In recent decades, the popularity of freeform shapes in contemporary architecture poses new challenges to digital design. One of them is the process of rationalization, i.e. to make freeform skins or structures affordable to manufacture, which draws the most attention from geometry researchers. In this thesis, we aim to realize this process with simple geometric primitives, circular arcs. We investigate architectural surfaces and structures consisting of circular arcs. Our focus is lying on how to employ them nicely and repetitively in architectural design, in order to decrease the cost in manufacturing. Firstly, we study Darboux cyclides, which are algebraic surfaces of order ≤ 4. We provide a computational tool to identify all families of circles on a given cyclide based on the spherical model of M ̈obius geometry. Practical ways to design cyclide patches that pass through certain inputs are presented. In particular, certain triples of circle families on Darboux cyclides may be suitably arranged as 3-webs. We provide a complete classification of all possible 3-webs of circles on Darboux cyclides. We then investigate the circular arc snakes, which are smooth sequences of circu- lar arcs. We evolve the snakes such that their curvature, as a function of arc length, remains unchanged. The evolution of snakes is utilized to approximate given surfaces by circular arcs or to generated freeform shapes, and it is realized by a 2-step pro- cess. More interestingly, certain 6-arc snake with boundary constraints can produce a smooth self motion, which can be employed to build flexible structures. Another challenging topic is approximating smooth freeform skins with simple panels. We contribute to this problem area by approximating a negatively-curved 5 surface with a smooth union of rational bilinear patches. We provide a proof for vertex consistency of hyperbolic nets using the CAGD approach of the rational B ́ezier form. Moreover, we use Darboux transformations for the generation of smooth sur- faces composed of Darboux cyclide patches. In this way we not only eliminate the restriction to surfaces with negative Gaussian curvature, but, also obtain surfaces consisting of circular arcs.

Les formes asphériques et les surfaces complexes sont une classe très avancée d'éléments optiques. Leur application a considérablement augmenté au cours des dernières années dans les systèmes d'imagerie, l'astronomie, la lithographie, etc. La métrologie de ces pièces est très difficile, en raison de la grande gamme dynamique d'information acquise et la traçabilité à l'unité SI mètre. Elle devrait faire usage de la norme infinie; (Méthode de zone minimum ou la méthode Min-Max) pour calculer l'enveloppe entourant les points dans le jeu de données en réduisant au minimum la différence entre l'écart maximum et l'écart minimal entre la surface et l'ensemble de données. Cette méthode a une grande complexité en fonction du nombre de points, enplus, les algorithmes impliqués sont non-déterministes. Bien que cette méthode fonctionne pour des géométries simples (lignes, plans, cercles, cylindres, cônes et sphères), elle est encore un défi majeur lorsqu' utilisée pour des géométries complexes (asphérique et surfaces complexes). Par conséquent, l'objectif de la thèse est le développement des algorithmes d'ajustement Min-Max pour les deux surfaces asphériques et complexes, afin de fournir des algorithmes de référence robustes pour la grande communauté impliquée dans ce domaine. Les algorithmes de référence à développer devraient être évalués et validés sur plusieurs données de référence (Softgauges) qui seront générées par la suite. / Aspheres and freeform surfaces are a very challenging class of optical elements. Their application has grown considerably in the last few years in imaging systems, astronomy, lithography, etc. The metrology for aspheres is very challenging, because of the high dynamic range of the acquired information and the traceability to the SI unit meter. Metrology should make use of the infinite norm; (Minimum Zone Method or Min-Max method) to calculate the envelope enclosing the points in the dataset by minimizing the difference between the maximum deviation and the minimum deviation between the surface and the dataset. This method grows in complexity as the number of points in the dataset increases, and the involved algorithms are non-deterministic. Despite the fact that this method works for simple geometries (lines, planes, circles, cylinders, cones and spheres) it is still a major challenge when used on complex geometries (asphere and freeform surfaces). Therefore, the main objective is to address this key challenge about the development of Min-Max fitting algorithms for both aspherical and freeform surfaces as well as least squares fitting algorithms, in order to provide robust reference algorithms for the large community involved in this domain. The reference algorithms to be developed should be evaluated and validated on several reference data (softgauges) that will be generated using reference data generators.

