Dissertations / Theses on the topic 'Cartesian coordinate system (3D)'

A single 3D virtual system in orthodontic-surgical and restorative patients has not been developed. The study aims were to evaluate: (1) the performance of a Cartesian coordinate system (CSYS) to measure translational and rotational tooth displacement, and (2) maxillary and mandibular tooth, occlusion and bone displacement with the CSYS and a lingual landmarks mathematical registration method. Eleven maxillary and mandibular dental models following an orthodontic surgical and restorative treatment sequence case were retrospectively collected. These were laser scanned and processed using reverse engineering computer aided design (CAD) technology to generate 3D models from which 6 paired models were evaluated. Phase one measured the errors in: (1) the correlation between CSYS – ABC landmarks (3 landmarks positioned on the anterior palate) selection and resulting tooth measurements, and (2) operator tooth landmark placement. Phase two: (1) used the maxillary and mandibular teeth landmarks to calculate the orthodontic, restorative and orthognathic translations and rotational displacement, (2) used additional registered landmarks on the lingual tori and mandibular teeth (when considered immobile) to isolate mandibular tooth displacement, and (3) calculated the inter-arch measurements. Relevant results indicated: (1) CSYS – ABC allocation mean translational deviation errors were small with 3D distance > anteroposterior > horizontal =vertical axes. Incisors were the least affected teeth, while molars the most affected. (2) Operator Landmarks selection error was negligible. (3) No statistical difference in tooth movement analysis for the mandible de-rotation methods was observed at the pre-surgical restorative stage. Customised CSYS and lingual landmarks of mandibular tori registration permits clinical orthodontic, restorative and orthognathic displacement measurement.

MRI (Magnetic Resonance Imaging) is a medical imaging method that uses magnetic fields in order to retrieve images of the human body. This thesis revolves around a novel acquisition method of 3D fMRI (functional Magnetic Resonance Imaging) called PRESTO-CAN that uses a radial pattern in order to sample the (kx,kz)-plane of k-space (the frequency domain), and a Cartesian sample pattern in the ky-direction. The radial sample pattern allows for a denser sampling of the central parts of k-space, which contain the most basic frequency information about the structure of the recorded object. This allows for higher temporal resolution to be achieved compared with other sampling methods since a fewer amount of total samples are needed in order to retrieve enough information about how the object has changed over time. Since fMRI is mainly used for monitoring blood flow in the brain, increased temporal resolution means that we can be able to track fast changes in brain activity more efficiently.The temporal resolution can be further improved by reducing the time needed for scanning, which in turn can be achieved by applying parallel imaging. One such parallel imaging method is SENSE (SENSitivity Encoding). The scan time is reduced by decreasing the sampling density, which causes aliasing in the recorded images. The aliasing is removed by the SENSE method by utilizing the extra information provided by the fact that multiple receiver coils with differing sensitivities are used during the acquisition. By measuring the sensitivities of the respective receiver coils and solving an equation system with the aliased images, it is possible to calculate how they would have looked like without aliasing.In this master thesis, SENSE has been successfully implemented in PRESTO-CAN. By using normalized convolution in order to refine the sensitivity maps of the receiver coils, images with satisfying quality was able to be reconstructed when reducing the k-space sample rate by a factor of 2, and images of relatively good quality also when the sample rate was reduced by a factor of 4. In this way, this thesis has been able to contribute to the improvement of the temporal resolution of the PRESTO-CAN method.
MRI (Magnetic Resonance Imaging) är en medicinsk avbildningsmetod som använder magnetfält för att framställa bilder av människokroppen. Detta examensarbete kretsar kring en ny inläsningsmetod för 3D-fMRI (functional Magnetic Resonance Imaging) vid namn PRESTO-CAN som använder ett radiellt mönster för att sampla (kx,kz)-planet av k-rummet (frekvensdomänen), och ett kartesiskt samplingsmönster i ky-riktningen. Det radiella samplingsmönstret möjliggör tätare sampling av k-rummets centrala delar, som innehåller den mest grundläggande frekvensinformationen om det inlästa objektets struktur. Detta leder till att en högre temporal upplösning kan uppnås jämfört med andra metoder eftersom det krävs ett mindre antal totala sampel för att få tillräcklig information om hur objektet har ändrats över tid. Eftersom fMRI framförallt används för att övervaka blodflödet i hjärnan innebär ökad temporal upplösning att vi kan följa snabba ändringar i hjärnaktivitet mer effektivt.Den temporala upplösningen kan förbättras ytterligare genom att minska scanningstiden, vilket i sin tur kan uppnås genom att tillämpa parallell avbildning. En metod för parallell avbildning är SENSE (SENSitivity Encoding). Scanningstiden minskas genom att minska samplingstätheten, vilket orsakar vikning i de inlästa bilderna. Vikningen tas bort med SENSE-metoden genom att utnyttja den extra information som tillhandahålls av det faktum att ett flertal olika mottagarspolar med sinsemellan olika känsligheter används vid inläsningen. Genom att mäta upp känsligheterna för de respektive mottagarspolarna och lösa ett ekvationssystem med de vikta bilderna är det möjligt att beräkna hur de skulle ha sett ut utan vikning.I detta examensarbete har SENSE framgångsrikt implementerats i PRESTO-CAN. Genom att använda normaliserad faltning för att förfina mottagarspolarnas känslighetskartor har bilder med tillfredsställande kvalitet varit möjliga att rekonstruera när samplingstätheten av k-rummet minskats med en faktor 2, och bilder med relativt bra kvalitet också när samplingstätheten minskats med en faktor 4. På detta sätt har detta examensarbete kunnat bidra till förbättrandet av PRESTO-CAN-metodens temporala upplösning.

