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Discrete differential geometry and its applications
Vidličková, Eva ; Šír, Zbyněk (advisor) ; Hron, Jaroslav (referee)
In this thesis we present an elementary introduction to the Discrete differen- tial geometry. We will work with both discrete curves and discrete surfaces. Firstly some basic definitions and theorems from classic Differential geometry will be mentioned and then we will translate these concepts to the discrete setting, so that some important global structures are still preserved. At the end we implement mean curvature flow defined on discrete surfaces and run it on two meshes, that show its area-minimizing feature. This can be used for denoising the discrete surfaces. 1
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Methods for generating computational grids suitable for finite element method
Langer, Lukáš ; Hron, Jaroslav (advisor) ; Šír, Zbyněk (referee)
This paper is focused on the problematic of the meshes and tries to introduce the basic kinds of these meshes with their advantages and disadvantages. The algorithm of choosen methods of generations is further described. Moreover it introduces in detail the method of generation quadrilateral mesh via the dual graph method presented in work Nowottny, 1999. There is described how to make the start dual graph, it's factorization and the final redualization into quadrilateral mesh. The own implementation of this method in the python programming language is included.
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Algorithms and Principles in Mathematics Education
Pazourek, Karel ; Šír, Zbyněk (advisor) ; Odvárko, Oldřich (referee) ; Šimša, Jaromír (referee)
The thesis is devoted to the use of the algorithms and the principles in mathe- matical education, especially in the education of the divisibility. It is based on study of the Czech secondary school textbooks which were published since 1852. We mention briefly the history of the system of education since 1848 as well as the theory of divisibility. There are analyzed the algorithmic methods connected with divisibility commented in these textbooks. Two applications of the algorithms are pointed out. The algorithms can be used in education of talented students as it is done in the mathematical course of the Talnet project. The principles are the fundaments of commentary of the textbooks, the principle of induction is the base of the divisibility. In the appendix, there is a demonstration of graduated development of algorithm in the case of the Euclid algorithm.
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Visually realistic modeling of dynamic objects deformations
Bulušek, Petr ; Boldyš, Jiří (advisor) ; Šír, Zbyněk (referee)
The present work deals with simulation methods for rigid bodies and deformable bodies. In the first chapter you can find research of some methods for simulation of rigid body physics with emphasis on method used in open source physics engine Bullet. In second chapter you can find methods for simulation of deformable bodies, again with emphasis on Bullet physics engine. In last chapter model order reduction technique is presented. This method enables to reduce system of ordinary differential equations. These equations come for example from applying finite element method to partial differential equations describing motion of elastic body. The technique is studied on bar truss systems. Powered by TCPDF (www.tcpdf.org)
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Properties and applications of low degree surfaces
Mirová, Aneta ; Šír, Zbyněk (advisor) ; Boček, Leo (referee)
My thesis "Properties and applications of low degree surfaces" deals mainly with properties of two quadrics. We introduce so called canonical forms of two quadrics and classification of intersection curve of two quadrics. There are used index and signature sequence to obtain classification of intersection curve of two quadrics. These sequence can determine the number of component of the intersection curve, their algebraic degrees and singularities. Our work also contains many examples with pictures. One part of the thesis also presents possible application of intersection curve of two quadrics in practice. This thesis also contains an enclosed CD, with the thesis in an electronic form and the source files of all pictures in the thesis.
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Generování a optimalizace meshů
Mokriš, Dominik ; Šír, Zbyněk (advisor) ; Hron, Jaroslav (referee)
This thesis is devoted to the problem of finding a suitable geometrical de- scription of the domain for the Finite Element Method (FEM). We present the most important methods used in generation and improvement of unstructured triangular meshes (grids) for two dimensional FEM. Possible measures of mesh quality are discussed with respect to their usage in linear Lagrange FEM. The relationship between mesh geometry (especially angles of particular triangles), discretization error and stiffness matrix condition number is examined. Two methods of mesh improvement, based on Centroidal Voronoi Tessellations (CVT) and Optimal Delaunay Triangulations (ODT), are discussed in detail and some results on convergence of CVT based methods are reviewed. Some aspects of these methods, e.g. the relation between density of boundary points and interior mesh vertices and the treatment of the boundary triangles is reconsidered in a new way. We have implemented these two methods and we discuss possible im- provements and new algorithms. A geometrically very interesting idea of recent alternative to FEM, Isogeometric Analysis (IGA), is outlined and demonstrated on a simple example. Several numerical tests are made in order to the compare the accuracy of solutions of isotropic PDEs obtained by FEM on bad mesh, mesh improved...
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Websites for teaching of geometric transformations at secondary school
Dobiášová, Kateřina ; Šír, Zbyněk (referee) ; Robová, Jarmila (advisor)
As a part of this thesis existing web pages about planar geometrical transformations were evaluated and new web pages were created. Pages in Czech and in English language were evaluated shortly verbally. There is a table summarizing findings about existing web pages at the end of this part. The second part of this thesis contains newly created web pages. Scope of subject matter of created web pages overreaches common secondary school curriculum. There are Java applets used for illustrating findings about geometrical transformations as an integral part of web pages. Solving of exercises is accompanied by step-by-step operation. Web pages contain definitions, theorems with proofs and explanations of constructions.
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Classical Problems of Ancient Greek Mathematicians
Švecová, Michaela ; Šír, Zbyněk (referee) ; Bečvář, Jindřich (advisor)
This diploma thesis deals with ve classical problems of ancient Greek mathematics. It is circle squaring, cube duplication, angle trisection, circle recti cation and regular polygons constructions. The proves of insolvability of these problems as well as various attempts to solve them are presented. These include, on one hand, exact techniques breaking the Euclidean constructions rules by using special tools, curves, etc., and, on the other hand, approximate methods using only compass and straightedge. Finally, several Czech contributions to the topic are mentioned.
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