How many triangles does it take to make a square? The answer is simple: two. This problem has a direct analogue in dimensions three and higher, but the answers are much harder to find. We provide new lower bounds in dimensions 4 through 13, an asymptotic lower bound which is inferior to the best known bound in high dimensions, and some new ideas which produce good upper bounds in both low and high dimensions.

Data warehouses store large volumes of consolidated and historized multidimensional data to be explored and analysed by various users. The data exploration is a process of searching relevant information in a dataset. In this thesis, the dataset to explore is a data cube which is an extract of the data warehouse that users query by launching sequences of OLAP (On-Line Analytical Processing) queries. However, this volume of information can be very large and diversified, it is thus necessary to help the user to face this problem by guiding him/her in his/her data cube exploration in order to find relevant information. The present work aims to propose recommendations, as OLAP queries, to a user querying a data cube. This proposal benefits from what the other users did during their previous explorations of the same data cube. We start by presenting an overview of the used framework and techniques in Information Retrieval, Web Usage Mining or e-commerce. Then, inspired by this framework, we present a state of the art on collaborative assistance for data exploration in (relationnal and multidimensional) databases. It enables us to release work axes in the context of multidimensional databases. Thereafter, we propose thus a generic framework to generate recommendations, generic in the sense that the three steps of the process are customizable. Thus, given a set of sequences of queries, corresponding to the previous explorations of various users, and given the sequence of queries of the current user, our framework proposes a set of queries as recommendations following his/her sequence. Then, various instantiations of our framework are proposed. Then, we present a Java prototype allowing a user to specify his/her current sequence of queries and it returns a set of recommendations. This prototype validates our approach and its effectiveness thanks to an experimentations collection. Finally, in order to improve this data cube exploration collaborative assistance and, in particular, to share, navigate or annotate the launched queries, we propose a framework to manage queries. Thus, an instantiation to manage recommendations is presented. / Les entrepôts de données stockent de gros volumes de données multidimensionnelles, consolidées et historisées dans le but d'être explorées et analysées par différents utilisateurs. L'exploration de données est un processus de recherche d'informations pertinentes au sein d'un ensemble de données. Dans le cadre de nos travaux, l'ensemble de données à explorer est un cube de données qui est un extrait de l'entrepôt de données que les utilisateurs interrogent en lançant des séquences de requêtes OLAP (On-Line Analytical Processing). Cependant, cette masse d'informations à explorer peut être très importante et variée, il est donc nécessaire d'aider l'utilisateur à y faire face en le guidant dans son exploration du cube de données afin qu'il trouve des informations pertinentes. Le travail présenté dans cette thèse a pour objectif de proposer des recommandations, sous forme de requêtes OLAP, à un utilisateur interrogeant un cube de données. Cette proposition tire parti de ce qu'ont fait les autres utilisateurs lors de leurs précédentes explorations du même cube de données. Nous commençons par présenter un aperçu du cadre et des techniques utilisés en Recherche d'Informations, Exploration des Usages du Web ou e-commerce. Puis, en nous inspirant de ce cadre, nous présentons un état de l'art sur l'aide à l'exploration des bases de données (relationnelles et multidimensionnelles). Cela nous permet de dégager des axes de travail dans le contexte des bases de données multidimensionnelles. Par la suite, nous proposons donc un cadre générique de génération de recommandations, générique dans le sens où les trois étapes du processus sont paramétrables. Ainsi, à partir d'un ensemble de séquences de requêtes, correspondant aux explorations du cube de données faites précédemment par différents utilisateurs, et de la séquence de requêtes de l'utilisateur courant, notre cadre propose un ensemble de requêtes pouvant faire suite à la séquence de requêtes courante. Puis, diverses instanciations de ce cadre sont proposées. Nous présentons ensuite un prototype écrit en Java. Il permet à un utilisateur de spécifier sa séquence de requêtes courante et lui renvoie un ensemble de recommandations. Ce prototype nous permet de valider notre approche et d'en vérifier l'efficacité avec un série d'expérimentations. Finalement, afin d'améliorer cette aide collaborative à l'exploration de cubes de données et de permettre, notamment, le partage de requêtes, la navigation au sein des requêtes posées sur le cube de données, ou encore de les annoter, nous proposons un cadre d'organisation de requêtes. Ainsi, une instanciation adaptée à la gestion des recommandations est présentée.

