Multiple Translational Containment Part I: An Approximate Algorithm

In Part I we present an algorithm for finding a solution to the two-dimensional translational approximate multiple containment problem: find translations for k polygons which place them inside a polygonal container so that no point of any polygon is more than 2ffl inside of the boundary of any other polygon. The polygons and container may be nonconvex. The value of ffl is an input to the algorithm. In industrial applications, the containment solution acts as a guide to a machine cutting out polygonal shapes from a sheet of material. If one chooses ffl to be a fraction of the cutter's accuracy, then the solution to the approximate containment problem is sufficient for industrial purposes. Given a containment problem, we characterize its solution and create a collection of containment subproblems from this characterization. We solve each subproblem by first restricting certain two-dimensional configuration spaces until a steady state is reached, and then testing for a solution inside the..