Optimally fast incremental Manhattan plane embedding and planar tight span construction

We describe a data structure, a rectangular complex, that can be used to represent hyperconvex metric spaces that have the same topology (although not necessarily the same distance function) as subsets of the plane. We show how to use this data structure to construct the tight span of a metric space given as an n x n distance matrix, when the tight span is homeomorphic to a subset of the plane, in time O(n^2), and to add a single point to a planar tight span in time O(n). As an application of this construction, we show how to test whether a given finite metric space embeds isometrically into the Manhattan plane in time O(n^2), and add a single point to the space and re-test whether it has such an embedding in time O(n).Comment: 39 pages, 15 figure

Dynamic Geodesic Convex Hulls in Dynamic Simple Polygons

We consider the geodesic convex hulls of points in a simple polygonal region in the presence of non-crossing line segments (barriers) that subdivide the region into simply connected faces. We present an algorithm together with data structures for maintaining the geodesic convex hull of points in each face in a sublinear update time under the fully-dynamic setting where both input points and barriers change by insertions and deletions. The algorithm processes a mixed update sequence of insertions and deletions of points and barriers. Each update takes O(n^2/3 log^2 n) time with high probability, where n is the total number of the points and barriers at the moment. Our data structures support basic queries on the geodesic convex hull, each of which takes O(polylog n) time. In addition, we present an algorithm together with data structures for geodesic triangle counting queries under the fully-dynamic setting. With high probability, each update takes O(n^2/3 log n) time, and each query takes O(n^2/3 log n) time

An Algorithmic Framework for Labeling Road Maps

Given an unlabeled road map, we consider, from an algorithmic perspective, the cartographic problem to place non-overlapping road labels embedded in their roads. We first decompose the road network into logically coherent road sections, e.g., parts of roads between two junctions. Based on this decomposition, we present and implement a new and versatile framework for placing labels in road maps such that the number of labeled road sections is maximized. In an experimental evaluation with road maps of 11 major cities we show that our proposed labeling algorithm is both fast in practice and that it reaches near-optimal solution quality, where optimal solutions are obtained by mixed-integer linear programming. In comparison to the standard OpenStreetMap renderer Mapnik, our algorithm labels 31% more road sections in average.Comment: extended version of a paper to appear at GIScience 201

Uncertain Curve Simplification

We study the problem of polygonal curve simplification under uncertainty, where instead of a sequence of exact points, each uncertain point is represented by a region, which contains the (unknown) true location of the vertex. The regions we consider are disks, line segments, convex polygons, and discrete sets of points. We are interested in finding the shortest subsequence of uncertain points such that no matter what the true location of each uncertain point is, the resulting polygonal curve is a valid simplification of the original polygonal curve under the Hausdorff or the Fr\'echet distance. For both these distance measures, we present polynomial-time algorithms for this problem.Comment: 25 pages, 5 figure

On the Minimum Ropelength of Knots and Links

The ropelength of a knot is the quotient of its length and its thickness, the radius of the largest embedded normal tube around the knot. We prove existence and regularity for ropelength minimizers in any knot or link type; these are $C^{1,1}$ curves, but need not be smoother. We improve the lower bound for the ropelength of a nontrivial knot, and establish new ropelength bounds for small knots and links, including some which are sharp.Comment: 29 pages, 14 figures; New version has minor additions and corrections; new section on asymptotic growth of ropelength; several new reference