Le sujet de thèse s’inscrit dans le cadre de la lutte contre le cancer. La radiothérapie externe est une technique de traitement de cibles tumorales par des particules radioactives, sous guidage d’un système d’imagerie. L’efficacité de ce traitement à ciblage macroscopique dépend en partie de l’évolution de la cible biologique au cours du traitement, qui se programme sur plusieurs séances. Dans notre travail, nous nous sommes concentrés sur l’évolution temporelle de la cible biologique séance par séance. Le but est d’estimer l’espace couvert par son mouvement, et permettant de prédire le positionnement futur de la cible et corriger ou adapter le plan de traitement. Notre cible biologique microscopique dans le cas du cancer ORL est décrite par la glande parotide qui est sensible aux radiations. L’objectif est de minimiser l’accessibilité des rayons à cet organe lorsque la cible tumorale se trouve dans sa périphérie. Nous avons proposé deux méthodes de reconstruction en 3D de la parotide, à savoir B-Spline de Bézier bi-cubique et Marching cubes. Une fois la reconstruction est effectuée, le volume de la glande parotide, à chaque session est calculé. Une zone de sécurité autour de la glande parotide a été définie. Enfin, une classification des patients a été réalisée en se basant sur le dosage associé au traitement. A un niveau macroscopique, nous avons développé une approche d'aide à la décision pour affiner la position relative du patient sur la table de traitement. Cette approche est basée sur la modélisation par réseaux de neurones et spécifiquement MLP et RBF. La performance entre ces deux techniques a été analysée, suivant les positionnements géométriques des patients. / The thesis subject is defined in the framework of the fight against cancer. External radiation therapy is a technical treatment of tumor targets by radioactive particles, under guidance of an imaging system. The effectiveness of this macroscopic targeting treatment depends in part on the evolution and movement of the biological target during the treatment, which is planned over several sessions. In our work, we focused to the temporal evolution of the biological target session by session. The goal is to estimate the space covered by the target movement in each session, and to predict the future position of the target and thus to correct or adapt the treatment plan. Our microscopic biological target in the case of head and neck cancer is described by the parotid gland which is sensitive to the radiation. The objective is to minimize avoid radiating the parotid, when the tumor is in its periphery. We proposed two 3D reconstruction methods of the parotid, namely B-Spline Bezier bi-cubic and Marching cubes. Once the reconstruction is performed, the volume of the parotid gland, at each session of treatment is calculated. A safety zone around the parotid gland was defined. Finally, a classification of the patients was performed based on the associated dose for the treatment.At a macroscopic level, we have developed a decision-support approach to refine the relative position of the patient on the treatment table. This approach is based on a modeling of Neural Network and especially MLP and RBF. The performance between two neuronal approximation techniques was analyzed, according to the geometrical positioning of patients.

Master of Science / Department of Mathematics / Natasha Rozhkovskaya / “Incomplete open cubes” is one of the major projects of the artist Sol Lewitt. It consists of a collection of frame structures and a presentation of their diagrams. Each structure in the project is a cube with some edges removed so that the structure remains three-dimensional and connected Structures are considered to be identical if one can be transformed into another by a space rotation (but not reflection). The list of incomplete cubes consists of 122 structures. In this project, the concept of incomplete cubes was formulated in the language of graph theory. This allowed us to compare the problem posed by the artist with the similar questions of graph theory considered during the last decades. Classification of Incomplete cubes was then refined using the language of combinatorics. The list produced by the artist was then checked to be complete. And lastly, properties of Incomplete cubes in the list were studied.

