TITLE: Tools for 3D modelling in teaching solid geometry AUTHOR: Bc. Jana Taušová DEPARTMENT: Department of Information Technology and Technology Education SUPERVISOR: PhDr. Josef Procházka, Ph.D. ABSTRACT: The key topic of the thesis is the possibility of enhancement of the teaching of solid geometry through a 3D modelling program of choice. The theoretical part of the work defines solid geometry as a branch of mathematics. It furthermore investigates spatial visualization ability as a crucial one for understanding this particular area; and also explains the possibilities of computer program usage. The practical part starts with an inquiry into the use of ICT tools at secondary schools in the Czech Republic. Then there is an overview of the programs suitable for teaching, and methodology for a particular section of solid geometry curriculum. The suggestions mentioned in the methodology section have been used during class work (the 7th year students of an 8-year comprehensive grammar school), as specified in the last part of the work. Thus we have researched the actual practical use of our theoretical inferences. KEY WORDS: Tools for 3D modelling, ICT support for education, spatial visualization ability, solid geometry

Stuttgart, Univ., Diss., 2007. / Druckausg. bei Shaker, Aachen erschienen.

Universiẗat, Diss., 2004--Bonn.

Thema dieser Arbeit sind echtzeitfähige 3D-Renderingverfahren, die 3D-Geometrie mit über der Standarddarstellung hinausgehenden Qualitäts- und Gestaltungsmerkmalen rendern können. Beispiele sind Verfahren zur Darstellung von Schatten, Reflexionen oder Transparenz. Mit heutigen computergraphischen Software-Basissystemen ist ihre Integration in 3D-Anwendungssysteme sehr aufwändig: Dies liegt einerseits an der technischen, algorithmischen Komplexität der Einzelverfahren, andererseits an Ressourcenkonflikten und Seiteneffekten bei der Kombination mehrerer Verfahren. Szenengraphsysteme, intendiert als computergraphische Softwareschicht zur Abstraktion von der Graphikhardware, stellen derzeit keine Mechanismen zur Nutzung dieser Renderingverfahren zur Verfügung.<br><br> Ziel dieser Arbeit ist es, eine Software-Architektur für ein Szenengraphsystem zu konzipieren und umzusetzen, die echtzeitfähige 3D-Renderingverfahren als Komponenten modelliert und es damit erlaubt, diese Verfahren innerhalb des Szenengraphsystems für die Anwendungsentwicklung effektiv zu nutzen. Ein Entwickler, der ein solches Szenengraphsystem nutzt, steuert diese Komponenten durch Elemente in der Szenenbeschreibung an, die die sichtbare Wirkung eines Renderingverfahrens auf die Geometrie in der Szene angeben, aber keine Hinweise auf die algorithmische Implementierung des Verfahrens enthalten. Damit werden Renderingverfahren in 3D-Anwendungssystemen nutzbar, ohne dass ein Entwickler detaillierte Kenntnisse über sie benötigt, so dass der Aufwand für ihre Entwicklung drastisch reduziert wird.<br><br> Ein besonderer Augenmerk der Arbeit liegt darauf, auf diese Weise auch verschiedene Renderingverfahren in einer Szene kombiniert einsetzen zu können. Hierzu ist eine Unterteilung der Renderingverfahren in mehrere Kategorien erforderlich, die mit Hilfe unterschiedlicher Ansätze ausgewertet werden. Dies erlaubt die Abstimmung verschiedener Komponenten für Renderingverfahren und ihrer verwendeten Ressourcen.<br><br> Die Zusammenarbeit mehrerer Renderingverfahren hat dort ihre Grenzen, wo die Kombination von Renderingverfahren graphisch nicht sinnvoll ist oder fundamentale technische Beschränkungen der Verfahren eine gleichzeitige Verwendung unmöglich machen. Die in dieser Arbeit vorgestellte Software-Architektur kann diese Grenzen nicht verschieben, aber sie ermöglicht den gleichzeitigen Einsatz vieler Verfahren, bei denen eine Kombination aufgrund der hohen Komplexität der Implementierung bislang nicht erreicht wurde. Das Vermögen zur Zusammenarbeit ist dabei allerdings von der Art eines Einzelverfahrens abhängig: Verfahren zur Darstellung transparenter Geometrie beispielsweise erfordern bei der Kombination mit anderen Verfahren in der Regel vollständig neuentwickelte Renderingverfahren; entsprechende Komponenten für das Szenengraphsystem können daher nur eingeschränkt mit Komponenten für andere Renderingverfahren verwendet werden.<br><br> Das in dieser Arbeit entwickelte System integriert und kombiniert Verfahren zur Darstellung von Bumpmapping, verschiedene Schatten- und Reflexionsverfahren sowie bildbasiertes CSG-Rendering. Damit stehen wesentliche Renderingverfahren in einem Szenengraphsystem erstmalig komponentenbasiert und auf einem hohen Abstraktionsniveau zur Verfügung. Das System ist trotz des zusätzlichen Verwaltungsaufwandes in der Lage, die Renderingverfahren einzeln und in Kombination grundsätzlich in Echtzeit auszuführen. / This thesis is about real-time rendering algorithms that can render 3D-geometry with quality and design features beyond standard display. Examples include algorithms to render shadows, reflections, or transparency. Integrating these algorithms into 3D-applications using today’s rendering libraries for real-time computer graphics is exceedingly difficult: On the one hand, the rendering algorithms are technically and algorithmically complicated for their own, on the other hand, combining several algorithms causes resource conflicts and side effects that are very difficult to handle. Scene graph libraries, which intend to provide a software layer to abstract from computer graphics hardware, currently offer no mechanisms for using these rendering algorithms, either.