ensure slow variation, will be im posed by adding the condition that variational classes be open with respect to these stronger metrics. Finally, it will be shown how the usual "perfect form" approach to tolerancing may be viewed as a special case of the variational class approach by using an appropriately strong slow variation constraint.