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Barycentric coordinates are functions on a polygon, one for each vertex, whose values are coefficients that provide an expression of a point of the polygon as a convex combination of the vertices. Wachspress barycentric coordinates are barycentric coordinates that are defined by rational functions of minimal degree. We study the rational map on P2 defined by Wachspress barycentric coordinates, the Wachspress map, and we describe polynomials that set-theoretically cut out the closure of the image, the Wachspress variety. The map has base points at the intersection points of non-adjacent edges. The Wachspress map embeds the polygon into projective space of dimension one less than the number of vertices. Adjacent edges are mapped to lines meeting at the image of the vertex common to both edges, and base points are blown-up into lines. The deformed image of the polygon is such that its non-adjacent edges no longer intersect but both meet the exceptional line over the blown-up corresponding base point. We find an ideal that cuts out the Wachspress variety set-theoretically. The ideal is generated by quadratics and cubics with simple expressions along with other polynomials of higher degree. The quadratic generators are scalar products of vectors of linear forms and the cubics are determinants of 3 x 3 matrices of linear forms. Finally, we conjecture that the higher degree generators are not needed, thus the ideal is generated in degrees two and three.

No / A methodology for shape morphing using partial differential equation (PDE) surfaces is presented in this work. The use of the PDE formulation shows how shape morphing can be based on a boundary-value approach by which intermediate shapes can be created. Furthermore, the mathematical properties of the method give rise to several alternatives in which morphing one shape into another can be achieved. Three of these alternatives are presented here. The first one is based on the gradual variation of the weighted sum of the boundary conditions for each surface, the second one consists of varying the Fourier mode for which the PDE is solved whilst the third results from a combination of the first two. Examples showing the efficiency of these methodologies are presented. Thus, it is shown that the PDE based approach for morphing, when combined with a parametric variation of the boundary conditions, is capable of obtaining smooth intermediate surfaces automatically.

Yes / Computer aided geometric design is an area where the improvement of surface generation techniques is an everlasting demand since faster and more accurate geometric models are required. Traditional methods for generating surfaces were initially mainly based upon interpolation algorithms. Recently, partial differential equations (PDE) were introduced as a valuable tool for geometric modelling since they offer a number of features from which these areas can benefit. This work summarises the uses given to PDE surfaces as a surface generation technique together

No / This title provides detailed description of how Partial Differential Equations are used in the field of geometric design, and supplies clear and concise explanations of how to implement the techniques described. It also offers extensive discussions (with examples) or practical applications of Partial Differential Equations in geometric design.

We consider the problem of moving a three dimensional solid object among polyhedral obstacles. The traditional formulation of configuration space for this problem uses three translational parameters and three angles (typically Euler angles), and the constraints between the object and obstacles involve transcendental functions. We show that a quaternion representation of rotation yields constraints which are purely algebraic in a higher-dimensional space. By simple manipulation, the constraints may be projected down into a six dimensional space with no increase in complexity. Using this formulation, we derive an efficient exact intersection test for an object which is translating and rotating among obstacles.

This thesis addresses different aspects of spatial data handling in connection with investigations of large rockslides. As such, most of the research was carried out in a cross disciplinary and highly applied context. The focus of the thesis is on spatial data handling methodology which directly or indirectly can be used to support in rockslide investigations. Rockslide investigation is a comprehensive term covering all aspects of the evaluation process; from the initial planning of field investigations to data analysis and communication of final results. Central topics addressed in this thesis are; a) How data reduction affect the geometrical accuracy of digital terrain models b) How interactive geometric modelling and geovisualization can be used in complex rockslide investigations and c) How statistical analyses can be used to evaluate displacement measurements of unstable rock slopes. Digital terrain modelling forms an important component of the methodology used for rockslide investigations. The first subtopic addressed in this thesis is related to the construction of Triangulated Irregular Networks (TINs) from Light Detection and Ranging (LIDAR) data. As the LIDAR technology tends to generate large data volumes, the resulting terrain models are generally too large to be efficiently handled by ordinary workstations. Therefore, comparisons of various data reduction (decimation) methods were conducted. Their performances were evaluated by means of deviations from terrain models constructed from full datasets. Evaluation criteria included deviations in volume, surface area and elevation. The results showed that the method using a vertical point selection threshold combined with a data dependent triangulation had the overall best performance when tested on 30 different test datasets. The main objective of the geovisualization part of this thesis was to determine the geometric shapes and locations of potential basal sliding surfaces, for the Åknes rockslide in western Norway, along with the volumes of unstable rock associated with different sliding scenarios. The Åknes rockslide is one of the world's most investigated rockslides due to its potentially catastrophic consequences. A custom written geovisualization application for the Åknes investigation provided the visual context needed for data interpretation and interactive geometric modelling of sliding surfaces. This geovisualization approach enabled geoscientists to develop different sliding scenarios. A scenario putting the basal sliding surface at a depth of 105m to 115m below the topographic surface, delineating an unstable rock volume of 43 million m3, was considered as the most realistic. Statistical approaches for analyzing displacement measurements were also addressed in this thesis. Several methods including regression analysis, spectral analysis and hypothesis testing were demonstrated to measurements obtained from Global Positioning System (GPS), total stations and extensometers at the Åknes rockslide. Displacement measurements obtained from lasers and crackmeters at the Nordnes rockslide in Northern Norway were also analysed. As with the Åknes rockslide, the Nordnes rockslide has the potential for devastating consequences in terms tsunami generation. Consequently, thorough statistical analyses of the available displacement data are crucial in order to obtain accurate estimates for the displacement rates as well as for gaining insight into the sliding processes. Displacement data from both sites clearly showed seasonal variations but the overall long term displacements were regarded constant. Prediction intervals were derived from the current monitoring data from the Nordnes site. These prediction intervals are considered useful for evaluation of future displacement measurements.

