Academic literature on the topic 'Intrinsic geometry'

An intrinsic geometric flow is an evolution of a Riemannian metric by a two-tensor. An extrinsic geometric flow is an evolution of an immersion of a manifold into Euclidean space. An extrinsic flow induces an evolution of a metric because any immersed manifold inherits a Riemannian metric from Euclidean space. In this paper we discuss the inverse problem of specifying an evolution of a metric and then seeking an extrinsic geometric flow which induces the given metric evolution. We limit our discussion to the case of manifolds that are rotationally symmetric and embeddable with codimension one. In this case, we reduce an intrinsic geometric flow to a plane curve evolution. In the specific cases we study, we are able to further simplify the evolution to an evolution of a function of one variable. We provide soliton equations and give proofs that some soliton metrics exist.

Le probleme du tolerancement des pieces mecaniques est decisif pour I'industrie de pointe. Ses incidences economiques sont importantes pour le secteur manufacturier qui subit des transformations profondes imposees par la globalisation des marches et revolution constante des technologies (CAO, CMM, 3D Scanner, etc.). U est admit aujourd'hui que I'optimisation des performances des produits requiert la prise en compte des variations inherentes aux processus de fabrication, d'ou le controle de la qualite a travers le processus de developpement et de fabrication. Dans le cas des composantes dites 'flexibles' (ou non rigides), par exemple des pieces mecanique a parol minces comme le revetement d'un avion ou d'une auto, I'usage industriel actuel est limite encore a I'utilisation de gabarit de conformite, relativement couteux, qui contraignent la geometric de la piece a un etat qui reflete 1'assemblage. Par la suite, des mesures par contact directe ou par scanner sont effectuees. C'est ainsi I'industrie elimine I'effet des deformations dues a la flexibilite de la piece pour tenter de detecter les defauts dus au procede de fabrication. Le projet propose a pour objectif de faciliter les operations d'inspection dimensionnelle et geometrique des composantes flexibles a partir d'un nuage de points, et ce, sans recours a un gabarit ou des operations secondaires de conformation. Plus specifiquement, nous visons le developpement d'une methodologie qui permettra de localiser et de quantifier les defauts de profil dans le cas des coques minces typique des industries aerospatiales et automobiles. Pour arriver a cet objectif, nous mettons en oeuvre une idee que nous appelons Numerical Inspection Fixture. Nous utilisons les distances geodesiques pour detecter la similarite intrinseque entre une piece a I'etat libre qui inclus les effets de gravite, des contraintes internes et des defauts de fabrication, et la meme piece telle que definie nominalement par un modele CAO. Ce memoire developpe le fondement theorique de cette methode et les algorithmes qu'y sont relies. Nous employons une approche, deja employe dans le domaine de I'imagerie medicale, pour identifier les distances geodesiques minimales (geometric metrique), les statistiques {Multidimensional Scaling - MDS) pour analyser les similarites et dissimilarites entre deux objets, ainsi que la methode d'elements finis (FE) pour aboutir a une approche generate et original pour I'inspection de pieces geometriques non rigides. Deux methodes y sont proposees avec des validations numeriques.

The aim of the present study was to investigate the potential for children to use the turtle methaphor to develop understandings of intrinsic, euclidean and cartesian geometrical ideas. Four aspects of the problem were investigated. a) the nature of the schema children form when they identify with the turtle in order to change its state on the screen; b) whether it is possible for them to use the schema to gain insights into certain basic geometrical principles of the cartesian geometrical system; c) how they might use the schema to form understandings of euclidean geometry developed inductively from specific experiences; d) the criteria they develop for choosing between intrinsic and euclidean ideas. Ten 11 to 12 year - old children participated in the research, previously having had 40 to 50 hours of experience with Turtle geometry. The research involved three case - studies of pairs of children engaging in cooperative activities, each case - study within a geometrical Logo microworld. The data included hard copies of everything that was said, typed and written. Issues a) and b) were investigated by means of the first case - study which involved three pairs of children and a microworld embedding intrinsic and coordinate ideas. A model of the children's intrinsic schema and a model of the coordinate schema which they formed during the study were devised. The analysis shows that the two schemas remained separate in the children's minds with the exception of a limited number of occasions of context specific links between the two. Issue c) was investigated in the second case - study involving one pair of children and a microworld where the turtle was equipped with distance and turn measuring instruments and a facility to mark positions. The analysis illustrates how a turtle geometric environment of a dynamic mathematical nature was generated by the children, who used their intrinsic schema and predominantly engaged in inductive thinking. The geometrical content available to the children within this environment was extended from intrinsic to both intrinsic and euclidean geometry. Issue d) was investigated by means of the third case - study involving a pair of children and a microworld where the children could choose among circle procedures embedding intrinsic and/or euclidean notions in order to construct figures of circle compositions. The analysis shows that the children employed their turtle schema in using both kinds of notions and did not seem to perceive qualitative differences between them. Their decisions on which type of notion to use were influenced by certain broader aspects of the mathematical situations generated in the study.

