Daniela Bímová | Technical University of Liberec (original) (raw)

Papers by Daniela Bímová

“TOPICAL ISSUES OF THERMOPHYSICS, ENERGETICS AND HYDROGASDYNAMICS IN THE ARCTIC CONDITIONS”: Dedicated to the 85th Birthday Anniversary of Professor E. A. Bondarev

The contribution describes the use of the freeware GeoGebra for solving geometric-combinatorial p... more The contribution describes the use of the freeware GeoGebra for solving geometric-combinatorial problems. We use the freeware not only during online teaching as one of the tools to develop the abilities of first-year students of the Bachelor cycle at the Faculty of Mechanical Engineering of TUL but also during teaching geometry courses determined for the second-year and third-year students of the Bachelor study cycle and the third-year and fourth-year students of the Master study cycle at the Faculty of Science, Humanities and Education to solve geometric tasks and geometriccombinatorial problems. We consider the mentioned areas to be significant from the perspective of future graduates of the Faculty of Mechanical Engineering and the Faculty of Science, Humanities and Education. In this contribution, we present and describe the solutions of selected geometric-combinatorial problems in the GeoGebra environment.

The contribution describes the usage of the software Rhinoceros for creating the 3D models of two... more The contribution describes the usage of the software Rhinoceros for creating the 3D models of two intersecting surfaces of revolution. We distinguish three different cases, in which there are couples of surfaces of revolution with one coincident axis, next to it with two non-coincident parallel axes and finally, with two intersecting axes. There exists the possibility to save the created 3D models of two intersecting surfaces of revolution in the *.stl format in the Rhinoceros. The files saved in the *.stl format are used as bases for 3D printing. We use 3D printing as a tool for creating realistic models of such types of spatial situations or constructions that are hard to image for the students of the first year of the Bachelor study at the Faculty of Mechanical Engineering at TUL. In the case of the contact way of teaching, we can lend the printed 3D models during the lectures or seminars to students. Consequently, the manipulative activities can lead to developing the imaginations of the real visibility of the spatial situations or constructions drawn on the paper by the ruler and the compass. We also introduce the possibilities of using 3D printing in the online way of teaching the course titled Constructive Geometry.

“TOPICAL ISSUES OF THERMOPHYSICS, ENERGETICS AND HYDROGASDYNAMICS IN THE ARCTIC CONDITIONS”: Dedicated to the 85th Birthday Anniversary of Professor E. A. Bondarev

The contribution describes the usage of the freeware GeoGebra for developing the skills of the st... more The contribution describes the usage of the freeware GeoGebra for developing the skills of the students of the first year of the bachelor study at the Faculty of Mechanical Engineering at the Technical University of Liberec. The freeware GeoGebra is used as one of the tools for a clearer projection of more demanding constructions of the surfaces of revolution in the Monge projection. Both the Graphics 2D and the Graphics 3D of the GeoGebra are utilized for the possibility of connecting the 2D construction in the Monge projection with modelling the spatial situation in the Graphics 3D. Consequently, the surfaces of revolution as well as the helpful objects and final curves of the intersections of two surfaces of revolution can be seen from different points of view in the Graphics 3D. Finally, the possibilities of using the GeoGebra in the online way of teaching within the course named Constructive Geometry are introduced.

Monge projection and particularly metric problems of Monge projection are very demanding in terms... more Monge projection and particularly metric problems of Monge projection are very demanding in terms of spatial imagination. Therefore, it is suitable to apply the dynamic applets in geometric software GeoGebra in the lessons of descriptive geometry. The above mentioned 3D applets are highly practical as they provide a better spatial imagination

A simulation-based study to calculate all the possible trajectories of differential drive mobile ... more A simulation-based study to calculate all the possible trajectories of differential drive mobile robot

In our contribution, we study different Riesz wavelet bases in Sobolev spaces based on cubic spli... more In our contribution, we study different Riesz wavelet bases in Sobolev spaces based on cubic splines satisfying homogeneous Dirichlet boundary conditions of the second order. These bases are consequently applied to the numerical solution of the biharmonic problem and their quantitative properties are compared.

Analytic geometry in the plane is a very important part of geometry, especially for the students ... more 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.