Dans cette thèse, nous etudions deux principaux problèmes de la théorie des graphes: problème d’homomorphisme des graphes planaires (signés) et problème de cycle hamiltonien.Comme une extension du théorème des quatre couleurs, il est conjecturé([80], [41]) que chaque graphe signé cohérent planaire de déséquilibré-maille d+1(d>1) admet un homomorphisme à cube projective signé SPC(d) de dimension d. La question suivant étalés naturelle:Est-ce que SPC(d) une borne optimale de déséquilibré-maille d+1 pour tous les graphes signés cohérente planaire de déséquilibré-maille d+1?Au Chapitre 2, nous prouvons que: si (B,Ω) est un graphe signé cohérente dedéséquilibré-maille d qui borne la classe des graphes signés cohérents planaires de déséquilibré-maille d+1, puis |B| ≥2^{d−1}. Notre résultat montre que si la conjecture ci-dessus est vérifiée, alors le SPC(d) est une borne optimale à la fois en terme du nombre des sommets et du nombre de arêtes.Lorsque d=2k, le problème est équivalent aux problème des graphes:est-ce que PC(2k) une borne optimale de impair-maille 2k+1 pour P_{2k+1} (tous les graphes planaires de impair-maille au moins 2k+1)? Notez que les graphes K_4-mineur libres sont les graphes planaires, est PC(2k) aussi une borne optimale de impair-maille 2k+1 pour tous les graphes K_4-mineur libres de impair-maille 2k+1? La réponse est négative, dans[6], est donné une famille de graphes d’ordre O(k^2) que borne les graphes K_4-mineur libres de impair-maille 2k+1. Est-ce que la borne optimale? Au Chapitre 3, nous prouvons que: si B est un graphe de impair-maille 2k+1 qui borne tous les graphes K_4-mineur libres de impair-maille 2k+1, alors |B|≥(k+1)(k+2)/2. La conjonction de nos résultat et le résultat dans [6] montre que l’ordre O(k^2) est optimal. En outre, si PC(2k) borne P_{2k+1}, PC(2k) borne également P_{2r+1}(r>k).Cependant, dans ce cas, nous croyons qu’un sous-graphe propre de P(2k) serait suffisant à borner P_{2r+1}, alors quel est le sous-graphe optimal de PC2k) qui borne P_{2r+1}? Le premier cas non résolu est k=3 et r= 5. Dans ce cas, Naserasr [81] a conjecturé que le graphe Coxeter borne P_{11}. Au Chapitre 4, nous vérifions cette conjecture pour P_{17}.Au Chapitres 5, 6, nous étudions les problèmes du cycle hamiltonien. Dirac amontré en 1952 que chaque graphe d’ordre n est hamiltonien si tout sommet a un degré au moins n/2. Depuis, de nombreux résultats généralisant le théorème de Dirac sur les degré ont été obtenus. Une approche consiste à construire un cycle hamiltonien à partir d'un ensemble de sommets en contrôlant leur position sur le cycle. Dans cette thèse, nous considérons deux conjectures connexes. La première est la conjecture d'Enomoto: si G est un graphe d’ordre n≥3 et δ(G)≥n/2+1, pour toute paire de sommets x,y dans G, il y a un cycle hamiltonien C de G tel que dist_C(x,y)=n/2.Notez que l’ ́etat de degre de la conjecture de Enomoto est forte. Motivé par cette conjecture, il a prouvé, dans [32], qu’une paire de sommets peut être posé des distances pas plus de n/6 sur un cycle hamiltonien. Dans [33], les cas δ(G)≥(n+k)/2 sont considérés, il a prouvé qu’une paire de sommets à une distance entre 2 à k peut être posé sur un cycle hamiltonien. En outre, Faudree et Li ont proposé une conjecture plus générale: si G est un graphe d’ordre n≥3 et δ(G)≥n/2+1, pour toute paire de sommets x,y dans G et tout entier 2≤k≤n/2, il existe un cycle hamiltonien C de G tel que dist_C(x,y)=k. Utilisant de Regularity Lemma et Blow-up Lemma, au chapitre 5, nous donnons une preuve de la conjeture d'Enomoto conjecture pour les graphes suffisamment grand, et dans le chapitre 6, nous donnons une preuve de la conjecture de Faudree et Li pour les graphe suffisamment grand. / In this thesis, we study two main problems in graph theory: homomorphism problem of planar (signed) graphs and Hamiltonian cycle problem.As an