The results of this thesis concern folding of one-dimensional objects in two dimensions: planar linkages. More precisely, a planar linkage consists of a collection of rigid bars (line segments) connected at their endpoints. Foldings of such a linkage must preserve the connections at endpoints, preserve the bar lengths, and (in our context) prevent bars from crossing. The main result of this thesis is that a planar linkage forming a collection of polygonal arcs and cycles can be folded so that all outermost arcs (not enclosed by other cycles) become straight and all outermost cycles become convex. A complementary result of this thesis is that once a cycle becomes convex, it can be folded into any other convex cycle with the same counterclockwise sequence of bar lengths. Together, these results show that the configuration space of all possible foldings of a planar arc or cycle linkage is connected. These results fall into the broader context of folding and unfolding <I>k</I>-dimensional objects in <i>n</i>-dimensional space, <I>k</I> less than or equal to <I>n</I>. Another contribution of this thesis is a survey of research in this field. The survey revolves around three principal aspects that have received extensive study: linkages in arbitrary dimensions (folding one-dimensional objects in two or more dimensions, including protein folding), paper folding (normally, folding two-dimensional objects in three dimensions), and folding and unfolding polyhedra (two-dimensional objects embedded in three-dimensional space).

Let d > 0 be a fixed integer and let A ⊆ ℝd be a collection of n ≥ d + 2 points which we lift into ℝd+1. Further let k be an integer satisfying 0 ≤ k ≤ n-(d+2) and assign to each k-subset of the points of A a (regular) triangulation obtained by deleting the specified k-subset and projecting down the lower hull of the convex hull of the resulting lifting. Next, for each triangulation we form the characteristic vector defined by Gelfand, Kapranov, and Zelevinsky by assigning to each vertex the sum of the volumes of all adjacent simplices. We then form a vector for the lifting, which we call the k-compound GKZ-vector, by summing all the characteristic vectors. Lastly, we construct a polytope Σk(A) ⊆ ℝ|A| by taking the convex hull of all obtainable k-compound GKZ-vectors by various liftings of A, and note that $\Sigma_0(\A)$ is the well-studied secondary polytope corresponding to A. We will see that by varying k, we obtain a family of polytopes with interesting properties relating to Minkowski sums, Gale transforms, and Lawrence constructions, with the member of the family with maximal k corresponding to a zonotope studied by Billera, Fillamen, and Sturmfels. We will also discuss the case k = d = 1, in which we can provide a combinatorial description of the vertices allowing us to better understand the graph of the polytope and to obtain formulas for the numbers of vertices and edges present.

Thesis (Ph. D.)--Florida State University, 2006. / Advisor: Philip Bowers, Florida State University, College of Arts and Sciences, Dept. of Mathematics. Title and description from dissertation home page (viewed Sept. 15, 2006). Document formatted into pages; contains ix, 98 pages. Includes bibliographical references.

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In an interval society, voters are represented by intervals on the real line, corresponding to their approval sets on a linear political spectrum. I imagine the society to be a representative democracy, and ask how to choose members of the society as representatives. Following work in mathematical psychology by Coombs and others, I develop a measure of the compatibility (political similarity) of two voters. I use this measure to determine the popularity of each voter as a candidate. I then establish local “agreeability” conditions and attempt to find a lower bound for the popularity of the best candidate. Other results about certain special societies are also obtained

L’objet de ce travail est l’utilisation de certains formalismes non classiques (analyses non standard, analyses constructives) afin de proposer des bases théoriques nouvelles autour des problèmes de discrétisations d’objets continus. Ceci est fait en utilisant un modèle discret du système des nombres réels appelé droite d’Harthong-Reeb ainsi que la méthode arithmétisation associée qui est un processus de discrétisation des fonctions continues. Cette étude repose sur un cadre arithmétique non standard. Dans un premier temps, nous utilisons une version axiomatique de l’arithmétique non standard. Puis, dans le but d’améliorer le contenu constructif de notre méthode, nous utilisons une autre approche de l’arithmétique non standard découlant de la théorie des Ω-nombres de Laugwitz et Schmieden. Cette seconde approche amène à une représentation discrète et multi-résolution de fonctions continues.Finalement, nous étudions dans quelles mesures, la droite d’Harthong-Reeb satisfait les axiomes de Bridges décrivant le continu constructif. / The aim of this work is to introduce new theoretical basis for the discretization of continuous objects using non classical formalisms. This is done using a discrete model of the continuum called the Harthong-Reeb line together with the related arithmetization method which is a discretisation process of continuous functions. This study stands on a nonstandard arithmetical framework. Firstly, we use an axiomatic version of nonstandard arithmetic. In order to improve the constructive content of our method, the next step is to use another approach of nonstandard arithmetic deriving from the theory of Ω-numbers by Laugwitzand Schmieden. This second approach leads to a discrete multi-resolution representation of continuous functions. Afterwards, we investigate to what extent the Harthong-Reeb line fits Bridges axioms of the constructive continuum.

