Modeling Spaces for Toleranced Objects

In this article we propose a modeling space for objects defined in terms of tolerances based on Requicha's sug gestion of using variational classes. Variational classes are subsets of the hyperspace 2 En , where E n is euclidean n-space. In order to motivate the ideas, the discussion involves the same simple example throughout: the specifi cation of a ball bearing defined by position, size, and form constraints. We begin by discussing the relationship between Reguicha's original proposal and our proposal for a definition of what should be viewed as a permissible variational class. We call such a permissible class, together with a nominal solid S, an R-class. We then introduce generalized versions of the regularized Boolean operations, which operate not on r-sets, but rather on R- classes. Just as the r-sets are closed under regularized Boolean operations, so the R-classes are closed under the generalized versions of the regularized Boolean opera tions. Finally, we discuss the relationship between the R- classes and the particular feature tolerancing methods proposed by Requicha.