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Complete Proceedings (17 Mb) more >>
| 1200 | Registration of participants | |
| 1300 | Lunch | |
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Chairpersons: Velichová, Lávička |
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| 1700 |
Symposium Opening |
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| 1710 |
Invited lecture |
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| 1800 | Dinner | |
1900 |
Welcome evening |
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| 800 | Breakfast | |
| Chairman: Lávička | ||
| 900 | Invited lecture Odehnal Boris |
Hermite-Interpolation of Ruled Surfaces and Canal Surfaces |
| 1000 | Coffee break | |
| 1030 | Stachel Hellmuth | Two Particular Quadratic Cones |
| 1050 | Weiss Gunter | Proofs by Visualisations Instead of Words |
| 1110 | Molnár Emil Szirmai Jenő |
Non-Euclidean Polyhedral Manifolds, Models and Visualization |
| 1130 | Notowidigdo Gennady | The Importance of Linear Algebra in Trigonometry |
| 1200 | Lunch | |
| 800 | Breakfast | |
| Chairman: Chalmovianský | ||
| 900 | Invited lecture Pech Pavel |
Investigation of Loci in Dynamic Geometric Environment |
| 1000 | Coffee break | |
| 1030 | Tomiczková Světlana | Geometrický software a výuka (nejen diferenciální) geometrie |
| 1050 | Kolcun Alexej Raunigr Petr |
Parametrická hladkosť pri plánovaní pohybu robota |
| 1110 | Bátorová Martina Kudličková Soňa |
Softvérová podpora výučby deskriptívnej geometrie a geometrického modelovania na FMFI UK |
| 1130 | Kudličková Soňa Mackovová Alžbeta Martina Bátorová |
Konštrukcie elipsy v interakcii s GeoGebrou |
| 1200 | Lunch | |
| 1300 | Trip to local pictoresque places - | Slovak glass museum in Lednické Rovne (with glass shop) Lednica Castle |
1900 |
Conference Dinner |
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| 800 | Breakfast | |
| Chairwoman: Vajsáblová | ||
| 900 | Králová Alice |
Zajímavé vlastnosti elipsy a hyperboly nad rámec běžného učivas |
| 920 | Ferdiánová Věra | Mascheroniho konstrukce |
| 940 | Šafařík Jan Slaběňáková Jana Sivčák Jozef | Výuka deskriptivni geometrie na Stavební fakultě VUT a nové studijní materiály vytvářené v dynamickém systému GeoGebra |
| 1000 | Coffee break | |
| 1030 | Holešová Michaela | Ovals in Technical Practice |
| 1050 | Plenary discussion | |
| 1110 | SLOVAK-CZECH GEOGEBRA WORKSHOP opening | |
| 1120 | Hašek Roman | Dynamická geometrie online |
| 1140 | Maťašovský Alexander, Visnyai Tomáš |
Možnosti zásuvného modulu GeoGebra systému Moodle |
| 1200 | Lunch | |
| 1300 | SLOVAK-CZECH GEOGEBRA WORKSHOP | |
| 1310 | Volná Jana Volný Petr |
CAS v GeoGebře |
| 1330 | Gergelitsová Šárka Holan Tomáš |
Jak rozpohybovat GeoGebru? |
| 1350 | Schreiberová Petra | Grafické znázornění dat v GeoGebře |
| 1410 | Morávková Zuzana | Základy skriptování v GeoGebře |
| 1430 | Paláček Radomír | Diferenciální a integrální počet v GeoGebře |
| 1450 | Čmelková Viera | GeoGebra a výučba zobrazovacích metód |
BÁTOROVÁ Martina, KUDLIČKOVÁ Soňa |
| BIZZARRI Michal, LÁVIČKA Miroslav, ŠÍR Zbyněk, VRŠEK Jan
" Linear Approach to Interpolations with Polynomial PN Surfaces. " We study polynomial surfaces which possess a polynomial area element which are equivalent to the PN surfaces. We show that for a rational surface the Gram determinant of its tangent space is a perfect square if and only if the Gram determinant of its normal space is a perfect square. Consequently the polynomial surfaces of a given degree with polynomial area element can be constructed from the prescribed normal fields solving a system of linear equations. The degree of the constructed surface depending on the degree and the properties of the prescribed normal field is investigated. We use the presented approach to interpolate a network of points and associated normals with piecewise polynomial surfaces with polynomial area element. |
