My thesis "Properties and applications of low degree surfaces" deals mainly with properties of two quadrics. We introduce so called canonical forms of two quadrics and classification of intersection curve of two quadrics. There are used index and signature sequence to obtain classification of intersection curve of two quadrics. These sequence can determine the number of component of the intersection curve, their algebraic degrees and singularities. Our work also contains many examples with pictures. One part of the thesis also presents possible application of intersection curve of two quadrics in practice. This thesis also contains an enclosed CD, with the thesis in an electronic form and the source files of all pictures in the thesis.

L'algèbre géométrique est un outil permettant de représenter et manipuler les objets géométriques de manière générique, efficace et intuitive. A titre d'exemple, l'Algèbre Géométrique Conforme (CGA), permet de représenter des cercles, des sphères, des plans et des droites comme des objets algébriques. Les intersections entre ces objets sont incluses dans la même algèbre. Il est possible d'exprimer et de traiter des objets géométriques plus complexes comme des coniques, des surfaces quadriques en utilisant une extension de CGA. Cependant due à leur représentation requérant un espace vectoriel de haute dimension, les implantations de l'algèbre géométrique, actuellement disponible, n'autorisent pas une utilisation efficace de ces objets. Dans ce manuscrit, nous présentons tout d'abord une implantation de l'algèbre géométrique dédiée aux espaces vectoriels aussi bien basses que hautes dimensions. L'approche suivie est basée sur une solution hybride de code pré-calculé en vue d'une exécution rapide pour des espaces vectoriels de basses dimensions, ce qui est similaire aux approches de l'état de l'art. Pour des espaces vectoriels de haute dimension, nous proposons des méthodes de calculs ne nécessitant que peu de mémoire. Pour ces espaces, nous introduisons un formalisme récursif et prouvons que les algorithmes associés sont efficaces en termes de complexité de calcul et complexité de mémoire. Par ailleurs, des règles sont définies pour sélectionner la méthode la plus appropriée. Ces règles sont basées sur la dimension de l'espace vectoriel considéré. Nous montrons que l'implantation obtenue est bien adaptée pour les espaces vectoriels de hautes dimensions (espace vectoriel de dimension 15) et ceux de basses dimensions. La dernière partie est dédiée à une représentation efficace des surfaces quadriques en utilisant l'algèbre géométrique. Nous étudions un nouveau modèle en algèbre géométrique de l'espace vectoriel $mathbb{R}^{9,6}$ pour manipuler les surfaces quadriques. Dans ce modèle, une surface quadrique est construite par l'intermédiaire de neuf points. Nous montrerons que ce modèle permet non seulement de représenter de manière intuitive des surfaces quadriques mais aussi de construire des objets en utilisant les définitions de CGA. Nous présentons le calcul de l'intersection de surfaces quadriques, du vecteur normal, du plan tangent à une surface en un point de cette surface. Enfin, un modèle complet de traitement des surfaces quadriques est détaillé / Geometric Algebra is considered as a very intuitive tool to deal with geometric problems and it appears to be increasingly efficient and useful to deal with computer graphics problems. The Conformal Geometric Algebra includes circles, spheres, planes and lines as algebraic objects, and intersections between these objects are also algebraic objects. More complex objects such as conics, quadric surfaces can also be expressed and be manipulated using an extension of the conformal Geometric Algebra. However due to the high dimension of their representations in Geometric Algebra, implementations of Geometric Algebra that are currently available do not allow efficient realizations of these objects. In this thesis, we first present a Geometric Algebra implementation dedicated for both low and high dimensions. The proposed method is a hybrid solution that includes precomputed code with fast execution for low dimensional vector space, which is somehow equivalent to the