Algorithms for computing intersection and union of toleranced polygons with applications

Because mechanical operations are performed only up to a certain precision, the geometry of parts involved in real-life products is never known precisely. But if tolerance models for specifying acceptable variations have received a substantial attention, operations on toleranced objects have not been studied extensively. That is the reason why we address in this paper the computation of the union and the intersection of toleranced simple polygons, under a simple and already known tolerance model. First, we provide a practical and efficient algorithm that stores in an implicit data structure the information necessary to answer a request for specific values of the tolerances without performing a computation from scratch. If the polygons are of sizes m and n, and s is the number of intersections between edges occurring for all the combinations of tolerance values, the preprocessed data structure takes O (s) space and the algorithm that computes a union/intersection from it takes O((rt + m)log s + k' + k log k) time, where k is the number of vertices of the union/intersection and k less than or equal to k' less than or equal to s. Although the algorithm is not output sensitive, we show that the expectations of k and k' remain within a constant factor tau, a function of the input geometry. Second, we define and study the stability of union or intersection features. Third, we list interesting applications of the algorithms related to feasibility of assembly and assembly sequencing of real assemblies.