| BLAŽKOVÁ Eva, ŠÍR Zbyněk " Multivalued Support Function at Inflection Points of Planar Curves " We study the behavior of the support function (see e.g. [1, 2]) in the neighborhood of a curve inflection. The gauss map at the inflection point is not regular and in the neighborhood is typically not injective. The support function is thus not regular and typically multivalued. We describe this function using an implicit algebraic equation and the rational Puiseux series of its branches. We show the correspondence between the degree of the approximation of the primary curve (using Taylor series) and the degree of the approximation of the support function (using Puiseux series). Based on this results we are able to approximate curve with inflections by curves with a simple support function which consequently possess rational offsets. We also analyze the approximation degree of this kind of dual approximation. In particular, we explain why the approximation order of the C^1 Hermite interpolation drops from 4 to 3 when an inflection occurs. Such a phenomenon was experienced e.g. when using segments of hypocycloids or epicycloids, see [3]. We propose an alternative adaptive subdivision scheme, which ensures the approximation degree 4 both for the inflection–free segments and the segments with inflections. References [1] Gruber, P.M., Wills, J.M.: Handbook of convex geometry. North–Holland, Amsterdam (1993). [2] Šír, Z., Gravesen, J., Jüttler, B.: Curves and surfaces represented by polynomial support functions. Theoretical Computer Science 392, 141–157 (2008). [3] Šír, Z., Bastl, B., Lávička, M.: Hermite interpolation by hypocycloids and epicycloids with rational offsets. Computer Aided Geometric Design 27, 405–417 (2010). |
| CHALMOVIANSKÝ Pavel " Local Intersection of Curves " We review some approaches for computation of intersection multiplicity of algebraic curves over complex number field. There are examples showing that the multiplicity of the intersection can be easily composed out of the intersection multiplicity of the corresponding curve and there are others where it is not so direct. The known algebraic description lacks geometric intuition or interpretation behind. We suggest certain explanations. |
| ČMELKOVÁ Viera " GeoGebra a výučba zobrazovacích metód " V príspevku si priblížime možnosti použitia GeoGebry pri výučbe zobrazovacích metód a geometrie. |
| FERDIÁNOVÁ Věra " Mascheroniho konstrukce " Příspěvek představí vybrané Mascheroniho konstrukce, při kterých se využívá pouze kružítko, jedná se o konstrukce s omezenými prostředky. Italský matematik Lorenzo Mascheroni (1750-1800) popsal tyto konstrukce ve své knize Geometria del compasso (1797). |
| GERGELITSOVÁ Šárka, Tomáš HOLAN " Jak rozpohybovat GeoGebru? " V příspěvku předvedeme applety vytvořené pro podporu prostorové představivosti, jejichž tvorba vyžaduje méně intuitivní nástroje. Ukážeme a okomentujeme tři vybrané postupy: 1) Přímé využití objektů GeoGebry a jejich vazby při využití tzv. seznamů, 2) Užití GeoGebra Scriptu (a jeho nevýhody) 3) Užití JavaScriptu |
| HAŠEK Roman " Dynamická geometrie online " Příspěvek je určen pro workshop programu GeoGebra, který je pořádán v rámci konference. Pojednává o nástrojích tohoto programu vhodných pro výuku geometrie. Zvláštní zřetel je při tom kladen na využití služeb webového prostředí, které je každému uživateli bezplatně k dispozici na portálu geogebra.org. Jedná se například o tvorbu a využití dynamických materiálů, strukturovaných online publikací, formování skupin uživatelů a online testování. |
| HOLEŠOVÁ Michaela " Ovals in Technical Practice " From the beginnings of architecture, we encounter elements and shapes that have a base in a circle or an ellipse. Very often, in technical practice, we meet the concept of oval. This term denotes a curve composed of circular arcs, but many times also the ellipse itself. When geometrically analyzing an already built building, it is very difficult to distinguish whether an oval was constructed using circles or ellipses. The quality of approximate constructions may be the reason. In the contribution, we will focus on some interesting constructions used by architects, theorists such as Sebastiano Serlio and Guarino Guarini. |
| KMEŤOVÁ Mária " Visualisation in Problem Solving " The contribution deals with various possibilities of visualisation of mathematical relationships in problem solving. A new viewpoint can give us new ideas and concepts that lead to a simplified solution of the mathematical problem. |