extension of the Four-Color Theorem, it is conjectured ([80],[41]) that every planar consistent signed graph of unbalanced-girth d+1(d>1) admits a homomorphism to signed projective cube SPC(d) of dimension d. It is naturally asked that:Is SPC(d) an optimal bound of unbalanced-girth d+1 for all planar consistent signed graphs of unbalanced-girth d+1?In Chapter 2, we prove that: if (B,Ω) is a consistent signed graph of unbalanced-girth d which bounds the class of consistent signed planar graphs of unbalanced-girth d, then |B|≥2^{d-1}. Furthermore,if no subgraph of (B,Ω) bounds the same class, δ(B)≥d, and therefore,|E(B)|≥d·2^{d-2}.Our result shows that if the conjecture above holds, then the SPC(d) is an optimal bound both in terms of number of vertices and number of edges.When d=2k, the problem is equivalent to the homomorphisms of graphs: isPC(2k) an optimal bound of odd-girth 2k+1 for P_{2k+1}(the class of all planar graphs of odd-girth at least 2k+1)? Note that K_4-minor free graphs are planar graphs, is PC(2k) also an optimal bound of odd-girth 2k+1 for all K_4-minor free graphs of odd-girth 2k+1 ? The answer is negative, in [6], a family of graphs of order O(k^2) bounding the K_4-minor free graphs of odd-girth 2k+1 were given. Is this an optimal bound? In Chapter 3, we prove that: if B is a graph of odd-girth 2k+1 which bounds all the K_4-minor free graphs of odd-girth 2k+1,then |B|≥(k+1)(k+2)/2. Our result together with the result in [6] shows that order O(k^2) is optimal.Furthermore, if PC(2k) bounds P_{2k+1},then PC(2k) also bounds P_{2r+1}(r>k). However, in this case we believe that a proper subgraph of PC(2k) would suffice to bound P_{2r+1}, then what’s the optimal subgraph of PC(2k) that bounds P_{2r+1}? The first case of this problem which is not studied is k=3 and r=5. For this case, Naserasr [81] conjectured that the Coxeter graph bounds P_{11} . Supporting this conjecture, in Chapter 4, we prove that the Coxeter graph bounds P_{17}.In Chapter 5,6, we study the Hamiltonian cycle problems. Dirac showed in 1952that every graph of order n is Hamiltonian if any vertex is of degree at least n/2. This result started a new approach to develop sufficient conditions on degrees for a graph to be Hamiltonian. Many results have been obtained in generalization of Dirac’s theorem. In the results to strengthen Dirac’s theorem, there is an interesting research area: to control the placement of a set of vertices on a Hamiltonian cycle such that thesevertices have some certain distances among them on the Hamiltonian cycle.In this thesis, we consider two related conjectures, one is given by Enomoto: if G is a graph of order n≥3, and δ(G)≥n/2+1, then for any pair of vertices x, y in G, there is a Hamiltonian cycle C of G such that dist_C(x, y)=n/2. Motivated by this conjecture, it is proved,in [32],that a pair of vertices are located at distances no more than n/6 on a Hamiltonian cycle. In [33], the cases δ(G) ≥(n+k)/2 are considered, it is proved that a pair of vertices can be located at any given distance from 2 to k on a Hamiltonian cycle. Moreover, Faudree and Li proposed a more general conjecture: if G is a graph of order n≥3, and δ(G)≥n/2+1, then for any pair of vertices x, y in G andany integer 2≤k≤n/2, there is a Hamiltonian cycle C of G such that dist_C(x, y) = k. Using Regularity Lemma and Blow-up Lemma, in Chapter 5, we give a proof ofEnomoto’s conjecture for graphs of sufficiently large order, and in Chapter 6, we give a proof of Faudree and Li’s conjecture for graphs of sufficiently large order.

The study of multicomputer interconnection networks is an important area of research in parallel processing. We introduce vertex-symmetric Hamming-group graphs as a model to design a wide variety of network topologies including the hypercube network.

by Robert M. Freund. / "April 1984." / Bibliography: leaf i.

