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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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Minimal surfaces and their applications
Beran, Filip ; Šír, Zbyněk (advisor) ; Slavík, Antonín (referee)
The aim of this bachelor thesis is to explain basic qualities of minimal surfaces and to demonstrate some significant examples. The first chapter summarizes clas- sic concepts of differential geometry of curves and surfaces, which are essential for formulation of the surface minimization problem. Solving of this variational problem brings us back to local property of surface, zero mean curvature. In the rest of the second chapter we reveal which other properties this condition implies; one of the most important is the conformity of the Gauss map. Emphasizing the geometric view, in the third chapter we derive minimal surfaces of revolution and ruled minimal surfaces. Finally we construct isometric deformation of these one parameter surface families, catenoids and helicoids, to show nontrivial case of local isometry which is also typical for minimal surfaces. 1
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Rational minimal surfaces
Bekrová, Martina ; Šír, Zbyněk (advisor) ; Šmíd, Dalibor (referee)
This bachelor thesis deals with rational surfaces with rational offsets and minimal surfaces. We will give a connection between these two classes of surfaces. We will introduce a method of finding all rational surfaces with rational offsets using dual representation of surface as an envelope of its own tangent surfaces. A connection will be established between minimal surfaces and functions of a complex variable. Furthermore, we will derive the known Weierstrass-Enneper representation and its modifications for generating minimal surfaces. By means of these two tools we will show that all rational minimal surfaces obtained from the Weierstrass-Enneper representation also have rational offsets. Powered by TCPDF (www.tcpdf.org)
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3D Texture deformation according to a polygonal model
Skřivan, Tomáš ; Křivánek, Jaroslav (advisor) ; Šír, Zbyněk (referee)
Aim of this bachelor thesis is to design and implement algorithm, which deforms 3D space according to the polygonal model. We focus on algorithm, where we calculate deformation as linear combination of vertices of deformed polygonal mesh. Coefficients of this linear combination are called generalized barycentric coordinated. In preceding literature are generalized barycentric coordinated defined only for triangular meshes, we propose further generalization to more objects such as polygonal meshes or parametric surfaces. In two dimensions it is possible to use complex numbers and obtain a bigger class of deformations, such as conformal mappings. We propose generalization to three dimensions with quaternions. We implement final algorithm to program Autodes Maya and Mental Ray. Powered by TCPDF (www.tcpdf.org)
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