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Spatial generalizations of the properties of the triangle
Šrubař, Jiří ; Karger, Adolf (advisor) ; Boček, Leo (referee) ; Lávička, Miroslav (referee)
TITLE Spatial generalizations of the properties of the triangle AUTHOR Jirˇı' Sřubarˇ SUPERVISOR Prof. RNDr. Adolf Karger, DrSc. DEPARTMENT Department of mathematics education ABSTRACT The present thesis describes various interesting properties of a triangle. The aim is to find and prove similar properties of its spatial generalization - a tetrahedron. Even though both synthetic and computational methods are used for proving spatial relations, synthetic approach is preferred whenever possible. The thesis is divided into two parts. In the first part, the properties of the tetrahedron analogous to the centroid and the orthocenter of the triangle are described. Also, conditions on the existence of the orthocenter of the tetrahedron are derived. Moreover, for tetrahedrons without an orthocenter, the so-called Monge point is introduced as its generalization. In the second part of the thesis, some further properties of the triangle are studied - - the Simson line, the de Longchamps point, the nine-point circle, the Euler line, the Lemoine point, the isodynamic points, the Lemoine axis and the Brocard axis. As the main contribution of the present thesis we define and prove the existence of spatial analogues of the above mentioned properties for the tetrahedron - the de Longchamps point, the twelve-point and...
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Descriptive Geometry Teaching in the Czech Lands
Moravcová, Vlasta ; Bečvář, Jindřich (advisor) ; Janyška, Josef (referee) ; Lávička, Miroslav (referee)
This thesis documents the history of descriptive geometry teaching in the Czech lands from its beginnings (i.e. from the 1830s) to the World War II. The center of the work is the description of development of descriptive geo- metry teaching at secondary schools, especially "Realschule�, supplemented with an analysis of Czech secondary school textbooks and a brief comparison between the situation in the Czech lands and foreign countries. Attention is also paid to the characterisation of descriptive geometry lectures at Czech and German polytechnics and universities in Prague, Brno and Příbram, inclusive of university textbooks. The work also mentions a brief global de- velopment of projection methods and a contribution of Czech geometers to the development of descriptive geometry. The thesis is supplemented with large factual and pictorial addenda.
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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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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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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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Descriptive Geometry Teaching in the Czech Lands
Moravcová, Vlasta ; Bečvář, Jindřich (advisor) ; Janyška, Josef (referee) ; Lávička, Miroslav (referee)
This thesis documents the history of descriptive geometry teaching in the Czech lands from its beginnings (i.e. from the 1830s) to the World War II. The center of the work is the description of development of descriptive geo- metry teaching at secondary schools, especially "Realschule�, supplemented with an analysis of Czech secondary school textbooks and a brief comparison between the situation in the Czech lands and foreign countries. Attention is also paid to the characterisation of descriptive geometry lectures at Czech and German polytechnics and universities in Prague, Brno and Příbram, inclusive of university textbooks. The work also mentions a brief global de- velopment of projection methods and a contribution of Czech geometers to the development of descriptive geometry. The thesis is supplemented with large factual and pictorial addenda.
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Analysis of Point Clouds Representing Surfaces of Engineering Practice
Surynková, Petra ; Voráčová, Šárka (advisor) ; Pokorný, Milan (referee) ; Lávička, Miroslav (referee)
Title: Analysis of Point Clouds Representing Surfaces of Engineering Practice Author: Petra Surynková Department: Department of Mathematics Education Supervisor: Mgr. Šárka Voráčová, Ph.D., Faculty of Transportation Sciences, Czech Technical University in Prague Abstract: The doctoral dissertation Analysis of Point Clouds Representing Surfaces of Engineering Practice addresses the development and application of methods of digital reconstruction of surfaces of engineering and construction practice from point clouds. The main outcome of the dissertation is a presentation of new procedures and methods that contribute to each of the stages of the reconstruction process from the input point clouds. The work is mainly focused on the analysis of input clouds that describe special types of surfaces. Several completely new algorithms and improvements of existing algorithms that contribute to individual steps of surface reconstruction are presented. New procedures are based on geometrical characteristics of the reconstructed object. An important result of the dissertation is an analysis of not only synthetically generated point clouds but above all an analysis of real point clouds that have been obtained from measurements of real objects. The significant contribution of the dissertation is also an...
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