Este trabalho descreve uma nova visão sobre um dos algoritmos mais importantes da Computação Gráfica, o algoritmo para a poligonização de isosuperfícies Marching Cubes (MC). Esta visão permite o estudo das situações onde são gerados triângulos de baixa qualidade, uma das grandes deficiências do MC, e serve como suporte à proposta de modificações no algoritmo. Neste trabalho também são propostas três modificações ao MC, com objetivo de gerar triângulos de boa qualidade e manter as características mais atraentes do MC, ou seja, a sua simplicidade, eficiência e robustez. O funcionamento do MC é analisado a partir da decomposição das células de amostragem em Grupos de Arestas. Os Grupos de Arestas permitem o estudo do modo como os triângulos são construídos no interior de cada célula de amostragem, e possibilitam identificar onde e como são gerados triângulos de baixa qualidade. As situações onde triângulos de baixa qualidade são gerados são resolvidas através da proposta de três modificações no algoritmo MC, sendo elas: (1) um critério para conduzir a construção da tabela de triangulações, (2) o reposicionamento das arestas ativas e (3) o deslocamento dos vértices de intersecção sobre as arestas ativas. A primeira modificação é a proposta de um novo critério para conduzir a construção da tabela de triangulações do MC. O critério tem como objetivo o aumento da qualidade da malha gerada, através do uso de triangulações que não resultem nos grupos de arestas que têm maior probabilidade de gerar triângulos de baixa qualidade. A segunda modificação propõe o reposicionamento das arestas ativas. Os grupos de arestas mostram que a posição das arestas ativas influencia a qualidade dos triângulos, o que motiva a criação de um procedimento que reposicione as arestas ativas em relação ao comportamento da isosuperfície. Este procedimento resulta na maximização da distância entre os vértices de intersecção e na melhora na qualidade da malha gerada. A terceira modificação propõe o deslocamento dos vértices de intersecção ao longo das arestas ativas. O movimento dos vértices é controlado por um parâmetro que oferece a oportunidade de comprometer a qualidade de aproximação (distância entre a malha gerada e a isosuperfície) em razão da qualidade da malha. O conjunto de modificações propostas resulta no algoritmo Macet (Marching Cubes with Edge Transformations). O Macet é simples e eficiente, gera malhas com qualidade de aproximação igual ou superior à do MC e onde os triângulos de baixa qualidade ainda têm qualidade comparável ou superior aos métodos disponíveis atualmente. / This work sheds a new light on one of the most important algorithms of Computer Graphics, the isosurface polygonization algorithm Marching Cubes (MC). The analysis described here allows the study of situations where low quality triangles are generated, one of the well-know MC drawbacks, and also supports the proposal of extensions for the algorithm. In this work three extensions for MC are also proposed, all with purpose of increase the quality of the resulting triangulations and keep the simplicity, efficiency and robustness of MC. The MC is studied with the breaking of its sampling cells in Edge Groups. The Edge Groups allow the study of how triangles are generated inside each sampling cell, and make possible to identify where and how low quality triangles are generated. The low quality triangle generation is tackled through the proposal of three modifications to the MC algorithm, namely: (1) a new rule to create the look-up table, (2) the warping of active edges and (3) the displacement of cut vertices along active edges. The first modification is the proposal of a new rule to create the triangulations of the MC look-up table. This rule aims the mesh quality improvement through the creation of triangulations that are based on edge groups with fewer chances of generate low quality triangles. The second modification is the proposal of the active edge warping. The edge groups analysis shows that active edges position have a high impact on the triangle quality, which motivates the proposal of a procedure that warps the active edges in response to the isosurface behavior. This procedure results in the maximization of the distance between cut edges, and, consequently, in the overall mesh quality improvement. The third modification is the proposal of the cut vertex displacement along active edges. The vertex displacement is controlled by a user-defined parameter which results in a tradeoff between approximation quality (the distance between the mesh and the “real” isosurface) and the mesh quality. The set of modifications results in the proposal of the Macet (Marching Cubes with Edge Transformations) algorithm. Macet is simple and efficient, and results in high quality meshes with the same (or higher) MC approximation quality.