Much research effort has been devoted to producing algorithms that contribute directly or indirectly to the extraction of 3D information from a wide variety of types of scenes and conditions of image capture. The research work presented in this thesis is aimed at three distinct applications in this area: interactively extracting 3D points from a pair of uncalibrated images in a flexible way; finding corresponding points automatically in high resolution images, particularly those of archaeological scenes captured from a freely moving light aircraft; and improving a correlation approach to dense disparity mapping leading to 3D surface reconstructions. The fundamental concepts required to describe the principles of stereo vision, the camera models, and the epipolar geometry described by the fundamental matrix are introduced, followed by a detailed literature review of existing methods. An interactive system for viewing a scene via a monochrome or colour anaglyph is presented which allows the user to choose the level of compromise between amount of colour and ghosting perceived by controlling colour saturation, and to choose the depth plane of interest. An improved method of extracting 3D coordinates from disparity values when there is significant error is presented. Interactive methods, while very flexible, require significant effort from the user finding and fusing corresponding points and the thesis continues by presenting several variants of existing scale invariant feature transform methods to automatically find correspondences in uncalibrated high resolution aerial images with improved speed and memory requirements. In addition, a contribution to estimating lens distortion correction by a Levenberg Marquard based method is presented; generating data strings for straight lines which are essential input for estimating lens distortion correction. The remainder of the thesis presents correlation based methods for generating dense disparity maps based on single and multiple image rectifications using sets of automatically found correspondences and demonstrates improvements obtained using the latter method. Some example views of point clouds for 3D surfaces produced from pairs of uncalibrated images using the methods presented in the thesis are included.
Al-Baath University
The appendices files and images are not available online.

Much research effort has been devoted to producing algorithms that contribute directly or indirectly to the extraction of 3D information from a wide variety of types of scenes and conditions of image capture. The research work presented in this thesis is aimed at three distinct applications in this area: interactively extracting 3D points from a pair of uncalibrated images in a flexible way; finding corresponding points automatically in high resolution images, particularly those of archaeological scenes captured from a freely moving light aircraft; and improving a correlation approach to dense disparity mapping leading to 3D surface reconstructions. The fundamental concepts required to describe the principles of stereo vision, the camera models, and the epipolar geometry described by the fundamental matrix are introduced, followed by a detailed literature review of existing methods. An interactive system for viewing a scene via a monochrome or colour anaglyph is presented which allows the user to choose the level of compromise between amount of colour and ghosting perceived by controlling colour saturation, and to choose the depth plane of interest. An improved method of extracting 3D coordinates from disparity values when there is significant error is presented. Interactive methods, while very flexible, require significant effort from the user finding and fusing corresponding points and the thesis continues by presenting several variants of existing scale invariant feature transform methods to automatically find correspondences in uncalibrated high resolution aerial images with improved speed and memory requirements. In addition, a contribution to estimating lens distortion correction by a Levenberg Marquard based method is presented; generating data strings for straight lines which are essential input for estimating lens distortion correction. The remainder of the thesis presents correlation based methods for generating dense disparity maps based on single and multiple image rectifications using sets of automatically found correspondences and demonstrates improvements obtained using the latter method. Some example views of point clouds for 3D surfaces produced from pairs of uncalibrated images using the methods presented in the thesis are included.