V poslednich letech se při výuce prostorove geometrie (stereometrie, analyticke geometrie v prost... more V poslednich letech se při výuce prostorove geometrie (stereometrie, analyticke geometrie v prostoru, konstruktivni geometrie, apod.) stale setkavam se situacemi, že si studenti nejsou schopni představit probirane prostorove situace a vzajemne prostorove vztahy. Jednou z možnosti, jak jim pomoci, může být trenovani jejich prostorove představivosti za pomoci netradicnich stereometrických uloh. Přispěvek je věnovan ukazce řeseni netradicnich stereometrických uloh za podpory geometrickeho výukoveho programu Cabri 3D.

The paper is devoted to Thomas Malton´s constructions of a regular pentagon. There are listed fiv... more The paper is devoted to Thomas Malton´s constructions of a regular pentagon. There are listed five different constructions of the regular pentagon in the Malton´s student-book. The Malton´s constructions are relatively easy to draw and they are very accurate. Unfortunately, most of them are not so widespread. We have checked the accuracy of all five Malton´s constructions by the usage of the modern technique, especially by geometric softwares GeoGebra and DesignCAD 3D Max.

PROCEEDINGS OF THE 45TH INTERNATIONAL CONFERENCE ON APPLICATION OF MATHEMATICS IN ENGINEERING AND ECONOMICS (AMEE’19), 2019

Spatial imagination belongs among the innate abilities of a man. This means that one is born with... more 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.

PROCEEDINGS OF THE 45TH INTERNATIONAL CONFERENCE ON APPLICATION OF MATHEMATICS IN ENGINEERING AND ECONOMICS (AMEE’19), 2019

Monge projection is taught at the Technical University of Liberec, among others, at the Faculty o... more 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.

AIP Conference Proceedings, 2018

Solving problems in geometry, whether planar or spatial, is for students generally more difficult... more 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.

The contribution presents the dynamic applets created in GeoGebra that show the origin and main p... more 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.

AIP Conference Proceedings, 2016

Many technical applications use objects having a shape of the torus. The aim of the paper is to s... more Many technical applications use objects having a shape of the torus. The aim of the paper is to show a new method how to calculate the volume of the torus. The method leads to an easier calculation because the double integral is used instead of the triple one, which is commonly found in conventional examples. The paper includes computation of the volumes of a ring torus and as well as of a horn torus by the mentioned method.

AIP Conference Proceedings, 2016

The paper is concentrated on technical applications using volumes of solids. These applications a... more The paper is concentrated on technical applications using volumes of solids. These applications are calculated by an unconventional method, which has been presented in AMEE (2013, 2014). Now the formulas for calculations of volumes in special cases are proved (solids of revolutions, fluid flow, etc.). Advantages and disadvantages of the method are compared.

AIP Publishing, 2015

GeoGebra 5.0 is the dynamic geometric and mathematic software which dynamic tools allow the user ... more 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.

The paper presents one important case of a canal surface called Dupin’s cyclide. Non-degenerate f... more The paper presents one important case of a canal surface called Dupin’s cyclide. Non-degenerate forms of Dupin’s cyclide are attractive for example for geometric design applications. In the paper, there is derived one possibility of a rational parameterization of the cyclide as a special case of a canal surface. An algorithm for that parameterization is also shown. The parameterization offers a good method how to approximate especially implicit blend canal surfaces.

ABSTRACT Jia and Zhao have recently proposed a construction of a cubic spline wavelet basis on th... more ABSTRACT Jia and Zhao have recently proposed a construction of a cubic spline wavelet basis on the interval which satisfies homogeneous Dirichlet boundary conditions of the second order. They used the basis for solving fourth order problems and they showed that Galerkin method with this basis has superb convergence. The stiffness matrices for the biharmonic equation defined on a unit square have very small and uniformly bounded condition numbers. In our contribution, we design wavelet bases with the same scaling functions and different wavelets. We show that our basis has the same quantitative properties as the wavelet basis constructed by Jia and Zhao and additionally the wavelets have vanishing moments. It enables to use this wavelet basis in adaptive wavelet methods and non-adaptive sparse grid methods. Furthermore, we even improve the condition numbers of the stiffness matrices by including lower levels.

AIP Conference Proceedings, 2014

The paper presents one important and interesting property of strictly convex functions. On the ba... more 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.

