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.

of a doctoral dissertation at the University of Miami.Dissertation supervised by Professor Victor J. Milenkovic. No. of pages in text: 162A new paradigm for rigid body simulation is presented and analyzed. Current techniques for rigid body simulation run slowly on scenes with many bodies in close proximity. Each time two bodies collide or make or break a static contact, the simulator must interrupt the numerical integration of velocities and accelerations. Even for simple scenes, the number of discontinuities per frame time can rise to the millions. An efficient optimization-based animation (OBA) algorithm is presented which can simulate scenes with many convex threedimensional bodies settling into stacks and other "crowded" arrangements. This algorithm simulates Newtonian (second order) physics and Coulomb friction, and it uses quadratic programming (QP) to calculate new positions, momenta, and accelerations strictly at frame times. The extremely small integration steps inherent to traditional simulation techniques are avoided.Contact points are synchronized at the end of each frame. Resolving contacts with friction is known to be a difficult problem. Analytic force calculation can have ambiguous or non-existing solutions. Purely impulsive techniques avoid these ambiguous cases, but still require an excessive and computationally expensive number of updates in the case of many simultaneous contacts. It is shown informally that even taking into account advances in stiff integration techniques, penalty force methods cannot overcome this issue of running time in highly crowded scenes. New algorithms are presented that calculate simultaneous impulses to resolve collisions and static contacts under the Coulomb friction model. The simultaneous impulses are the solution to a QP.In addition, the algorithms apply "bouncing at distance" and "freezing of bodies" to further speed up the simulation. These new QP algorithms are hybridized with a traditional priority queue momentum update scheme to allow sequential impulses when they are required for realism, such as in the office toy pendulum. When added to the implementation of OBA, these new algorithms increase the speed of the simulation by a factor of up to 30.The position update has been hybridized with retroactive detection (RD) to prevent fast and thin bodies from passing through each other. Due to the modular design of the OBA simulator, the described techniques can be used as components in any existing simulator that follows a modular design of position update, finding contacts, and resolving contacts. Non-convex bodies are simulated as unions of convex bodies. Links and joints are simulated with bi-directional constraints. Analysis of the algorithm and discussion of example simulations are provided. DedicationIn loving memory of my father.iii

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An algorithm and a robust floating point implementation is given for rotational polygon containment: given polygons P 1 , P 2 , P 3 ,. .. , P k and a container polygon C, find rotations and translations for the k polygons that place them into the container without overlapping. A version of the algorithm and implementation also solves rotational minimum enclosure: given a class C of container polygons, find a container C ∈ C of minimum area for which containment has a solution. The minimum enclosure is approximate: it bounds the minimum area between (1 − ε)A and A. Experiments indicate that finding the minimum enclosure is practical for k = 2, 3 but not larger unless optimality is sacrificed or angles ranges are limited (although these solutions can still be useful). Important applications for these algorithm to industrial problems are discussed. The paper also gives practical algorithms and numerical techniques for robustly calculating polygon set intersection, Minkowski sum, and range intersection: the intersection of a polygon with itself as it rotates through a range of angles. In particular, it introduces nearest pair rounding, which allows all these calculations to be carried out in rounded floating point arithmetic.