Cubical complexes are defined in a manner analogous to that for simplicial complexes, the chief difference being that cubical complexes are unions of cubes rather than of simplices. A very natural cubical complex to consider is the complex C(k_1,...,k_n) where k_1,...,k_n are nonnegative integers. This complex has as its underlying space [0,k_1]x...x[0,k_n] subset of R^n with vertices at all points having integer coordinates and higher dimensional cubes formed by the vertices in the natural way. The genus of a cubical complex is defined to be the maximum genus of all surfaces that are subcomplexes of the cubical complex. A formula is given for determining the genus of the cubical complex C(k_1,...,k_n) when at least three of the k_i are odd integers. For the remaining cases a general solution is not known. When k_1=...=k_n=1 the genus of C(k_1,...,k_n) is shown to be (n-4)2^{n-3}+1 which is equivalent to the genus of the graph of the n-cube. Indeed, the genus of the complex and the genus of the graph of the 1-skeleton of the complex, are shown to be equal when at least three of the k_i are odd, but not equal in general.

Differential geometry is about space (a manifold) and a geometric structure on that space. In Riemann’s lecture (see [17]), he stated that “Thus arises the problem, to discover the matters of fact from which the measure-relations of space may be determined...”. It is key then to understand how manifolds differ from one another geometrically. The results of this dissertation concern how the geometry of a manifold changes when we alter metrical connections. We investigate how diverse geodesics are in different metrical connections. From this, we investigate a new class of metrical connections which are dependent on the class of smooth functions. Specifically, we fix a Riemannian metric and investigate the geometry of the manifold when we change the metrical connections associated with the fixed Riemannian metric. We measure the change in the Riemannian curvatures associated with this new class of metrical connections, and then give uniqueness and existence criterion for curvature of compact 2-manifolds. These results depend on the use of Hodge Theory and ultimately on the function f we choose to define a metrical connection.

[pt] Este trabalho apresenta uma estratégia de modelagem aberta e extensível, desenvolvida em Python, para a criação de modelos de subdivisões planares. A estratégia se dá na forma de uma biblioteca de modelagem geométrica, denominada HETOOL, desenvolvida no trabalho e de uso genérico, baseada na conhecida e consagrada estrutura de dados topológica Half-Edge. Além de considerar os aspectos topológicos e geométricos da modelagem, a estratégia também permite a configuração pelo usuário final dos atributos de simulação. Essas características, somadas à disponibilização do código fonte, conferem um caráter útil e relevante para o desenvolvimento de ferramentas educacionais para modelagem em mecânica computacional. Para demonstrar a aplicabilidade da estratégia proposta, foi desenvolvido um aplicativo, denominado de Finite Element Method Educational Computer Program (FEMEP), que permite a criação de modelos bidimensionais de elementos finitos, com geração de malhas por região, para diversos tipos de simulação de mecânica computacional. O pacote desenvolvido apresenta uma modelagem iterativa e dinâmica que realiza a interseção automática entres os elementos geométricos modelados. O HETOOL oferece várias funcionalidades e facilidades ao usuário, permitindo o uso do pacote mesmo sem o usuário ter conhecimento sobre os conceitos topológicos envolvidos na implementação dessa estrutura de dados. O pacote possibilita a criação e configuração atributos de forma simples e rápida a partir de um arquivo no formato JSON. Essa versatilidade na criação atributos permite a aplicação deste pacote na resolução de vários problemas presentes na engenharia e em outras áreas do meio científico. / [en] This work presents an open and extensible modeling strategy, developed in Python, for creating planar subdivision models. The strategy takes the form of a geometric modeling library called HETOOL, developed in the work and of general use, based on the well-known and renowned Half-Edge topological data structure. In addition to considering the topological and geometric aspects of the modeling, a strategy also allows for an end-user configuration of simulation attributes. These characteristics, added to the availability of the source code, provide a useful and relevant tool for the development of educational tools for modeling computational mechanics. To demonstrate the applicability of the proposed strategy, an application was developed, called the Finite Element Method Educational Computer Program (FEMEP), which allows the creation of two-dimensional finite element models, with mesh generation per region, for various types of mechanics simulation computational. The developed package presents iterative and dynamic modeling that performs an automatic intersection between the modeled geometric elements. HETOOL offers several functions and facilities to the user, allowing the use of the package even without the user having knowledge about the topological concepts involved in the implementation of this data structure. The package makes it possible to create and configure attributes simply and quickly from a file in JSON format. This versatility in creating attributes allows the application of this package to solve several problems present in engineering and in other areas of the scientific environment.