state of the art method. For high dimensional vector spaces, we propose runtime computations with low memory requirement. For these high dimensional vector spaces, we introduce new recursive scheme and we prove that associated algorithms are efficient both in terms of computationnal and memory complexity. Furthermore, some rules are defined to select the most appropriate choice, according to the dimension of the algebra and the type of multivectors involved in the product. We will show that the resulting implementation is well suited for high dimensional spaces (e.g. algebra of dimension 15) as well as for lower dimensional spaces. The next part presents an efficient representation of quadric surfaces using Geometric Algebra. We define a novel Geometric Algebra framework, the Geometric Algebra of $mathbb{R}^{9,6}$ to deal with quadric surfaces where an arbitrary quadric surface is constructed by merely the outer product of nine points. We show that the proposed framework enables us not only to intuitively represent quadric surfaces but also to construct objects using Conformal Geometric Algebra. In the proposed framework, the computation of the intersection of quadric surfaces, the normal vector, and the tangent plane of a quadric surface are provided. Finally, a computational framework of the quadric surfaces will be presented with the main operations required in computer graphics

Many applications in computer graphics and related fields can benefit fromautomatic simplification of complex polygonal surface models. Applications areoften confronted with either very densely over-sampled surfaces or models toocomplex for the limited available hardware capacity. An effective algorithmfor rapidly producing high-quality approximations of the original model is avaluable tool for managing data complexity. In this dissertation, I present my simplification algorithm, based on iterativevertex pair contraction. This technique provides an effective compromisebetween the fastest algorithms, which often produce poor quality results, andthe highest-quality algorithms, which are generally very slow. For example, a1000 face approximation of a 100,000 face model can be produced in about 10seconds on a PentiumPro 200. The algorithm can simplify both the geometryand topology of manifold as well as non-manifold surfaces. In addition toproducing single approximations, my algorithm can also be used to generatemultiresolution representations such as progressive meshes and vertex hierarchiesfor view-dependent refinement. The foundation of my simplification algorithm, is the quadric error metricwhich I have developed. It provides a useful and economical characterization oflocal surface shape, and I have proven a direct mathematical connection betweenthe quadric metric and surface curvature. A generalized form of this metric canaccommodate surfaces with material properties, such as RGB color or texturecoordinates. I have also developed a closely related technique for constructing a hierarchyof well-defined surface regions composed of disjoint sets of faces. This algorithminvolves applying a dual form of my simplification algorithm to the dual graphof the input surface. The resulting structure is a hierarchy of face clusters whichis an effective multiresolution representation for applications such as radiosity.

This thesis deals with construction of polygons with use of a didactic aid a geoboard at primary education. I have done analysis of three series of mathematics books in order to know, if they include possibility of mechanical work during construction of polygons. Based on study of professional literature related to teaching mathematics, mainly geometry at primary education, I have created patterns that can be used altogether with geoboard in mathematics. The patterns had been used in three classes thank to this I could have assess their practical use. Then I have corrected wrong instruction of patterns in order to be clear. Next I described three stages of beginning to work with geoboard that I have personally done and that are written in details in protocol. I have concluded didactic potential work with geoboard and theoretical obstacles while geoboard is using.

