Current feature models do not explicitly represent the relation between the parameters and the topology of the model. For theoretical and practical purposes, it is important to make this relation more explicit. A method is presented here that determines parameter values for which the topology of a feature model changes, i.e. the critical values of a given variant parameter. The considered feature model consists of a system of geometric constraints, relating parameters to feature geometry, and a cellular model. The cellular model partitions Euclidean space into quasi-disjoint cells, determined by the intersections of the feature geometry. Our method creates a new system of geometric constraints to relate the parameters of the model to topological entities in the cellular model. For each entity that is dependent on the variant parameter, degenerate cases are enforced by specific geometric constraints. Solving this system of constraints yields the critical parameter values. Critical values can be used to compute parameter ranges corresponding to families of objects, e.g. all parameter values which correspond to models that satisfy given topological constraints.

Parametric and feature-based CAD models can be thought of to represent families of similar objects. In current modelling systems, however, model semantics is unclear and ambiguous in the context of families of objects.We present the Declarative Family of Objects Model (DFOM), which enables to adequately specify and maintain family semantics. In this model, not only geometry, but also topology is specified declaratively, by means of constraints. A family of objects is modelled by a DFOM with multiple realisations. A member of the family is modelled by adding constraints, e.g. to set dimension variables, until a single realisation remains. The declarative approach guarantees that the realisation of a family member is also a realisation of the family.The realisation of a family member is found by solving first the geometric constraints, then the topological constraints. From the geometric solution, a cellular model is constructed. Topological constraints indirectly specify which combinations of cellular model entities are allowed in the realisation. The system of topological constraints is translated into a boolean constraint satisfaction problem. The realisation is found by solving this problem. The feasibility of solving topological constraints has been investigated using an existing boolean satisfiability solver.

Geometric constraints are at the heart of parametric and feature-based CAD systems. Changing values of geometric constraint parameters is one of the most common operations in such systems. However, because allowable parameter values are not known to the user beforehand, this is often a trial-and-error process. We present a solution for automatically determining the allowable range for parameters of geometric constraints. Considered are systems of distance and angle constraints on points in 3D that can be decomposed into triangular and tetrahedral subproblems, by which most practical situations in parametric and feature-based CAD systems can be represented. Our method uses the decomposition to find critical parameter values for which subproblems degenerate. By solving one problem instance for each interval between two subsequent critical values, the exact parameter ranges are determined for which a solution exists.