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Concept# Forme (géométrie)

Résumé

En géométrie classique, la forme permet d’identifier ou de distinguer des figures selon qu’elles peuvent ou non être obtenues les unes à partir des autres par des transformations géométriques qui préservent les angles en multipliant toutes les longueurs par un même coefficient d’agrandissement.
Au sens commun, la forme d’une figure est en général décrite par la donnée combinatoire d’un nombre fini de points et de segments ou d’autres courbes délimitant des surfaces, des comparaisons de longueurs ou d’angles, d’éventuels angles droits et éventuellement du sens de courbure. Ceci permet notamment de distinguer parmi les triangles ceux de forme équilatérale, isocèle et/ou rectangle, et de caractériser la présence d’un angle obtus. Cette acception permet aussi d’écrire « deux rectangles ont tous deux la forme d’un... rectangle [mais] peuvent être ou ne pas être semblables ».
Pour un objet dans l’espace, la forme décrit la frontière externe de l’objet — abstraction faite de son emplacement

Source officielle

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Aire (géométrie)

thumb|L'aire du carré vaut ici 4.
En mathématiques, l'aire est une grandeur relative à certaines figures du plan ou des surfaces en géométrie dans l'espace.
Le développement de cette notion mathémati

Mathématiques

thumb|upright|Raisonnement mathématique sur un tableau.
Les mathématiques (ou la mathématique) sont un ensemble de connaissances abstraites résultant de raisonnements logiques appliqués à des objets

Polygone

Un polygone, en géométrie euclidienne, est une figure géométrique plane formée d'une ligne brisée (appelée aussi ligne polygonale) fermée, c'est-à-dire d'une suite cyclique de segments consécutifs.

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The course's objectivs are: Learning several advenced methods in experimental physics, and critical reading of experimental papers.

The course allows students to get familiarized with the basic tools and concepts of modern microeconomic analysis. Based on graphical reasoning and analytical calculus, it constantly links to real economic issues.

Computer Vision aims at modeling the world from digital images acquired using video or infrared cameras, and other imaging sensors.
We will focus on images acquired using digital cameras. We will introduce basic processing techniques and discuss their field of applicability.

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The distance from self-intersection of a (smooth and either closed or infinite) curve q in three dimensions can be characterised via the global radius of curvature at q(s), which is defined as the smallest possible radius amongst all circles passing through the given point and any two other points on the curve. The minimum value of the global radius of curvature along the curve gives a convenient measure of curve thickness or normal injectivity radius. Given the utility of the construction inherent to global curvature, it is natural to consider variants defined in related ways. The first part of the thesis considers all possible circular and spherical distance functions and the associated, single argument, global radius of curvature functions that are constructed by minimisation over all but one argument. It is shown that among all possible global radius of curvature functions there are only five independent ones. And amongst these five there are two particularly useful ones for characterising thickness of a curve. We investigate the geometry of how these two functions, ρpt and ρtp, can be achieved. Properties and interrelations of the divers global radius of curvature functions are illustrated with the simple examples of ellipses and helices. It is known that any Lipschitz continuous curve with positive thickness actually has C1,1-regularity. Accordingly, C1,1 is the natural space in which to carry out computations involving self-avoiding curves. The second part of the thesis develops the mathematical theory of biarcs, which are a geometrically elegant way of discretizing C1,1 space curves. A biarc is a pair of circular arcs joined in a C1 fashion according to certain matching rules. We establish a self-contained theory of the geometry of biarc interpolation of point-tangent data sampled from an underlying base curve, and demonstrate that such biarc curves have attractive convergence properties in both a pointwise and function-space sense, e.g. the two arcs of the biarc interpolating a coalescent point-tangent data pair on a C2-curve approach the osculating circle of the curve at the limit of the data points, and for a C1,1-base curve and a sequence of (possibly non-uniform) meshes, the interpolating biarc curves approach the base curve in the C1-norm. For smoother base curves, stronger convergence can be obtained, e.g. interpolating biarc curves approach a C2 base curve in the C1,1-norm. The third part of the thesis concerns the practical utility of biarcs in computation. It is shown that both the global radius of curvature function ρpt and thickness can be evaluated efficiently (and to an arbitrarily small, prescribed precision) on biarc curves. Moreover, both the notion of a contact set, i.e. the set of points realising thickness, and an approximate contact set can be defined rigorously. The theory is then illustrated with an application to the computation of ideal shapes of knots. Informally ideal knot shapes can be described as the configuration allowing a given knot to be tied with the shortest possible piece of rope of prescribed thickness. The biarc discretization is combined with a simulated annealing code to obtain approximate ideal shapes. These shapes provide rigorous upper bounds for rope length of ideal knots. The approximate contact set and the function ρpt evaluated on the computed shapes allow us to assess closeness of the computations to ideality. The high accuracy of the computations reveal various, previously unrecognized, features of ideal knot shapes.

