Orientador: Alice Kimie Miwa Libardi / Banca: Suzinei Aparecida Siqueira Marconato / Banca: Karina Schiabel Silva / Resumo: O objetivo deste trabalho é apresentar uma demonstração combinatória dos teore- mas de Índice de Poincaré, a saber: "Sejam D um disco e γ seu bordo. Seja V um campo vetorial contínuo sobre D com pontos críticos isolados P1, P2, . . . , Pn pertencentes ao interior de D. Se V nunca se anula em γ, então W(γ) = I(P1) + I(P2) + . . . + I(Pn), onde I(Pi) é o índice do ponto crítico Pi e W(γ) o número de voltas de V sobre γ." "Seja V um campo vetorial tangente contínuo sobre uma superfície compacta, co- nexa e orientável S. Então a soma dos índices dos pontos críticos de V é igual à característica de Euler de S." / Abstract: bstract In this work we present a combinatorial proof for the Poincaré index theorems. "Let V be a continuous vector field. Let D be a cell and γ its boundary. Supposing that V is not zero on γ, then W(γ) = I(P1) + I(P2) + . . . + I(Pn) where P1, P2, . . . , Pn are the critical points of V inside D, I(Pi) is the index of Pi, and W(γ) is the winding number of V on γ." "Let V be a continuous tangent vector field on a compact, connected, orientable surface S. Then the sum of the indexes of the critical points of V equals the Euler characteristic of S." / Mestre

Neste trabalho propomos uma nova maneira de visualizar campos vetoriais, dados de considerável importância em vários ramos da ciência. Fizemos uma revisão bibliográfica sobre segmentação de campos vetoriais e desenvolvemos nosso próprio método. Neste método são extraídas informações do campo e, de distribuições de frequências dos dados coletados são formados vetores multidimensionais. Esses vetores são projetados em duas dimensões e os agrupamentos destes pontos são utilizados para formar a segmentação do campo original. Os profissionais que fazem uso de ferramentas de visualização científica possuem, em geral, informações relevantes sobre o domínio do campo vetorial, mas essa informação é raramente aproveitada nas técnicas de segmentação. A técnica desenvolvida permite que o usuário interaja com os resultados, de maneira intuitiva, corrigindo e explorando a segmentação usando seu próprio conhecimento. Como contribuições desta pesquisa podemos citar o mecanismo de interação com o usuário para o auxílio da segmentação e uma nova maneira para representar os dados colhidos de campos vetoriais em dimensão alta / In this research we introduce a novel method for visualizing vector fields, data of considerable importance in several branches of science. We did a literature review targeting vector fields and developed our own method. In this method information is extracted from the field and, from frequency distributions of the collected data multidimensional vectors are created. These vectors are projected in two dimensions and clusters of these points are used to form a segmentation of the original field. The professionals that make use of scientific visualization tools have, in general, relevant information about the domain of the vector field, but this information is rarely exploited by segmentation techniques. The developed technique allows the user to interact with the results, intuitively, exploring and correcting the segmentation using his own knowledge. As contributions of this research include the mechanism of interaction with the user to aid the segmentation and a new method to represent the collected data from vector fields in high dimension

A section of the total tangent space of a scheme X of finite type over a field k, i.e. a vector field on X, corresponds to an X-valued 1-jet on X. In the language of jets the notion of a vector field becomes functorial, and the total tangent space constitutes one of an infinite family of jet schemes Jm(X) for m ≥ 0. We prove that there exist families of “logarithmic” jet schemes JDm(X) for m ≥ 0, in the category of k-schemes of finite type, associated to any given X and its family of divisors D = (D₁, . . . ,Dr). The sections of JD₁(X) correspond to so-called vector fields on X with logarithmic poles along the family of divisors D = (D₁, . . . ,Dr). To prove this, we first introduce the categories of pairs (X,D) where D is as mentioned, an r-tuple of (effective Cartier) divisors on the scheme X. The categories of pairs provide a convenient framework for working with only those jets that pull back families of divisors. / Science, Faculty of / Mathematics, Department of / Graduate

京都大学 / 0048 / 新制・課程博士 / 博士(人間・環境学) / 甲第22521号 / 人博第924号 / 新制||人||221(附属図書館) / 2019||人博||924(吉田南総合図書館) / 京都大学大学院人間・環境学研究科共生人間学専攻 / (主査)准教授 木坂 正史, 教授 角 大輝, 教授 足立 匡義 / 学位規則第4条第1項該当 / Doctor of Human and Environmental Studies / Kyoto University / DFAM

Cooperative control for miniature air vehicles (MAVs) is currently a highly researched topic. There are many application for which MAVs are well suited, including fire monitoring, surveillance and reconaissance, and search and rescue missions. All of these applications can be carried out more effictively by a team of MAVs than by a single vehicle. As technologies for microcontrollers and small sensors have improved so have the capabilities of MAVs. This improvement in MAV performance abilities increases the possibility for cooperative missions. The focus of this research was on cooperative timing missions. The issues faced when dealing with multi-MAV flight include information transfer, real time path planning, and maintenance of a fleet of flight-worthy MAVs. Additional challenges associated with timing missions include path following and velocity control. Two timing scenarios were studied and both of these scenarios were flight tested. The first scenario was a sequenced arrival of the MAVs over a target at a predetermined fly-through heading. The second scenario was a simultaneous arrival of the team ofMAVs over a known target location. The ideas of coordination functions and coordination variables have been employed to achieve coordination. Experimental results verify the feasibility of real time coperative control for a team of MAVs. Initial cooperative timing tests revealed the need for more accurate path following. Accordingly, a new method for path following using vector fields was developed. A vector field of desired ground track headings is calculated and commanded ground track headings are calculated such that ground track heading error and lateral following error decay asymptotically even in the presence of constant wind disturbances. The utilization of ground track heading and ground speed in the path following control, in combination with the vector field methods is what makes this zero-error following possible. Methods for following straight lines and orbits as well as combinations of the lines and circular arcs are presented. The assertions that minimal following errors result when using these methods have been verified experimentally.

