Michael Barton

BCAM
Bilbao, Spain

Zoltan Kovacs

JKU
Linz, Austria

Ján Brajerčík

Prešovská univerzita
Prešov, Slovakia

Josef Mikeš

Univerzita Palackého v Olomouci
Olomouc, Czechia

Abstracts information

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Michael Barton

BCAM, Spain

5-axis flank CNC machining of free-form surfaces using custom-shaped tools – abstract

Computer numerically controlled (CNC) machining is the leading subtractive manufacturing technology and even though it is in use since decades, it is far from being fully solved and still offers a rich source of challenging problems in geometric computing and motion planning. While geometric modeling of free-form surfaces is a relatively easy task for a moderately experienced modeler, manufacturing (aka rationalization) and the related problems such as optimal tool selection and its motion planning are very difficult due to very complex nature of a general free-form surface.
In this talk, I will discuss our recent advances in rationalization of free-form surfaces in the context of 5-axis flank CNC machining. In particular, I will discuss initialization strategies for flank milling of free-form surface using curved tools and on an example of spiral bevel gears will demonstrate a more efficient variant of flank machining called double-flank.

Zoltan Kovacs

JKU, Austria

Stepwise discovery of geometrical knowledge in GeoGebra – abstract

GeoGebra, a well-known dynamic geometry software tool, has recently been extended to be able to help discover interesting geometric properties in a planar construction. To collect information on generally true properties like parallelism of lines, perpendicularity, equality of lengths of segments, collinearity or concyclicity of points, were already achievable by using the Discover tool or command in an experimental version of GeoGebra, "GeoGebra Discovery". The presentation introduces a further improvement of the Discover command, the "stepwise discovery" mode:
It supports discovering generally true statements "on the fly", that is, every new point being added to the construction will be checked by the program automatically. The presentation explains how stepwise discovery mode can be helpful in teaching planar geometry at secondary level, or in the researcher's work.

Ján Brajerčík

Prešovská univerzita v Prešove

Problems of Global Variational Geometry – abstract

The global variational geometry is a branch of mathematics, devoted to extremal problems on the frontiers of differential geometry, topology, global analysis, algebra, the calculus of variations, and mathematical physics. It generalizes classical calculus of variations, where underlying Euclidean spaces are replaced by smooth manifolds and fibered spaces, and Lagrange functions are replaced by Lagrange differential forms. The subject of global variational geometry is to study extremals of integral variational functionals for sections of fibered manifolds, corresponding differential equations, and objects invariant under transformations of underlying geometric structures. In the contribution, we introduce basic concepts concerning global variational geometry such as smooth manifold, fibered manifold, jet, Lagrangian, Euler-Lagrange equation. Examples of problems solved by methods of global variational geometry are discussed.

Josef Mikeš

Univerzita Palackého v Olomouci

The 190th anniversary of Janos Bolyai in Olomouc and non-Euclidean geometry – abstract

This year, we celebrate the 190th anniversary of János Bolyai's presence in Olomouc. He stayed in Olomouc for a short period (1832-1833) during his military service. His studies brought a complete concept of non-Euclidean and hyperbolic geometry (whose origin is also attributed to Lobachevski and Gauss) that had, and still have, an impact on mathematics in general and various physics processes. The generalization of his studies also leads to the development of the theory of relativity.

Daniela Velichová

Strojnícka fakulta, Slovenská technická univerzita v Bratislave

Pselical surfaces – abstract

Pselical surfaces form a specific group of surfaces that can be regarded as special group of two-axial surfaces of revolution, namely surfaces of Euler type. These surfaces can be generated by simultaneous revolution of a basic curve about two skew perpendicular axes in the space. Armiloid, one typical representative of this group of surfa, has been defined and presented by professor František Kadeřávek (1885–1961) in his paper Zovšeobecnění rotačních ploch printed in the scientific journal Věstník Královské české společnosti nauk (1939), třída matematicko-přírodovědná, no. 17, pp. 1-3. Armiloid can be generated by a composite movement of a basic curve (a circle or an ellipse), which is composed from two systems of central collinear transformations determined bycharacteristics, while their axes are in two skew and perpendicular lines in the projective space. Furthermore, determining elements of the two collineations are in a special relation. Centres of one system of these central collineations are located always on the axes of the other system of central collineations involved, while characteristics of the two systems are inverse real numbers. Surfaces generated by such generating principle form the group of pselical surfaces

