Compaction Algorithms for Non- Convex Polygons and Their Applications The Harvard community has made this article openly available. Please share how this access benefits you. Your story matters Citation Li, Zhenyu. 1994. Compaction Algorithms for Non-Convex Polygons and Their Applications. Harvard Computer Science Group Technical Report TR-15-94. Citable link http://nrs.harvard.edu/urn-3:HUL.InstRepos:25619464 Terms of Use This article was downloaded from Harvard University’s DASH repository, and is made available under the terms and conditions applicable to Other Posted Material, as set forth at http:// nrs.harvard.edu/urn-3:HUL.InstRepos:dash.current.terms-of- use#LAA Compaction Algorithms for NonConvex Polygons and Their Applications A thesis presented by Zhenyu Li to The Division of Applied Sciences in partial fulllment of the requirement for the degree of Do ctor of Philosophy in the sub ject of Computer Science Harvard University Cambridge Massachusetts May c by Zhenyu Li All rights reserved Contents List of Figures Introduction Automated Marker Making Cutting and Packing Problems Twodimensional Packing and Marker Making Pro ject Background Problem Representation for Marker Making Compaction Description and Examples Motivation Denitions and a Lower Bound Related Work Contribution A PositionBased Optimization Mo del A VelocityBased Optimization Mo del Complexity Organization of the Thesis Compaction Using a VelocityBased Optimization Mo del High Level Description of the Compaction Algorithm Collision Detection Collision Detection For Two Translating Polygons A Collision Detection Algorithm Finding Vertexedge Touching Pairs Type of Contact A Simple Algorithm A Sweepline Algorithm An Alternative Algorithm Nonp enetration Constraints Bounds on the Velocities NonPenetration Constraints for VertexVertex Contacts Performance The Theory of Minkowski Sum and Dierence Background Polygon Intersection and Containment Problems Conguration Space Approach Denitions and Prop erties Applications Intersection and Containment Intersection Containment Algorithms for Computing Minkowski Sums Convex Polygons Simple Polygons Starshap ed Polygons Monotone Polygons Compaction Using a PositionBased Optimization Mo del The Theory of a PositionBased Optimization Mo del NonOverlapping Conditions for Two Translating Polygons A Lo cality Heuristic Linear Constraints with the Boundaries of the Container The Positionbased Compaction Algorithm Running Time and Robustness Compaction Functions Leftward Compaction Vector Compaction Bumping Gravity Compaction Separation of Overlapping Polygons and Database Driven Marker Mak ing Denition and Complexity Algorithm for Separating Overlapping Polygons Layout Made Easy Database Driven Automated Marker Making Shap e Matching Criteria An Example Cut Planning Compaction with Small Rotations Rotational Compaction by Relaxation Translational Relaxation of a Single Polygon Rotational Compaction of a Single Polygon Algorithm for Rotational Compaction Using Relaxation Rotational Compaction Using Linearization Formulation for Translation Only Compaction Other VertexEdge Supp orting Pair Linearization The Algorithm Examples Comparison of the Two Rotational Compaction Metho ds Floating Distance Between Polygons Controlling the Distance Between Polygons Linear Constraints for Floating Maximize the Minimum Distance Between Polygons Separating Polygons by a Sp ecic Distance Maximizing the Overall Distance b etween Polygons Uniform Distribution of Free Areas Maximize the Minimum Distance Between Polygons An Alternative Metho d Floating For Overlapped Layouts Mixed Integer Programming Mo del for Compaction and TwoDimensional Packing Limitation of the Lo cality Heuristic MIP Formulation for Optimal TwoDimensional PackingCompaction Algorithms for Finding Convex Covering MIP Formulation for Multiple Containment Problem The Complexity of the Compaction Problem Introduction The PSPACEHardness of Compaction Review of the Warehouseman Problem PSPACEhardness Pro of Compaction in an Exp onential Number of Moves Finding a Lo cal Minimum Requires an Exp onential Number of Moves Conclusion A Vectors and Cross Pro ducts Bibliography List of Figures A human generated pants marker in apparel manufacturing Marker co ordinate systems and p oints on the b oundary of a piece Another leftward compaction example General compaction minimizing the area of the b ounding rectangle Strip compaction minimizing the length of a xed width b ounding rectangle Reduction of strip compaction to general compaction Reduction of PARTITION to strip compaction An example of vertexedge contact An example of vertexvertex contact a The vertex A of P touches the edge BC of Q P moves with velocity u and Q moves with velocity v b The p osition of P and Q after time interval t A vertexedge supp orting pair The Minkowski sum for a circle and a square The Minkowski sum and intersection detection Minkowski sum and nonoverlapping placement The Minkowski dierence and p olygon containment problem The Minkowski sum of two starshap ed p olygons The nearest convex region in the exterior of the Minkowski sum An example of leftward compaction input An example of leftward compaction output The human generated pants marker in Figure after leftward compaction Left A human generated pants marker RightThe human generated marker after compaction An example of vector compaction An example of op ening a gap An example of gravity compaction Reduction of PARTITION to separation of overlapping p olygons Minkowski sum of two slightly overlapped p olygons Marker generated by matching and substitution Marker after elimination of overlaps and leftward compaction Length in eciency Marker after adjustment on small p olygons Length in eciency The corresp onding marker generated by a human Length eciency An example of translational relaxation for p olygon An example of rotational relaxation for p olygon An example of rotational compaction using relaxation a human generated marker and the result rotational compaction using relaxation An.