<br><br> The objective of this thesis is to design and to implement a software architecture for a scene graph library that models real-time rendering algorithms as software components allowing an effective usage of these algorithms for 3D-application development within the scene graph library. An application developer using the scene graph library controls these components with elements in a scene description that describe the effect of a rendering algorithm for some geometry in the scene graph, but that do not contain hints about the actual implementation of the rendering algorithm. This allows for deploying rendering algorithms in 3D-applications even for application developers that do not have detailed knowledge about them. In this way, the complexity of development of rendering algorithms can be drastically reduced.<br><br> In particular, the thesis focuses on the feasibility of combining several rendering algorithms within a scene at the same time. This requires to classify rendering algorithms into different categories, which are, each, evaluated using different approaches. In this way, components for different rendering algorithms can collaborate and adjust their usage of common graphics resources.<br><br> The possibility of combining different rendering algorithms can be limited in several ways: The graphical result of the combination can be undefined, or fundamental technical restrictions can render it impossible to use two rendering algorithms at the same time. The software architecture described in this work is not able to remove these limitations, but it allows to combine a lot of different rendering algorithms that, until now, could not be combined due to the high complexities of the required implementation. The capability of collaboration, however, depends on the kind of rendering algorithm: For instance, algorithms for rendering transparent geometry can be combined with other algorithms only with a complete redesign of the algorithm. Therefore, components in the scene graph library for displaying transparency can be combined with components for other rendering algorithms in a limited way only.<br><br> The system developed in this work integrates and combines algorithms for displaying bump mapping, several variants of shadow and reflection algorithms, and image-based CSG algorithms. Hence, major rendering algorithms are available for the first time in a scene graph library as components with high abstraction level. Despite the required additional indirections and abstraction layers, the system, in principle, allows for using and combining the rendering algorithms in real-time.

CoordenaÃÃo de AperfeÃoamento de Pessoal de NÃvel Superior / It has been the Principle of Cavalieri , an effective axiom for demonstration areas calculation formulas and volumes , one math course content present throughout the Basic Education Brazil and always also found in external evaluations . The goal here is to show that the Cavalieri principle is very efficient and simple in the statement formulas areas of plane figures and volumes of solids , as this axiom simplifies the calculation of areas to measure segments and volumes in areas . Exposes the importance of such content in the basic training of students so that , we need to remedy the difficulties that they have , for about 30% of the race of Mathematics issues of external evidence is about geometry. Finally , we conclude that one should recognize the importance of this study for the construction of knowledge of the student and not try a ready-made thing , because only then it is more concrete learning. / Tem-se no PrincÃpio de Cavalieri, um axioma eficiente para demonstraÃÃo de fÃrmulas de cÃlculo de Ãreas e volumes, um conteÃdo da disciplina de MatemÃtica presente em todo o Ensino BÃsico do Brasil e sempre encontrado tambÃm nas avaliaÃÃes externas. O objetivo aqui à mostrar que o PrincÃpio de Cavalieri à muito eficiente e simples na demonstraÃÃo de fÃrmulas de Ãreas de figuras planas e volumes de sÃlidos, pois este axioma simplifica o cÃlculo de Ãreas ao medir segmentos e o de volumes em Ãreas. ExpÃe-se a importÃncia destes conteÃdos na formaÃÃo bÃsica dos alunos de maneira que, precisa-se sanar as dificuldades que eles tenham, pois cerca de 30% das questÃes de MatemÃtica das provas externas à sobre geometria. Por fim, conclui-se que se deve reconhecer a importÃncia desse estudo para construÃÃo do conhecimento do aluno e nÃo tentar uma coisa jà pronta, pois sà assim ele terà um aprendizado mais concreto.