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This work has the purpose contribute to the improvement in some teaching contents of analytic geometry and trigonometry in high school . The content work was based on the National Curriculum Parameters, highlighting de nitions, theorems and properties necessary for the development of student learning. The theme was chosen after a practical experience involving the Naval Battle game in order to reduce the students' di culties. The playful work, as the game, has a practical application that does the student become familiar with the content. That's an interesting way to propose problems and solutions involving the content. Thus becomes something attractive to the student and encourages creativity in nding problems solutions.
O trabalho tem como objetivo contribuir para o aprimoramento no ensino de alguns conteúdos de Geometria Analítica e Trigonometria no Ensino Médio. Dentro dos Parâmetros Curriculares Nacionais, trabalhamos o conteúdo destacando de nições, teoremas e propriedades necessárias para o desenvolvimento de aprendizagem do aluno. O tema foi escolhido depois de uma experiência prática envolvendo o jogo Batalha Naval a m de diminuir as di culdades dos alunos. O trabalho lúdico, como o jogo, tem uma aplicação prática que faz o aluno se familiarizar com os conceitos. É uma forma interessante de propor problemas e soluções envolvendo o conceitos. Assim se torna algo atrativo para o aluno e favorece a criatividade na busca de soluções para os problemas.

The geometrical structure of a tree can consist of thousands, even millions, of branches, twigs and leaves in complex arrangements. The structure contains a lot of useful information and can be used for example to assess a tree's health or calculate parameters such as total wood volume or branch size distribution. Because of the complexity, capturing the structure of an entire tree used to be nearly impossible, but the increased availability and quality of particularly digital cameras and Light Detection and Ranging (LIDAR) instruments is making it increasingly possible. A set of digital images of a tree, or a point cloud of a tree from a LIDAR scan, contains a lot of data, but the information about the tree structure has to be extracted from this data through analysis. This work presents a method of reconstructing 3D models of trees from point clouds. The model is constructed from cylindrical segments which are added one by one. Bayesian inference is used to determine how to optimize the parameters of model segment candidates and whether or not to accept them as part of the model. A Hough transform for finding cylinders in point clouds is presented, and used as a heuristic to guide the proposals of model segment candidates. Previous related works have mainly focused on high density point clouds of sparse trees, whereas the objective of this work was to analyze low resolution point clouds of dense almond trees. The method is evaluated on artificial and real datasets and works rather well on high quality data, but performs poorly on low resolution data with gaps and occlusions.

L'acide rétinoïque (AR) est un morphogène dérivé de la vitamine A, qui intervient dans le contrôle de l'organogenèse, de la prolifération et de la différenciation cellulaires chez les Chordés. Dans ce contexte, nous avons étudié les régulations spatio-temporelles de la voie de signalisation de l’AR au cours du développement de l’amphioxus, en mettant l'accent sur l’espèce européenne Branchiostoma lanceolatum.Nous avons tout d'abord inhibé ou activé la voie de signalisation de l’AR lors du développement embryonnaire en traitant des embryons d’amphioxus à des doses variables de composés pharmacologiques interférant avec le métabolisme des rétinoïdes. Grâce à l’utilisation d’outils mathématiques spécifiques, nous avons établi un schéma détaillé des effets des traitements effectués sur le développement du système nerveux central (SNC) et du pharynx chez l’amphioxus en nous basant sur l’expression de gènes marqueurs de tissus spécifiques. À l’issue de cette première analyse, nous avons par la suite étudié les effets d’une perturbation de la signalisation de l’AR à des points clés du développement chez l’amphioxus lors de la régionalisation du SNC et du pharynx. Nous avons ainsi montré que la voie de signalisation de l’AR intervient dans la régionalisation de l’axe antéro-postérieur via le contrôle des gènes hox dès le stade gastrula et jusqu’aux stades larvaires. En outre, nous avons réalisé l'étude préliminaire du gène homologue chez l’amphioxus du gène aldh1a2 des Vertébrés, et avons démontré que la régulation du niveau de synthèse de l’AR au cour du développement est conservée entre l’amphioxus et les Vertébrés. Finalement, nous avons montré que la voie de l’AR participe également à la morphogenèse caudale chez l’amphioxus, et que le mécanisme impliqué semble différent de celui proposé chez les Vertébrés où l’AR contrôle la structuration de la nageoire caudale par le ciblage des tissus mésenchymateux
Retinoic acid (RA) is an endogenous vitamin A-derived morphogen. In this context, we studied the spatiotemporal roles of RA signaling in amphioxus development, focusing on the European amphioxus species: Branchiostoma lanceolatum. We first created excess and insufficiency models of RA signaling by exposing amphioxus embryos to series of doses of different pharmacological compounds targeting either the RA receptors or the RA metabolism machinery. By introducing the important mathematical concept of a Cartesian coordinate system founded by René Descartes, we created detailed diagrams of the concentration-dependent defects caused by RA signaling in the central nervous system (CNS) and pharynx of amphioxus by evaluating the statistical significances of tissue-specific marker gene expression in labeled embryos. This analysis yielded a very detailed description of the sensitivities of the developing amphioxus CNS and pharynx to altered RA signaling levels. Following this initial challenge, we correlated the effects of altered RA signaling levels with key amphioxus developmental stages characterized by structural transitions in CNS and pharynx. We show that hox-mediated RA signaling in axial patterning is active beyond the gastrula stage and might be maintained until at least early larval stage, with possible roles in more regionalized axis formation and organ induction. In addition, we carried out a preliminary study on a RA synthesizing gene in amphioxus, called aldh1a, a possible homolog of the vertebrate aldh1a2 gene, demonstrating that the feedback between RA signaling and RA synthesizing levels has emerged before the split of the cephalochordate and vertebrate lineages. Moreover, we are able to show that RA signaling also participates in tail fin morphogenesis in amphioxus by a mechanism that is probably not comparable to that in vertebrates, where RA modulates caudal fin patterning through targeting mesenchymal derivatives

