Compaction and Separation Algorithms for Non-Convex Polygons and Their Applications

Given a two dimensional, non-overlapping layout of convex and non-convex polygons, compaction can be thought of as simulating the motion of the polygons as a result of applied \forces." We apply compaction to improve the material utilization of an already tightly packed layout. Compaction can be modeled as a motion of the polygons that reduces the value of some functional on their positions. Optimal compaction, planning a motion that reaches a layout that has the global minimum functional value among all reachable layouts, is shown to be NP-complete under certain assumptions. We rst present a compaction algorithm based on existing physical simulation approaches. This algorithm uses a new velocity-based optimization model. Our experimental results reveal the limitation of physical simulation: even though our new model improves the running time of our algorithm over previous simulation algorithms, the algorithm still can not compact typical layouts of one hundred or more polygons in a reasonable amount of time. The essential di culty of physical based models is that they can only generate velocities for the polygons, and the nal positions must be generated by numerical integration. We present a new position-based optimization model that allows us to calculate directly new polygon positions via linear programming that are at a local minimum of the objective. The new model yields a translational compaction algorithm that runs two orders of magnitude faster than physical simulation methods. We also consider the problem of separating overlapping polygons using a minimal amount of motion and show it to be NP-complete. Although this separation problem looks quite di erent from the compaction problem, our new model also yields an e cient algorithm to solve it. The compaction/separation algorithms have been applied to marker making: the task of packing polygonal pieces on a sheet of cloth of xed width so that total length is minimized. The compaction algorithm has improved cloth utilization of human generated pants markers. The separation algorithm together with a database of human-generated markers can be used for automatic generation of markers that approach human performance.