| KOLCUN Alexej, RAUNIGR Peter " Parametrická hladkosť pri plánovaní pohybu robota " V príspevku demonštrujeme ovládanie robota SPHERO nástrojami NURBS. Na jednoduchom príklade je prakticky ukázaná závislosť kvality výslednej trajektórie na zvolenej hladkosti použitých kriviek. |
| KOLOMAZNÍK Ivan, ČERVENKA František " Modelování ploch v technické praxi a jejich 3D tisk " Modelování těles v programech OpenScad a OnShape a jejich tisk na 3D tiskárně. |
| KRÁLOVÁ Alice " Zajímavé vlastnosti elipsy a hyperboly nad rámec běžného učiva " Konstrukce elipsy a hyperboly s využitím poměrové definice, odvození parametrických rovnic kuželoseček s osami v pootočené poloze vzhledem k osám x a y, využití těchto param. rovnic elipsy pro její konstrukci při zadaných bodech A,C,M. |
| KUDLIČKOVÁ Soňa, MACKOVOVÁ Alžbeta, BÁTOROVÁ Martina " Konštrukcie elipsy v interakcii s GeoGebrou " V príspevku sa zameriame na doplnenie klasických konštrukcií elipsy pomocou rysovacích potrieb o konštrukcie elipsy realizované v softvéri GeoGebra. Riešené úlohy využívajú interakciu s GeoGebrou, ide o vykreslenie elíps pri meniacich sa hodnotách vstupných dát. Príspevok je zameraný na inovatívne vyučovanie s využitím digitálnych technológií. |
| LÁVIČKA Miroslav, BIZZARI Michal, VRŠEK Jan " Rational Approximation of Square-Root Parameterizable Curves " We study situations when non-rational parameterizations of planar or space curves as results of certain geometric operations or constructions are obtained. We focus especially on such cases in which one can identify a~rational mapping which is a double cover of a rational curve. Hence, we deal with rational, elliptic or hyperelliptic curves that are birational to plane curves in the Weierstrass form and thus they are square-root parameterizable. We design a simple algorithm for computing an approximate (piecewise) rational parametrization using topological graphs of the Weierstrass curves. Predictable shapes reflecting a number of real roots of a univariate polynomial and a possibility to approximate easily the branches separately play a crucial role in the approximation algorithm. |
| MAŤAŠOVSKÝ Alexander, Tomáš VISNYAI " Možnosti zásuvného modulu GeoGebra systému Moodle " V tomto článku si ukážeme niekoľko možností využitia zásuvného modulu GeoGebra systému na riadenie výučby Moodle vo vyučovacom procese. |
| MOLNÁR Emil, SZIRMAI Jenő " Non-Euclidean Polyhedral Manifolds, Models and Visualization " As a byproduct of our recent papers [1], [2], and the previous initiative of the first author, we have recently found an infinite sequence of hyperbolic polyhedra Cw(2z, 2z, 2z) (6 =< 2z, 3 =< z odd integer) which can be equipped with a fixed point free face pairing, as a gluing procedure, so that the polyhedron become a compact hyperbolic manifold. That means each point has a ball-like neighbourhood. The visualization of such “finite Worlds” seems to be a timely task, and we try to involve our students as well. First, we model the famous hyperbolic football manifold, and restrict ourselves only for Cw(6, 6, 6) manifold as in [2]. |
| MORÁVKOVÁ Zuzana " Základy skriptování v GeoGebře " Seznámíme se se základy skriptování a ukážeme si, jak se skriptování dá využít při tvorbě studijního materiálu nebo ke zpestření přednášek. |
| NOTOWIDIGDO Gennady " The Importance of Linear Algebra in Trigonometry " Abstract : In this talk, I present the tools used in linear algebra that are applicable in the study of trigonometry, specifically Wildberger´s framework of Rational trigonometry. Starting with a re-formulation of metrical quantities in n-dimensional (affine) space using linear algebraic tools, we can then talk about visualising geometric objects in n-dimensional space. Marrying these two concepts together with the concept of a symmetric bilinear form, we can make significant progress with regards to the understanding of trigonometry in higher dimensions over a general metrical framework. We will pay specific attention to the three-dimensional space and the most important object in it: the tetrahedron. |
