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Space curves with Pythagorean Hodograph
Schneider, Petr ; Šír, Zbyněk (advisor) ; Bizzarri, Michal (referee)
The works discusses the basics of motions along curves in 3D space and especially those that are both rational and whose frames have the least rotation during their movement. These rotation minimizing rational frames open up avenues in both computer modelling and adjecent fields. We also introduce an alternative way of defining what a Pythagorean hodograph is, which could allow a new path for research. 1
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Tessellation of trimmed NURBS surfaces.
Safko, Martin ; Šír, Zbyněk (advisor) ; Pelikán, Josef (referee)
Tessellation of trimmed NURBS surfaces is classical problem in CAD/CAM, with long history and huge amount of research developed so far. We present and describe a tessellation algorithm suitable for visualization purposes in either offline or online setting and present our results. We also provide pointers to litera- ture and to tessellation algorithms for simulation. We discuss relevant definitions and procedures necessary to work with CAD data and try to familiarize it for people outside the industry. 1
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Tessellation of trimmed NURBS surfaces.
Safko, Martin ; Šír, Zbyněk (advisor) ; Pelikán, Josef (referee)
Tessellation of trimmed NURBS surfaces is classical problem in CAD/CAM, with long history and huge amount of research developed so far. We present and describe a tessellation algorithm suitable for visualization purposes in either offline or online setting and present our results. We also provide pointers to litera- ture and to tessellation algorithms for simulation. We discuss relevant definitions and procedures necessary to work with CAD data and try to familiarize it for people outside the industry. 1
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Subdivision curves and their convergence
Molnár, Michal ; Šír, Zbyněk (advisor) ; Surynková, Petra (referee)
In this thesis we summarize the basic terms of subdivision and accompany them with examples of widely used schemes. We also show our own implementations of those schemes. In further parts we define the so-called Exponential scheme, which will be a subject of our research, and we illustrate it's behavior using our own images. We focus on properties of curves this scheme produces. As we will see, those curves are C1 -continuous. We also sum up the terms necessary for our research of subdivision curves, which will form the basis of our proof justifying said property of Exponential scheme. 1
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Selected problems in differential geometry
Paclt, Jan ; Slavík, Antonín (advisor) ; Šír, Zbyněk (referee)
This thesis covers an overview and solution to selected problems in the differential geometry of plane curves. It focuses mainly on calculating areas of regions bounded by plane curves and also on evolutes, involutes and related trochoids and their properties. The work also provides a self-contained theoretical introduction to the differential geome- try of plane curves. Some ideas and mathematical derivations obtained from the original publications were further expanded and generalized by the author. All derivations men- tioned in the work are given in a uniform convention, which should make it easier for the reader to orientate and find relations between the topics discussed. The thesis can be used as a study support for students of bachelor's courses in geometry or specifically for students with a focus on descriptive geometry. 1
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Important theorems of affine geometry
Kundratová, Lucie ; Šír, Zbyněk (advisor) ; Halas, Zdeněk (referee)
V této práci se seznámíme s pojmy a nástroji afinní geometrie. To jsou zejména barycentrické souřadnice, afinní zobrazení a dělicí poměr. Ty pak použijeme k dokázání afinních vět. K Menelaově větě uvidíme vícero důkazů. Dále ukážeme Menelaovu větu z Cevovy a naopak. Uvedeme projektivní tvrzení, ze kterých obě věty plynou. Nakonec se budeme zabývat překladem Menelaových Sférik z latiny a spletitou historií Menelaovy věty. 1
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Websites for teaching of geometric transformations at secondary school
Dobiášová, Kateřina ; Robová, Jarmila (advisor) ; Šír, Zbyněk (referee)
As a part of this thesis existing web pages about planar geometrical transformations were evaluated and new web pages were created. Pages in Czech and in English language were evaluated shortly verbally. There is a table summarizing findings about existing web pages at the end of this part. The second part of this thesis contains newly created web pages. Scope of subject matter of created web pages overreaches common secondary school curriculum. There are Java applets used for illustrating findings about geometrical transformations as an integral part of web pages. Solving of exercises is accompanied by step-by-step operation. Web pages contain definitions, theorems with proofs and explanations of constructions.
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Generování a optimalizace meshů
Mokriš, Dominik ; Šír, Zbyněk (advisor) ; Hron, Jaroslav (referee)
This thesis is devoted to the problem of finding a suitable geometrical de- scription of the domain for the Finite Element Method (FEM). We present the most important methods used in generation and improvement of unstructured triangular meshes (grids) for two dimensional FEM. Possible measures of mesh quality are discussed with respect to their usage in linear Lagrange FEM. The relationship between mesh geometry (especially angles of particular triangles), discretization error and stiffness matrix condition number is examined. Two methods of mesh improvement, based on Centroidal Voronoi Tessellations (CVT) and Optimal Delaunay Triangulations (ODT), are discussed in detail and some results on convergence of CVT based methods are reviewed. Some aspects of these methods, e.g. the relation between density of boundary points and interior mesh vertices and the treatment of the boundary triangles is reconsidered in a new way. We have implemented these two methods and we discuss possible im- provements and new algorithms. A geometrically very interesting idea of recent alternative to FEM, Isogeometric Analysis (IGA), is outlined and demonstrated on a simple example. Several numerical tests are made in order to the compare the accuracy of solutions of isotropic PDEs obtained by FEM on bad mesh, mesh improved...
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Rational minimal surfaces
Bekrová, Martina ; Šír, Zbyněk (advisor) ; Šmíd, Dalibor (referee)
This bachelor thesis deals with rational surfaces with rational offsets and minimal surfaces. We will give a connection between these two classes of surfaces. We will introduce a method of finding all rational surfaces with rational offsets using dual representation of surface as an envelope of its own tangent surfaces. A connection will be established between minimal surfaces and functions of a complex variable. Furthermore, we will derive the known Weierstrass-Enneper representation and its modifications for generating minimal surfaces. By means of these two tools we will show that all rational minimal surfaces obtained from the Weierstrass-Enneper representation also have rational offsets. Powered by TCPDF (www.tcpdf.org)
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