碩士
淡江大學
教育科技學系數位學習在職專班
100
The aim of the study is to design and construct a teaching material for learning the mathematical Scratch Programming Language developed specific for junior high school students. Design-Based Research was used as the research theory and the basis of teaching material development. By utilizing seventh graders’ mathematics lessons, students were guided to use the Visualization Tool Scratch Programming Language. Through the problem-solving based learning process, it ignited students’ learning motivations as well as increased learning effectiveness. As the theme of the developed material integrates with “Cartesian Coordinate System” and “Linear Equation” of seventh graders’ Mathematics curriculum, it did not only enable students to enjoy the fun of designing IT programming language, but also repetitively revise, test and verify the mathematical concepts of “Cartesian Coordinate System” and “Linear Equation”, which were often considered difficult for students to grasp, during the learning process. Experimental teaching sessions were carried out for this particular teaching material, and conclusions are listed as below: 1. Students had Scratch integrated in their mathematics curriculum performed better in their learning achievement test compared to students in the control group. 2. Students had Scratch integrated in their mathematics curriculum performed better in their learning achievement procrastinated test compared to students in the control group. 3. Positive feedbacks were received from students on the integration of Scratch in their mathematics curriculum.

碩士
國立彰化師範大學
特殊教育學系所
101
The purpose of this study was to examine the effects of Scratch programming on enhancing and maintaining Cartesian coordinate system and graphs of linear equation for students with learning disabilities. A single-subject experimental design of multiple probes design across subjects was used to evaluate the results. Data collected during the study were analyzed by using visual inspection techniques. The results of this study were as following: 1. After teaching-experiment, all of the three subjects had progress on Cartesian coordinate system and graphs of linear equation. 2. After removing Scratch programming, all of the three subjects could remember what they had learned. 3. Positive feedbacks were received from three subjects on the integration of Scratch in the mathematics curriculum.

碩士
國立交通大學
土木工程學系
99
This study develops a semi-3D model based on a vertical-horizontal splitting (VHS) method to analyze the flow in open-channel bends. In horizontal, the surface elevation and depth-averaged velocity components are computed by 2D depth-averaged model. In vertical, assume the 3D velocity profile of Navier-Stokes equations is equal to the depth-averaged velocity plus the deviation of velocity profile, and then the vertical governing equations can be derived by subtracting the 2D depth-averaged equations to the 3D Navier-Stokes equations. In order to fit the complex geometry in both side wall and bed slope of channel, the orthogonal curvilinear coordinate system is used in horizontal gird, and sigma coordinate system is used in vertical grid. As for the numerical solution procedure, the two-step split-operator approach, which includes dispersion process (advection and diffusion terms) and propagation process (bed shear stress and pressure terms), is adopted to solve the 2D depth-averaged flow equations to improve the application flexibility. Implicit difference methods are adopted to relax the time step restriction allowing large time steps. Finally, three sets of experimental data including mildly curved, sharply curved and meandering channel are used to demonstrate the capability and accuracy of the semi-3D model, and the results of 2D depth-averaged model are also compared. The simulation results of semi-3D model show well agreement with experimental data considering different curved channels, bend lengths, secondary current and transverse mixing conditions.

