19 research outputs found

    Planar panels and planar supporting beams in architectural structures

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    In this article, we investigate geometric properties and modeling capabilities of quad meshes with planar faces whose mesh polylines enjoy the additional property of being contained in a single plane. This planarity is a major benefit in architectural design and building construction: If a structural element is contained in a plane, it can be manufactured on the ground without scaffolding and put into place as a whole. Further, the plane it is contained in serves as part of a so-called support structure. We discuss design of meshes under the requirement that one half of mesh polylines are planar (“P meshes”), and we also investigate the geometry and design of meshes where all polylines enjoy this property (“PP meshes”). We work in the space of planes and with appropriate transformations of that space. We also incorporate further properties relevant for architectural design, such as near-rectangular panels and repetitive nodes. We provide geometric insights, give explicit constructions, and show an approach to geometric modeling of both P meshes and PP meshes, in particular, the case of nearly rectangular panels

    Constructive Lattice Geometry

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    Lattice structures are widespread in product and architectural design. Recent work has demonstrated the printing of nano-scale lattices. However, an anticipated increase in product complexity will require the storage, processing, and design of lattices with orders of magnitude more elements than current Computer-Aided Design (CAD) software can manage. To address this, we propose a class of highly regular lattices called Steady Lattices, which due to their regularity, provide opportunities for highly compressed storage, accelerated processing, and intuitive design. Special cases of steady lattices are also presented, which provide varying degrees of compromise between design freedom and geometric regularity. For example, the commonly used regular lattices, which provide little design freedom but offer maximum regularity, are the least general form of steady lattice. We propose the 2-directional, Bent Corner-Operated Trans-Similar (BeCOTS) lattices as a useful compromise between regular lattices and fully general steady lattices. A BeCOTS lattice may be controlled by four non-coplanar points, which represent four corners of the lattice. The Trans-Similar property ensures that a BeCOTS lattice is composed of groups of beams such that each consecutive pair of groups of beams along a particular direction is related by the same similarity. Trans-Similarity also enables constant-time queries such as surface area calculation, volume calculation, and point-membership classification. We take advantage of the regularity in steady lattices to efficiently produce and query highly complex lattice structures that we call Constructive Lattice Geometry (CLG), where CLG is an extension of traditional Constructive Solid Geometry (CSG). CLG models are periodic CSG models for which regular patterns of primitives are combined into many repeating CSG microstructures that are ultimately combined into one CSG macrostructure. We provide strategies for designing and processing recursively defined CLG models to enable the creation of CLG models composed of smaller CLG models. Parameterized steady lattices and CLG models may be defined by a few lines of code, which facilitates lazy (on-demand) evaluation, massively parallel processing, interactive editing, and algorithmic optimization.Ph.D

    Le contrôle des inflexions et des extremums de courbure portés par les courbes et les surfaces B-Splines