Submitted by Automa??o e Estat?stica (sst@bczm.ufrn.br) on 2017-02-22T20:15:09Z No. of bitstreams: 1 JoaoGutembergBarbosaDeFariasFilho_DISSERT.pdf: 3897527 bytes, checksum: ee9805fc353cbac5ca3716f673a21d0c (MD5) / Approved for entry into archive by Arlan Eloi Leite Silva (eloihistoriador@yahoo.com.br) on 2017-03-08T22:09:44Z (GMT) No. of bitstreams: 1 JoaoGutembergBarbosaDeFariasFilho_DISSERT.pdf: 3897527 bytes, checksum: ee9805fc353cbac5ca3716f673a21d0c (MD5) / Made available in DSpace on 2017-03-08T22:09:44Z (GMT). No. of bitstreams: 1 JoaoGutembergBarbosaDeFariasFilho_DISSERT.pdf: 3897527 bytes, checksum: ee9805fc353cbac5ca3716f673a21d0c (MD5) Previous issue date: 2016-09-30 / Coordena??o de Aperfei?oamento de Pessoal de N?vel Superior (CAPES) / O trabalho trata da aplica??o do Controle Preditivo com o proposto M?todo da Compensa??o de N?o-Linearidades para o controle de um quadric?ptero. Esse m?todo consiste de utilizar o modelo n?o-linear do sistema para prever e inserir n?o-linearidades na formula??o do controle preditivo por meio de uma estrutura afim. Dentro do contexto de controle de um quadric?ptero, o controlador proposto ? comparado, atrav?s de simula??es num?ricas, com o controlador preditivo linear tradicional. Posteriormente experimentos s?o feitos com um quadric?ptero real montado em uma bancada de testes. Como resultado o uso do m?todo proposto trouxe um ganho de desempenho consider?vel quando o estado do sistema se distanciou do ponto de opera??o em que foi linearizado. / This study analises the application of Model Predictive Control (MPC) with the proposed Nonlinear Compensation Method in order to control a quadcopter. This method is based on the use of the system?s nonlinear model to predict and insert nonlinearities in the predictive control formulation by means of an affine structure. With respect to the control of a quadcopter, the proposed controller is compared, using numerical simulations, to the traditional linear predictive controller. After that, experiments were made with a real quadcopter mounted in a test bench. As a result, the use of the proposed method brought a considerable perfomance improvement when the system?s state were farther apart from the operation point in which it was linearized.

L’étude topologique des variétés algébriques réelles remonte au moins aux travaux de Harnack, Klein, et Hilbert au 19éme siecle; en particulier, la classification des types d’isotopie réalisés par les courbes algébriques réelles d’un degré fixé dans RP2 est un sujet qui a connu un essor considérable jusqu'à aujourd'hui. En revanche, en dehors des études concernants les surfaces de Hirzebruch et les surfaces de degré au plus 3 dans RP3, à peu près rien n’est connu dans le cas de surfaces ambiantes plus générales. Cela est du en particulier au fait que les variétés construites en utilisant le "patchwork" sont des hypersurfaces de variétés toriques. Or, il existe de nombreuses autre surfaces algébriques réelles. Parmi celles-ci se trouvent les surfaces rationnelles réelles, et plus particulièrement les surfaces rèelles minimales. Dans cette thèse, on élargit l’étude des types d’isotopie réalisés par les courbes algébriques réelles aux surfaces réelles minimales de del Pezzo de degré 1 et 2. En outre, on termine la classification des types topologiques réalisés par les courbes algébriques réelles séparantes et non-séparantes de bidegré (5,5) sur la quadrique ellipsoide. / The study of the topology of real algebraic varieties dates back to the work of Harnack, Klein and Hilbert in the 19th century; in particular, the isotopy type classification of real algebraic curves with a fixed degree in RP2 is a classical subject that has undergone considerable evolution. On the other hand, apart from studies concerning Hirzebruch surfaces and at most degree 3 surfaces in RP3, not much is known for more general ambient surfaces. In particular, this is because varieties constructed using the patchworking method are hypersurfaces of toric varieties. However, there are many other real algebraic surfaces. Among these are the real rational surfaces, and more particularly the $mathbb{R}$-minimal surfaces. In this thesis, we extend the study of the topological types realized by real algebraic curves to the real minimal del Pezzo surfaces of degree 1 and 2. Furthermore, we end the classification of separating and non-separating real algebraic curves of bidegree $(5,5)$ in the quadric ellipsoid.

To decrease the rendering time of a mesh, Level of Detail can be generated by reducing the number of polygons based on some geometrical error. While this works well for most meshes, it is not suitable for meshes with an associated texture atlas. By iteratively collapsing edges based on an extended version of Quadric Error Metric taking both spatial and texture coordinates into account, textured meshes can also be simplified. Results show that constraining edge collapses in the seams of a mesh give poor geometrical appearance when it is reduced to a few polygons. By allowing seam edge collapses and by using a pull-push algorithm to fill areas located outside the seam borders of the texture atlas, the appearance of the mesh is better preserved.