碩士
國立高雄應用科技大學
電子工程系
99
In wireless sensor network, routing protocol is an important issue. Many geographic routing protocols have been proposed for three-dimensional (3D) wireless sensor networks. However, each of methods requires assistance from a global positioning system (GPS), which is not always available. In this thesis, we propose a method of constructing an axis-based virtual coordinate assignment in 3D wireless sensor networks (ABVCap3D) that requires no GPS assistance. The routing protocol based on ABVCap3D is also proposed, which guarantees packet delivery in 3D networks. Using simulations, we evaluate the performance of the proposed routing protocol (ABVCap3D routing), of routing based on axis-based virtual coordinate assignment protocol (ABVCap routing), of greedy-random-greedy routing (GRG routing), and of greedy-hull-greedy routing (GHG routing). Simulations show that ABVCap3D routing requires significantly relative lower cost for guaranteeing packet delivery in comparison with ABVCap routing. Simulations also demonstrate that ABVCap3D routing ensures a moderate ratio for routing path length to shortest (ideal) path length.

碩士
國立臺南大學
應用數學系碩士班
105
The purpose of the study is to assess whether the strategy of applying mathematical games in teaching cartesian coordinate plane and the graph of a linear system of two variables has positive learning effects on students and to explore if the mathematical games enhance retention effects . A Quasi-experimental method is adopted in the study. From one secondary school , two first grade classes were chosen to be participants . One of them was the experimental class and the other was the control class . According to the class average score of the the final examination , the class which had the similar average score with the experimental class was selected to be the control class . The research taught the experimental class by using mathematical games but another teacher taught the control class by adopting traditional pedagogy . After the experimental courses , a post-test in both classes was conducted and analyzed by the SPSS statistical software to have a clear picture of the effectiveness in the students’ learning . Two weeks after the post-test , a delayed- post-test was conducted in the experimental class . The post-test , and the delayed- post-test were analyzed by the SPSS statistical software to explore if the mathematical games enhanced retention effects . The findings of the study are as follows： 1.The result of the post-test revealed that the average score of the experimental class is higher then the score of the control class . There is a significant difference between them . It is assumed that the teaching startegy of mathematical games is better than traditional pedagogy . 2.The average score of the experimental class is higher in the delayed- post-test than in the post-test . It shows that there is not a significant difference between the post-test and the delayed- post-test . It is assumed that the teaching startegy of mathematical games helps improve the retention effects .

碩士
大同大學
機械工程學系(所)
99
In this research, a revised similar triangle algorithm based upon the pinhole imaging principle is proposed to conduct a 3D optical measurement system with a vertical moving camera, The algorithm can be applied to obtain 3D-information by using simple linear relationships, in the beginging ,the measurement results can’t achieve the desired value of z coordinate values. By using the tiangle algorithm , the relationship of the height of the object can be infered. This study focus on the datas which is collected from the position on the target that we interest in, at the mean time, the accuracy can be improved by using these datas to find the main factor that effecting the measuring result.

碩士
國立臺北科技大學
技術及職業教育研究所
107
The purpose of this study was to investigate the impact of playing tabletop games on students’ mathematics learning achievement for planar Cartesian coordinate system unit learning in junior high school. All participants were selected from two separate classes of the seventh grade students of a school in New Taipei City. This study adopted a quasi-experimental design of which the subjects were two seventh-grader classes, and sixty-one students in total. The experimental group students were taught with the method of collaborative mathematic game, and the control group students were taught with didactic teaching. The experimental teaching lasted for six weeks and the design contained pre- and post-tests. The research tools included the “mathematics learning accomplishment test”, “mathematics learning attitude scale”,and the “mathematics learning retention test” for quantitative statistics, and then the “course feedback form” was used for qualitative analysis to investigate the performance of students’ mathematics learning. This study explored the differences between the game's integration of mathematics teaching and didactic teaching methods on students’ learning achievement, and compared the differences in gender, mathematical ability, and parental education. The conclusions of the research were as follows: 1.The integration of games into mathematics teaching had made significant progress in students' learning attitude, and there was no significant difference in didactic teaching methods. The integration of games into mathematics teaching had not significantly improved students' learning accomplishment and learning retention as expected. 2.The integration of games into mathematics teaching had significantly improved the learning accomplishment of the “high-achieving” students, but there was no significant difference between the “middle achievers” and the “low achievers”. 3.The integration of games into mathematics teaching had made significant progress in the learning attitude of the students whose parents were " above the university " and " junior college " education level ; there was no obvious impact on students whose parents’ educational were “high school level below”. 4.Students’ love and identification of the game was integrated into mathematics teaching. They thought that the operations could help them to understand the concept. There was a positive and pleasant learning atmosphere in class, but it was necessary to pay attention to the fact that the repeated same game would bore the students. Finally, based on the results of this study, some suggestions for math teaching and further studies were proposed.