Line geometry is a foundation of screw algebra in line coordinates that were created by Plücker as ray coordinates taking a line as a ray between two points and axis coordinates taking a line as the intersection of two planes. This Thesis reveals the geometrical meaning and intrinsic relationship between these ray coordinates and axis coordinates, leading to an in-depth understanding of conformability and duality of these two sets of screw coordinates, and their related vector space and dual vector space. Based on the study of screw algebra, the resultant twist of a serial manipulator is presented geometrically by an assembly of unit joint screws with the corresponding velocity amplitudes. This leads to the geometrical interpretation for the resultant twist with its instantaneous screw axis (ISA) that is formulated by a combination of weighted position vectors of joint screws. The screw-based Jacobian is then derived after recognizing the resultant twist of a serial manipulator. The case leads to a revelation for the first time the relationship of a Jacobian matrix acquired by using screw algebra and a derivative Jacobian matrix using differential functions, and to an in-depth investigation of transformation between these two Jacobians. To extend the application of screw algebra and this derivative Jacobian, kinematics analysis of a novel reconfigurable base-integrated parallel mechanism is proposed and its screw-based Jacobian is derived, leading to its equivalent model, the Metamorphic hand with a reconfigurable palm. The method is then applied to the investigation of the Metamorphic Hand, while manipulating an object, based on the product sub-manifolds and the exponential method. Evaluation of the functionality of the Metamorphic hand is further analysed, with the Anthropomorphism Index (AI) and palmar shape modulation as the criteria, evaluating performance enhancement of the Metamorphic hand in comparison to other robotic hands with a fixed palm. The Thesis presents novel discoveries in the intrinsic geometry of screw coordinates and the coherent connection between Jacobian formed by screw algebra and the Jacobian using the derivative method. This intrinsic geometry insight is then used to investigate for the first time the parallel mechanism with a reconfigurable base, paving a way for an in-depth investigation of the Metamorphic hand on its reconfigurability and grasp affordability and for the first time using Anthropomorphic Index to evaluate the Metamorphic hand.

Surface topography is, generally, composed of many length scales starting from its physical geometry, to its microscopic or atomic scales known by roughness. The spatial and geometrical evolution of the roughness topography of engineering surfaces avail comprehensive understanding, and interpretation of many physical and engineering problems such as friction, and wear mechanisms during the mechanical contact between adjoined surfaces. Obviously, the topography of rough surfaces is of random nature. It is composed of irregular hills/valleys being spatially correlated. The relation between their densities and their geometric properties are the fundamental topics that have been developed, in this research study, using the theory of random fields and the integral geometry.An appropriate random field model of a rough surface has been defined by the most significant parameters, whose changes influence the geometry of its excursion. The excursion sets were quantified by functions known as intrinsic volumes. These functions have many physical interpretations, in practice. It is possible by deriving their analytical formula to estimate the parameters of the random field model being applied on the surface, and for statistical analysis investigation of its excursion sets. These subjects have been essentially considered in this thesis. Firstly, the intrinsic volumes of the excursion sets of a class of mixture models defined by the linear combination of Gaussian and t random fields, then for the skew-t random fields are derived analytically. They have been compared and tested on surfaces generated by simulations. In the second stage, these random fields have been applied to real surfaces measured from the UHMWPE component, involved in application of total hip implant, before and after wear simulation process. The primary results showed that the skew-t random field is more adequate, and flexible for modelling the topographic roughness. Following these arguments, a statistical analysis approach, based on the skew-t random field, is then proposed. It aims at estimating, hierarchically, the significant levels including the real hills/valleys among the uncertain measurements. The evolution of the mean area of the hills/valleys and their levels enabled describing the functional behaviour of the UHMWPE surface over wear time, and indicating the predominant wear mechanisms.