Dana Kolářová, Stanislava Čečáková

Fakulta architektury, České vysoké učení technické v Praze

3D Modeling of Geometry Surfaces in Education of Future Architects – abstract

Příspěvek poukazuje na význam 3D modelování ve výuce deskriptivní geometrie oboru Architektrura a urbanismus. Změny v modelování ploch v průběhu století, od prvních modelů T. Oliviera (1793-1853) až po nejnovější technologie.

Bohuslava Bezstarostová, Marie Chodorová

Přírodovědecká fakulta, Univerzita Palackého v Olomouci

Tvorba nástrojů v GeoGebře při řešení střech – abstract

Cílem příspěvku je ukázat v GeoGebře tvorbu nástrojů, které mají efektivní využití při řešení střech nad zadaným půdorysem.

Hellmuth Stachel

Vienna University of Technology

Plücker's conoid and related quadrics – abstract

Plücker's conoid (cylindroid) C is a ruled surface of degree three with a finite double line. Using particular cylinder coordinates (r, φ, z), the conoid satisfies the equation z = c sin 2φ with constant c. Two properties of the cylindroid play a major role in the geometric literature:
1) The bisector of two skew lines l₁, l₂ in the Euclidean 3-space is an orthogonal hyperbolic paraboloid P. All generators of P are axes of hyperboloids of revolution H which pass through l₁ and l₂ . Conversely, the locus of pairs of skew lines l₁, l₂ for which a given orthogonal hyperbolic paraboloid P is the bisector, is a Plücker conoid C.
2) In spatial kinematics, Plücker's conoid C is well-known as the locus of axes l₂₁ of the relative screw motion for two wheels Σ₁ , Σ₂ which rotate about fixed skew axes l₁ and l₂ with constant. The axodes of the relative screw motion are hyperboloids of revolution H with mutual contact along l₂₁. The common surface normals along l₂₁ form an orthogonal hyperbolic paraboloid P which passes through the axes l₁ and l₂.
We discuss these two main properties. Though both deal with the same three families of surfaces, there seem to be no direct bridges between them.

Emil Molnár

Budapest University of Technology and Economics

Non-Euclidean Crystal Geometry To Honour of János BOLYAI on 220 Anniversary of His Birth – abstract

The new discovery, hyperbolic geometry H3 of János BOLYAI and Nikolay Ivanovich LOBACHEVSKY open also new directions in material sciences. Besides of 3-spaces of constant curvature: E3, S3, H3, other five homogeneous 3-spaces: S2×R, H2×R, Nil, ~SL2R, Sol (so-called Thurston geometries) come into considerations. In analogies of classical crystallography, ball-packing modells deserve investigations and yield interesting new results for comparing with the real crystals, extremal arrangements, and open problems.
joint work with my colleague Jenő Szirmai

Alice Králová

Lesnická a dřevařská fakulta, Mendelova univerzita v Brně

Parametrizace parabolického konoidu a jeho zobrazení v programech GeoGebra a Maple – abstract

Na příkladu parabolického konoidu jako jedné z ploch technické praxe je ukázán postup, jak lze odvodit jeho parametrické rovnice, díky čemuž je možné vytvořit jeho prostorový model v programech GeoGebra a Maple.

