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Geometric transformations in secondary school mathematics
Smetana, Adam ; Surynková, Petra (advisor) ; Hromadová, Jana (referee)
The current availability of computers and mobile devices in school facilities and the affordability of mobile devices for everyone provide for an increasing use of interactive geometry in the curriculum. The aim of this thesis is to summarize the printed and electronic resources used for teaching geometry, to study them in detail and to use them to build an electronic textbook and a collection of exercises in interactive geometry environment. GeoGebra is used among the interactive geometry environments, mainly because it offers the greatest variability for the purposes of this publication. The electronic publication can be used by both students and their teachers. Electronic publications were an essential part of the Czech education system during the period of compulsory distance education in years 2020 and 2021. The thesis includes a short questionnaire survey that aims to find out what textbooks or exercise collections are being used by teachers, what transformations are taught in secondary schools and to what extent. The results show that among teachers specific printed textbooks and collections are still more popular than interactive resources. Keywords: geometric transformations, rotation, translation, homothety, GeoGebra
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Lobachevskian geometry
Neubauerová, Alžběta ; Halas, Zdeněk (advisor) ; Hromadová, Jana (referee)
Title: Lobachevskian geometry Author: Alžběta Neubauerová Department: The Department of Mathematics Education Supervisor: Mgr. Zdeněk Halas, DiS., Ph.D., The Department of Mathematics Education Abstract: The aim of this bachelor's thesis is to introduce the topic of Lobachev- skian geometry to secondary school students. In the first chapter, we focus on the history of the discovery of Lobachevskian geometry due to attempts to prove Euclid's fifth postulate. In the second chapter we explain the basic terms, in the third chapter we list and prove chosen theorems from absolute geometry. The fourth chapter deals with theorems that are equivalent to the fifth postulate. By negating them, together with the facts from the chapter on absolute geometry, we obtain several theorems from Lobachevskian geometry in the fifth chapter. In the final chapter, we introduce Poincaré's model of the half-plane and thus gain more vivid idea about the theorems that we built in the previous chapter. Keywords: non-Euclidean geometry; Lobachevskian geometry; Euclidean geome- try
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Deeper properties of conics in the projective plane
Dvořák, Pavel ; Krump, Lukáš (advisor) ; Hromadová, Jana (referee)
The bachelor thesis presents alternative constructions and procedures in order to deepen the understanding of conic sections. Several types of procedures were used while solving certain problems, all of which led to the same result. Thesis also includes a more complicated construction of a conic section that does not lead to a unique solution. The main outcome of this thesis is reminiscent of the basic projective geometry course with an extended focus on conic sections. 1
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Projections in engineering drawings
Kukučík, Martin ; Surynková, Petra (advisor) ; Hromadová, Jana (referee)
This master's thesis focuses on illustrative axonometric projections and representations of elementary solid figures and other objects in these projections. The first chapter is an introduction to projections and the implementation of axonometry, including its analytical expression. The second, third and fourth chapters, which form the core of the thesis, are devoted to the representation of elementary solid figures and proper components in axonometric projections. An important part of the work are the appendices containing projections of all objects in sizes corresponding to the specifications. The thesis aims to present an overview of illustrative projection methods applicable in practice and to show the advantages and disadvantages of each projection.
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Complex projective line
Šteflová, Pavlína ; Krump, Lukáš (advisor) ; Hromadová, Jana (referee)
This thesis is about the expansion of the real projective line to the complex projective line. The first chapter is dedicated to a brief history of projective geometry. The next chapter introduces definitions and basic facts about the real projective line, plane, projective transformation and the cross-ratio. The third chapter concerns the complex projective line and related themes such as m¨obius transformation, cocircularity or stereographic projection. The chapter also briefly summarizes basic facts about complex numbers.
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Stereoscopic projection
Vlachová, Jana ; Hromadová, Jana (advisor) ; Surynková, Petra (referee)
Title: Stereoscopic Projection Author: Jana Vlachová Department: Department of Mathematics Education Supervisor: RNDr. Jana Hromadová, Ph.D. Supervisor's e-mail address: Jana.Hromadova@mff.cuni.cz Abstract: This graduation thesis deals with a special case of a double-central projection - the stereoscopic projection, whereas the position of the centres of projection and the projection plane is adjusted to the conditions of the human vision. The thesis introduces a brief historical development of the imaging and the stereoscopy itself, basic biological and optical characteristics of a human eye and vision and the principles of stereoscopic projection. Furthermore it occupies with the procedures of making stereoscopic drawings and photographs along with the methods of their observation and creating some necessary tools using generally available materials. The end of this thesis is devoted to the possibilities of not only practical usage of the stereoscopy, but mainly of its application in the descriptive geometry teaching. The thesis includes many stereoscopic pictures, some of them are viewable with the lens- glasses or the anaglyph-glasses accompanied in the end of this thesis. Keywords: double-central projection, stereoscopic projection, anaglyph
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Secondary school polyhedrons with internet
Helm, Jan ; Hromadová, Jana (advisor) ; Šarounová, Alena (referee)
The thesis is destined mainly for high school teachers and students of descriptive geometry. Above all it deals with the intersection and the construction of pyramids and prisms in projections. Students can meet with these phenomena at high schools during lessons of descriptive geometry. The constructions of the intersections of figures are demonstrated on solved tasks. The tasks are processed in graphic programmes GeoGebra and Cabri 3D prospering from the following advantages and facilities of these programmes: a stepping of the construction, a contour accentuation or a secretion of auxiliary lines etc. Besides these solved tasks, there are also some unsolved tasks for practice at the ends of chapters. The introductory chapter contains definitions and characters of common polyhedrons and regular (Platonic) figures. The thesis consists of web sites, a printed version and an enclosed printed version in .pdf format.
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The Golden Section
Chmelíková, Vlasta ; Šarounová, Alena (advisor) ; Hromadová, Jana (referee)
This text has been written especially as an educational material for secondary school teachers of mathematics and despriptive geometry, however it can be interesting also for students of secondary schools and universities and for another persons interested in problems of the golden section. The thesis includes a calculation and properties of the golden number, several construction methods of the golden section, its occurence and use in plane and solid geometry, history of the golden section and its connections with art, architecture, nature, psychology, etc. In addition, there are added examples from older textbooks and proposals of the jobsheets for improvement of the mathematics lessons.
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Nonlinear perspectives
Michalik, Jindřich ; Hromadová, Jana (advisor) ; Surynková, Petra (referee)
This thesis characterizes nonlinear perspectives and describes the most used ones - cylindrical and spherical. Their advantages and disadvantages are compared to each other, as well as with respect to linear perspective. For each of them there is an analytical expression deduced and the perspective image of a general line described. The thesis contains illustrative pictures created in the modeling software Rhinoceros. As a part of the thesis, a program created as a Matlab script file is enclosed, which demonstrates the mapping process of a convex polyhedron in described perspectives.
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