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    RÉSUMÉ Le contrôle des propriétés différentielles des courbes et des surfaces B-splines est un enjeu important, en particulier pour le domaine de la conception géométrique assistée par ordinateur. Un enjeu qui sollicite autant les méthodes analytiques que numériques dans le but de permettre au concepteur de manipuler les formes avec une aisance toujours croissante. Ce texte explore les possibilités offertes lorsqu'on combine des méthodes numériques de pointe aux travaux de grands géomètres du 19e siècle. Ainsi, de nouveaux algorithmes pour l’optimisation sous contraintes des B-splines ont été développés. Ensuite, ces algorithmes ont été combinés à la théorie des groupes de transformations comme elle a été développée à l’origine par des pionniers comme Sophus Lie, Gaston Darboux et Felix Klein. Ceci permet d’ouvrir des portes vers de nouveaux horizons. Il devient possible de générer de larges espaces de formes sur lesquels on contrôle les propriétés différentielles. Il devient également possible d’éliminer des oscillations de façon sélective ou de manipuler les formes sans introduire d’oscillations indésirables. Avant de progresser vers cet objectif ambitieux, il faut d’abord être en mesure de bien comprendre et de bien visualiser ces propriétés différentielles que l’on souhaite contrôler. L’histoire de la géométrie différentielle classique des courbes et des surfaces est très riche. Cette histoire est revisitée avec une perspective nouvelle. Soit la perspective du contrôle des inflexions et des extremums de courbure. Ceci permet de faire émerger des liens importants entre la géométrie différentielle, la théorie des singularités, les groupes de transformations et l’optique géométrique. Ensuite viennent les algorithmes d’optimisation des B-splines sous contraintes. Les variables indépendantes sont les positions des points de contrôle de la B-spline alors que les contraintes portent sur la position des points de contrôle d’une fonction qui représente les propriétés différentielles de la B-spline. Les algorithmes sont d’abord développés pour les fonctions B-splines à une et deux variables. Une fois ces algorithmes développés, plusieurs possibilités nouvelles s’offrent à nous. Il devient possible, par exemple d’obtenir la courbe qui s’approche le plus d’une autre courbe quelconque sous la contrainte de posséder certaines propriétés différentielles. De cette manière, il devient possible de travailler avec un plus grand nombre de points de contrôle et ainsi dans un espace de forme plus riche sans avoir à se soucier d’oscillations arbitraires. Ceci permet en particulier d’éliminer de façon sélective des oscillations indésirables sur des profils aérodynamiques.----------ABSTRACT Control of B-spline differential properties is an important stake, especially for the field of computer-aided geometric design. An issue that calls for analytical and numerical skills to allow the designer to manipulate shapes in an increasingly efficient way. This text explores possibilities offered by combining new numerical methods with works of 19th century great geometers. Thus, new algorithms for constrained optimization of B-splines are selected and then grafted to the group theory of transformations as it was originally developed by pioneers such as Sophus Lie, Gaston Darboux and Felix Klein. This opens doors to new horizons. It becomes possible to generate large spaces of shapes with a control over their differential properties. This also gives us a selective eraser of curvature extrema and the option to manipulate shapes without introducing undesirables oscillations. Before progressing towards this ambitious goal, one must first be able to understand and visualize these differential properties that one wishes to control. This rich history of the classical differential geometry is revisited with a new perspective. This new perspective is the one of the control of inflections and extrema of curvature. A perspective that allows to establish important links between differential geometry, the theory of singularities, groups of transformations and geometric optics. Next comes the B-splines optimization algorithms with constraints. The independent variables are the B-spline position of the control points, while the constraints are applied to the control points of a function which represents the differential properties of the B-spline. The algorithms are first developed for B-spline functions. Once these algorithms have been developed, several new possibilities open up to us. It becomes possible, for example, to find the closest curve to another one under specified differential properties. This particular algorithm is introduced as an extension to the standard B-spline least squares method to approximate a series of points. The extension consists in adding constraints to produce curve segments with monotonously increasing or decreasing curvature. The interior point method is used to solve the constrained optimization problem. The method requires gradients and those are provided by symbolic B-spline operators. Therefore, the algorithm relies on the arithmetic, differential and variation diminishing properties of the Bsplines to apply the constraints. Thereby, it becomes possible to work with a greater number of control points and thus in a richer shape space without having to manage undesired oscillations

    Squelettes pour la reconstruction 3D : de l'estimation de la projection du squelette dans une image 2D à la triangulation du squelette en 3D

