Solid sweeping: Properties, computations and applications

Sweeps are considered to be one of the basic representation schemes used in solid modeling. Sweep, as an infinite union operation, is used extensively in the modeling of many practical problems in engineering design, manufacturing and computer graphics for the purposes of shaping and simulation of moving shapes. The volume swept by an object is defined by envelopes, and the envelopes of families of both rigid and non-rigid objects play a fundamental role in the corresponding applications. Despite their relevance, computing the boundary of the set swept by an arbitrary moving object according to an arbitrary affine motion is largely an open problem due to well known theoretical and computational difficulties of envelopes. Geometric singularities in these envelopes are known to induce malfunctions or unintended system behavior, such as undercutting or overcutting. Although the concept of envelope singularities has been known for a long time, it has not yet been formally defined or studied except for some specific applications. ^ We describe a generic approach to compute points on the boundary of the swept volume and fold regions, and to predict, quantify and eliminate potential malfunctions induced by geometric singularities. The approach characterizes the geometric singularities in the envelopes of families of both rigid and non-rigid 2- or 3-dimensional solids by reframing the problem in terms fold points and fold regions in the neighborhood of these singularities. We also show that our method can generate the subsets of the moving objects that generate fold regions, and identify the moving boundaries that generate boundary of the swept set, which has important applications in mechanical design and manufacturing. Furthermore, the problem of computing boundary points can be performed by Point Membership Classification test for sweeps which can provide complete geometric information about the set swept by the moving solid object. ^ We show that our classification establishes the foundations for developing new generation computational tools for sweep boundary evaluation including self intersections, detection and quantification of geometric singularities in the boundary of swept volume, as well as number of practical applications such as shape synthesis, contact analysis, collision detection and path planning.