Topological framework for part families

One of the fundamental unsolved problems in geometric design of mechanical solids has been the lack of a proper notion of family or class. Numerous heuristic and often incompatible definitions are used throughout the CAD industry, and it is usually not clear how to generate members of a family or, to decide if a given object belongs to an assumed family. Until these difficulties are resolved, no guarantees or standards for parametric modeling are possible, and all efforts to allow exchange of parametric representations between different CAD systems are likely to remain futile. Standardizing on a particular definition may be difficult, because parametric families depend intrinsically not only on shape but also on its representation. We classify families into parameter-space and representation-space, and show that both types are representation-induced families. We propose a formal framework for families based on the notion of topological categories. Every parametric family is defined by the representation-induced topological space of solids that are closed under the continuous maps in the assumed topology. We illustrate several well defined families and formally define a special but important case of CSG-induced family that generalizes to the more general case of feature-induced families.