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    La reconstruction 3D consiste à acquérir des images d’un objet, et à s’en servir pour en estimer un modèle 3D. Dans ce manuscrit, nous développons une méthode de reconstruction basée sur la modélisation par squelette. Cette méthode a l’avantage de renvoyer un modèle 3D qui est un objet virtuel complet (i.e. fermé) et aisément éditable, grâce à la structure du squelette. Enfin, l’objet acquis n’a pas besoin d’être texturé, et entre 3 et 5 images sont suffisantes pour la reconstruction. Dans une première partie, nous étudions les aspects 2D de l’étude. En effet, l’estimation d’un squelette 3D nécessite d’étudier la formation de la silhouette de l’objet à partir de son squelette, et donc les propriétés de sa projection perspective, appelée squelette perspectif. Cette étude est suivie par notre première contribution : un algorithme d’estimation de la projection perspective d’un squelette 3D curviligne, constitué d’un ensemble de courbes. Cet algorithme a toutefois tendance, comme beaucoup d’algorithmes estimant un squelette, à générer des branches peu informatives, notamment sur une image rastérisée. Notre seconde contribution est donc un algorithme d’estimation de squelette 2D, capable de prendre en compte la discrétisation du contour de la forme 2D, et d’éviter ces branches peu informatives. Cet algorithme, d’abord conçu pour estimer un squelette classique, est ensuite généralisé à l’estimation d’un squelette perspectif. Dans une seconde partie, nous estimons le squelette 3D d’un objet à partir de ses projections. Tout d’abord, nous supposons que le squelette de l’objet 3D à reconstruire est curviligne. Ainsi, chaque squelette perspectif estimé correspond à la projection du squelette 3D de l’objet, sous différents points de vue. La topologie du squelette étant affectée par la projection, nous proposons notre troisième contribution, l’estimation de la topologie du squelette 3D à partir de l’ensemble de ses projections. Une fois celle-ci estimée, la projection d’une branche 3D du squelette est identifiée sur chaque image, i.e. sur chacun des squelettes perspectifs. Avec cette identification, nous pouvons trianguler les branches du squelette 3D, ce qui constitue notre quatrième contribution : nous sommes donc en mesure d’estimer un squelette curviligne associé à un ensemble d’images d’un objet. Toutefois, les squelettes 3D ne sont pas tous constitués d’un ensemble de courbes : certains d’entre eux possèdent aussi des parties surfaciques. Notre dernière contribution, pour reconstruire des squelettes 3D surfaciques, est une nouvelle approche pour l’estimation d’un squelette 3D à partir d’images : son principe est de faire grandir le squelette 3D, sous les contraintes données par les images de l’objet

    Design paramétrico a partir da digitalização 3D de geometrias da natureza com padrão de crescimento espiral

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    A modelagem de geometrias da natureza pode ser um processo complexo devido ás características orgânicas dos elementos. Propõe-se com essa dissertação identificar geometrias espaciais que sigam o padrão de crescimento espiral observado na natureza, utilizando as Tecnologias 3D como ferramentas para o processo de projeto. Para a execução do trabalho foram investigadas os Métodos de Biônica, Crescimento Espiral e a Sequência de Fibonacci, Engenharia Reversa e Design Paramétrico. O processo de representação dos elementos foi realizado em conformidade com a Metodologia para o Desenvolvimento de Produtos Baseados no Estudo da Biônica com o acréscimo das tecnologias de digitalização tridimensional e de processamento de nuvem de pontos, complementado pela parametrização de superfícies à base de curvas. Foram utilizados três processos para modelagem de curvas paramétricas representadas (i) pelo desenho de linhas sobre a malha digitalizada em 3D, (ii) por programação visual no software Grasshopper e (iii) por programação com scripts Python. Foi avaliada como melhor alternativa para o Design Paramétrico a utilização da programação visual otimizada com a programação por scripts, a qual apresentou melhor aproximação entre as curvas analisadas. Estudos de casos realizados com elementos da natureza (abacaxi e pinha) demonstraram a viabilização do método. Desta maneira a sistematização do conhecimento permitirá a proposição de um modelo paramétrico baseado na Biônica para fase inicial de inspiração e concepção de alternativas do projeto de produto.Modeling the geometries of nature can be a complex process due to the organic characteristics of the elements. It is proposed with this dissertation to identify spatial geometries that follow the pattern of spiral growth observed in nature, using 3D Technologies as tools for the design process. For the execution of the work were investigated the Bionics, Spiral Growth and Fibonacci Sequence, Reverse Engineering and Parametric Design. The process of representation of the elements was carried out in accordance with the Methodology for the Development of Products Based on the Study of the Bionics with the addition of the technologies of three-dimensional digitization and processing of cloud of points, complemented by the parameterization of surfaces based on curves. Three methods were used for modeling parametric curves represented by (i) the drawing of lines on the 3D scanned mesh, (ii) by visual programming in the Grasshopper software and (iii) by programming with Python scripts. It was evaluated as the best alternative for Parametric Design the use of optimized visual programming with programming by scripts, which presented better approximation between the analyzed curves. Case studies carried out with nature elements (pineapple and pine cone) demonstrated the viability of the method. In this way the systematization of the knowledge will allow the proposition of a parametric model based on the Bionics for the initial phase of inspiration and design of alternatives of the product design

    Surface fitting with cyclide splines

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    Discrete Differential Geometry

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    This is the collection of extended abstracts for the 26 lectures and the open problem session at the fourth Oberwolfach workshop on Discrete Differential Geometry
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