Este trabalho descreve uma nova visão sobre um dos algoritmos mais importantes da Computação Gráfica, o algoritmo para a poligonização de isosuperfícies Marching Cubes (MC). Esta visão permite o estudo das situações onde são gerados triângulos de baixa qualidade, uma das grandes deficiências do MC, e serve como suporte à proposta de modificações no algoritmo. Neste trabalho também são propostas três modificações ao MC, com objetivo de gerar triângulos de boa qualidade e manter as características mais atraentes do MC, ou seja, a sua simplicidade, eficiência e robustez. O funcionamento do MC é analisado a partir da decomposição das células de amostragem em Grupos de Arestas. Os Grupos de Arestas permitem o estudo do modo como os triângulos são construídos no interior de cada célula de amostragem, e possibilitam identificar onde e como são gerados triângulos de baixa qualidade. As situações onde triângulos de baixa qualidade são gerados são resolvidas através da proposta de três modificações no algoritmo MC, sendo elas: (1) um critério para conduzir a construção da tabela de triangulações, (2) o reposicionamento das arestas ativas e (3) o deslocamento dos vértices de intersecção sobre as arestas ativas. A primeira modificação é a proposta de um novo critério para conduzir a construção da tabela de triangulações do MC. O critério tem como objetivo o aumento da qualidade da malha gerada, através do uso de triangulações que não resultem nos grupos de arestas que têm maior probabilidade de gerar triângulos de baixa qualidade. A segunda modificação propõe o reposicionamento das arestas ativas. Os grupos de arestas mostram que a posição das arestas ativas influencia a qualidade dos triângulos, o que motiva a criação de um procedimento que reposicione as arestas ativas em relação ao comportamento da isosuperfície. Este procedimento resulta na maximização da distância entre os vértices de intersecção e na melhora na qualidade da malha gerada. A terceira modificação propõe o deslocamento dos vértices de intersecção ao longo das arestas ativas. O movimento dos vértices é controlado por um parâmetro que oferece a oportunidade de comprometer a qualidade de aproximação (distância entre a malha gerada e a isosuperfície) em razão da qualidade da malha. O conjunto de modificações propostas resulta no algoritmo Macet (Marching Cubes with Edge Transformations). O Macet é simples e eficiente, gera malhas com qualidade de aproximação igual ou superior à do MC e onde os triângulos de baixa qualidade ainda têm qualidade comparável ou superior aos métodos disponíveis atualmente. / This work sheds a new light on one of the most important algorithms of Computer Graphics, the isosurface polygonization algorithm Marching Cubes (MC). The analysis described here allows the study of situations where low quality triangles are generated, one of the well-know MC drawbacks, and also supports the proposal of extensions for the algorithm. In this work three extensions for MC are also proposed, all with purpose of increase the quality of the resulting triangulations and keep the simplicity, efficiency and robustness of MC. The MC is studied with the breaking of its sampling cells in Edge Groups. The Edge Groups allow the study of how triangles are generated inside each sampling cell, and make possible to identify where and how low quality triangles are generated. The low quality triangle generation is tackled through the proposal of three modifications to the MC algorithm, namely: (1) a new rule to create the look-up table, (2) the warping of active edges and (3) the displacement of cut vertices along active edges. The first modification is the proposal of a new rule to create the triangulations of the MC look-up table. This rule aims the mesh quality improvement through the creation of triangulations that are based on edge groups with fewer chances of generate low quality triangles. The second modification is the proposal of the active edge warping. The edge groups analysis shows that active edges position have a high impact on the triangle quality, which motivates the proposal of a procedure that warps the active edges in response to the isosurface behavior. This procedure results in the maximization of the distance between cut edges, and, consequently, in the overall mesh quality improvement. The third modification is the proposal of the cut vertex displacement along active edges. The vertex displacement is controlled by a user-defined parameter which results in a tradeoff between approximation quality (the distance between the mesh and the “real” isosurface) and the mesh quality. The set of modifications results in the proposal of the Macet (Marching Cubes with Edge Transformations) algorithm. Macet is simple and efficient, and results in high quality meshes with the same (or higher) MC approximation quality.

