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Projective perspective on planar euclidean geometry
Řada, Jakub ; Krump, Lukáš (advisor) ; Šír, Zbyněk (referee)
In this thesis we study projective perspective on planar euclidean geometry. First we take an euclidean construction and transform it into the projective language. Then we discover and show principles of this transformation. We show equivalence between complex points I, J and some euclidean structures. Moreover we study conics, triangles, polygons and circles. We build this thesis on examples. 1
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Discrete connection on triangular meshes
Vráblíková, Jana ; Šír, Zbyněk (advisor) ; Souček, Vladimír (referee)
Abstract. In this thesis we are going to deal with constructing parallel tangent vector fields on discrete surfaces. Ať first, we are going to present theory of tangent vector fields on smooth surfaces in R3 , define notion of connection, which will help us describe tangent vector fields, and we will formulate corollary of Poincare-Hopf theorem, that will tell us that on most surfaces smooth tangent vector field which is nonzero at every point does not exist. Then we are going to introduce analogies of notions from differential geometry for discrete surfaces, which we represent by triangular meshes, and we are going to explain how to use these concepts when constructing tangent vector fields that are parallel at the whole surface. At the end we are going to describe algorithm for constructing these vector fields, which can be found in the electronic attachement, implemented using software Wolfram Mathematica, and we will show its results on several examples.
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Construction of G^1 continuous surfaces.
Kostelecká, Adéla ; Šír, Zbyněk (advisor) ; Bizzarri, Michal (referee)
This thesis introduces an algorithm that connects two Bézier patches indis- tinguishtably. The algorithm modifies patches to have a common tangent plane. We use the Chiyokura Kimura method to a tensor product Bézier surfaces and Bé- zier triangles. We ensure this type of continuity for multiple patches by replacing the control points with rational functions. These are called the Gregory patches. We prove that both of the methods connect two patches with G1 continuity. Fi- nally, we present the results of the algorithm on asymmetric icosahedron and on real geometric objects such as Standford Bunny. 1
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Structure and approximation of real planar algebraic curves
Blažková, Eva ; Šír, Zbyněk (advisor)
Finding a topologically accurate approximation of a real planar algebraic curve is a classic problem in Computer Aided Geometric Design. Algorithms describing the topology search primarily the singular points and are usually based on algebraic techniques applied directly to the curve equation. In this thesis we propose a more geometric approach, taking into account the subsequent high-precision approximation. Our algorithm is primarily based on the identification and approximation of smooth monotonous curve segments, which can in certain cases cross the singularities of the curve. To find the characteristic points we use not only the primary algebraic equation of the curve but also, and more importantly, its implicit support function representation. Using the rational Puiseux series, we describe local properties of curve branches at the points of interest and exploit them to find their connectivity. The support function representation is also used for an approximation of the segments. In this way, we obtain an approximate graph of the entire curve with several nice properties. It approximates the curve within a given Hausdorff distance. The actual error can be measured efficiently. The ap- proximate curve and its offsets are piecewise rational. And the question of topological equivalence of the...
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Structure and approximation of real planar algebraic curves
Blažková, Eva ; Šír, Zbyněk (advisor) ; Lávička, Miroslav (referee) ; Surynková, Petra (referee)
Finding a topologically accurate approximation of a real planar algebraic curve is a classic problem in Computer Aided Geometric Design. Algorithms describing the topology search primarily the singular points and are usually based on algebraic techniques applied directly to the curve equation. In this thesis we propose a more geometric approach, taking into account the subsequent high-precision approximation. Our algorithm is primarily based on the identification and approximation of smooth monotonous curve segments, which can in certain cases cross the singularities of the curve. To find the characteristic points we use not only the primary algebraic equation of the curve but also, and more importantly, its implicit support function representation. Using the rational Puiseux series, we describe local properties of curve branches at the points of interest and exploit them to find their connectivity. The support function representation is also used for an approximation of the segments. In this way, we obtain an approximate graph of the entire curve with several nice properties. It approximates the curve within a given Hausdorff distance. The actual error can be measured efficiently. The ap- proximate curve and its offsets are piecewise rational. And the question of topological equivalence of the...
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Depth of two-dimensional data
Dočekalová, Denisa ; Šír, Zbyněk (advisor) ; Hlubinka, Daniel (referee)
In this paper we summarize the basic information about halfplane depth function. It consists of two parts. In the first part we deal with the halfplane depth based on the distribution function, we describe its basic properties and define the concepts of depth contours, central regions and the halfplane median. We also deal with these concepts in the rest of the paper with the main focus on the halfplane median. In the second part of this work we deal with the halfplane depth based on the random choice with the main focus on data visualization. The used methods for visualization are the display of depth contours and the bagplot. This work includes pictures of depth contours for specific distributions which were gained by implementation of an algorithm in the software Mathematica. 1
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Rectagles inscribed in Jordan curves.
Ye, Tomáš ; Šír, Zbyněk (advisor) ; Vršek, Jan (referee)
We will introduce quotients, which are very special kinds of continuous maps. We are going to study their nice universal properties and use them to for- malize the notion of topological gluing. This concept will allow us to define interesting topological structures and analyze them. Finally, the developed theory will be used for writing down a precise proof of the existence of an inscribed rectangle in any Jordan curve. 1
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Isogeometric analysis in applications
Bekrová, Martina ; Šír, Zbyněk (advisor) ; Hron, Jaroslav (referee)
Isogeometric analysis (IGA) is a numerical method for solving partial differential equations (PDE). In this master thesis we explain a concept of IGA with special emphasis on problems on closed domains created by a single NURBS patch. For them we show a process how to modify the NURBS basis to ensure the highest possible continuity of the function space. Then we solve the minimal surface problem using two different Newton type methods. The first one is based on the classical approach using PDE, in the second one we use unique advantages of IGA to directly minimize the area functional.
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Curves with pythagorean hodograph
Kadlec, Kryštof ; Šír, Zbyněk (advisor) ; Šmíd, Dalibor (referee)
In the thesis we will look at curves with pythagorean hodograph (PH curves) whose speed is polynomial with respect to parameter. We will consider planar PH curves of degree 3 (PH cubics) exclusively. We will present their complex representation and preimage. Preimage is a simpler curve from which a PH curve is created and which determines its properties. First we will look at the basic properties of PH curves with respect to their preimage. The main aim of the thesis is determining continuousness of joints of PH curves on the basis of the shape of their preimage. We will give specific conditions on preimage for achieving certain types of continousness. Finally we will give some examples in order to illustrate the results. 1
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