905 research outputs found

    On the geometric dilation of closed curves, graphs, and point sets

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    The detour between two points u and v (on edges or vertices) of an embedded planar graph whose edges are curves is the ratio between the shortest path in in the graph between u and v and their Euclidean distance. The maximum detour over all pairs of points is called the geometric dilation. Ebbers-Baumann, Gruene and Klein have shown that every finite point set is contained in a planar graph whose geometric dilation is at most 1.678, and some point sets require graphs with dilation at least pi/2 = 1.57... We prove a stronger lower bound of 1.00000000001*pi/2 by relating graphs with small dilation to a problem of packing and covering the plane by circular disks. The proof relies on halving pairs, pairs of points dividing a given closed curve C in two parts of equal length, and their minimum and maximum distances h and H. Additionally, we analyze curves of constant halving distance (h=H), examine the relation of h to other geometric quantities and prove some new dilation bounds.Comment: 31 pages, 16 figures. The new version is the extended journal submission; it includes additional material from a conference submission (ref. [6] in the paper

    Algorithmic and Combinatorial Results on Fence Patrolling, Polygon Cutting and Geometric Spanners

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    The purpose of this dissertation is to study problems that lie at the intersection of geometry and computer science. We have studied and obtained several results from three different areas, namely–geometric spanners, polygon cutting, and fence patrolling. Specifically, we have designed and analyzed algorithms along with various combinatorial results in these three areas. For geometric spanners, we have obtained combinatorial results regarding lower bounds on worst case dilation of plane spanners. We also have studied low degree plane lattice spanners, both square and hexagonal, of low dilation. Next, for polygon cutting, we have designed and analyzed algorithms for cutting out polygon collections drawn on a piece of planar material using the three geometric models of saw, namely, line, ray and segment cuts. For fence patrolling, we have designed several strategies for robots patrolling both open and closed fences

    Moduli of Tropical Plane Curves

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    We study the moduli space of metric graphs that arise from tropical plane curves. There are far fewer such graphs than tropicalizations of classical plane curves. For fixed genus gg, our moduli space is a stacky fan whose cones are indexed by regular unimodular triangulations of Newton polygons with gg interior lattice points. It has dimension 2g+12g+1 unless g≀3g \leq 3 or g=7g = 7. We compute these spaces explicitly for g≀5g \leq 5.Comment: 31 pages, 25 figure

    Lower bounds on the dilation of plane spanners

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    (I) We exhibit a set of 23 points in the plane that has dilation at least 1.43081.4308, improving the previously best lower bound of 1.41611.4161 for the worst-case dilation of plane spanners. (II) For every integer n≄13n\geq13, there exists an nn-element point set SS such that the degree 3 dilation of SS denoted by ÎŽ0(S,3) equals 1+3=2.7321
\delta_0(S,3) \text{ equals } 1+\sqrt{3}=2.7321\ldots in the domain of plane geometric spanners. In the same domain, we show that for every integer n≄6n\geq6, there exists a an nn-element point set SS such that the degree 4 dilation of SS denoted by ÎŽ0(S,4) equals 1+(5−5)/2=2.1755
\delta_0(S,4) \text{ equals } 1 + \sqrt{(5-\sqrt{5})/2}=2.1755\ldots The previous best lower bound of 1.41611.4161 holds for any degree. (III) For every integer n≄6n\geq6 , there exists an nn-element point set SS such that the stretch factor of the greedy triangulation of SS is at least 2.02682.0268.Comment: Revised definitions in the introduction; 23 pages, 15 figures; 2 table

