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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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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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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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Classical Problems of Ancient Greek Mathematicians
Švecová, Michaela ; Bečvář, Jindřich (advisor) ; Šír, Zbyněk (referee)
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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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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Envelopes of implicit surfaces
Vráblíková, Jana ; Šír, Zbyněk (advisor) ; Lávička, Miroslav (referee)
The aim of the thesis is to study envelopes and characteristic curves of one parameter systems of quadratic surfaces in real three dimensional space. We define one parameter systems and their envelopes generally and present algebraic geometry approach for en- velope computation using Gröbner bases and elimination theory. We convey a proof of rationality of envelopes of rational one parameter systems of spheres, cones and cylin- ders of revolution using dual space and different models of Laguerre geometry. Then we present a new approach to one parameter systems and their envelopes. We introduce the systems as curves in homogeneous spaces which allows us to study all characteristic curves at a single surface. This approach allows us to prove rationality and even provide an explicit parameterization of characteristic curves and the envelope of a one parameter system of isometric cones of revolution. We provide several other examples illustrating the concepts and results. 1
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From computer 3D modelling to reality and back
Zdražil, Michal ; Surynková, Petra (advisor) ; Šír, Zbyněk (referee)
Technological advancements are faster than ever and on the frontier are applications and mechanisms entwined with 3D computer aided modelling, such as 3D printing, scan- ning and extended reality technologies. This work gives a peek behind the veil of mystery surrounding these technologies. We aim to give a brief look into each of the mentioned areas and let the reader experience them practically, mathematically and algorithmically in hope to bring these three so often separated views closer together and to the reader. 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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Pythagorean hodograph splines
Kadlec, Kryštof ; Šír, Zbyněk (advisor) ; Lávička, Miroslav (referee)
In this thesis the main object of our concern is a Pythagorean hodograph B- spline curve. We recall notions of both Pythagorean hodograph (PH) curves and B-spline functions separately first. Then we put these fields together to generalize PH curves to their B-spline instances. We encapsulate these curves in various spaces under one algebraic structure using the formalism of Clifford algebras. We consider both Euclidean and Minkowski spaces of lower dimensions which give room for real applications and use of these curves. We support our results by giving numerous examples. 1
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