There is a real need for methods to allow the measurement of the form and deformation of large objects. For example, for maintenance and production costs as well as for security. Even though there are many methods of measuring the form and deformation of small objects (up to 1 m2), currently none of them are able to quickly measure larger objects (i.e. at a large number of points at the same time). Among the existing techniques, the fringe projection seem to us one of the most adequate to deal with these kinds of problems. In its classical form, this technique is very simple, since it consists of projecting equispaced rectilinear fringes on an object from one direction and of observing the scene from another with a CCD camera. The displacement of the fringes distorted by the object contains the desired shape information. The phase shifting and phase unwrapping procedures allow automatic, rapid acquisition of an optical print (called a "phase map") of the object. In the case of small objects (the classical approach), the extraction of the shape information from this optical print is quite simple. A phase map of the object as well as a phase map of a reference surface plane is acquired. Then, basically, the desired shape information is obtained by subtracting these two maps from each other. For larger objects, this approach is not possible anymore. On the one hand, such a reference surface does not exist. On the other hand, in order to measure the whole surface at once, it must be fully illuminated. This suggests the use of interferometrically generated fringes, in divergent beams, which implies that the fringes are no longer rectilinear and equispaced. For these reasons the classical approach is no longer valid. It is therefore necessary to find another method to be able to extract the object shape information from the optical print. In the frame of this work, three methods have been conceived and developed in order to extract the shape information of large objects from their optical print. The first two methods proposed here are dedicated to quasi-planar objects that are parallel to the imaging plane of the camera. These assumptions allow the simplification of the equations describing the system to "mimic" the classical approach, where the desired shape information is proportional to the difference between the measured and reference phase maps. The next step is to determine the parameters of the projection head. The first method is based on two coupled interferometers (of the Mach-Zehnder and Young's type), and the other uses least squares calculations with a small number of calibration points, aiming at minimizing the difference between the theoretical and measured phase at more than four calibration points. Finally, the desired reference phase map is artificially generated. These two techniques are simple but their application is limited only to planar object parallel to the imaging plane of the camera. In the last technique, which is based on a new approach, the system is described by the interferometric equation and the central perspective equations. Solving them simultaneously allows the determination of the coordinates (x,y,z) of all measured points from the optical print. This approach is general and offers the advantage of allowing the measurement of the shape and deformation of large objects. In addition, it also makes the system more flexible. In this report, these different techniques are presented and their feasibility is shown; examples of measurements give a first evaluation of their precision, and assess the new possibilities offered in terms of object shape and configuration of the measurement system, as well as their limitations. Finally, the three methods are compared one to another and advice for their optimization is proposed.

In contemporary architecture there is an increasing demand for transparency and curved shape. The Cold-Bending process is an effective and relatively inexpensive way of creating double curved surface glass. This thesis focuses on the shaping of a plate with the application of an out-of-plane displacement at its corner, also called cold-warping.

2015