    Geometric Dilation and Halving Distance

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    Let us consider the network of streets of a city represented by a geometric graph G in the plane. The vertices of G represent the crossroads and the edges represent the streets. The latter do not have to be straight line segments, they may be curved. If one wants to drive from a place p to some other place q, normally the length of the shortest path along streets, d_G(p,q), is bigger than the airline distance (Euclidean distance) |pq|. The (relative) DETOUR is defined as delta_G(p,q) := d_G(p,q)/|pq|. The supremum of all these ratios is called the GEOMETRIC DILATION of G. It measures the quality of the network. A small dilation value guarantees that there is no bigger detour between any two points. Given a finite point set S, we would like to know the smallest possible dilation of any graph that contains the given points on its edges. We call this infimum the DILATION of S and denote it by delta(S). The main results of this thesis are - a general upper bound to the dilation of any finite point set S, delta(S) - a lower bound for a specific set P, delta(P)>(1+10^(-11))pi/2, which approximately equals 1.571 In order to achieve these results, we first consider closed curves. Their dilation depends on the HALVING PAIRS, pairs of points which divide the closed curve in two parts of equal length. In particular the distance between the two points is essential, the HALVING DISTANCE. A transformation technique based on halving pairs, the HALVING PAIR TRANSFORMATION, and the curve formed by the midpoints of the halving pairs, the MIDPOINT CURVE, help us to derive lower bounds to dilation. For constructing graphs of small dilation, we use ZINDLER CURVES. These are closed curves of constant halving distance. To give a structured overview, the mathematical apparatus for deriving the main results of this thesis includes - upper bound: * the construction of certain Zindler curves to generate a periodic graph of small dilation * an embedding argument based on a number theoretical result by Dirichlet - lower bound: * the formulation and analysis of the halving pair transformation * a stability result for the dilation of closed curves based on this transformation and the midpoint curve * the application of a disk-packing result In addition, this thesis contains - a detailed analysis of the dilation of closed curves - a collection of inequalities, which relate halving distance to other important quantities from convex geometry, and their proofs; including four new inequalities - the rediscovery of Zindler curves and a compact presentation of their properties - a proof of the applied disk packing result.Geometrische Dilation und Halbierungsabstand Man kann das von den Straßen einer Stadt gebildete Netzwerk durch einen geometrischen Graphen in der Ebene darstellen. Die Knoten dieses Graphen reprĂ€sentieren die Kreuzungen und die Kanten sind die Straßen. Letztere mĂŒssen nicht geradlinig sein, sondern können beliebig gekrĂŒmmt sein. Wenn man nun von einem Ort p zu einem anderen Ort q fahren möchte, dann ist normalerweise die LĂ€nge des kĂŒrzesten Pfades ĂŒber Straßen, d_G(p,q), lĂ€nger als der Luftlinienabstand (euklidischer Abstand) |pq|. Der (relative) UMWEG (DETOUR) ist definiert als delta_G(p,q) := d_G(p,q)/|pq|. Das Supremum all dieser BrĂŒche wird GEOMETRISCHE DILATION (GEOMETRIC DILATION) von G genannt. Es ist ein Maß fĂŒr die QualitĂ€t des Straßennetzes. Ein kleiner Dilationswert garantiert, dass es keinen grĂ¶ĂŸeren Umweg zwischen beliebigen zwei Punkten gibt. FĂŒr eine gegebene endliche Punktmenge S wĂŒrden wir nun gerne bestimmen, was der kleinste Dilationswert ist, den wir mit einem Graphen erreichen können, der die gegebenen Punkte auf seinen Kanten enthĂ€lt. Dieses Infimum nennen wir die DILATION von S und schreiben kurz delta(S). Die Haupt-Ergebnisse dieser Arbeit sind - eine allgemeine obere Schranke fĂŒr die Dilation jeder beliebigen endlichen Punktmenge S: delta(S) - eine untere Schranke fĂŒr eine bestimmte Menge P: delta(P)>(1+10^(-11))pi/2, was ungefĂ€hr der Zahl 1.571 entspricht Um diese Ergebnisse zu erreichen, betrachten wir zunĂ€chst geschlossene Kurven. Ihre Dilation hĂ€ngt von sogenannten HALBIERUNGSPAAREN (HALVING PAIRS) ab. Das sind Punktpaare, die die geschlossene Kurve in zwei Teile gleicher LĂ€nge teilen. Besonders der Abstand der beiden Punkte ist von Bedeutung, der HALBIERUNGSABSTAND (HALVING DISTANCE). Eine auf den Halbierungspaaren aufbauende Transformation, die HALBIERUNGSPAARTRANSFORMATION (HALVING PAIR TRANSFORMATION), und die von den Mittelpunkten der Halbierungspaare gebildete Kurve, die MITTELPUNKTKURVE (MIDPOINT CURVE), helfen uns untere Dilationsschranken herzuleiten. Zur Konstruktion von Graphen mit kleiner Dilation benutzen wir ZINDLERKURVEN (ZINDLER CURVES). Dies sind geschlossene Kurven mit konstantem Halbierungspaarabstand. Die mathematischen Hilfsmittel, mit deren Hilfe wir schließlich die Hauptresultate beweisen, sind unter anderem - obere Schranke: * die Konstruktion von bestimmten Zindlerkurven, mit denen periodische Graphen kleiner Dilation gebildet werden können * ein Einbettungsargument, das einen zahlentheoretischen Satz von Dirichlet benutzt - untere Schranke: * die Definition und Analyse der Halbierungspaartransformation * ein StabilitĂ€tsresultat fĂŒr die Dilation geschlossener Kurven, das auf dieser Transformation und der Mittelpunktkurve basiert * die Anwendung eines Kreispackungssatzes ZusĂ€tzlich enthĂ€lt diese Dissertation - eine detaillierte Analyse der Dilation geschlossener Kurven - eine Sammlung von Ungleichungen, die den Halbierungsabstand zu anderen wichtigen GrĂ¶ĂŸen der Konvexgeometrie in Beziehung setzen, und ihre Beweise; inklusive vier neuer Ungleichungen - die Wiederentdeckung von Zindlerkurven und eine kompakte Darstellung ihrer Eigenschaften - einen Beweis des angewendeten Kreispackungssatzes

    Computer tool for maximizing the placement of congruent polyhedra

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    Given multiple identical polyhedral objects and a parallelepiped container, how should one place the objects so that the largest number fits inside the container? This simple question is important in many applications, yet the answer is elusive. In fact, we know of no published solution for this very general formulation. Still, in many circumstances, further restrictions apply, resulting in a large number of variations requiring different algorithmic strategies. This paper is the continuation of [12] and focus on the fundamental concepts and tools that are used for this kind of problem, such as the no-fit polygon. We also present some of its many variations, giving in particular one that applies to the stereolithographic rapid prototyping technology
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