Approximate convex decomposition (ACD) is a technique that partitions an input object into "approximately convex" components. Decomposition into approximately convex pieces is both more efficient to compute than exact convex decomposition and can also generate a more manageable number of components. It can be used as a basis of divide-and-conquer algorithms for applications such as collision detection, skeleton extraction and mesh generation. In this paper, we propose a new method called Fast Approximate Convex Decomposition (FACD) that improves the quality of the decomposition and reduces the cost of computing it for both 2D and 3D models. In particular, we propose a new strategy for evaluating potential cuts that aims to reduce the relative concavity, rather than absolute concavity. As shown in our results, this leads to more natural and smaller decompositions that include components for small but important features such as toes or fingers while not decomposing larger components, such as the torso that may have concavities due to surface texture. Second, instead of decomposing a component into two pieces at each step, as in the original ACD, we propose a new strategy that uses a dynamic programming approach to select a set of n_c non-crossing (independent) cuts that can be simultaneously applied to decompose the component into n_c + 1 components. This reduces the depth of recursion and, together with a more efficient method for computing the concavity measure, leads to significant gains in efficiency. We provide comparative results for 2D and 3D models illustrating the improvements obtained by FACD over ACD and we compare with the segmentation methods given in the Princeton Shape Benchmark.

Contact structures on 3-manifolds are 2-plane fields satisfying a set of conditions. The study of contact structures can be traced back for over two-hundred years, and has been of interest to mathematicians such as Hamilton, Jacobi, Cartan, and Darboux. In the late 1900's, the study of these structures gained momentum as the work of Eliashberg and Bennequin described subtleties in these structures that could be used to find new invariants. In particular, it was discovered that contact structures fell into two classes: tight and overtwisted. While overtwisted contact structures are relatively well understood, tight contact structures remain an area of active research. One area of active study, in particular, is the classification of tight contact structures on 3-manifolds. This began with Eliashberg, who showed that the standard contact structure in real three-dimensional space is unique, and it has been expanded on since. Some major advancements and new techniques were introduced by Kanda, Honda, Etnyre, Kazez, Matić, and others. Convex decomposition theory was one product of these explorations. This technique involves cutting a manifold along convex surfaces (i.e. surfaces arranged in a particular way in relation to the contact structure) and investigating a particular set on these cutting surfaces to say something about the original contact structure. In the cases where the cutting surfaces are fairly nice, in some sense, Honda established a correspondence between information on the cutting surfaces and the tight contact structures supported by the original manifold. In this thesis, convex surface theory is applied to the case of handlebodies with a restricted class of dividing sets. For some cases, classification is achieved, and for others, some interesting patterns arise and are investigated.

Geometric computations are essential in many real-world problems. One important issue in geometric computations is that the geometric models in these problems can be so large that computations on them have infeasible storage or computation time requirements. Decomposition is a technique commonly used to partition complex models into simpler components. Whereas decomposition into convex components results in pieces that are easy to process, such decompositions can be costly to construct and can result in representations with an unmanageable number of components. In this work, we have developed an approximate technique, called Approximate Convex Decomposition (ACD), which decomposes a given polygon or polyhedron into "approximately convex" pieces that may provide similar benefits as convex components, while the resulting decomposition is both significantly smaller (typically by orders of magnitude) and can be computed more efficently. Indeed, for many applications, an ACD can represent the important structural features of the model more accurately by providing a mechanism for ignoring less significant features, such as wrinkles and surface texture. Our study of a wide range of applications shows that in addition to providing computational efficiency, ACD also provides natural multi-resolution or hierarchical representations. In this dissertation, we provide some examples of ACD's many potential applications, such as particle simulation, mesh generation, motion planning, and skeleton extraction.

This work focuses on the problem that pertains to area searching with UAVs. Specifically developing algorithms that generate flight paths that are short with- out sacrificing flyability. For instance, very sharp turns will compromise flyability since fixed wing aircraft cannot make very sharp turns. This thesis provides an analysis of different types of search methods, area decompositions, and combi- nations thereof. The search methods used are side to side searching and spiral searching. In side to side searching the aircraft goes back and forth only making 90-degree turns. Spiral search searches the shape in a spiral pattern starting on the outer perimeter working its way in. The idea being that it should generate flight paths that are easy to fly since all turns should be with a large turn radii. Area decomposition is done to divide complex shapes into smaller more manage- able shapes. The report concludes that with the implemented methods the side to side scanning method without area decomposition yields good and above all very reliable results. The reliability stems from the fact that all turns are 90 degrees and that algorithm never get stuck or makes bad mistakes. Only having 90 degree turns results in only four different types of turns. This allows the airplanes behav- ior along the route to be predictable after flying the first four turns. Although this assumes that the strength of the wind is a greater influence than the turbulences effect on the aircraft’s flight characteristics. This is a very valuable feature for an operator in charge of a flight. The other tested methods and area decompositions often yield a shorter flight path, however, despite extensive adjustments to the algorithms they never came to handle all cases in a satisfactory manner. These methods may also generate any kind of turn at any time, including turns of nearly 180 degrees. These turns can lead to an airplane missing the intended flight path and thus missing to scan the intended area properly. Area decomposition proves to be really effective only when the area has many protrusions that stick out in different directions, think of a starfish shape. In these cases the side to side algo- rithm generate a path that has long legs over parts that are not in the search area. When the area is decomposed the algorithm starts with, for example, one arm of the starfish at a time and then search the rest of the arms and body in turn.

Baigiamajame magistro darbe nagrinėjama paprasto skylėto daugiakampio skaidymo į dalis, kurių viršūnių skaičius neviršyja nustatyto skaičiaus problema. Apibrėžiamas uždavinys ir jo svarba. Apžvelgiami egzistuojantys skaidymo algoritmai, padedantys išspręsti uždavinį, bei jų realizacijos. Pateikiamos trianguliacijos ir padalinimo į apytiksliai iškilius daugiakampius algoritmų modifikacijos, jų privalumai ir trūkumai. Įvertinamas šių modifikuotų algoritmų sudėtingumas. Eksperimentinėje dalyje pateikiami skaičiavimo eksperimentų rezultatai, jų analizė ir palyginimas su teoriniais algoritmų sudėtingumo įverčiais. Remiantis skaičiavimo eksperimentų rezultatais pateikiamos išvados ir siūlymai. / This study deals with decomposition of simple polygon with holes into components so that every piece does not exceed some defined number of vertices. We define the problem and its appliances. Existing studies and algorithms for polygon decomposition are covered. We propose modifications of polygon triangulation and approximate convex decomposition algorithms. Also the complexity analysis of both algorithms is made. In the experimental part of the work results of computing experiments are presented, analyzed and compared to the theoretical complexity bounds.