Constructive representations, such as Constructive Solid Geometry (CSG) and its various feature-based extensions are inherently parametric in nature and are well suited for defining parametric family of solids. On the other hand, cell complex representations contain explicit shape elements (cells) and also their topology. However they are non-constructive, difficult to parameterize, and it is extremely difficult to enforce continuity in the usual cell complex topology (considered as a sub-space of Euclidean space). When cell complexes are used in conjunction with constructive representations, even if we can enforce continuity in limited cases, it is impossible to relate cellular operations with global semantics of constructive operations (such as Boolean and feature operations).Using the framework developed in our earlier work for defining part families on any solid representation, we propose constructive topological representations by identifying every constructive representation with its corresponding unique spatial decomposition. By applying the proposed definitions to spatial CSG representations we will systematically develop algorithms for topologizing a CSG and also enforcing continuity between two given topologized CSG representations. These algorithms have been implemented in a prototype system about which we will briefly discuss. Finally, we will illustrate some interesting applications of the proposed constructive topological representations: specification of constructive families and the enforcement of global semantics of feature operations.

One of the major unsolved problems in parametric solid modeling is a robust update (regeneration) of the solid's boundary representation, given a specified change in the solid's parameter values. The fundamental difficulty lies in determining the mapping between boundary representations for solids in the same parametric family. Several heuristic approaches have been proposed for dealing with this problem, but the formal properties of such mappings are not well understood. We propose a formal definition for boundary representation. (BR-)deformation for solids in the same parametric family, based on the assumption of continuity: small changes in solid parameter values should result in small changes in the solid's boundary reprentation, which may include local collapses of cells in the boundary representation. The necessary conditions that must be satisfied by any BR-deforming mappings between boundary representations are powerful enough to identify invalid updates in many (but not all) practical situations, and the algorithms to check them are simple. Our formulation provides a formal criterion for the recently proposed heuristic approaches to “persistent naming,” and explains the difficulties in devising sufficient tests for BR-deformation encountered in practice. Finally our methods are also applicable to more general cellular models of pointsets and should be useful in developing universal standards in parametric modeling.

One of the key unsolved problems in parametric solid modeling is a robust update (regeneration) of the solid's boundary representation, given a specified change in solid's parameter values.The fundamental difficulty lies in determining the mapping between boundary representations for solids in the same parametric family, also known as "persistent naming."We formulate the notion of Boundary Representation (BR-)variance for solids in the same parametric family based on the assumption of continuity: small changes in solid's parameter values should result in small changes in solid's boundary rep reeentation, which may include local collapses of cells in boundary representations.We show that our formulation gives the necessary conditions that must be satisfied by any BR-variant mappings between boundary representations.These conditions are powerful enough to identify invalid updates in many practical situations and provide a formal criteria for the recently proposed heuristic approaches to the "persistent naming" problem. Motivation Ambiguity of parametric solid modeling