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There is a very rapid development of modern technologies in the last few years. Teaching and learning processes at all levels of schools must reflect on it. Nevertheless, some topics are timeless because their practical usages are essential all over the years. E.g. designing new models of cars is impossible to do without knowing different types of surfaces. But ways of teaching such topics should differ as time goes by years. The handmade drawings were constructed for several decades of previous years. On the contrary, modern technologies such as special kinds of geometric software, and 3D printers can be included nowadays. Consequently, virtual models of surfaces or 3D objects can be constructed more quickly and easily in geometric software than by hand on paper and some of them can be even printed on 3D printers. This contribution deals with the possibilities of using various modern technologies in teaching helicoidal surfaces. [ABSTRACT FROM AUTHOR]

Playing with wooden or plastic cubes, with the various educational kits, children obtain the first experience with the geometric solids. Starting the school attendance, the pupils learn their correct names and find out the important properties of the basic solids. Continuing their study, the students learn the principles of spatial geometry in the connection with the basic solids, as well as with the geometric polyhedrons. Besides, the knowledge of properties of the basic solids and the geometric polyhedrons is very important for constructing them and projecting them in different kinds of projections and for constructing their projections in geometric or CAD software. The contribution introduces the possibilities of training the properties as well as constructions of basic solids and geometric polyhedrons via GeoGebra applets. [ABSTRACT FROM AUTHOR]

Analytic geometry in the plane is a very important part of geometry, especially for the students of mechanical engineering study field. They start studying the analytic geometry in the plane within the secondary school study. Although the study of analytic geometry in the plane is important for the future study at university, students often have insufficient knowledge of the above mentioned part of geometry when entering university. The basics, such as the parametric equations of the straight-line, coordinates of the point lying on the straight-line, the parameter of the point etc. cause students major problems. Therefore, the GeoGebra book containing the chapters of analytic geometry in the plane has been created for the students as a learning tool for the repetitive lessons in the first term of the university study. There is also a quick test for checking the acquired knowledge created in the GeoGebra book. [ABSTRACT FROM AUTHOR]

Monge projection is taught at the Technical University of Liberec, among others, at the Faculty of Mechanical Engineering in the course titled Constructive geometry. However, students often encounter the descriptive geometry in the first year of their Bachelor study at the university and it is very complicated and demanding in terms of spatial imagination for them. Therefore, when it was decided to reduce the number of hours of exercises of the Constructive geometry course from two hours per week to one hour per week, we have created worksheets for Monge projection. Moreover, solutions of the problems occurring in these worksheets were created using the dynamic 3D geometry software GeoGebra. The students can practice as much as possible Monge projection and they can also check the particular steps of solution and the solution itself. These worksheets and the solutions are easy accessible for students as GeoGebra book on the internet. [ABSTRACT FROM AUTHOR]

Spatial imagination belongs among the innate abilities of a man. This means that one is born with certain assumptions of spatial imagination. These are needed in everyday life, but especially in the pursuit of various occupations, such as engineering, construction, design, medicine and many others. On the other hand, a number of researches have been conducted to investigate whether the level of human spatial imagination can be developed throughout life. It has been found it can be, but it was also realized that there are so about 3 periods of human development in which the level of spatial imagination can be developed best. If we focus on two of these periods that are in the school-age and give appropriate incentives, it is very likely that the level of spatial imagination of the students will increase. For example, the inclusion of stereometric warm-ups in Mathematics lessons may be suitable stimuli. This article lists several suitable stereometric warm-ups created in software GeoGebra. [ABSTRACT FROM AUTHOR]

Solving problems in geometry, whether planar or spatial, is for students generally more difficult than solving problems in other branches of mathematics, e.g. in arithmetic or in algebra. Therefore, it is necessary to consider how students’ geometric thinking and spatial imagination can be practised and developed. One of the options is to provide students with appropriate examples that would practise and develop their geometric thinking and spatial imagination in a relatively short time. Such examples we can denote "warm-ups". There are organized the preparatory courses for students of the Faculty of mechanical engineering by our department before the first semester of their bachelor study starts. Practising geometry is one part of the preparatory course. The inclusion of the planimetric warm-ups in one lesson of the preparatory course seems to be useful. There are mentioned several planimetric warm-ups drawn in software GeoGebra, creating the base of so-called online GeoGebra book named Planimetric Warm-ups that is used as e-learning material during the preparatory course, in this contribution. [ABSTRACT FROM AUTHOR]

The contribution presents the dynamic applets created in GeoGebra that show the origin and main properties of a helix and it also presents some constructive problems connected with the helix. There are created the step by step algorithms of some constructions in the chosen applets. Three-dimensional applets include illustrative helix samples and spatial animations that help students better see problems concerning the helix spatially. There is mentioned the website in the contribution on which there is situated GeoGebra book dedicated to the topic "Helix" and containing the mentioned applets. The created applets and materials of the GeoGebra book "Helix" help in teaching and studying the course Constructive Geometry determined for the students of the Faculty of Mechanical Engineering of the Technical University of Liberec. [ABSTRACT FROM AUTHOR]

GeoGebra 5.0 is the dynamic geometric and mathematic software which dynamic tools allow the user to solve dynamic problems in a plane and as well as in the three-dimensional space. The contribution presents the dynamic applets constructed in GeoGebra 5.0 that show the origin as well as some properties of various kinds of canal surfaces. There are described the applets for constructing the canal surfaces in the paper. Some of the created canal surfaces are compared with the real life canal surface examples. [ABSTRACT FROM AUTHOR]

The paper presents one important and interesting property of strictly convex functions. On the base of previously proved properties of these functions, we can generalize some global-geometric conclusions, without the even assumption of their differentiability, and also point out some geometric characteristics of the sets of all solutions of parametric optimization problems. Two specific nonlinear parametric optimizations of nonlinear problems -- the minimization and the maximization problems -- exist for them. They express a pair of dual optimization problems. The set of all optimal points has an important geometric interpretation. This paper connects the proved properties of a specific case and a geometric sense. [ABSTRACT FROM AUTHOR]

The paper connects the alternative theory published in AMEE'13 (D. Bittnerova: Alternative Method for Calculations of Volumes by Using Parameterizations of Surface Areas) and presents some applications of it in Euclidean space of dimension 2, especially for cycloids. According to that theory, we can calculate areas of figures respectively volumes of solids in En, if we know suitable descriptions of surface areas of computed figures, solids. Applying the alternative theory in E2, we obtain the corollary of Green's theorem for the curvilinear integral. [ABSTRACT FROM AUTHOR]