“TOPICAL ISSUES OF THERMOPHYSICS, ENERGETICS AND HYDROGASDYNAMICS IN THE ARCTIC CONDITIONS”: Dedicated to the 85th Birthday Anniversary of Professor E. A. Bondarev

The contribution describes the use of the freeware GeoGebra for solving geometric-combinatorial p... more The contribution describes the use of the freeware GeoGebra for solving geometric-combinatorial problems. We use the freeware not only during online teaching as one of the tools to develop the abilities of first-year students of the Bachelor cycle at the Faculty of Mechanical Engineering of TUL but also during teaching geometry courses determined for the second-year and third-year students of the Bachelor study cycle and the third-year and fourth-year students of the Master study cycle at the Faculty of Science, Humanities and Education to solve geometric tasks and geometriccombinatorial problems. We consider the mentioned areas to be significant from the perspective of future graduates of the Faculty of Mechanical Engineering and the Faculty of Science, Humanities and Education. In this contribution, we present and describe the solutions of selected geometric-combinatorial problems in the GeoGebra environment.

The contribution describes the usage of the software Rhinoceros for creating the 3D models of two... more The contribution describes the usage of the software Rhinoceros for creating the 3D models of two intersecting surfaces of revolution. We distinguish three different cases, in which there are couples of surfaces of revolution with one coincident axis, next to it with two non-coincident parallel axes and finally, with two intersecting axes. There exists the possibility to save the created 3D models of two intersecting surfaces of revolution in the *.stl format in the Rhinoceros. The files saved in the *.stl format are used as bases for 3D printing. We use 3D printing as a tool for creating realistic models of such types of spatial situations or constructions that are hard to image for the students of the first year of the Bachelor study at the Faculty of Mechanical Engineering at TUL. In the case of the contact way of teaching, we can lend the printed 3D models during the lectures or seminars to students. Consequently, the manipulative activities can lead to developing the imaginations of the real visibility of the spatial situations or constructions drawn on the paper by the ruler and the compass. We also introduce the possibilities of using 3D printing in the online way of teaching the course titled Constructive Geometry.

“TOPICAL ISSUES OF THERMOPHYSICS, ENERGETICS AND HYDROGASDYNAMICS IN THE ARCTIC CONDITIONS”: Dedicated to the 85th Birthday Anniversary of Professor E. A. Bondarev

The contribution describes the usage of the freeware GeoGebra for developing the skills of the st... more The contribution describes the usage of the freeware GeoGebra for developing the skills of the students of the first year of the bachelor study at the Faculty of Mechanical Engineering at the Technical University of Liberec. The freeware GeoGebra is used as one of the tools for a clearer projection of more demanding constructions of the surfaces of revolution in the Monge projection. Both the Graphics 2D and the Graphics 3D of the GeoGebra are utilized for the possibility of connecting the 2D construction in the Monge projection with modelling the spatial situation in the Graphics 3D. Consequently, the surfaces of revolution as well as the helpful objects and final curves of the intersections of two surfaces of revolution can be seen from different points of view in the Graphics 3D. Finally, the possibilities of using the GeoGebra in the online way of teaching within the course named Constructive Geometry are introduced.

Monge projection and particularly metric problems of Monge projection are very demanding in terms... more Monge projection and particularly metric problems of Monge projection are very demanding in terms of spatial imagination. Therefore, it is suitable to apply the dynamic applets in geometric software GeoGebra in the lessons of descriptive geometry. The above mentioned 3D applets are highly practical as they provide a better spatial imagination

A simulation-based study to calculate all the possible trajectories of differential drive mobile ... more A simulation-based study to calculate all the possible trajectories of differential drive mobile robot

In our contribution, we study different Riesz wavelet bases in Sobolev spaces based on cubic spli... more In our contribution, we study different Riesz wavelet bases in Sobolev spaces based on cubic splines satisfying homogeneous Dirichlet boundary conditions of the second order. These bases are consequently applied to the numerical solution of the biharmonic problem and their quantitative properties are compared.

Analytic geometry in the plane is a very important part of geometry, especially for the students ... more 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.