Com o aumento da capacidade de armazenamento de informação em bancos de dados e em computadores pessoais, surge a necessidade de algoritmos computacionais capazes de realizar o processamento automático para recuperação desses dados. Esse fato não é diferente para objetos tridimensionais armazenados em formato de arquivos. Nesta Dissertação de Mestrado foram estudadas novas técnicas para processamento desses objetos utilizando uma abordagem não comum à área: técnicas para análise de séries temporais, tais como scattering wavelets e gráficos de recorrência. No caso de recuperação total de objetos, ou seja, dado uma malha tridimensional encontrar outras malhas que são visualmente semelhantes, uma única característica é extraída curvatura gaussiana e variação de superfície, por exemplo e ordenada como uma série com a informação provida pelo vetor de Fiedler. Então processa-se essa série utilizando a técnica scattering wavelets, que são capazes de analisar o comportamento temporal de conjunto de dados seriais. Para esse problema, os resultados obtidos são comparáveis com outras abordagens apresentadas na literatura que utilizam várias características para se chegar ao resultado. Já no caso de recuperação parcial de objetos, em que apenas uma parte do objeto é dado como parâmetro de busca, é necessário realizar uma segmentação das malhas para se encontrar outras partes que são visualmente semelhantes. Ao utilizarmos um gráfico de recorrência para analisar os objetos, é possível encontrar não apenas a região mais semelhante dentro da mesma (ou de outra) malha, mas também se obtém todas as regiões que são similares ao parâmetro de busca. / With the increasing data storage capacity of databases and personal computers, arises the necessity of computer algorithms capable of performing processing for automatic recovery of data and information. This fact is no different for three-dimensional objects stored in files. In this Masters Thesis we studied new techniques for processing such data objects using an unusual approach to the geometric processing area: techniques for analyzing time series, such as scattering wavelets and recurrence plots. For shape retrieval problem, i.e., given a tridimensional mesh try finding other meshes that are visually similar, our method extract only one feature Gaussian curvature and surface variation, for example and organize it as a series using information given by Fiedler vector. Then, the next step is to process the resulting series using a technique called scattering wavelets, that is capable of analyzing the temporal behavior of a set of serial data. For this problem, the results are comparable with other approaches reported in the literature that use multiple characteristics to find a matching mesh. In the case of partial retrieval of objects, in which only a part of the object is given as search parameter, it is necessary to perform a segmentation of the meshes in order to find other parts that are visually similar to the query. By using Recurrence Plot to analyze the objects, our method can find not only the most similar region within the same (or other) object, but also get all the regions that are similar to the search parameter.

Com o aumento da capacidade de armazenamento de informação em bancos de dados e em computadores pessoais, surge a necessidade de algoritmos computacionais capazes de realizar o processamento automático para recuperação desses dados. Esse fato não é diferente para objetos tridimensionais armazenados em formato de arquivos. Nesta Dissertação de Mestrado foram estudadas novas técnicas para processamento desses objetos utilizando uma abordagem não comum à área: técnicas para análise de séries temporais, tais como scattering wavelets e gráficos de recorrência. No caso de recuperação total de objetos, ou seja, dado uma malha tridimensional encontrar outras malhas que são visualmente semelhantes, uma única característica é extraída curvatura gaussiana e variação de superfície, por exemplo e ordenada como uma série com a informação provida pelo vetor de Fiedler. Então processa-se essa série utilizando a técnica scattering wavelets, que são capazes de analisar o comportamento temporal de conjunto de dados seriais. Para esse problema, os resultados obtidos são comparáveis com outras abordagens apresentadas na literatura que utilizam várias características para se chegar ao resultado. Já no caso de recuperação parcial de objetos, em que apenas uma parte do objeto é dado como parâmetro de busca, é necessário realizar uma segmentação das malhas para se encontrar outras partes que são visualmente semelhantes. Ao utilizarmos um gráfico de recorrência para analisar os objetos, é possível encontrar não apenas a região mais semelhante dentro da mesma (ou de outra) malha, mas também se obtém todas as regiões que são similares ao parâmetro de busca. / With the increasing data storage capacity of databases and personal computers, arises the necessity of computer algorithms capable of performing processing for automatic recovery of data and information. This fact is no different for three-dimensional objects stored in files. In this Masters Thesis we studied new techniques for processing such data objects using an unusual approach to the geometric processing area: techniques for analyzing time series, such as scattering wavelets and recurrence plots. For shape retrieval problem, i.e., given a tridimensional mesh try finding other meshes that are visually similar, our method extract only one feature Gaussian curvature and surface variation, for example and organize it as a series using information given by Fiedler vector. Then, the next step is to process the resulting series using a technique called scattering wavelets, that is capable of analyzing the temporal behavior of a set of serial data. For this problem, the results are comparable with other approaches reported in the literature that use multiple characteristics to find a matching mesh. In the case of partial retrieval of objects, in which only a part of the object is given as search parameter, it is necessary to perform a segmentation of the meshes in order to find other parts that are visually similar to the query. By using Recurrence Plot to analyze the objects, our method can find not only the most similar region within the same (or other) object, but also get all the regions that are similar to the search parameter.