La réalité augmentée (RA) consiste en l’insertion d’éléments virtuels dans une scène réelle, observée à travers un écran ou en utilisant un système de projection sur la scène ou l’objet d’intérêt. Les systèmes de réalité augmentée peuvent prendre des différentes formes pour obtenir l’équilibre désiré entre trois critères : précision, latence et robustesse. Il est possible d’identifier trois composants principaux à ces systèmes : localisation, reconstruction et affichage. Les contributions de cette thèse se concentrent essentiellement sur l’affichage et plus particulièrement le rendu des applications de réalité augmentée. À l’opposé des récentes avancées dans le domaine de la localisation et de la reconstruction, l’insertion d’éléments virtuels de façon plausible et esthétique reste une problématique compliquée, mal-posée et peu adaptée à un contexte temps réel. En effet, cette insertion requiert une reconstruction de l’illumination de la scène afin d’appliquer les conditions lumineuses adéquates à l’objet inséré. L’illumination de la scène peut être divisée en plusieurs catégories. Nous pouvons modéliser l’environnement de façon à décrire l’interaction de la lumière incidente et réfléchie pour chaque point 3D d’une surface. Il est également possible d’expliciter l’environnement en calculant la position des sources de lumière, leur type (lampe de bureau, néon, ampoule, ….), leur intensité et leur couleur. Pour insérer un objet de façon cohérente et réaliste, il est primordial d’avoir également une connaissance de la surface recevant l’illumination. Cette interaction lumière/matériaux est dépendante de la géométrie de la surface, de sa composition chimique (matériau) et de sa couleur. Pour tous ces aspects, le problème de reconstruction de l’illumination est difficile, car il est très complexe d’isoler l’illumination sans connaissance a priori de la géométrie, des matériaux de la scène et de la pose de la caméra observant la scène. De manière générale, sur une surface, une source de lumière laisse plusieurs traces telles que les ombres, qui sont créées par l’occultation de rayons lumineux par un objet, et les réflexions spéculaires ou spécularités qui se manifestent par la réflexion partielle ou totale de la lumière. Bien que ces spécularités soient souvent considérées comme des éléments parasites dans les applications de localisation de caméra, de reconstruction ou encore de segmentation, ces éléments donnent des informations cruciales sur la position et la couleur de la source lumineuse, mais également sur la géométrie de la surface et la réflectance du matériau où elle se manifeste. Face à la difficulté du problème de modélisation de la lumière et plus particulièrement du calcul de l’ensemble des paramètres de la lumière, nous nous sommes focalisés, dans cette thèse, sur l’étude des spécularités et sur toutes les informations qu’elles peuvent fournir pour la compréhension de la scène. Plus particulièrement, nous savons qu’une spécularité est définie comme la réflexion d’une source de lumière sur une surface réfléchissante. Partant de cette remarque, nous avons exploré la possibilité de considérer la spécularité comme étant une image issue de la projection d’un objet 3D dans l’espace. Nous sommes partis d’un constat simple, mais peu traité par la littérature qui est que les spécularités présentent une forme elliptique lorsqu’elles apparaissent sur une surface plane. À partir de cette hypothèse, pouvons-nous considérer un objet 3D fixe dans l’espace tel que sa projection perspective dans l’image corresponde à la forme de la spécularité ? Plus particulièrement, nous savons qu’un ellipsoïde projeté perspectivement donne une ellipse. Considérer le phénomène de spécularité comme un phénomène géométrique a de nombreux avantages. (...) / Augmented Reality (AR) consists in inserting virtual elements in a real scene, observed through a screen or a projection system on the scene or the object of interest. The augmented reality systems can take different forms to obtain a balance between three criteria: precision, latency and robustness. It is possible to identify three main components to these systems: localization, reconstruction and display. The contributions of this thesis focus essentially on the display and more particularly the rendering of augmented reality applications. Contrary to the recent advances in the field of localization and