碩士
國立臺灣海洋大學
通訊與導航工程學系
103
In this study three webcams are applied to a 5DoF robotic arm system that is applied to smartphone testing operation. One of the cameras is used to recognize words and numbers from the control panel by the use of Optical Character Recognize (OCR) and pattern matching process; it is an indication or command sending to human's brain for decision. In here the computer is the robot's brain that receives the command and executes decision. Another two cameras are used for catching the left and right images for 3D coordinates of object, and they are similar to human's eyes that can tell the position of object. We can easily see that the robotic arm system can catch the 3D coordinates of object and perform testing operations by the command from visual recognition. In the first step, we need to process the calibration procedure and get the relative internal/external parameters by two webcams. Then the values from image plan can be compared and transformed to 3D coordinates by the Q matrix. The coordinates can be translated to 4096 precision values in robotic arm system. In here we also use the inverse kinematics and translation between pixels and distance in the real world to check the relative position and further to execute the tested smartphone functions. In control panel, for receiving command, we use another webcam to catch the message from the monitor of the control PC by the OCR and pattern matching process. The words of command can be obtained after image translation from RGB to HSL color space. The message will then be sent to robotic arm. Movements of the robotic arm are based on fuzzy logic theory that can drive the robot arm to the relative point and position of object. The robot will execute operations that are requested. The feedback values of arm movement are applied to correct the position error in real time.

This thesis didactically maps and describes new didactical mathematic environment and the possibilities of its application in teaching in the first grade of primary education. In this thesis I describe the process of an experiment conducted with pupils of the 1st - 5th year of elementary school, with whom I have tried and verified a series of new environment exercises. Based on the reflection of this experiment, and the analysis of pupils' solutions, I formulated a final version of these exercises. I led the process of the whole experiment in line with the constructivist educational style. In the theory part, I am mentioning the important terms of mathematics and its didactics, that are necessary for the purposes of this thesis, and that I refer to the relevant professional literature. This thesis also contains the analysis of chosen textbook collections that are related to its topic. The last part of this thesis is the final and complete series of exercises useful for teaching mathematics in the 1st grade of elementary school, especially if the teacher aims to purposefully develop the pupils' understanding of the terminology and relations from the mathematical areas of analytic geometry, functions and sequences.

Кравченко А. В. Аналіз методів розв’язання афінних та метричних задач : кваліфікаційна робота магістра спеціальності 111 "Математика" / наук. керівник Є. В. Стєганцев. Запоріжжя : ЗНУ, 2020. 43 с.
UA : Кваліфікаційна робота магістра : 43 с., 12 рис., 12 джерел. Об’єкт дослідження – афінні та метричні задачі. Мета роботи: дослідити ефективність методів розв’язання афінних та метричних задач. Метод дослідження – аналітичний. У кваліфікаційній роботі наведені традиційний, векторний, координатний методи розв’язання метричних та афінних задач. Досліджено, які методи є більш ефективними.
EN : The master’s qualifying paper : 43 pages, 12 figures, 12 references. The object of study: the affine and metrical problems. The aim of the work is the investigation of the efficiency of the methods of the solution of the affine and metrical problems. The method of the study is analytical. The diploma work deals with the traditional, vectorial and coordinate methods of the solution of the affine and the metrical problems. One studies which of the method is the most effective.

The diploma thesis deals with metric tasks in space, using synthetic and analytical geometry. In addition to explaining the different approaches, there is a set of examples to practice. The solution of the examples is part of the Portal of High School Mathematics (Portál středoškolské matematiky), where we can and analytical solutions, synthetic numerical solutions and synthetic constructional solutions.