Gunter Weiss

TU Dresden & TU Vienna

Symmetry – a natural phenomenon causing homo sapiens to become a homo mathematicus – abstract

The term “symmetry” usually translates this Greek composed word as “equal measure” and interprets it as “Euclidean isometry”. Such a restrictive interpretation is much too narrow, and even it covers a wide field in Geometry and Mathematics it pays to have a broader view on it. Mathematically restricted it mainly deals with the infinite Euclidean plane or n-space and the symmetry groups of infinite or finite objects in such a space. It is global and absolute. In contrast to this, the detection of what is subsumed as “symmetry” acts locally on different abstraction levels of our natural (and artificial) environment. It means to discover a certain law and structure from a sample of objects in space and time. From a few footprints of an animal trail it is not far to imagine an infinite frieze and apply this insight to ornaments. The detection of structures and similarities requires abstraction abilities. Finally, these abstraction abilities can also be applied to men-created abstractions, as e.g. Geometry and Mathematics, but also other fields of science and art. From finding the same abstract structure within different realisations it sems natural to ask for all objects with that structure and to modify that structure to gain a bigger family of related objects. In other words, we deal with and handle “classification-problems”. The lecture will deal with some explicit examples to above mentioned topics, focussing on Geometry and esthetical aspects of different symmetry-levels.

Jan Vršek

Západočeská Univerzita v Plzni

Pythagorean-hodograph projections of spatial polynomial curves – abstract

Pythagorean-hodograph (PH) curves form a class of curves with special properties, which are widely used in applied geometry. In our talk, we will investigate the existence of projection of spatial polynomial curve onto a PH curve. To achieve this, we will build a vocabulary enabling to translate questions about polynomial PH curves into geometric questions about rational projective curves. The method will be applied afterwards to counting orthogonal an oblique PH projections of spatial cubics and quintics respectively.

Miroslav Lávička

University of West Bohemia

Exact and approximate symmetries of discrete curves – abstract

We present a new direct method for determining exact symmetries of planar discrete curves. The basic strategy is to decompose a given curve into a family of suitable components (simpler discrete curves) whose symmetries can be found more easily. We determine all rotational and reflectional symmetries if they exist. We then modify the formulated approach for perturbed discrete curves for which approximate symmetries are to be computed. We illustrate the functionality of the proposed algorithm with several examples.

Šárka Voráčová

Faculty of Transportation, CTU in Prague

Geometer’s Guide to the Olomouc (map) – abstract

Olomouc had always been among the most important cities of the Kingdom of Bohemia. With its convenient location, ancient university, and spiritual, cultural, and craft traditions Olomouc has been a natural center of Moravia. Baroque reconstruction after the Thirty Years' War gave it a new monumental appearance enclosed on the walled mound. The long, rich history of Olomouc is narrated by the ancient structures together with religious iconography and symbolism. The construction of cathedrals was often based on geometries intended to make the viewer see the world through mathematics and through this understanding, gain a better understanding of the divine. Ornamentation gives an imagery of grandeur and royalty, and with that it made a mark of its symbolic, functional, and even cultural value.

Klaudia Hamajová

Univerzita Komenského v Bratislave

Nonlinear Subdivision Schemes Using Conic Sections – abstract

Subdivision is a well-known and established method for generating smooth curves and surfaces from discrete data by repeated refinements. We describe a modifications of nonlinear method for constructing a sequence of refined polygons, which starts with a sequence of points and associated normals. The newly generated points are sampled from conics which approximate the given points and the corresponding normals according to the proposed refinement rules.

Michal Zamboj

Faculty of Education, Charles University

Higher-dimensional spaces arising from robot kinematice – abstract

In this paper, we look at some simple robot mechanisms dealing mainly with rotational movement. Our focus lies in visual interpretations of higher-dimensional configuration spaces and workspaces. Several examples of robot mechanisms will bring us to visualizations of Cartesian products of circles and spheres embedded in the four-dimensional Euclidean space.

Pavel Chalmoviansky

Faculty of Mathematics, Physics and Informatics, Comenius University

Famous intersection examples – abstract

We go through several examples of intersection theory which played an important role during the evolution of the field.