reconstruction, the insertion of virtual elements in a plausible and aesthetic way remains a complicated problematic, ill-posed and not adapted to a real-time context. Indeed, this insertion requires a good understanding of the lighting conditions of the scene. The lighting conditions of the scene can be divided in several categories. First, we can model the environment to describe the interaction between the incident and reflected light pour each 3D point of a surface. Secondly, it is also possible to explicitly the environment by computing the position of the light sources, their type (desktop lamps, fluorescent lamp, light bulb, . . . ), their intensities and their colors. Finally, to insert a virtual object in a coherent and realistic way, it is essential to have the knowledge of the surface’s geometry, its chemical composition (material) and its color. For all of these aspects, the reconstruction of the illumination is difficult because it is really complex to isolate the illumination without prior knowledge of the geometry, material of the scene and the camera pose observing the scene. In general, on a surface, a light source leaves several traces such as shadows, created from the occultation of light rays by an object, and the specularities (or specular reflections) which are created by the partial or total reflection of the light. These specularities are often described as very high intensity elements in the image. Although these specularities are often considered as outliers for applications such as camera localization, reconstruction or segmentation, these elements give crucial information on the position and color of the light source but also on the surface’s geometry and the material’s reflectance where these specularities appear. To address the light modeling problem, we focused, in this thesis, on the study of specularities and on every information that they can provide for the understanding of the scene. More specifically, we know that a specularity is defined as the reflection of the light source on a shiny surface. From this statement, we have explored the possibility to consider the specularity as the image created from the projection of a 3D object in space.We started from the simple but little studied in the literature observation that specularities present an elliptic shape when they appear on a planar surface. From this hypothesis, can we consider the existence of a 3D object fixed in space such as its perspective projection in the image fit the shape of the specularity ? We know that an ellipsoid projected perspectivally gives an ellipse. Considering the specularity as a geometric phenomenon presents various advantages. First, the reconstruction of a 3D object and more specifically of an ellipsoid, has been the subject to many publications in the state of the art. Secondly, this modeling allows a great flexibility on the tracking of the state of the specularity and more specifically the light source. Indeed, if the light is turning off, it is easy to visualize in the image if the specularity disappears if we know the contour (and reciprocally of the light is turning on again). (...)

Submitted by Automa??o e Estat?stica (sst@bczm.ufrn.br) on 2016-12-15T16:27:02Z No. of bitstreams: 1 ManoelPedroDeMedeirosNeto_DISSERT.pdf: 3664081 bytes, checksum: f1856f73174bde3b90b40604d7d1ae0e (MD5) / Approved for entry into archive by Arlan Eloi Leite Silva (eloihistoriador@yahoo.com.br) on 2016-12-15T16:36:27Z (GMT) No. of bitstreams: 1 ManoelPedroDeMedeirosNeto_DISSERT.pdf: 3664081 bytes, checksum: f1856f73174bde3b90b40604d7d1ae0e (MD5) / Made available in DSpace on 2016-12-15T16:36:27Z (GMT). No. of bitstreams: 1 ManoelPedroDeMedeirosNeto_DISSERT.pdf: 3664081 bytes, checksum: f1856f73174bde3b90b40604d7d1ae0e (MD5) Previous issue date: 2016-02-29 / Ve?culos n?o tripulados est?o cada vez mais presentes no cotidiano das empresas e das pessoas, pois esse tipo de ve?culo est? de forma crescente desempenhando atividades que anteriormente eram apenas executadas por seres humanos. No entanto, para se compreender melhor o potencial de ve?culos n?o tripulados, ? importante conhecer seus tipos, caracter?sticas, aplica??es, limita??es e desafios, pois somente com esse conhecimento pode-se entender as potencialidades do uso de ve?culos dessa natureza em aplica??es variadas. Nesse contexto, na primeira parte desta pesquisa foram estudados