The thesis deals with the representation of objects in the solid geometry. The first part contains a summary of theorems and definitions which are needed for understanding problems of mutual position of lines and planes, cut of objects with a plane and intersection of the object with the line. The second part contains a collection of worksheets for students. It has been created a solution for each worksheet. This solution can be used in printed version or in electronic version which is created in the software GeoGebra. This thesis can be used in tuition or self-studying.

Made available in DSpace on 2015-03-26T14:00:07Z (GMT). No. of bitstreams: 1 texto completo.pdf: 1123753 bytes, checksum: b46764bbd1d7d69bdb596dd568e0887d (MD5) Previous issue date: 2014-04-04 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / This work addresses the Euler s theorem for polyhedra. We present historical facts related to Euler s theorem, some proofs of the theorem and a notion of the Euler-Poincaré characteristic. We also present a material aimed for High School students, with a proof of Euler s theorem using only basic mathematics, discuss a version of the theorem for the plane and use Euler s theorem to show the existence of only five regular convex polyhedra. / Este trabalho aborda o Teorema de Euler para poliedros. Apresentamos fatos históricos relacionados ao Teorema de Euler, algumas demonstrações do teorema e uma noção da característica de Euler-Poincaré. Apresentamos também um material voltado para o Ensino Médio, com uma demonstração para o Teorema de Euler usando apenas Matemática básica, discutimos o caso plano do teorema e usamos o Teorema de Euler para mostrar a existência de apenas cinco poliedros convexos regulares.

This work deals with an efficient and robust technique of performing Boolean operations on polygonal models. Full robustness is achieved within an internal representation based on planes and BSP (binary space partitioning) trees, in which operations can be carried out exactly in mere fixed precision arithmetic. Necessary conversions from the usual representation to the inner one and back, including their consequences are analyzed in detail. The performance of the method is optimized by a localization scheme in the form of an adaptive octree. The resulting implementation RazeCSG is experimentally compared with implementations used in practice Carve and Maya, which are not fully robust. For large models, RazeCSG shows only twice lower performance in the worst case than Carve, and is at least 130 times faster than Maya.

A new approach to image-space hardware-based rendering of Constructive Solid Geometry (CSG) models is presented. The work is motivated by the evolving functionality and performance of computer graphics hardware. This work is also motivated by a specific industrial application --- interactive verification of five axis grinding machine tool programs. The goal is to minimise the amount of time required to render each frame in an animation or interactive application involving boolean combinations of three dimensional shapes. The Sequenced Convex Subtraction (SCS) algorithm utilises sequenced subtraction of convex objects for the purpose of interactive CSG rendering. Concave shapes must be decomposed into convex shapes for the purpose of rendering. The length of Permutation Embedding Sequences (PESs) used as subtraction sequences are shown to have a quadratic lower bound. In many situations shorter sequences can be used, in the best case linear. Approaches to s ubtraction sequence encoding are presented including the use of object-space overlap information. The implementation of the algorithm is experimentally shown to perform better on modern commodity graphics hardware than previously reported methods. This work also examines performance aspects of the SCS algorithm itself. Overall performance depends on hardware characteristics, the number and spatial arrangement of primitives, and the structure and boolean operators of the CSG tree.

This work aims at rendering 3D scene using ray tracing. It describes advantages and disadvantages of this technology and its alternation known as distributed ray tracing. Other part deals with method from different branch, which are closely related to distributed ray tracing - constructive solid geometry and procedural texturing.