This thesis deals with k-irreducible triangulations of closed, compact 2-manifolds without boundary. A triangulation is k-irreducible, if all its closed cycles of length less than k are nullhomotopic and no edge can be contracted without losing this property. k-irreducibility is a generalization of the well-known concept of irreducibility, and can be regarded as a measure of how closely the triangulation approximates a smooth version of the underlying surface. Research follows three main questions: What are lower and upper bounds for the minimum and maximum size of a k-irreducible triangulation? What are the smallest and biggest explicitly constructible examples? Can one achieve complete classifications for specific 2-manifolds, and fixed k?

Un problème toujours d'actualité est la reconstruction automatique de la surface d'une scène à partir du flot d'images prises par une caméra en mouvement. Il se résout en général en deux étapes : le calcul de la géométrie où les poses de la caméra et un nuage épars de points 3D de la scène sont simultanément estimés, et un calcul de stéréo dense qui permet d'obtenir une surface en estimant la profondeur de tous les pixels. L' approche que nous proposons se distingue des précédentes en cumulant les caractéristiques suivantes. La surface est une 2-variété, ce qui est utile pour les traitements ou utilisations ultérieurs. Elle est calculée directement à partir du nuage épars donné par la première étape, afin d'éviter la seconde étape coûteuse et pour obtenir une modélisation compacte d'une scène complexe. Le calcul est incrémental afin d'avoir un résultat pendant la lecture de la vidéo. Le principe est le suivant. A chaque itération, de nouveaux points 3D sont estimés et insérés dans une triangulation de Delaunay 3D. Celle-ci partitionne l'espace en tétraèdres vides et pleins grâce à l'information de visibilité également fournie par la première étape. On met aussi à jour une seconde partition en tétraèdres intérieurs et extérieurs dont le bord est la 2-variété recherchée. Sous certaines hypothèses, et contrairement à la seule méthode précédente ayant les même propriétés et hypothèses, la complexité d'une itération est bornée. Notre méthode a été expérimentée sur des séquences synthétiques et réelles, dont une séquence longue de 2;5 km prise en milieu urbain avec une caméra omnidirectionnelle. La qualité du résultat est proche de celle obtenue par la méthode globale (non incrémentale) qui a servi d'inspiration, mais le temps de calcul ne permet pas actuellement une utilisation en-ligne sur un PC standard. On a aussi étudié l'intérêt d'ajouter des contours dans le processus de reconstruction. / The automatic reconstruction of a scene surface from images taken by a moving camera is still an active research topic. This problem is usually solved in two steps : first estimate the camera poses and a sparse cloud of 3D points using Structure-from-Motion, then apply dense stereo to obtain the surface by estimating the depth for all pixels. Compared to the previous approaches, ours accumulates the following properties. The output surface is a 2-manifold, which is useful for applications and postprocessing. It is computed directly from the sparse point cloud provided by the first step, so as to avoid the second and time consuming step and to obtain a compact model of a complex scene. The computation is incremental to allow access to intermediary results during the processing. The principle is the following. At each iteration, new 3D points are estimated and added to a 3D Delaunay triangulation; the tetrahedra are labeled as free-space or matter thanks to the visibility information provided by the first step. We also update a second partition of outside and inside tetrahedra whose boundary is the target 2-manifold. Under some assumptions, the time complexity of one iteration is bounded (there is only one previous method with the same properties, but its complexity is greater than that). Our method is experimented on synthetic and real sequences, including a 2:5 km. long urban sequence taken by an omnidirectional camera. The surface quality is similar to that of the batch method which inspired us. However, the computations are not yet real-time on a commodity PC. We also study the use of contours in thereconstruction process.