O presente trabalho tem por objeto fazer uma introdução ao estudo das subvariedades isoparamétricas do espaço Euclidiano. Começamos com uma introdução ao desenvolvimento histórico desses objetos. A seguir apresentamos os conceitos básicos da teoria de subvariedades de formas espaciais. Deduzimos as equações fundamentais de primeira e segunda ordem e demonstramos o teorema fundamental da teoria de subvariedades. Em seguida damos a definição de subvariedade isoparamétrica e desenvolvemos conceitos elementares para o caso do espaço Euclidiano como são normais de curvatura, grupo de Coxeter, câmera de Weyl e variedades paralelas e focais. Provamos dois teoremas referentes à decomposição de subvariedades isoparamétricas do espaço Euclidiano adaptando ferramentas usadas em [HL97] para ocaso de subvariedades isoparamétricas de espaços de Hilbert. Demonstramos o teorema da fatia e discutimos sobre subvariedades isoparamétricas desde o ponto de vista clássico, a saber, aplicações isoparamétricas. Concluímos com alguns exemplos: hipersuperfécies isoparamétricas da esfera e órbitas principais da ação adjunta de um grupo de Lie sobre a respectiva álgebra de Lie. / The goal of this dissertation is to present an introduction to the study of isoparametric submanifolds of Euclidean space. We begin with an introduction to the history of the subject. Then we present the basic results of submanifold theory of space forms. We compute the fundamental equations of first and second order, and we prove the fundamental theorem of submanifold theory. Next, we define isoparametric submanifolds and discuss some basic constructions, as curvature normals, Coxeter groups, Weyl chambers and parallel and focal submanifolds. We prove two decomposition theorems about isoprametric submanifolds using techniques that we learnt from [HL97], paper in which the case of submanifolds of Hilbert spaces is studied. Then we prove slice theorem. We also discuss those submanifold from the classical point of view, namely, isoparametric maps. We finish by explaining some examples: isoparametric hipersurfaces of spheres and principal orbits of the adjoint action of a Lie group on its Lie algebra.

O presente trabalho tem por objeto fazer uma introdução ao estudo das subvariedades isoparamétricas do espaço Euclidiano. Começamos com uma introdução ao desenvolvimento histórico desses objetos. A seguir apresentamos os conceitos básicos da teoria de subvariedades de formas espaciais. Deduzimos as equações fundamentais de primeira e segunda ordem e demonstramos o teorema fundamental da teoria de subvariedades. Em seguida damos a definição de subvariedade isoparamétrica e desenvolvemos conceitos elementares para o caso do espaço Euclidiano como são normais de curvatura, grupo de Coxeter, câmera de Weyl e variedades paralelas e focais. Provamos dois teoremas referentes à decomposição de subvariedades isoparamétricas do espaço Euclidiano adaptando ferramentas usadas em [HL97] para ocaso de subvariedades isoparamétricas de espaços de Hilbert. Demonstramos o teorema da fatia e discutimos sobre subvariedades isoparamétricas desde o ponto de vista clássico, a saber, aplicações isoparamétricas. Concluímos com alguns exemplos: hipersuperfécies isoparamétricas da esfera e órbitas principais da ação adjunta de um grupo de Lie sobre a respectiva álgebra de Lie. / The goal of this dissertation is to present an introduction to the study of isoparametric submanifolds of Euclidean space. We begin with an introduction to the history of the subject. Then we present the basic results of submanifold theory of space forms. We compute the fundamental equations of first and second order, and we prove the fundamental theorem of submanifold theory. Next, we define isoparametric submanifolds and discuss some basic constructions, as curvature normals, Coxeter groups, Weyl chambers and parallel and focal submanifolds. We prove two decomposition theorems about isoprametric submanifolds using techniques that we learnt from [HL97], paper in which the case of submanifolds of Hilbert spaces is studied. Then we prove slice theorem. We also discuss those submanifold from the classical point of view, namely, isoparametric maps. We finish by explaining some examples: isoparametric hipersurfaces of spheres and principal orbits of the adjoint action of a Lie group on its Lie algebra.