Barbora Sováková, Martin Plinta

Ostravská univerzita

Architektonické prvky a jejich využití ve výuce geometrie – abstract

Již v učebnici rýsování od Boženy Ritschlové-Vanečkové z roku 1934 se objevuje speciální kapitola ornamentálních příkladů. Příspěvek bude prioritně určen na prvky, jenž se vyskytují v architektuře Socialistického realismu, která je velmi typická pro město Havířov a porubskou část města Ostravy. S využitím softwaru GeoGebra představíme ukázky vhodné do výuky geometrie.

Věra Ferdiánová

Katedra matematiky, Přírodovědecká fakulta, Ostravské univerzity

Výuka geometrie ve Vietnamu – abstract

Na první pohled se může jevit, že výuka matematika na celém světě musí být obdobná. U některých oblastí tomu tak je, ale například výuka geometrie ve Vietnamu je velmi odlišná. Cílem příspěvku je představení několika zajímavých geometrických úloh a jejich řešení, jenž se učí v této zemi.

Zbyněk Šír

MFF UK

Polynomiality versus rationality of Pythagorean hodograph curves – abstract

We will fully describe the linear space of rational PH curves and identify the polynomial ones as their subspace.

Maria Trnková

University of California in Davis, USA; Charles University, Czech Republic

Approximating embedded surfaces by triangular meshes – abstract

We will talk about problems which arise when visualizing a surface embedded in R3. Such problems appear in numerous applications such as MRI or CT scans dat. Their goal is to replace a surface of a body by a mesh, a piecewise-flat triangulated surface, used in Computer Graphics. Many algorithms are written on the presentation of a surface as a mesh that approximate it. We describe desirable features of a mesh. Then we present a new algorithm of approximating a surface by a high quality mesh whose triangles are as close as possible to equilateral. The main advantage of our method is improved mesh quality guaranteed for smooth surfaces. The GradNormal algorithm generates a triangular mesh that gives a piecewise-differentiable approximation of F, with angles between 35.2 and 101.5 degrees. As the mesh size approaches 0, the mesh converges to F through surfaces that are isotopic to F. This is joint work with J.Hass.

Juliana Beganová

SvF STU Bratislava

Vytvorenie animácie v programe Geogebra – abstract

Cieľom príspevku je prezentácia postupu vytvorenia videa, v ktorom je zobrazený pohyb objektov v rôznych zobrazeniach deskriptívnej geometrie. Zobrazenia v deskriptívnej geometrii sú pre študentov náročné na priestorovú predstavivosť. Animačné videá podporujú videnie objektov v priestore a tým pomáhajú študentom lepšie pochopiť detaily konštrukcií. Grafický program Geogebra nám pomáha jednotlivé zobrazenia lepšie vysvetliť. Celý objekt je nakreslený v 3D prostredí programu Geogebra, kde je zadefinovaná samotná animácia. Video je potom vytvorené a upravené pomocou ďalších programov a vložené do prezentácií.

Pavel Pech

University of South Bohemia

A spatial generalization of Wallace--Simson theorem on four lines – abstract

Planar version of the Wallace--Simson theorem is as follows: Given a triangle ABC. The locus of the point P such that symmetrical images of P with respect to the sides lines of ABC are collinear is the circumcircle of the triangle ABC. Moreover, all the lines formed by symmetrical images of P intersect at one point, the orthocenter of ABC. 3D generalization of the Wallace--Simson theorem might be as follows: Given four straight lines which are parallel to a fixed plane. Determine the locus of the point P such that symmetrical images of P with respect to these four lines are coplanar. Surprisingly the locus of P is a cylinder of revolution with the axis which is perpendicular to the fixed plane. Moreover, all the planes formed by symmetrical images of the point P have a common line.