| ODEHNAL Boris " Hermite Interpolation with Ruled and Canal Surfaces " We show an algebraic way to interpolate Hermite data of ruled or canal surfaces. For that we construct rational (indeed polynomial) curves within Plücker’s quadric M42 and within Lie’s quadric L42 which are point models for the geometries of lines and spheres. The technique we use applies to both types of surfaces, because they can be represented as curves within the afore mentioned quadrics. The Bézier ansatz for a curve in either quadric involves some design parameters guiding the shape of the ruled or canal surfaces. These parameters are to be determined by solving a system of algebraic equations. The degrees of the equations admit a prediction of the number of possible solutions. Together with geometric criteria, useful solutions, i.e., solutions that meet practical requirements can be selected. Our main goal is the interpolation of Gk data at the boundaries of ruled surfaces and canal surfaces. Depending on k, the degree n of the curve in the Bézier ansatz has to be chosen: the higher k, the higher the degree of the ansatz. Nevertheless, we aim at low degree interpolants, and therefore, we choose the lowest possible n in any case. |
| PALÁČEK Radomír " Diferenciální a integrální počet v GeoGebře " V tomto příspěvku si ukážeme několik appletů vytvořených v programu GeoGebra, které lze využít ve výuce diferenciálního a integrálního počtu. Zaměříme se hlavně na derivace, Taylorův polynom a Riemannův integrál. |
| PECH Pavel " Investigation of Loci in Dynamic Geometric Environment " A classical problem in plane geometry consists of searching for the path of a point, that is subject to given constraints. Except for the most simple loci such as lines, circles or possibly conics, this topic is not contained in most geometry texts. The reason might be difficulties when visualizing various objects with different movements. The use of dynamic geometry software (Cabri, GeoGebra, Sketchpad,...) considerably facilitates the loci investigation. Whereas in the past the study of loci by DGS was based on numerical methods, now we are facing the introduction of symbolic methods based on the theory of automated theorem proving into DGS. The result is the implicit equation of the locus. In the talk a few concrete examples, including drawbacks which can occur in some cases, are given. "Vyšetrovanie množín bodov v dynamickom geometrickom prostredí" Klasický problém rovinnej geometrie je úloha nájsť dráhu pohybu bodu, ktorý je viazaný danou podmienkou. Okrem určovania najjednoduchších takýchto množín bodov, ako sú priamky, kružnice, či prípadne kužeľosečky, väčšina učebníc geometrie uvedenú tému neobsahuje. Príčinou môžu byť ťažkosti s vizualizáciou rozličných objektov podrobených rôznym pohybom. Použitie dynamických geometrických softvérov (Cabri, GeoGebra, Sketchpad,...) výrazne napomáha vyšetrovaniu spomínaných množín bodov. Kým v minulosti bolo štúdium takýchto množín bodov pomocou DGS postavené na numerických metódach, dnes sa stretávame s uvádzaním symbolických metód teórie automatického dokazovania tvrdení do DGS. Výsledkom je implicitná rovnica hľadanej množiny bodov. V prednáške bude uvedených niekoľko konkrétnych príkladov, včítane problémov, ktoré môžu vzniknúť v niektorých prípadoch. |
| ŠAFAŘÍK Jan, SLABĚŇÁKOVÁ Jana, SIVČÁK Jozef " Vyuka deskriptivni geometrie na Stavební fakultě VUT a nové studijní materiály vytvářené v dynamickém systému GeoGebra " Příspěvek má za cíl seznámit s historií a současností výuky deskriptivní geometrie na Stavební fakultě VUT v Brně a nové směry při vytváření vyukových materiálů, především za pomoci dynamického systému GeoGebra. |
| SCHREIBEROVÁ Petra " Grafické zobrazení dat v GeoGebře " Ukážeme si možnosti využití GeoGebry ve výuce statistiky. Seznámíme se nejen s nástroji a funkcemi pro tvorbu grafů potřebných pro přehledné zobrazení dat, ale i s hotovými pomůckami. |