Les ``C-arm'' sont des systémes de radiologie interventionnelle fréquemment utilisés en salle d'opération ou au lit du patient. Des images 3D des structures anatomiques internes peuvent être calculées à partir de multiples radiographies acquises sur un ``C-arm mobile'' et isocentrique décrivant une trajectoire généralement circulaire autour du patient. Pour cela, la géométrie conique d'acquisition de chaque radiographie doit être précisément connue. Malheureusement, les C-arm se déforment en général au cours de la trajectoire. De plus leur motorisation engendre des oscillations non reproductibles. Ils doivent donc être calibrés au cours de l'acquisition. Ma thèse concerne la calibration intrinsèque d'un C-arm à partir de la détection de la projection du collimateur de la source dans les radiographies.Nous avons développé une méthode de détection de la projection des bords linéaires du collimateur. Elle surpasse les méthodes classiques comme le filtre de Canny sur données simulées ou réelles. La précision que nous obtenons sur l'angle et la position (phi,s) des droites est de l'ordre de: phi{RMS}=+/- 0.0045 degrees et s{RMS}=+/- 1.67 pixels. Nous avons évalué nos méthodes et les avons comparés à des méthodes classiques de calibration dans le cadre de la reconstruction 3D<br>Mobile isocentric x-ray C-arm systems are an imaging tool used during a variety of interventional and image guided procedures. Three-dimensional images can be produced from multiple projection images of a patient or object as the C-arm rotates around the isocenter provided the C-arm geometry is known. Due to gravity affects and mechanical instabilities the C-arm source and detector geometry undergo significant non-ideal and possibly non reproducible deformation which requires a process of geometric calibration. This research investigates the use of the projection of the slightly closed x-ray tube collimator edges in the image field of view to provide the online intrinsic calibration of C-arm systems.A method of thick straight edge detection has been developed which outperforms the commonly used Canny filter edge detection technique in both simulation and real data investigations. This edge detection technique has exhibited excellent precision in detection of the edge angles and positions, (phi,s), in the presence of simulated C-arm deformation and image noise: phi{RMS} = +/- 0.0045 degrees and s{RMS} = +/- 1.67 pixels. Following this, the C-arm intrinsic calibration, by means of accurate edge detection, has been evaluated in the framework of 3D image reconstruction