Dans cette thèse, nous nous intéressons à la reconnaissance des primitives discrètes multi-échelles. Nous considérons qu'une primitive discrète multi-échelles est une superposition de primitives discrètes de différentes échelles ; et nous proposons des approches qui permettent de déterminer les caractéristiques d'une primitive discrète ou d'une partie d'une primitive discrète.Nous proposons une nouvelle approche de reconnaissance de sous-segment discret qui se base sur des propriétés portant sur l'ordre des restes arithmétiques de la droite discrète. Nous établissons des liens entre les points d'appuis du sous-segment discret et les points ayant des restes arithmétiques minimaux et maximaux sur la droite discrète. D'après les résultats de nos comparaisons, cette approche se relève être plus efficace que des approches existantes.Nous nous intéressons ensuite à des approches de reconnaissance d'arcs et de cercles discrets par le centre généralisé. Nous étudions le dual de la médiatrice généralisée et proposons de calculer le centre généralisé par des calculs de visibilité dans l'espace dual afin de réduire son temps de calcul. Cette approche est valide aussi bien dans une grille régulière que dans une grille irrégulière isothétique.Finalement, nous nous intéressons à des approches de reconnaissance de droite discrète par la préimage généralisée. Nous utilisons la notion de frontière afin de diminuer le nombre d'éléments rentrant dans le calcul de la préimage généralisée ; ce qui simplifie le calcul et réduit le temps de calcul. Cette approche s'applique aussi dans une grille régulière comme dans une grille irrégulière isothétique. / This thesis is about discrete geometry and particularly recognition of multi-scale discrete primitives. We consider that a multiscale discrete primitive is a superimposition of many discrete primitives of different scales. Then we propose approaches of recognition of discrete primitives or parts of a discrete primitives.Firstly we propose a new approach for the recognition of digital subsegment that is based on properties of the sequence of arithmetic remainders of the digital straight line. We show there are sorne links between the leaning points of the digital subsegment and the points that have the minimal and maximal arithmetic remainders on the digital straight line. Based on the results of comparisons with others approaches, the approach seems more efficient. Secondly we present sorne work on improving digital rings and circles recognition by general circumcenter. We use the dual of the generalized bissector in order to simplify the computation of the intersections of generalized bissectors as a polygon stabbing problem. The dual of the generalized bissector is computed likely for pixels of a regular grid or paves of an irregular isothetic grid. Finaly we present some work on improving digital straight line recogrutlon by generalized preimage. To reduce the number of elements to take into account for the computation of the generalized preimage we introduce the concept of boundary. The approach based on boundary could be used in a regular grid or an irregular isothetic grid.

