Computing Straight Skeletons and Motorcycle Graphs: Theory and Practice

The straight skeleton is a geometric structure that is similar to generalizedVoronoi diagrams. Straight skeletons were introduced to the field of computational geometry one and a half decades ago. Since then many industrial and academical applications emerged, such as the computation of mitered offset curves, automatic roof construction, solving fold-and-cut problems and the reconstruction of surfaces, to name the most prominent ones. However, there is a significant gap between the most efficient straight-skeleton algorithms and implementations, on one hand, and the best known lower runtime bounds on the other hand. The primary goal of this thesis is the development of an algorithm that is suitable for implementation and efficient in terms of time and space in order to make the advantages of straight skeletons available to real-world applications. We start with investigations concerning upper and lower bounds on the number of socalled flip events that occur in the triangulation-based straight-skeleton algorithm by Aichholzer and Aurenhammer. In particular, we prove the existence of Steiner triangulations that are free of flip events. This result motivates a novel straight-skeleton algorithm for non-degenerate simple polygons that is based on the so-called motorcycle graph. In order to extend this algorithm to arbitrary planar straight-line graphs, we carefully generalize the motorcycle graph. This generalization leads to practical and theoretical applications: Firstly, we obtain an extension of the alternative characterization of straight skeletons by Cheng and Vigneron to planar straight-line graphs. Secondly, this characterization motivates a straight-skeleton algorithm that is based on 3D graphics hardware. Thirdly, the generalized motorcycle graph leads to a wavefront-type straight-skeleton algorithm for arbitrary planar straight-line graphs. Our algorithm is easy to implement, has a theoretical worstcase time complexity of O(n2 log n) and operates in O(n) space. Extensive runtime tests with our implementation Bone exhibit an actual runtime of O(n log n) on a database containing more than 13 500 datasets of different characteristics. In practice, this constitutes an improvement of a linear factor in time and space compared to the current state-of-the-art straight-skeleton code, which is shipped with the CGAL library. In particular, Bone performs up to 100 times faster than the current CGAL code on datasets with a few thousand vertices, requires significant less memory and accepts more general input. Our straight-skeleton algorithm motivates the investigation of motorcycle graphs and their practical computation. We start with stochastic considerations of the average length of a motorcycles trace. The results obtained motivate a simple yet fast algorithm that employs geometric hashing. Runtime tests with our implementation Moca exhibit an O(n log n) runtime on the vast majority of our datasets. Finally, we revisit the geometric relation of straight skeletons and motorcycle graph. We present an algorithm that constructs planar straight-line graphs whose straight skeleton approximates any given motorcycle graph to an arbitrary precision. This algorithm finally leads to a P-completeness proof for straight skeletons of polygons with holes that is based on the P-completeness of motorcycle graphs. A C K N O W L E D G M E N T S I would like to thank my parents, Stefan and Gabriele, for their unlimited and unconditional support throughout my entire life. Special thanks go to my girlfriend Christina for her constant support and understanding during my doctoral studies. I also thank my supervisor Martin Held for his guidance and his extensive support. Further, I thank my colleagues and co-workers. In particular I thank Gerhard Mitterlechner and Roland Kwitt for their proofreading and valuable discussions. Eventually, I thank theAustrian Science Fund (FWF) for supporting this thesis by funding project no. L367-N15.

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