Roman Hašek

University of South Bohemia

Introduction to dynamic construction in GeoGebra – abstract

The contribution is intended for the GeoGebra workshop. It deals with the basic principles of creating planimetric geometric figures in the dynamic geometric environment of GeoGebra and preparing students for their correct implementation. The key feature of a dynamic geometric construction, which has an obvious didactic potential, is to preserve the geometrical properties given by the assignment of the task when its independent points are dragged. We are talking about the so-called robust construction. The practice of using dynamic geometric software (DGS) with school pupils as well as university students of mathematics education shows that the creation of robust constructions is not self-evident for many of them and they need to be prepared for it. Selected Euclidean constructions are a suitable means for this preparation. The contribution is primarily devoted to an illustrative demonstration of the creation and use of the environment for the realization of Euclidean constructions. The editing of the GeoGebra toolbar, the creation of online applets and their use in the GeoGebra Classroom environment will be presented. Real examples of the use of the GeoGebra program in the teaching of mathematics at primary school will also be shown.

Jaroslav Cibulka

Fakulta strojní, ČVUT v Praze

Parametrický 3D model kompresorové lopatky NASA rotor 37 – abstract

NASA rotor 37 je letadlový transsonický axiální kompresor sestavený ze 36 lopatek vytvořený a testovaný ve výzkumném středisku NASA Lewis Research Center v 70. letech. Geometrie lopatek a experimentální měření aerodynamických vlastností provedené v NASA se později staly základem pro vytvoření testovací úlohy pro ověření vlastností numerických metod v aerodynamice. Předmětem práce byla matematická analýza geometrie lopatky a následné použití vhodného softwaru pro sestrojení parametrického 3D modelu lopatky, včetně implementovaných geometrických modifikací 3D geometrie, vhodného jako testovací geometrie pro numerické analýzy.

Jakub Müller

Fakulta strojní, ČVUT v Praze

Analýza odchylky bodu od plochy – abstract

Program Rhinoceros patří mezi CAD softwary, které umožňují uživateli velmi precizní práci s modelem, zejména pak s jeho povrchem. Jedním z hlavních prvků analýzy povrchu je funkce ,,Point Deviation“ sloužící k určení odchylek mezi dvěma geometriemi, např. bodů měřených na reálně vyrobeném modelu od nominálního CAD modelu. Výsledkem je vizualizace odchylek měřených bodů od CAD modelu. Problémem této vizualizace je její nejednoznačnost způsobená nerozlišením kladné a záporné orientace odchylek a tím viditelné rozlišení na podříznutí, nebo přídavek materiálu.
Cílem skriptu, tvořeného v programovacím jazyku Python, je tak vytvoření vlastní funkce, schopné výpočtu jednotlivých odchylek bodů reálného modelu s následným určením jejich orientací na základu normál sledované plochy. Výsledkem je textový a grafický výstup zahrnující vizualizaci na CAD modelu a základní statistické hodnoty.

Zuzana Tereňová

SvF STU Bratislava

Geometria pre odbor Matematicko-počítačové modelovanie na SvF STU v Bratislave – abstract

Príspevok sa zameriava na predmet Geometria na Stavebnej fakulte STU v Bratislave, ktorý je určený pre študentov študijného odboru Matematicko-počítačové modelovanie a tomu je prispôsobený aj jeho obsah. Prepája sa v ňom klasická deskriptívna geometria s analytickým vyjadrením kriviek a plôch, čo umožňuje študentom zobraziť tieto krivky a plochy v nejakom matematickom softvéri, napr. v softvéri Mathematica. Analytické vyjadrenia sa týkajú kužeľosečiek a niektorých plôch technickej praxe a to priamkových nerozvinuteľných plôch, rotačných a skrutkových plôch.

Marta Hlavová, Ivana Linkeová

Fakulta strojní, ČVUT v Praze

Construction of interpolation B-spline curve for technical application – abstract

Interpolating B-spline curve of degree n is a piecewise polynomial curve passing through a set of given definition points. In our case, the interpolation curve of the fifth degree is constructed to respect the unit normal vector at each definition point. Definition points and normal vectors are obtained by means of real measurement on a coordinate measuring machine.