L'ensemble d'excursion EX(u) d'un champ aléatoire réel X sur R^d à un niveau de seuil u ∈ R est le sous-ensemble du domaine R^d où X dépasse u. Ainsi, l'ensemble d'excursion est aléatoire, et sa distribution à un niveau fixe u est déterminée par la distribution de X. Étant des sous-ensembles de R^d, les ensembles d'excursions peuvent être étudiés en termes de leurs propriétés géométriques comme moyen d'obtenir des informations partielles sur les propriétés de distribution des champs aléatoires sous-jacents.Cette thèse examine :(a) comment les mesures géométriques d'un ensemble d'excursion peuvent être déduites à partir d'un échantillon discret de l'ensemble d'excursion, et(b) comment ces mesures peuvent être liées aux propriétés distributionnelles du champ aléatoire à partir duquel l'ensemble d'excursion a été obtenu. Chacun de ces points est examiné en détail dans le Chapitre 1, qui fournit un aperçu global des résultats trouvés tout au long du reste de ce manuscrit. Les mesures géométriques que nous étudions (pour les ensembles d'excursion et les sous-ensembles déterministes de R^d) lors de l'adressage du point (a) sont la mesure de la surface de dimension (d−1), le reach, et le rayon de r-convexité. Chacune de ces quantités peut être liée à la régularité de la frontière de l'ensemble, ce qui est souvent difficile à déduire à partir d'échantillons discrets de points.Pour résoudre ce problème, nous apportons les contributions suivantes au domaine de la géométrie computationnelle :- Dans le Chapitre 2, nous identifions le facteur de biais qui correspond aux algorithmes de comptage local pour calculer la mesure de la surface de dimension (d − 1) des ensembles d'excursion sur une grande classe de pavages de R^d. Le facteur de biais dépend uniquement de la dimension d et non de la géométrie précise du pavage.- Dans le Chapitre 3, nous introduisons un algorithme de comptage pseudo-local pour calculer le périmètre des ensembles d'excursion en deux dimensions. L'algorithme proposé est convergent multigrille (multigrid convergent en anglais) et comporte un hyper paramètre réglable pouvant être choisi automatiquement à partir d'informations accessibles.- Dans le Chapitre 4, nous introduisons le β-reach en tant que généralisation du reach, et l'utilisons pour prouver la cohérence d'un estimateur du reach des sous-ensembles fermés de R^d. De même, nous définissons un estimateur cohérent du rayon de r-convexité des sous-ensembles fermés de R^d. De nouvelles relations théoriques sont établies entre le reach et le rayon de r-convexité. Nous étudions également comment ces mesures géométriques des ensembles d'excursion sont liées à la distribution du champ aléatoire.- Dans le Chapitre 5, nous introduisons l'extremal range : une statistique géométrique locale qui caractérise l'étendue spatiale des dépassements de seuil à un niveau fixe u ∈ R. La distribution de l'extremal range est entièrement déterminée par la distribution de l'ensemble d'excursion au niveau u. Nous montrons comment l'extremal range est liée distributionnellement aux volumes intrinsèques de l'ensemble d'excursion. De plus, le comportement limite de l'extremal range aux grands seuils est étudié en relation avec la stabilité des peaks-over-threshold du champ aléatoire sous-jacent. Enfin, la théorie est appliquée à des données climatiques réelles pour mesurer le degré d'indépendance asymptotique présent et sa variation dans l'espace.Des perspectives sur la manière dont ces résultats peuvent être améliorés et étendus sont fournies dans le Chapitre 6<br>The excursion set EX(u) of a real-valued random field X on R^d at a threshold level u ∈ R is the subset of the domain R^d on which X exceeds u. Thus, the excursion set is random, and its distribution at a fixed level u is determined by the distribution of X. Being subsets of R^d, excursion sets can be studied in terms of their geometrical properties as a means of obtaining partial information about the distributional properties of the underlying random fields.This thesis investigates(a) how the geometric measures of an excursion set can be inferred from a discrete sample of the excursion set, and(b) how these measures can be related back to the distributional properties of the random field from which the excursion set was obtained.Each of these points are examined in detail in Chapter 1, which provides a broad overview of the results found throughout the remainder of this manuscript. The geometric measures that we study (for both excursion sets and deterministic subsets of R^d) when addressing point (a) are the (d − 1)-dimensional surface area measure, the reach, and the radius of r-convexity. Each of these quantities can be related to the smoothness of the boundary of the set, which is often difficult to infer from discrete samples of points. To address this problem, we make the following contributions to the field of computational geometry:• In Chapter 2, we identify the bias factor in using local counting algorithms for computing the (d − 1)-dimensional surface area of excursion sets over a large class of tessellations of R^d. The bias factor is seen to depend only on the dimension d and not on the precise geometry of the tessellation.• In Chapter 3, we introduce a pseudo-local counting algorithm for computing the perimeter of excursion sets in two-dimensions. The proposed algorithm is multigrid convergent, and features a tunable hyperparameter that can be chosen automatically from accessible information.• In Chapter 4, we introduce the β-reach as a generalization of the reach, and use it to prove the consistency of an estimator for the reach of closed subsets of R^d. Similarly, we define a consistent estimator for the radius of r-convexity of closed subsets of R^d. New theoretical relationships are established between the reach and the radius of r-convexity.We also study how these geometric measures of excursion sets relate to the distribution of the random field.• In Chapter 5, we introduce the extremal range: a local, geometric statistic that characterizes the spatial extent of threshold exceedances at a fixed level threshold u ∈ R. The distribution of the extremal range is completely determined by the distribution of the excursion set at the level u. We show how the extremal range is distributionally related to the intrinsic volumes of the excursion set. Moreover, the limiting behavior of the extremal range at large thresholds is studied in relation to the peaks-over-threshold stability of the underlying random field. Finally, the theory is applied to real climate data to measure the degree of asymptotic independence present, and its variation throughout space.Perspectives on how these results may be improved and expanded upon are provided in Chapter 6