os diferentes tipos de ve?culos n?o tripulados, i.e. terrestres, aqu?ticos, a?reos e h?bridos. Durante a segunda fase da pesquisa, foi realizado um aprofundamento tendo como foco as interfaces de usu?rio para controle dos ve?culos a?reos n?o tripulados. Esses dois levantamentos iniciais do dom?nio, permitiram a identifica??o de desafios e oportunidades para o desenvolvimento de novas aplica??es para esse contexto. Com base no conhecimento adquirido com esses estudos, ent?o, foi desenvolvido um sistema de entrega automatizada de objetos para o campus de Universidades, denominado de PostDrone University, e desenvolvido um ve?culo a?reo n?o tripulado para realizar as entregas, denominado de PostDrone University UAV K-263. O sistema possui uma interface de usu?rio de f?cil uso, que n?o requer conhecimentos de dom?nios espec?ficos como avia??o ou controle de aeronaves para sua opera??o. Por fim, diversos testes foram realizados com o intuito de validar e identificar as limita??es da solu??o desenvolvida nesta pesquisa. / Unmanned vehicles are increasingly present in the daily of companies and people, because this kind of vehicle is performing ever more tasks that were previously only executed by human beings. However, to better understand the potential of unmanned vehicles, it is important to know their types, features, applications, limitations and challenges, thus with this knowledge it is possible to comprehend the possibilities of use of these vehicles in several applications. In this context, the first step of the present research consists in studying the different kinds of unmanned vehicles, i.e., ground, surface and underwater, aerial, and hybrid. During the second step of the research, a deepening study was accomplished, with focus on user interfaces of unmanned aerial vehicles. These two initial reviews of the domain allowed the identification of challenges and opportunities to the development of new applications for this context. Based on the acquired knowledge from these studies, then, an automated goods delivery system was developed for universities? campuses, called PostDrone University, and an unmanned vehicle to make the deliveries, called PostDrone University UAV K-263, was also developed. The system has an easy use UI, which does not require the user to have knowledge about specific domains, as aviation or aircraft control, for the operation of the system. Lastly, several test were accomplished aiming to validate the solution proposed in the present research and identify its limitations / 2018-03-31

[pt] A reconstrução de superfícies é um problema que recebe bastante atenção em Computação Gráfica dada a importância de suas aplicações. Uma solução comum é representar essas superfícies por malhas triangulares. Neste trabalho é proposta uma estrutura de tetraedros com cúbicas definidas em seu interior que são utilizadas para aproximar a superfície. Essas cúbicas, chamadas de TetraQuads, são superfícies implícitas de grau 3 definidas como interpolação de quádricas posicionadas nos vértices dos tetraedros. Esses elementos foram idealizados de forma que seja rápido o processamento para visualização dessa estrutura pelo hardware gráfico. Os objetos definidos dessa maneira carregam mais informações do ponto de vista da geometria diferencial que uma malha triangular. Por esse motivo, têm uma modelagem mais complexa a ser resolvida. Esse problema é discutido ao apresentar os passos para reconstrução de superfícies por TetraQuads a partir de nuvens de pontos. / [en] Surface reconstruction is a problem that receives a lot of attention in Computer Graphics due to the importance and the number of its applications. A common solution is to represent these surfaces through triangular meshes. This work introduces an alternative structure of tetrahedrons with cubics defined in its interior, which are used to approximate the surface. Those cubics, called TetraQuads, are third-degree implicit surfaces defined as an interpolation of quadrics positioned at the tetrahedrons vertices. These elements are constructed for an efficient visualization by the graphics hardware. Objects defined in this manner contain more information from the differential geometry point of view than a triangle mesh, which entails a more complex modeling problem. This problem is discussed throughout the steps of surface reconstruction from point clouds through TetraQuads.