Ivana Linkeová, Marta Hlavová

Fakulta strojní, ČVUT v Praze

CTU freeform standard Pharaoh – abstract

The paper presents a new freeform standard – the CTU freeform standard Pharaoh, with a complex freeform geometry developed and manufactured at the Czech Technical University in Prague. The standard was developed for the purpose of research in the field of applications of mathematical-geometrical modelling in the design, CNC machining and coordinate measurement of engineering components with freeform surfaces.

Jan Šafařík

Fakulta stavební, VUT v Brně

Počátky kartografie – Od nejstarších památek po středověk – abstract

Kartografie je nauka o mapách. Lidstvo dovedlo vyjádřit svoje zeměpisné znalosti mapou mnohem dříve nežli písmem. Mapa a její tvorba vždy byla a je obrazem doby, v níž vznikla a ukazují nám myšlenkovou a technickou úroveň v jednotlivých obdobích dějin. Cílem tohoto příspěvku je seznámit s vývojem kartografie od nejstarších kartografických památek (Egypt, Čína a Amerika), starověkou řeckou a římskou kartografii, středověkou kartografii církevní a kartografii arabskou, počátky námořních map.

Tatiana Rückschlossová

Katedra matematiky a Deskriptívnej geometrie, SvF STU, Bratislava

Plochy technickej praxe – proces tvorby tlačených 3D modelov – abstract

Pod plochami technickej praxe rozumieme plochy (resp. ich časti), ktorých aplikácie nachádzame v technických odvetviach ako napr. v stavebníctve, strojníctve, architektúre a dizajne. Z geometrického hľadiska sem patria plochy rotačné, skrutkové či priamkové. V záujme ešte viac sprístupniť študentom poznatky a názornú predstavu o týchto typoch plôch a ich aplikáciách, doplnili sme statické a interaktívne grafické modely sériou tlačených 3D modelov. Príspevok popisuje proces ich modelovania pomocou softvéru a následné spracovanie a úskalia s tým spojené v záujme optimálnej 3D tlače. Vytlačené 3D modely slúžia ako jedna z vizuálnych didaktických pomôcok pri výuke predmetov, ktoré sú zamerané na geometriu či počítačovú grafiku.

Margita Vajsáblová

Faculty of Civil Engineering, Slovak University of Technology in Bratislava

Vedutes and their geometric character – abstract

Vedutes of cities, castles and military fortresses are a frequent part of the historical maps, however, they also form separate works that have an artistic, cartographic, but also geometrical character. From the point of view of archaeological research, vedutes have an important informative character documenting objects of the past times. In the contribution, we will present samples of the analysis of the vedutes of defensive fortresses on the territory of Slovakia, from the point of view of the correctness of the use of imaging methods to express geometric elements.

Michal Bizzarri

ZČU

A note on spherical macro-elements and polynomial surfaces with Pythagorean normals – abstract

Various interpolation/approximation methods arising in different practical applications deal, at a particular step, with the problem of computing suitable rational patches (of low degree) on the unit sphere. Therefore, we are concerned with the construction of a system of spherical triangular patches with prescribed vertices that meet with a C⁰ continuity along their common pairwise edges. We give some scenarios leading to a reduction in the degree of the whole spherical macro-element. We in turn use these spherical patches to construct polynomial surfaces with Pythagorean normals.

Ilja Velich

Slovenská spoločnosť pre geometriu a grafiku

Jana Bulantová

Fakulta stavební, VUT v Brně

Lucie Zrůstová

Fakulta stavební, VUT v Brně

Anne Weiss-Torjussen

Dagmar Szarková

SSGG - Slovenská spoločnosť pre Geometriu a Grafiku, Bratislava

Marie Koktavá

Mendelova univerzita v Brně

Daniela Richtárková

SjF STU, SSGG

Organizing committee