V poslednich letech se při výuce prostorove geometrie (stereometrie, analyticke geometrie v prost... more V poslednich letech se při výuce prostorove geometrie (stereometrie, analyticke geometrie v prostoru, konstruktivni geometrie, apod.) stale setkavam se situacemi, že si studenti nejsou schopni představit probirane prostorove situace a vzajemne prostorove vztahy. Jednou z možnosti, jak jim pomoci, může být trenovani jejich prostorove představivosti za pomoci netradicnich stereometrických uloh. Přispěvek je věnovan ukazce řeseni netradicnich stereometrických uloh za podpory geometrickeho výukoveho programu Cabri 3D.

The paper is devoted to Thomas Malton´s constructions of a regular pentagon. There are listed fiv... more The paper is devoted to Thomas Malton´s constructions of a regular pentagon. There are listed five different constructions of the regular pentagon in the Malton´s student-book. The Malton´s constructions are relatively easy to draw and they are very accurate. Unfortunately, most of them are not so widespread. We have checked the accuracy of all five Malton´s constructions by the usage of the modern technique, especially by geometric softwares GeoGebra and DesignCAD 3D Max.

PROCEEDINGS OF THE 45TH INTERNATIONAL CONFERENCE ON APPLICATION OF MATHEMATICS IN ENGINEERING AND ECONOMICS (AMEE’19), 2019

Spatial imagination belongs among the innate abilities of a man. This means that one is born with... more 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.

PROCEEDINGS OF THE 45TH INTERNATIONAL CONFERENCE ON APPLICATION OF MATHEMATICS IN ENGINEERING AND ECONOMICS (AMEE’19), 2019

Monge projection is taught at the Technical University of Liberec, among others, at the Faculty o... more 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.

AIP Conference Proceedings, 2018

Solving problems in geometry, whether planar or spatial, is for students generally more difficult... more 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.

The contribution presents the dynamic applets created in GeoGebra that show the origin and main p... more 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.

AIP Conference Proceedings, 2016

Many technical applications use objects having a shape of the torus. The aim of the paper is to s... more Many technical applications use objects having a shape of the torus. The aim of the paper is to show a new method how to calculate the volume of the torus. The method leads to an easier calculation because the double integral is used instead of the triple one, which is commonly found in conventional examples. The paper includes computation of the volumes of a ring torus and as well as of a horn torus by the mentioned method.

AIP Conference Proceedings, 2016

The paper is concentrated on technical applications using volumes of solids. These applications a... more The paper is concentrated on technical applications using volumes of solids. These applications are calculated by an unconventional method, which has been presented in AMEE (2013, 2014). Now the formulas for calculations of volumes in special cases are proved (solids of revolutions, fluid flow, etc.). Advantages and disadvantages of the method are compared.

AIP Publishing, 2015

GeoGebra 5.0 is the dynamic geometric and mathematic software which dynamic tools allow the user ... more 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.

The paper presents one important case of a canal surface called Dupin’s cyclide. Non-degenerate f... more The paper presents one important case of a canal surface called Dupin’s cyclide. Non-degenerate forms of Dupin’s cyclide are attractive for example for geometric design applications. In the paper, there is derived one possibility of a rational parameterization of the cyclide as a special case of a canal surface. An algorithm for that parameterization is also shown. The parameterization offers a good method how to approximate especially implicit blend canal surfaces.

ABSTRACT Jia and Zhao have recently proposed a construction of a cubic spline wavelet basis on th... more ABSTRACT Jia and Zhao have recently proposed a construction of a cubic spline wavelet basis on the interval which satisfies homogeneous Dirichlet boundary conditions of the second order. They used the basis for solving fourth order problems and they showed that Galerkin method with this basis has superb convergence. The stiffness matrices for the biharmonic equation defined on a unit square have very small and uniformly bounded condition numbers. In our contribution, we design wavelet bases with the same scaling functions and different wavelets. We show that our basis has the same quantitative properties as the wavelet basis constructed by Jia and Zhao and additionally the wavelets have vanishing moments. It enables to use this wavelet basis in adaptive wavelet methods and non-adaptive sparse grid methods. Furthermore, we even improve the condition numbers of the stiffness matrices by including lower levels.

AIP Conference Proceedings, 2014

The paper presents one important and interesting property of strictly convex functions. On the ba... more 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.