| SROKA-BIZON Monika, Piotr POLINCEUSZ " Tensegrity structures - the idea and realization " The idea of tensigrity structures are very intersting. And the history of developing this idea is interesting either. Who is the original author of the concept? Master or pupil? The first tensegrity structure was constructed in 1948 by Kenneth Snelson, the young student of art. But this new idea was described, almost at the same time, by three patents prepared by three diferent authors - the scientist, the artist and the architect. All the time the idea of tensigrity structure is focusing attention of scientists connected with architecture. Nowadays new groups of scientists are looking for the best and precisly definition for tensigrity and they are trying to defined true or false tensigrity structures. Authors want to present the basic concepts of tensigrity structures which are described in the original patents and on the base of this present the realization of tensigrity structures which were realized in architecture. |
| STACHEL Hellmuth " Two Particular Quadratic Cones " The Euclidean geometry of quadratic cones is equivalent to the study of spherical conics. The normal or orthogonal quadratic cones have circular sections being orthogonal to vertex generators. These cones can be generated by congruent pencils of planes with intersecting axes. The corresponding conics are the spherical analogues of Thales circles. Equilateral quadratic cones are characterized by a vanishing trace. The associated equilateral spherical conics have the property that the three vertices of a regular right-angled spherical triangle can simultaneously move along. Dualization yields cones which are the envelopes of triples of mutually orthogonal planes. If cones of this type are tangent to a regular quadric then their apices are located on a sphere. This reveals that in general ellipsoids are still movable within a fixed circumscribed box. Literature: G. Glaeser, H. Stachel, B. Odehnal: The Universe of Conics. Springer Spektrum, Heidelberg 2016 G. Glaeser, H. Stachel, B. Odehnal: The Universe of Quadrics. Springer Spektrum, in preparation |
| TOMICZKOVÁ Světlana " Geometrický software a výuka (nejen diferenciální) geometrie " Možnosti využití geometrického software ve výuce geometrie a úskalí s tím spojená. |
| TYTKOWSKI Krzysztof " The Addition of Two Imaging Parallel Projection " In article „Dwuobrazowy rzut równoległy” (Two image parallel projection) A. Zawadzki, K. Bolek have submitted a new projection method. The given method is based on imaging of Euclidean space on a plane. A projection plane and two projection directions, not parallel to the plane, are at first assumed. Each point of space has two points assigned, that come from projections of the given point in the defined directions. The method does not enable restitution of the given point basing on its imaging neither application of metrical constructions. Addition of possibility to save the directions of projection broadens way of using the imaging. The satellite picture of the same object taken from two different directions may be treated as two-image parallel projection. Thank to the addition, basing on the image, it will be possible to figure out geometrical features of the photographed object. |
| VAJSÁBLOVÁ Margita " Geometric Tools in a Precision of Image Elements on Maps " Maps in analog and digital form affect every area of life. An actual problem affected by the precision of positioning geodetic points using new GNSS technologies in coordinate systems is the precision of the map projection. Map projections are coming out of geometric expression of properties of reference surfaces of Earth using methods of differential geometry, as well as of relation between two linear manifolds - reference ellipsoid and map plane. Choice of cartographic projection is determined by the geometrical characteristics of the territory and choice criteria for distortion of map elements. The aim of this paper is to show the role of geometry and mathematics in the cartography and different options for access to the distortions of the territory, such as optimization of extreme value of distortion, summing and integral criterion on area territory, in some case using criterion with the requirement of a minimum mean value of scale distortion in a given area. "Nástroje geometrie v spresňovaní obrazu prvkov na mapách": Mapy v analógovej a digitálnej forme zasahujú do každej oblasti života. Spresňovanie