Mechanical grinding processes is the most popular way to fabricate the endface of optical fibers, although there are some other methods like chemical etching and leaser machining. Mechanical grinding has its uniqueness in cases of grinding Conical-Wedge-Shaped Fiber Endface¡]CWSFE¡^, polygon-cone-shape fiber endface and fiber endface with double-variable curvatures. Despite Mechanism Design Lab of National Sun Yat-sen university has successfully developed Unsymmetrical Fiber Endface Grinding Mechanism and Torque-Control Fiber Endface Grinding Mechanism in NSC 94-2212-E-230-005 and NSC 95-2221-E-230-020, it still face difficulties when fabricating the fiber endface with double-variable curvatures due to the mechanism constraints. In this study, the focus are concentrated on both the designing and implementing of a mechanism for grinding optical-fiber endface with double-variable curvatures, which controls Material Removal Rate by simultaneously and periodically adjusting the relative positions, as well as the normal pressure between the fiber endface and the grinding film. This study is composed of first, the reviewing the anterior references, both the papers and the patents, and then a series of engineering design methods, which involve the design requirements and constraints, conceptual design, evaluating alternatives, detail design, assembly and calibration. The mechanism, the research result, and those needed to be ameliorated will be demonstrated in the conclusion and discussion part, so as to offer the investigating direction in the future. It¡¦s believed that the grinding machine system developed in this study can be successfully applied to fabricating optical fiber lenses as well as different types of micro probes , micro electrodes, and micro spectroscopefors for other applications, with little adjustment of the jig and ferrule of this machine .

Discrete Differential Geometry (DDG) concerns discrete counterparts of notions and methods in differential geometry. This thesis deals with a core subject in DDG, discrete curvature theories on various types of polyhedral surfaces that are practically important for free-form architecture, sunlight-redirecting shading systems, and face recognition. Modeled as polyhedral surfaces, the shapes of free-form structures may have to satisfy different geometric or physical constraints. We study a combination of geometry and physics – the discrete surfaces that can stand on their own, as well as having proper shapes for the manufacture. These proper shapes, known as circular and conical meshes, are closely related to discrete principal curvatures. We study curvature theories that make such surfaces possible. Shading systems of freeform building skins are new types of energy-saving structures that can re-direct the sunlight. From these systems, discrete line congruences across polyhedral surfaces can be abstracted. We develop a new curvature theory for polyhedral surfaces equipped with normal congruences – a particular type of congruences defined by linear interpolation of vertex normals. The main results are a discussion of various definitions of normality, a detailed study of the geometry of such congruences, and a concept of curvatures and shape operators associated with the faces of a triangle mesh. These curvatures are compatible with both normal congruences and the Steiner formula. In addition to architecture, we consider the role of discrete curvatures in face recognition. We use geometric measure theory to introduce the notion of asymptotic cones associated with a singular subspace of a Riemannian manifold, which is an extension of the classical notion of asymptotic directions. We get a simple expression of these cones for polyhedral surfaces, as well as convergence and approximation theorems. We use the asymptotic cones as facial descriptors and demonstrate the practicability and accuracy of their applications in face recognition.

Mechanical grinding process is the most popular way to fabricate the fiber micro lenses, although there are some other methods, such as chemical etching, laser machining and focused ion beam micro-cutting. Mechanical grinding has its uniqueness in grinding Conical-Wedge-Shaped Fiber Endface, fiber endface with polygon-cone-shape, and fiber endface with double-variable curvatures. The double-variable curvatures fiber endface polisher, designed and manufactured by Mechanism Design Lab of NSYSU, is employed in this study. The normal force of the fiber endface is derived firstly and then the experimental parameters and data are substituted into the material removal rate (M.R.R.) formula to obtain M.R.R. and the Preston¡¦s constant K. The process parameters of the feed rate and polishing time on the fabrication of the fiber endface are analyzed. The polisher is calibrated and adjusted to improve the precision of the optical-fiber endface. A fiber endface with double-variable curvature is successfully fabricated in a single grinding process by properly controlling the fiber rotation angle, inclining angle, and the distant between the endface and the grinding film simultaneously. The grinding process developed in this study can be applied for fabricating optical fiber lenses in fiber optics communication as well as different types of micro probes, and micro spectroscopefors in other applications.