Este trabalho descreve uma nova visão sobre um dos algoritmos mais importantes da Computação Gráfica, o algoritmo para a poligonização de isosuperfícies Marching Cubes (MC). Esta visão permite o estudo das situações onde são gerados triângulos de baixa qualidade, uma das grandes deficiências do MC, e serve como suporte à proposta de modificações no algoritmo. Neste trabalho também são propostas três modificações ao MC, com objetivo de gerar triângulos de boa qualidade e manter as características mais atraentes do MC, ou seja, a sua simplicidade, eficiência e robustez. O funcionamento do MC é analisado a partir da decomposição das células de amostragem em Grupos de Arestas. Os Grupos de Arestas permitem o estudo do modo como os triângulos são construídos no interior de cada célula de amostragem, e possibilitam identificar onde e como são gerados triângulos de baixa qualidade. As situações onde triângulos de baixa qualidade são gerados são resolvidas através da proposta de três modificações no algoritmo MC, sendo elas: (1) um critério para conduzir a construção da tabela de triangulações, (2) o reposicionamento das arestas ativas e (3) o deslocamento dos vértices de intersecção sobre as arestas ativas. A primeira modificação é a proposta de um novo critério para conduzir a construção da tabela de triangulações do MC. O critério tem como objetivo o aumento da qualidade da malha gerada, através do uso de triangulações que não resultem nos grupos de arestas que têm maior probabilidade de gerar triângulos de baixa qualidade. A segunda modificação propõe o reposicionamento das arestas ativas. Os grupos de arestas mostram que a posição das arestas ativas influencia a qualidade dos triângulos, o que motiva a criação de um procedimento que reposicione as arestas ativas em relação ao comportamento da isosuperfície. Este procedimento resulta na maximização da distância entre os vértices de intersecção e na melhora na qualidade da malha gerada. A terceira modificação propõe o deslocamento dos vértices de intersecção ao longo das arestas ativas. O movimento dos vértices é controlado por um parâmetro que oferece a oportunidade de comprometer a qualidade de aproximação (distância entre a malha gerada e a isosuperfície) em razão da qualidade da malha. O conjunto de modificações propostas resulta no algoritmo Macet (Marching Cubes with Edge Transformations). O Macet é simples e eficiente, gera malhas com qualidade de aproximação igual ou superior à do MC e onde os triângulos de baixa qualidade ainda têm qualidade comparável ou superior aos métodos disponíveis atualmente. / This work sheds a new light on one of the most important algorithms of Computer Graphics, the isosurface polygonization algorithm Marching Cubes (MC). The analysis described here allows the study of situations where low quality triangles are generated, one of the well-know MC drawbacks, and also supports the proposal of extensions for the algorithm. In this work three extensions for MC are also proposed, all with purpose of increase the quality of the resulting triangulations and keep the simplicity, efficiency and robustness of MC. The MC is studied with the breaking of its sampling cells in Edge Groups. The Edge Groups allow the study of how triangles are generated inside each sampling cell, and make possible to identify where and how low quality triangles are generated. The low quality triangle generation is tackled through the proposal of three modifications to the MC algorithm, namely: (1) a new rule to create the look-up table, (2) the warping of active edges and (3) the displacement of cut vertices along active edges. The first modification is the proposal of a new rule to create the triangulations of the MC look-up table. This rule aims the mesh quality improvement through the creation of triangulations that are based on edge groups with fewer chances of generate low quality triangles. The second modification is the proposal of the active edge warping. The edge groups analysis shows that active edges position have a high impact on the triangle quality, which motivates the proposal of a procedure that warps the active edges in response to the isosurface behavior. This procedure results in the maximization of the distance between cut edges, and, consequently, in the overall mesh quality improvement. The third modification is the proposal of the cut vertex displacement along active edges. The vertex displacement is controlled by a user-defined parameter which results in a tradeoff between approximation quality (the distance between the mesh and the “real” isosurface) and the mesh quality. The set of modifications results in the proposal of the Macet (Marching Cubes with Edge Transformations) algorithm. Macet is simple and efficient, and results in high quality meshes with the same (or higher) MC approximation quality.