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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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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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