Approximate convex decomposition and its applications

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

The Skyrme Model: Curved Space, Symmetries and Mass

The presented thesis contains research on topological solitons in (2+1) and (3+1) dimensional classical field theories, focusing upon the Skyrme model. Due to the highly non-linear nature of this model, we must consider various numerical methods to find solutions. We initially consider the (2+1) baby Skyrme model, demonstrating that the currently accepted form of minimal energy solutions, namely straight chains of alternating phase solitons, does not hold for higher charge. Ring solutions with relative phases changing by pi for even configurations or pi-pi/B for odd numbered configurations, are demonstrated to have lower energy than the traditional chain configurations above a certain charge threshold, which is dependant on the parameters of the model. Crystal chunk solutions are then demonstrated to take a lower energy but for extremely high values of charge. We also demonstrate the infinite charge limit of each of the above configurations. Finally, a further possibility of finding lower energy solutions is discussed in the form of soliton networks involving rings/chains and junctions. The dynamics of some of these higher charge solutions are also considered. In chapter 3 we numerically simulate the formation of (2+1)-dimensional baby Skyrmions from domain wall collisions. It is demonstrated that Skyrmion, anti-Skyrmion pairs can be produced from the interaction of two domain walls, however the process can require quite precise conditions. An alternative, more stable, formation process is proposed and simulated as the interaction of more than two segments of domain wall. Finally domain wall networks are considered, demonstrating how Skyrmions may be produced in a complex dynamical system. The broken planar Skyrme model, presented in chapter 4, is a theory that breaks global O(3) symmetry to the dihedral group D_N. This gives a single soliton solution formed of N constituent parts, named partons, that are topologically confined. We show that the configuration of the local energy solutions take the form of polyform structures (planar figures formed by regular N-gons joined along their edges, of which polyiamonds are the N=3 subset). Furthermore, we numerically simulate the dynamics of this model. We then consider the (3+1) SU(2) Skyrme model, introducing the familiar concepts of the model in chapter 5 and then numerically simulating their formation from domain walls. In analogue with the planar case, it is demonstrated that the process can require quite precise conditions and an alternative, more stable, formation process can be achieved with more domain walls, requiring far less constraints on the initial conditions used. The results in chapter 7 discuss the extension of the broken baby Skyrme model to the 3-dimensional SU(2) case. We first consider the affect of breaking the isospin symmetry by altering the tree level mass of one of the pion fields breaking the SO(3) isospin symmetry to an SO(2) symmetry. This serves to exemplify the constituent make up of the Skyrme model from ring like solutions. These rings then link together to form higher charge solutions. Finally the mass term is altered to allow all the fields to have an equivalent tree level mass, but the symmetry of the Lagrangian to be broken, firstly to a dihedral symmetry D_N and then to some polyhedral symmetries. We now move on to discussing both the baby and full SU(2) Skyrme models in curved spaces. In chapter 8 we investigate SU(2) Skyrmions in hyperbolic space. We first demonstrate the link between increasing curvature and the accuracy of the rational map approximation to the minimal energy static solutions. We investigate the link between Skyrmions with massive pions in Euclidean space and the massless case in hyperbolic space, by relating curvature to the pion mass. Crystal chunks are found to be the minimal energy solution for increased curvature as well as increased mass of the model. The dynamics of the hyperbolic model are also simulated, with the similarities and differences to the Euclidean model noted. One of the difficulties of studying the full Skyrme model in (3+1) dimensions is a possible crystal lattice. We hence reduce the dimension of the model and first consider crystal lattices in (2+1)-dimensions. In chapter 9 we first show that the minimal energy solutions take the same form as those from the flat space model. We then present a method of tessellating the Poincare disc model of hyperbolic space with a fundamental cell. The affect this may have on a resulting Skyrme crystal is then discussed and likely problems in simulating this process. We then consider the affects of a pure AdS background on the Skyrme model, starting with the massless baby Skyrme model in chapter 10. The asymptotics and scale of charge 1 massless radial solutions are demonstrated to take a similar form to those of the massive flat space model, with the AdS curvature playing a similar role to the flat space pion mass. Higher charge solutions are then demonstrated to exhibit a concentric ring-like structure, along with transitions (dubbed popcorn transitions in analogy with models of holographic QCD) between different numbers of layers. The 1st popcorn transitions from an n layer to an n+1-layer configuration are observed at topological charges 9 and 27 and further popcorn transitions for higher charges are predicted. Finally, a point-particle approximation for the model is derived and used to successfully predict the ring structures and popcorn transitions for higher charge solitons. The final chapter considers extending the results from the penultimate chapter to the full SU(2) model in a pure AdS_4 background. We make the prediction that the multi-layered concentric ring solutions for the 2-dimensional case would correlate a multi-layered concentric rational map configuration for the 3-dimensional model. The rational map approximation is extended to consider multi-layered maps and the energies demonstrated to reduce the minimal energy solution for charge B=11 which is again dubbed a popcorn transition. Finally we demonstrate that the multi shell structure extends to the full field solutions which are found numerically. We also discuss the affect of combined symmetries on the results which (while likely to be important) appear to be secondary to the dominant effective potential of the metric which simulates a packing problem and hence forces the popcorn transitions to act accordingly with the 2-dimensional model