Les appareils d'acquisition d'image 3D sont désormais omniprésents dans plusieurs domaines scientifiques, dont l'imagerie biomédicale, la science des matériaux ou encore l'industrie. La plupart de ces appareils (IRM, scanners à rayons X, micro-tomographes, microscopes confocal, PET scans) produisent un ensemble de données organisées sur une grille régulière que nous nommerons des données digitales, plus couramment des pixels sur des images 2D et des voxels sur des images 3D. Lorsqu'elles sont bien récupérées, ces données approchent la géométrie de la forme capturée (comme des organes en imagerie biomédicale ou des objets dans l'ingénierie). Dans cette thèse, nous nous sommes intéressés à l'extraction de la géométrie sur ces données digitales, et plus précisément, nous nous concentrons à nous approcher des quantités géométriques différentielles comme la courbure sur ces objets. Ces quantités sont les ingrédients critiques de plusieurs applications comme la reconstruction de surface ou la reconnaissance, la correspondance ou la comparaison d'objets. Nous nous focalisons également sur les preuves de convergence asymptotique de ces estimateurs, qui garantissent en quelque sorte la qualité de l'estimation. Plus précisément, lorsque la résolution de l'appareil d'acquisition est augmenté, notre estimation géométrique est plus précise. Notre méthode est basée sur les invariants par intégration et sur l'approximation digitale des intégrations volumiques. Enfin, nous présentons une méthode de classification de la surface, qui analyse les données digitales dans un système à plusieurs échelles et classifie les éléments de surface en trois catégories : les parties lisses, les parties planes, et les parties singulières (discontinuités de la tangente). Ce type de détection de points caractéristiques est utilisé dans plusieurs algorithmes géométriques, comme la compression de maillage ou la reconnaissance d'objet. La stabilité aux paramètres et la robustesse au bruit sont évaluées en fonction des méthodes de la littérature. Tous nos outils pour l'analyse de données digitales sont appliqués à des micro-structures de neige provenant d'un tomographe à rayons X, et leur intérêt est évalué et discuté. / 3D image acquisition devices are now ubiquitous in many domains of science, including biomedical imaging, material science, or manufacturing. Most of these devices (MRI, scanner X, micro-tomography, confocal microscopy, PET scans) produce a set of data organized on a regular grid, which we call digital data, commonly called pixels in 2D images and voxels in 3D images. Properly processed, these data approach the geometry of imaged shapes, like organs in biomedical imagery or objects in engineering. In this thesis, we are interested in extracting the geometry of such digital data, and, more precisely, we focus on approaching geometrical differential quantities such as the curvature of these objects. These quantities are the critical ingredients of several applications like surface reconstruction or object recognition, matching or comparison. We focus on the proof of multigrid convergence of these estimators, which in turn guarantees the quality of estimations. More precisely, when the resolution of the acquisition device is increased, our geometric estimates are more accurate. Our method is based on integral invariants and on digital approximation of volumetric integrals. Finally, we present a surface classification method, which analyzes digital data in a multiscale framework and classifies surface elements into three categories: smooth part, planar part, and singular part (tangent discontinuity). Such feature detection is used in several geometry pipelines, like mesh compression or object recognition. The stability to parameters and the robustness to noise are evaluated with respect to state-of-the-art methods. All our tools for analyzing digital data are applied to 3D X-ray tomography of snow microstructures and their relevance is evaluated and discussed.

In 1946, Erdös posed the distinct distances problem, which asks for the minimum number of distinct distances that any set of n points in the real plane must realize. Erdös showed that any point set must realize at least &Omega(n1/2) distances, but could only provide a construction which offered &Omega(n/&radic(log(n)))$ distances. He conjectured that the actual minimum number of distances was &Omega(n1-&epsilon) for any &epsilon > 0, but that sublinear constructions were possible. This lower bound has been improved over the years, but Erdös' conjecture seemed to hold until in 2010 Larry Guth and Nets Hawk Katz used an incidence theory approach to show any point set must realize at least &Omega(n/log(n)) distances. In this thesis we will explore how incidence theory played a roll in this process and expand upon recent work by Adam Sheffer and Cosmin Pohoata, using geometric incidences to achieve bounds on the bipartite variant of this problem. A consequence of our extensions on their work is that the theoretical upper bound on the original distinct distances problem of &Omega(n/&radic(log(n))) holds for any point set which is structured such that half of the n points lies on an algebraic curve of arbitrary degree.

[pt] No presente trabalho apresentamos um estudo de curvas silhuetas discretas sobre alguns tipos particulares de malhas, com o objetivo de avaliar propriedades dessas curvas. Nosso objeto de estudo são malhas quadrangulares, ou seja, onde todas as faces sejam quadriláteros e também sejam planares. Em particular dois tipos de malhas são discutidas: circular e cônica. Essas malhas são particularmente interessantes em arquitetura para modelagem de estrutura de vidros. A geração das malhas é feita aplicando-se um processo de otimização e em seguida, sobre essas malhas, definimos curvas discretas como candidatas a silhuetas e buscamos medidas de qualidade para essas curvas. / [en] In this work we study discrete silhouette curves on Planar Quad meshes (PQ meshes), with the objective of evaluate some properties of these curves. PQ meshes correspond to planar quadrilaterals meshes, and our interest is focused particularly on two kinds of meshes: Conical and Circular. They are interesting in architecture for design with glass structures. An optimization process is applied for the mesh generation and we follow defining discrete curves on the meshes to obtain silhouette and to measure their quality.