určenia bodov pomocou nových GNSS technológií je motiváciou k spresňovaniu zobrazenia mapových prvkov. Kartografické zobrazenia vychádzajú z geometrického vyjadrenia vlastností referenčnej plochy Zeme metódami diferenciálnej geometrie a zo vzťahu medzi dvomi lineárnymi varietami - referenčným elipsoidom a rovinou zobrazenia. Výber kartografického zobrazenia je determinovaný geometrickými vlastnosťami územia a voľbou kritéria na skreslenie mapových prvkov. Cieľom príspevku je ukázať úlohu geometrie a matematiky v kartografickom zobrazovaní, a tiež rôzne varianty prístupu k skresleniam na ploche územia, ako optimalizácia krajných hodnôt skreslenia, súčtové a integrálne kritérium na ploche územia, príp. kritérium s požiadavkou na minimalizáciu strednej kvadratickej hodnoty skreslenia na zobrazovanej ploche. |
| VELICHOVÁ Daniela " 3D Data Reconstruction " Information about results achieved in the project GEOCRIM - Reconstruction of real 3D dimensions and position of selected objects from the criminalistically relevant records taken by stable camera systems. |
| VOLNÁ Jana, VOLNÝ Petr " CAS v GeoGebře " Ukážeme si použití modulu symbolických manipulací CAS v GeoGebře. |
| VRŠEK Jan, LÁVIČKA Miroslav, ALCÁZAR J. G. " Hledání symetrií algebraických křivek " Symetrií křivky rozumíme izometrii Eukleidovské roviny, jež tuto křivku zobrazuje na sebe samu. Snadno se nahlédne, že grupa symetrií algebraické křivky je podgrupou symetrií nějakého mnohoúhelníku ( pokud křivka není tvořena rovnoběžnými přímkami, nebo soustřednými kružnicemi). Postačí nám tedy nalézt střed a směry os tohoto mnohoúhelníku. Problém vyřešíme nalezením nové, tzv. harmonické, křivky jejíž grupa symetrií bude snadno nalezitelná a úzce svázána s grupou symetrií křivky původní. |
| WEISS Gunter " Proofs by Visualisations Instead of Words " Some more or less well-known geometric theorems can be visualised such that their proofs are obvious. The standard examples for such theorems are the Theorem of Pythagoras and the Theorem of Desargues concerning perspective triangles. But there are many others, too. For example, the Apollonius definition of say an ellipse allows the construction of its tangents as angle bisectors of the “focal segments”. This construction is based on a mechanical interpretation of the Apollonius definition an E.W. v. Tschirnhaus used this idea for constructing tangents of curves defined by a (finite) set of focal points and constant weighted sum of distances from these focal points. But even the figures in his famous book “medicina mentis” from 1695 are quite clear to a geometer, the mathematician, who made the explaining footnotes to the German edition from 1963, did not understand them. Therefore, the question arises, what are the conditions which a figure should fulfil to visualise what an author aims at. As an answer it has to be stated that a proper visualisation is only one part, the other, more important, is the addressee. He/she must understand the problem, be creative enough to guess the idea of proof, have imagination, and have some experience in problem solving, too. Shortly said, he/she must be educated. Therefore, the lecture takes up the cudgels on behalf of a solid education in Geometry, including Descriptive Geometry. |
| ZAMBOJ Michal " On Methods of Synthetic Projective Geometry " In the contribution, we talk about synthetic methods in the projective extension of the real plane or three-dimensional space for solving problems of projective incidence and affine geometry. We use the concept of von Staudt´s "Wurf", defined in his Beiträge zur Geometrie der Lage, and derived property that cross-ratios are invariant under projective transformations. The concept of choosing an infinite hyperplane is used for making hypothesis in an affine space to solve projective problems and vice-versa. Their mixtures with the analytic use of homogenous coordinates is applied on projective theorems. An insight into the von Staudt´s constructions on the projective scale is given. The methods are shown on some examples in elementary planimetry and stereometry, on proofs of Menelaus´ and Ceva´s theorems and on applications of Pappus´s theorem. |