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Geodésicas em superfícies poliedrais e elipsóides / Geodesics in polyhedral surfaces and ellipsoidsPlaza, Luis Felipe Narvaez 14 March 2016 (has links)
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Previous issue date: 2016-03-14 / Conselho Nacional de Pesquisa e Desenvolvimento Científico e Tecnológico - CNPq / This work is divided in four parts, in the first chapter we give an introduction. In the
next chapter we study basic theory of geometry and differential equations, we study some
results of geodesics theory on surfaces in R3; based in the works of R. Garcia and J. Sotomayor
in [10] and W. Klingenberg in [15]. These ones provide a study of the behavior
of the geodesic in the ellipsoid.
The third chapter is inspired by the famous question given in 1905, in his famous article
“Sur les lignes géodésiques des surfaces convexes” H. Poincaré posed a question
on the existence of at least three geometrically different closed geodesics without
self-intersections on any smooth convex two-dimensional surface (2-surface) M homeomorphic
to the two-dimensional sphere (2-sphere) S2. We study this question for convex
polyhedral surfaces following the paper [9] by G. Galperin and the books [1],[4].
In the last topic we will address the behavior of geodesics on Lorentz surfaces, focusing
our study on the ellipsoid based mainly on the book of Tilla Weinstein [25] and in the
paper [11] by S. Tabachnikov, Khesin and Genin. / Este trabalho se divide em quatro partes principais, no primeiro capítulo fazemos uma
breve introdução. No segundo capítulo estudamos teoria básica de geometria e equações
diferenciais, estudamos também geodésicas em superfícies no R3 baseados nos trabalhos
de R. Garcia e J. Sotomayor em [10] e de W. Klingenberg em [15], estes fornecem um
estudo rigoroso do comportamento das geodésicas no elipsóide.
O terceiro capítulo é inspirado na famosa conjectura dada em em 1905 em seu artigo “Sur
les lignes géodésiques des surfaces convexes” H. Poincaré fez uma pergunta sobre a existência
de pelo menos três geodésicas simples fechadas sobre superfícies suaves convexas
homeomorfas à esfera S2, neste capítulo estudamos esta conjectura em superfícies poliedrais
baseado em [9] e os textos [1],[4].
No último tema de abordagem analisamos o comportamento de geodésicas em superfícies
no espaço de Lorentz, focando nosso estudo no elipsóide, este estudo é baseado principalmente
no livro de Tilla Weinstein [25] e no artigo [11] de S. Tabachnikov, Khesin e
Genin.
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Simplification, approximation and deformation of large modelsParadinas Salsón, Teresa 13 October 2011 (has links)
The high level of realism and interaction in many computer graphic applications requires techniques for processing complex geometric models. First, we present a method that provides an accurate low-resolution approximation from a multi-chart textured model that guarantees geometric fidelity and correct preservation of the appearance attributes. Then, we introduce a mesh structure called Compact Model that approximates dense triangular meshes while preserving sharp features, allowing adaptive reconstructions and supporting textured models. Next, we design a new space deformation technique called *Cages based on a multi-level system of cages that preserves the smoothness of the mesh between neighbouring cages and is extremely versatile, allowing the use of heterogeneous sets of coordinates and different levels of deformation. Finally, we propose a hybrid method that allows to apply any deformation technique on large models obtaining high quality results with a reduced memory footprint and a high performance. / L’elevat nivell de realisme i d’interacció requerit en múltiples aplicacions gràfiques fa que siguin necessàries tècniques pel processament de models geomètrics complexes. En primer lloc, presentem un mètode de simplificació que proporciona una aproximació precisa de baixa resolució d'un model texturat que garanteix fidelitat geomètrica i una correcta preservació de l’aparença. A continuació, introduïm el Compact Model, una nova estructura de dades que permet aproximar malles triangulars denses preservant els trets més distintius del model, permetent reconstruccions adaptatives i suportant models texturats. Seguidament, hem dissenyat *Cages, un esquema de deformació basat en un sistema de caixes multi-nivell que conserva la suavitat de la malla entre caixes veïnes i és extremadament versàtil, permetent l'ús de conjunts heterogenis de coordenades i diferents nivells de deformació. Finalment, proposem un mètode híbrid que permet aplicar qualsevol tècnica de deformació sobre models complexes obtenint resultats d’alta qualitat amb una memòria reduïda i un alt rendiment.
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