A well-constrained geometric system seldom occurs in practice, especially at the sketch-based initial conceptual design stage. Usually, it is either under- or overconstrained because design is a progressive process and it is difficult for a designer to specify all involved constraints in a consistent way. This paper presents a priority-based graph-reduction solution, in which each constraint is assigned with a priority to guide the reduction of a geometric constraint graph. The advantage of this method lies in its ability to find the optimal solutions to a geometric constraint system automatically, without requiring interactive intervention from users.

The use of multiresolution control toward the editing of freeform curves and surfaces has already been recognized as a valuable modeling tool [1–3]. Similarly, in contemporary computer aided geometric design, the use of constraints to precisely prescribe freeform shape is considered an essential capability [4,5]. This paper presents a scheme that combines multiresolution control with linear constraints into one framework, allowing one to perform multiresolution manipulation of nonuniform B-spline curves, while specifying and satisfying various linear constraints on the curves. Positional, tangential, and orthogonality constraints are all linear and can be easily incorporated into a multiresolution freeform curve editing environment, as will be shown. Moreover, we also show that the symmetry as well as the area constraints can be reformulated as linear constraints and similarly incorporated. The presented framework is extendible and we also portray this same framework in the context of freeform surfaces.

Many applications of geometric nature can be modeled by geometric problems defined by constraints in which the constraint parameters have interval uncertainty. In a previous work, we developed a method for solving geometric constraint problems where parameters are narrow intervals in the domain of the geometric problem. Based on this work, we present a new approach to solve more general problems with non-trivial-width interval parameters that may not necessarily be in the domain of the problem. We show how our approach is successfully applied to a number of problems like solving geometric problems with tolerances, checking constraint feasibility and analyzing link motion of planar mechanisms.

This paper introduces topological constraints to robustly and comprehensively process interference calculation in solid modeling and feature modeling, and describes a method of symbolic notation of their expressions and algorithms to handle them. The interference calculation should be processed consistently against the contradictions in numerical values and should output the result model in the same form of representation as that in the input. To satisfy this, the topological constraints are relied on in the processing rather than numerical values, and represented symbolically and explicitly in the solid model, which is based on the Face-based representation using a table of sequences of face names around each face of the solid. The topological constraints represent degeneracy and connectedness among faces, edges and vertices, and are used against errors derived from the ambiguities caused in the numerical calculation as well as in the input data, while the errors are deliberately kept within a given tolerance. The constraints are also specified by a designer for representing his intention. When the intersection is regarded as degenerate at a point such as vertex-vertex coincidence, its topological constraint is represented by a cluster of multiple basic intersection points between edges and faces, and the name of the cluster is expressed with face names around the degenerate point. In the calculation for the set operation, the symbols of cluster names and non-cluster intersection points are used to make intersection line loops, to divide the faces of both solids along the intersection lines and to reconnect the divided ones to make the result solid. Examples are shown to demonstrate that the algorithms generate output solids even when they are subtly intersected.

In current modeling systems, all dimensions in a model have to be fully specified by the user. It is desirable that systems become more flexible in this respect, i.e. that non-critical dimensions in a model can be declared variant, and that the model can be automatically adjusted to enforce its validity when it is invalid. A method to realize this in feature modeling systems is described. The underlying feature model definition and validation approach are introduced. Validation is done by a collection of constraint solvers. An overview of invalid situations in which automatic model adjustment can be applied is given. The constraint solving scheme and, in particular, the automatic model adjustment strategies for different types of constraints are elaborated. Applications to enforce model validity are given for the areas of design by features, creating a member of a family of products, and feature conversion. These illustrate that automatic model adjustment is a very useful concept.