The topic of this thesis is motivated by recent developments in Architectural Geometry, namely Eike Schling’s asymptotic gridshells and progress in solutions for paneling freeform facades. An asymptotic gridshell is fabricated from flat straight lamellas of bendable material such as sheet metal. These strips are then arranged in a grid-like spatial structure, such that the lamellas are orthogonal to a reference surface, which however is not materialized. Differential geometry then tells us that the strips must follow asymptotic curves of that reference surface. The actual construction is simplified if angles at nodes are constant. If that angle is a right angle, one gets minimal surfaces as reference surfaces. If the angle is constant, one obtains negatively curved surfaces with a constant ratio of principal curvatures (CRPC surfaces). Their characteristic parameterizations are equi-angular asymptotic parameterizations. We are also interested in the positively curved CRPC surfaces. Due to the relation between curvatures, they have a one-parameter family of curvature elements, which facilitates cost-effective paneling solutions through mold-reuse. Our approach to positively curved CRPCS surfaces is again based on equi-angular characteristic parameterizations. These characteristic parameterizations are conjugate and symmetric with respect to the principal curvature directions. After a review of the required results from classical surface theory, we first present a derivation of rotational CRPC surfaces. By simple geometric considerations one can find their characteristic parameterizations. In this way we add some new insight to this known class of surfaces. However, it turns out to be very hard to come up with explicit results on non-rotational CRPC surfaces. This is in big contrast to the special case of minimal surfaces which are characterized be the constant principal curvature ratio -1. Due to the difficulties in handling smooth CRPC surfaces, we turn to discrete models in form of constrained quad meshes. The discrete models belong to the area of Discrete Differential Geometry. There, one does not discretize equations from the smooth theory, but fundamental concepts of the theory. We introduce the basic structures needed in this context: asymptotic nets, conjugate nets and principal symmetric nets. The latter are a recent development in discrete differential geometry and characterized by spherical vertex stars. This means that a vertex of the quad mesh and its four connected neighbors lie on a sphere. If that sphere degenerates to a plane at all vertices, one has the classical discrete asymptotic parameterization as an A-net. Several ways to discretize the constant intersection angle are presented. The actual computation of discrete CRPC surfaces is performed with numerical optimization with an appropriately regularized Gauss-Newton algorithm for solving a nonlinear least squares problem. Optimization requires initial configurations. Those can come from the known classes of CRPC surfaces such as rotational surfaces of minimal surfaces. The latter case yields some surprising results on negatively curves CRPC surfaces of nontrivial topology. In general, such discrete models can serve as a guiding line for future research on the theoretical side. This is briefly indicated in the final discussion on future research directions.

L'objectif de la thèse est l'étude des propriétés géométriques des ensembles définissables dans les structures o-minimales et de ses applications. Il existe trois principaux résultats présentés dans cette thèse. Le premier est une preuve géométrique de l'existence de stratifications vérifiant les conditions (a) et (b) de Whitney d'ensembles définissables. Ce résultat fut d'abord prouvé par T. L. Loi en 1994 par une autre méthode. Le second est une preuve de l'existence de stratifications de Lipschitz (dans le sens de Mostowski) pour les ensembles définissables dans une structure o-minimale polynomialement bornée. Ceci est une généralisation de résultats de Parusin'ski en 1994 pour les ensembles sous-analytiques. Le troisième résultat est au sujet de la continuité des variations de géométrie intégrale appelées courbures de Lipschitz Killing locales, qui ont été introduites par A. Bernig et L. Broker en 2002. Nous prouvons que les courbures de Lipschitz Killing locales sont continues le long de strates de stratifications de Whitney d'ensembles définissable dans une structure o-minimale polynomialement bornée, et si les stratifications sont (w) régulières alors les courbures de Lipschitz Killing locales sont localement lipschitziennes le long des strates. / The thesis focus on study geometric properties of definable sets in o-minimal structures and its applications. There are three main results presented in this thesis. The first is a geometric proof of the existence of Whitney (a) and (b)-regular stratifications of definable sets. The result was initially proved by T. L. Loi in 1994 by using another method. The second is a proof of existence of Lipschitz stratifications (in the sense of Mostowski) of definable sets in a polynomially bounded o-minimal structure. This is a generalization of Parusinski's 1994 result for subanalytic sets. The third result is about the continuity of of variations of integral geometry called local Lipschitz Killing curvatures which were introduced by A. Bernig and L. Broker in 2002. We prove that Lipschitz Killing curvatures are continuous along strata of Whiney stratifications of definable sets in a polynomially bounded o-minimal structure. Moreover, if the stratifications are (w)-regular the Lipspchitz Killing curvatures are locally Lipschitz.

Neste trabalho, classificamos localmente as hipersuperfcies dos espaços produto S n × R e H n × R, n 6 = 3, com g curvaturas principais constantes e distintas, g {1, 2, 3}. Verifi- camos que tais hipersuperfcies são isoparamétricas de Q nc × R. Além disso, encontramos uma condição necessária e suficiente para que uma hipersuperfcie isoparamétrica de Q nc × R que possui fibrado normal plano, quando observada como uma subvariedade de codimensão dois de R n+2 contendo S n × R e de L n+2 contendo H n × R, tenha curvaturas principais constantes. / In this work, we classify locally the hypersurfaces in product spaces S n × R and H n × R, n 6 = 3, with g distinct constant principal curvatures, g {1, 2, 3}. We verify that such hy- persurfaces are isoparametric in Q nc × R. Furthermore, we find a necessary and sufficient condition for an isoparametric hypersurface in Q nc × R with flat normal bundle, when re- garded as a submanifold with codimension two of the flat spaces R n+2 containing S n × R and L n+2 containing H n × R, having constant principal curvatures.

