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Statistics of directional data with an application in crystallography
Karafiátová, Iva ; Šedivý, Ondřej (advisor) ; Hlubinka, Daniel (referee)
The main purpose of this thesis is to derive methods used to estimate the mean orientation of a grain in polycrystalline material. The estimation is highly impor- tant in crystallography. However, it is often expressed by only one representative value regardless of its variability within the grain. In this thesis we present the basic concepts of crystallography, including the principle of the Electron Backs- catter Diffraction, which is used to explore the microstructure of a polycrystalline materials. Subsequently, we present six of the most commonly used descriptors of three-dimensional orientation of a crystal lattice. This overview is completed by the summary of the particular relationships between those descriptors. In the practical part of this thesis we apply derived methods on real data within the scope of research of microstructure of an aluminum alloy. 1
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Quaternions and Möbius transformations in dimension 4
Kosina, Jan ; Lávička, Roman (advisor) ; Krump, Lukáš (referee)
In this work we describe transformations of the 3-dimensional and the 4- dimensional Euclidean space. First we show how one can elegantly describe re- flections and rotations in these dimensions using quaternions and we prove 2 structural theorems concerning the connection between the group of unit qua- ternions and the special orthogonal groups SO(3) and SO(4). Next we recall a part of the conformal mapping theory, which we use later in the description of the Möbius transformations. We define the Möbius transformations in dimension 4 as compositions of an even number of spherical inversions and reflections. We show that one can describe them also in dimension 4 as linear fractional trans- formations in an analogous way as in dimension 2, if we use quaternions instead of complex numbers. We then outline a classification of Möbius transformations into elliptic, loxodromic and parabolic classes and in dimension 4, we describe what each class looks like. 1
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Algebra of dual quaternions in image analysis
Hrubý, Jan ; Návrat, Aleš (referee) ; Hrdina, Jaroslav (advisor)
This work has two goals. Firstly it is to acquaint the reader with the classical use of quaternions and dual quaternions in geometry. Secondly the generalization of the Fourier transform into the set of dual quaternions. At first it goes into algebraic properties and structure of quaternions and ways of their inscriptions. Later dual numbers are introduced and consecutively with their help dual quaternions. Then the work deals with description of rotations and translations using quaternions and dual quaternions, that enable their easy description. Finally the discreet dual quaternion Fourier transform is defined, and for its effective calculation the algorithm is derived, which is then brought into effect as a code in program environment MATLAB.
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3D Photo Slideshow Using OpenGL
Ondrejíček, Marián ; Navrátil, Jan (referee) ; Seeman, Michal (advisor)
The aim of this thesis was to design a photo slideshow in 3D space using OpenGL and to implement functional application. Also there is description of Qt toolkit, which was used to design graphical user interface of the program. Furthermore there are explained basics of quaternion interpolation, which is the core of animation and camera system. The second part describes practical realization and each part of the project in details. Application was successfully implemented. In the end, the results are discussed, including possibilities of future development.
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Fusion of Procedural and Keyframe Animation
Klement, Martin ; Pečiva, Jan (referee) ; Polok, Lukáš (advisor)
The goal of this work is to create an application, which will combine procedural and keyfram animations with subsequent visualization. Composition of this two different animations techniques is used to animate a virtual character. To combine this two techniques one starts with interpolations from keyframe animation and then enchance them by procedural animations to properly fit into the characters surroundings. This procedural part of animation is obtained by using forward and inverse kinematics. Whole application is written in C++, uses GLM math library for computations and OpenGL and GLUT for final visualization.
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Mathematical principles of Robotics
Pivovarník, Marek ; Kureš, Miroslav (referee) ; Hrdina, Jaroslav (advisor)
Táto diplomová práca sa zaoberá matematickými aparátmi popisujúcimi doprednú a inverznú kinematiku robotického ramena. Pre popis polohy koncového efektoru, teda doprednej kinematiky, je potrebné zaviesť špeciálnu Euklidovskú grupu zobrazení. Táto grupa môže byť reprezentovaná pomocou matíc alebo pomocou duálnych kvaterniónov. Problém inverznej kinematiky, kedy je potrebné z určenej polohy koncového efektoru dopočítať kĺbové parametre robotického ramena, je v tejto práci riešený pomocou exponenciálnych zobrazení a Grobnerovej bázy. Všetky spomenuté popisy doprednej a inverznej kinematiky sú aplikované na robotické rameno s troma rotačnými kĺbami. Odvodené postupy sú následne implementované a vizualizované v prostredí programu Mathematica.
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Geometric structures based on quaternions.
Floderová, Hana ; Vašík, Petr (referee) ; Hrdina, Jaroslav (advisor)
A pair (V, G) is called geometric structure, where V is a vector space and G is a subgroup GL(V), which is a set of transmission matrices. In this thesis we classify structures, which are based on properties of quaternions. Geometric structures based on quaternions are called triple structures. Triple structures are four structures with similar properties as quaternions. Quaternions are generated from real numbers and three complex units. We write quaternions in this shape a+bi+cj+dk.
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Integration of inertial navigation with global navigation satellite system
Štefanisko, Ivan ; Otava, Lukáš (referee) ; Blaha, Petr (advisor)
This paper deals with study of inertial navigation, global navigation satellite system, and their fusion into the one navigation solution. The first part of the work is to calculate the trajectory from accelerometers and gyroscopes measurements. Navigation equations calculate rotation with quaternions and remove gravity sensed by accelerometers. The equation’s output is in earth centred fixed navigation frame. Then, inertial navigation errors are discussed and focused to the bias correction. Theory about INS/GNSS inte- gration compares different integration architecture. The Kalman filter is used to obtain navigation solution for attitude, velocity and position with advantages of both systems.
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Complex numbers, quaternions and their applications
BRDLÍK, Pavel
The bachelor thesis is dedicated to the topic of complex numbers and quaternions and their application. The main goal of this thesis is to familiarize with the subject of complex numbers and quaternions and their important properties and to present these terms with suitably selected examples. Complex numbers can be used among others to represent a planar rotational movement, quaternions can be used to represent rotations in three-dimensional space. The aim of this thesis is to provide a well arranged summary of the theory and a sample solution of practical examples that would illustratively demonstrate the application of complex numbers and quaternions by description of planar or spatial movements or their further applications.
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