This thesis deals with the velocity distribution in space between the cylinders. All measurement conducted in labs of VUT Brno Faculty of Civil Engineering. Cause of measurement is research, which is being done by Masarykova unverzita within the thesis of one of the students. It is a research of behavior of certain animals in the aquatic enviroment.

The goal of this thesis is the development of a fast and robust algorithm that is able to detect patterns in flow fields independent from their orientation and adequately visualize the results for a human user. This thesis is an interdisciplinary work in the field of vector field visualization and the field of pattern recognition. A vector field can be best imagined as an area or a volume containing a lot of arrows. The direction of the arrow describes the direction of a flow or force at the point where it starts and the length its velocity or strength. This builds a bridge to vector field visualization, because drawing these arrows is one of the fundamental techniques to illustrate a vector field. The main challenge of vector field visualization is to decide which of them should be drawn. If you do not draw enough arrows, you may miss the feature you are interested in. If you draw too many arrows, your image will be black all over. We assume that the user is interested in a certain feature of the vector field: a certain pattern. To prevent clutter and occlusion of the interesting parts, we first look for this pattern and then apply a visualization that emphasizes its occurrences. In general, the user wants to find all instances of the interesting pattern, no matter if they are smaller or bigger, weaker or stronger or oriented in some other direction than his reference input pattern. But looking for all these transformed versions would take far too long. That is why, we look for an algorithm that detects the occurrences of the pattern independent from these transformations. In the second part of this thesis, we work with moment invariants. Moments are the projections of a function to a function space basis. In order to compare the functions, it is sufficient to compare their moments. Normalization is the act of transforming a function into a predefined standard position. Moment invariants are characteristic numbers like fingerprints that are constructed from moments and do not change under certain transformations. They can be produced by normalization, because if all the functions are in one standard position, their prior position has no influence on their normalized moments. With this technique, we were able to solve the pattern detection task for 2D and 3D flow fields by mathematically proving the invariance of the moments with respect to translation, rotation, and scaling. In practical applications, this invariance is disturbed by the discretization. We applied our method to several analytic and real world data sets and showed that it works on discrete fields in a robust way.

<div>Volumetric cloud generation and rendering algorithms are well-developed to meet the need for a realistic sky performance in animation or games. However, it is challenging to create a stylized or designed animation for volumetric clouds using physics-based generation and simulation methods in real-time.</div><div>The problem raised by the research is the current volumetric cloud animation controlling methods are not art-directable. Making a piece of volumetric cloud move in a specific way can be difficult when using only a physics-based simulation method. The purpose of the study is to implement an animating method for volumetric clouds and with art-directable controllers. Using this method, a designer can easily control the cloud's motion in a reliable way. The program will achieve interactive performance using parallel processing with CUDA. Users will be able to animate the cloud by input a few vectors inside the cloud volume. </div><div>After reviewing the literature related to the real-time simulation method of clouds, texture advection algorithms, fluid simulation, and other processes to achieve the results, the thesis offers a feasible design of the algorithm and experiments to test the hypotheses. The study uses noise textures and fractional Brownian motion (fBm) to generate volumetric clouds and render the clouds by the ray marching technique. The program will render user input vectors and a three-dimension interpolation vector field with OpenGL. By adding or changing input vectors, the user will gain a divergence minimization interpolation field. The cloud volume could be animated by the texture advection technique based on the interpolation vector field in real-time. By inputting several vectors, the user could plausibly animate the volume cloud in an art-directable way.</div>

In this thesis we develop methods that facilitate an aerial rendezvous between two air vehicles. The objective of this research is to produce a method that can be used to insert a miniature air vehicle behind a rendezvous vehicle and then track that vehicle to enable a visual rendezvous. For this research we assume the rendezvous vehicle is following a relatively stable and roughly elliptical orbit. Path planners and controllers have been developed that can be used to effectively intercept the rendezvous vehicle by inserting the MAV onto the orbit of interest. A method for planning and following time-optimal Dubins airplane interception paths between a miniature air vehicle and the rendezvous vehicle is presented. We demonstrate how a vector field path following a scheme can be used for navigation along these time-optimal Dubins airplane paths. A post-orbit insertion tracking method is also presented which can be used to track the target vehicle on an arbitrarily oriented elliptical orbit while maintaining a specified following distance. We also present controllers that can be used for disturbance rejection during the orbit-insertion and interception operations. All of these methods were implemented in simulation and with hardware. Results from these implementations are presented and analyzed.

Vector field topology has a long tradition as a visualization tool. The separatrices segment the domain visually into canonical regions in which all streamlines behave qualitatively the same. But application scientists often need more than just a nice image for their data analysis, and, to best of our knowledge, so far no workflow has been proposed to extract the critical points, the associated separatrices, and then provide the induced segmentation on the data level. We present a workflow that computes the segmentation of the domain of a 2D vector field based on its separatrices. We show how it can be used for the extraction of quantitative information about each segment in two applications: groundwater flow and heat exchange.