Nous étudions les relations entre des propriétés géométriques et des propriétés métriques dans les domaines de C^n.Plus précisément, nous nous intéressons au comportement des courbures bisectionnelles holomorphes de métriques de Kähler invariantes, la métrique de Bergman et la métrique de Kähler-Einstein, au voisinage du bord des domaines pseudoconvexe bornés à bord lisse.Nous prouvons qu'aux points de stricte pseudoconvexité ou tels que la fonction squeezing du domaine tend vers 1 les courbures bisectionnelles holomorphes de la métrique de Kähler-Einstein du domaine tendent vers les courbures bisectionnelles holomorphes de la métrique de Kähler-Einstein de la boule.Nous étudions également les courbures de la métrique de Kähler-Einstein et de la métrique de Bergman dans certains domaines polynomiaux (notamment les domaines tubes et les domaines de Thullen de C^2) qui servent de modèles locaux aux points du bord qui sont de type fini. A partir de ces études nous prouvons qu'en certains points du bord de domaines convexes bornés lisse de type fini dans C^2 il existe un voisinage non tangentiel tel que les courbures bisectionnelles holomorphes de la métrique de Kâhler-Einstein sont pincées négativement. Nous prouvons également que pour tout domaine pseudoconvexe borné de type fini qui est Reinhardt complet il existe un voisinage du bord relatif au domaine tel que les courbures bisectionnelles holomorphes de la métrique de Bergman sont comprises entre deux constantes strictement négatives. / We study the relationships between geometric properties and metric properties of domains in C^n.More precisely, we are interested in the behavior of holomorphic bisectional curvatures of invariant Kähler metrics, namely the Bergman metric and the Kähler-Einstein metric, near the boundary of bounded pseudoconvex domains with smooth boundary.We prove that at boundary points that are either strictly pseudoconvex or such that the squeezing function of the domain tends to one the holomorphic bisectional curvatures of the Kähler-Einstein metric of the domain tends to the holomorphic bisectional curvatures of the Kähler-Einstein metric of the ball.We also study the holomorphic bisectional curvatures of the Kähler-Einstein metric and of the Bergman metric in some polynomial domains (namely tube and Thullen domains in C^2) which serve as local models at boundary point of finite type. Using these studies we prove that at certain boundary points of smoothly bounded convex domains of finite type there exists a non tangential neighbourhood such the holomorphic bisectional curvatures of the Kähler-Einstein metric are pinched between two negative constants. We also prove that for every smoothly bounded pseudoconvex complete Reinhardt domain of finite type inf C^2 there exists a neighbourhood of the boundary relative to the domain in which the holomorphic bisectional curvatures of the Bergman metric are pinched between two negative constants.

This Ph.D thesis work is dedicated to 3D facial surface analysis, processing as well as to the newly proposed 3D face recognition modality, which is based on mapping techniques. Facial surface processing and analysis is one of the most important steps for 3Dface recognition algorithms. Automatic anthropometric facial features localization also plays an important role for face localization, face expression recognition, face registration ect., thus its automatic localization is a crucial step for 3D face processing algorithms. In this work we focused on precise and rotation invariant landmarks localization, which are later used directly for face recognition. The landmarks are localized combining local surface properties expressed in terms of differential geometry tools and global facial generic model, used for face validation. Since curvatures, which are differential geometry properties, are sensitive to surface noise, one of the main contributions of this thesis is a modification of curvatures calculation method. The modification incorporates the surface noise into the calculation method and helps to control smoothness of the curvatures. Therefore the main facial points can be reliably and precisely localized (100% nose tip localization using 8 mm precision)under the influence of rotations and surface noise. The modification of the curvatures calculation method was also tested under different face model resolutions, resulting in stable curvature values. Finally, since curvatures analysis leads to many facial landmark candidates, the validation of which is time consuming, facial landmarks localization based on learning technique was proposed. The learning technique helps to reject incorrect landmark candidates with a high probability, thus accelerating landmarks localization. Face recognition using 3D models is a relatively new subject, which has been proposed to overcome shortcomings of 2D face recognition modality. However, 3Dface recognition algorithms are likely more complicated. Additionally, since 3D face models describe facial surface geometry, they are more sensitive to facial expression changes. Our contribution is reducing dimensionality of the input data by mapping3D facial models on to 2D domain using non-rigid, conformal mapping techniques. Having 2D images which represent facial models, all previously developed 2D face recognition algorithms can be used. In our work, conformal shape images of 3Dfacial surfaces were fed in to 2D2 PCA, achieving more than 86% recognition rate rank-one using the FRGC data set. The effectiveness of all the methods has been evaluated using the FRGC and Bosphorus datasets.