Orientador: Maria Gabriela Caffarena Celani / Dissertação (mestrado) - Universidade Estadual de Campinas, Faculdade de Engenharia Civil, Arquitetura e Urbanismo / Made available in DSpace on 2018-08-10T23:07:07Z (GMT). No. of bitstreams: 1 Vieira_EricaPinheiro_M.pdf: 5079109 bytes, checksum: ed92461f98d5f0d28c0fb6e2b52c30df (MD5) Previous issue date: 2007 / Resumo: Este trabalho consiste em um estudo exploratório sobre a produção digital de maquetes arquitetônicas. Inicialmente, foi realizada uma revisão bibliográfica com a finalidade de conhecer os novos métodos de produção digital de maquetes, incluindo software de modelagem e equipamentos de prototipagem rápida. Nesse estudo inicial, além de explorar os principais equipamentos, processos, aplicações e materiais, identificou-se dois arquitetos renomados que fazem uso dessa tecnologia nos seus processos de projeto: Frank Gehry e Norman Foster. Deste estudo verificou-se processos distintos de projeto e diferentes abordagens sobre a utilização dessas ferramentas, o que motivou a realização de experimentos para exemplificar como produzir digitalmente maquetes arquitetônicas. O Museu Guggenheim de Bilbao, de Frank Gehry, foi escolhido como variável fixa para realização dos experimentos de produção digital de maquetes, por ser um modelo de grande complexidade, permitindo testar os limites dos equipamentos de prototipagem rápida disponíveis no Laboratório de Prototipagem para Arquitetura e Construção (LAPAC) da FEC ¿ Unicamp e no Centro de Pesquisas Renato Archer (CenPRA). Além disso, foram testadas diferentes técnicas e materiais, inclusive materiais alternativos, com o objetivo de viabilização econômica. Os resultados comprovaram que o processo de produção digital de maquetes arquitetônicas é viável em termos de procedimentos, de custo, de materiais disponíveis no mercado, qualidade das maquetes produzidas e rapidez na execução. A partir das conclusões obtidas nos experimentos realizados foi elaborado um caderno de recomendações para a confecção de maquetes que será utilizado pelos usuários do LAPAC e que servirá como importante ferramenta de auxílio para os iniciantes na produção digital de maquetes arquitetônicas. Espera-se que os resultados desta pesquisa possam auxiliar a estabelecer diretrizes para a incorporação dessas técnicas e equipamentos em disciplinas de projeto e na prática de arquitetura / Abstract: The present work is an exploratory study about the digital fabrication of architectural models. It started with a literature review, with the aim of getting in contact with the new digital methods for making models and prototypes, from modeling software to rapid prototyping equipment, processes, materials and applications. Still in this initial study the work of two well-known architects, Frank Gehry and Norman Foster, who use rapid prototyping techniques in their design process, was analyzed. From this part of the research it was possible to conclude that the different approaches that architects have to the design process is reflected in the way they use digital techniques for making their models. The second part of the research consisted of a series of experiments with the objective of illustrating the digital production of architectural models. For these experiments, Frank Gehry's Guggenheim Museum in Bilbao was chosen as a fixed variable for the production of models, due to its geometric complexity, which allowed to push the use of the available rapid prototyping equipment to their limits. Only the equipment available at FEC-UNICAMP's (Laboratório de Prototipagem para Arquitetura e Construção - LAPAC) and CENPRA's (Laboratório de Prototipagem Rápida do Centro de Pesquisas Renato Archer) laboratories were used. They consisted of a 3d printer, a fusion deposition modeller (FDM) machine, and a laser cutter. Different techniques and materials were tested in these machines, with the objective of evaluating the quality and economic viability of the resulting models. The results showed that the digital production of architectural models is viable for use in Brazilian architecture schools, in terms of procedures, cost, availability of materials, time spent and quality of the models. Finally, a manual with recommendations and tips was produced, with the aim of helping students build their own models using rapid prototyping equipment. We hope that the results of this research will help guiding the incorporation of these techniques in architecture education and practice in Brazil / Mestrado / Arquitetura e Construção / Mestre em Engenharia Civil

No / Parameterisation of facial expressions on PDE surface representations of human faces are presented in this work. Taking advantage of the boundary-value approach inherent to Bloor-Wilson PDE method, facial expressions are achieved by manipulating the original boundary curves. Such curves are responsible for generating a surface representation of a human face in its neutral configuration, so that regions on these curves represent a given facial expression in a fast and realistic manner. Additionally, the parameterisation proposed here is carried out by applying different mathematical transformations to the affected curves according to the corresponding facial expression. Full analytic expressions parameterising some of the most common facial expressions such as smiling and eyebrow raising are in this work. Some graphical examples of these facial expressions are used to illustrate the results obtained using Bloor-Wilson PDE method as the foundations of the parameterisation scheme proposed here. Thus, it is shown that an efficient, intuitive and realistic parameterisation of facial expressions is attainable using Bloor-Wilson PDE method in along with a suitable mathematical expression.