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Complete Proceedings (17 Mb) more >>
| 1200 | Registration of participants | |
| 1300 | Lunch | |
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Chairpersons: Daniela Velichová, Miroslav Lávička |
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| 1700 |
Conference Opening |
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1710 |
Invited lecture |
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1750 |
Invited lecture |
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| 1830 | Dinner | |
1900 |
Welcome evening |
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| 800 | Breakfast | |
| Chairman: Pavel Pech | ||
| 900 | Invited lecture Jan Vršek |
Computing Projective Equivalences of Algebraic Varieties |
| 1000 | Coffee break | |
| 1030 | Miroslav Lávička | Approximate Reconstructions of Rational Cubics with Inexact Input |
| 1050 | Zbyněk Šír | Rational Curves of Given Direction |
| 1110 | Alexej Kolcun | Right-Triangulated Irregular Network Meshes |
| 1130 | Adéla Kostelecká | Konstrukce G1 spojitých ploch |
| 1200 | Lunch | |
| 800 | Breakfast | |
| Chairwoman: Daniela Velichová | ||
| 900 | Invited lecture Zlatan Magajna |
Automated Observation of Dynamic Geometric Constructions in School Geometry |
| 1000 | Coffee break | |
| 1050 | Hellmuth Stachel |
Movement of Conics on Quadrics |
| 1050 | Gunter Weiss |
Similarfaced and Equifaced Polyhedrons |
| 1110 | Emil Molnár | E5 › E2 Animation of Regular and Other Nice Solids with Visibility |
| 1130 | Monika Sroka-Bizoń | DIAD-Tools Project - Presentation of Didactic Materials to Support the Teaching of Construction Drawing |
| 1200 | Lunch | |
| 1300 | Excursion | |
1900 |
Conference Dinner |
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| 800 | Breakfast | |
| 900 | Slovak-Czech GeoGebra workshop | |
| Chairman: Roman Hašek | ||
| 900 | Daniela Velichová |
Výukové vídeá - produkty projektu DIAD-Tools |
| 920 | Jana Bulantová | Topografické plochy v GeoGebra |
| 940 | Šárka Voráčová | Slovník těles v GeoGebře |
| 1000 | Šárka Gergelitsová | Dělení příček a drobnosti v planimetrických konstrukcích |
| 1030 | Coffee break | |
| Chairwoman: Michaela Holešová | ||
| 1100 | Martin Billich | Geometrická interpretácia riešenia rovníc a ich sústav v prostredí programu GeoGebra |
| 1120 | Renáta Vágová | GeoGebra - prepojenie digitálneho a materiálneho sveta |
| 1140 | Martina Bátorová
Soňa Kudličková Alžbeta Mackovová |
Logické prvky v didaktickej výstavbe a praxi |
| 1200 | Roman Hašek | Řešení historických geometrických úloh pomocí počítače |
| 1220 | Martina Bímová | Studijní materiály v GeoGebře pro výuku pravoúhlé axonometrie |
| 1240 | Workshop discussions and closure | |
| 1300 | Lunch | |
| Departure | ||
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| VRŠEK Jan
" Computing Projective Equivalences of Algebraic Varieties " The talk is devoted to the investigation of the computation of projective (and other) equivalences of algebraic varieties. In particular, we will focus on the cases when the problem can be reduced to the detection of equivalences between finite sets on the projective line. The functionality of the designed method is presented for computing projective equivalences of rational curves, on determining projective equivalences of rational ruled surfaces, on the detection of affine transformations between planar curves, and on computing similarities between two implicitly given algebraic surfaces. When possible, symmetries of given shapes are also discussed as special cases. |
| MAGAJNA Zlatan " Automated Observation of Dynamic Geometric Constructions in School Geometry " It is very hard to predict the role of new technologies once they are introduced in school mathematics. Dynamic geometry systems are by no means an example of successful implementation of a new technology, for today they are widely used in schools all over the world. In last decade new types of software with potentials in school geometry came to light, in particular programs for automated proving and, less known, programs for automated observation. While dynamic geometry software is aimed at conceptual understanding and exploration in geometry, the other two types of programs are focused on proving and argumentation in geometry. We shall present the very basic principles of automatic proving in geometry and, in greater detail, OK Geometry, a software for automated observation of dynamic constructions. OK Geometry generates plausible hypotheses, but does not prove them. The research on the use of these programs in school setting (at any school level) is very limited as is their use in schools. These programs, in particular those for automated proving of geometry statements, introduce new paradigms in proving, thus it is hard to predict their trajectory in school mathematics. However, they raise important questions and dilemmas on concepts related to proofs and proving, ways of working out proofs, ways of of presenting proofs. |
| ÁBRAHÁMOVÁ Andrea, VAJSÁBLOVÁ Margita " Kinetic Curves in Cartographic Projections " The creation of map projections are highly variable in using various geometric objects. In this paper we will introduce the use of cycloids, epicycloids and evolvents in cartographic projections. Jervis´s cycloidal projection, which was published in 1895, originally uses cycloids as an image of parallels. August´s epicycloidal projection (1873) refines Lagrange´s circular projection (1779), whereby preserves the conformity of map projection. The outline meridians in August´s projection are displayed in an epicycloid called nephroid and the other meridians and parallels are displayed into the evolvents of epicycloids of the same type. The description of the constructability of the images of the points of the reference sphere in the Jervis´s and August´s projection and their geometric properties was also supplemented by a distortion analysis. The aim of comparing the scale distortion was to show the advantages and disadvantages of using cycloids and epicycloids in mentioned cartographic projections. Jervis´s cycloidal projection was compared with using circles in conical projection equidistant on the meridians. The use of epicycloids and evolvents in August´s projection we compared with an alternative to using circles in the Lagrange´s projection. |
| BILLICH Martin " Geometrická interpretácia riešenia rovníc a ich sústav v prostredí programu GeoGebra " V príspevku sa zameriame na niektoré možnosti použitia programu dynamickej geometrie pri vizualizácii pojmov z algebry, a to pri riešení rovníc prvého a druhého stupňa a ich sústav. Pre vybrané úlohy zo školskej matematiky ukážeme rôzne pohľady na geometrické či grafické hľadanie koreňov kvadratickej rovnice, prepojené s ich algebraickou podstatou. V procese riešenia úloh budú použité Vietove vzťahy pre korene kvadratickej rovnice, jedna z Euklidových viet, ako aj mocnosť bodu vzhľadom na kružnicu. |
| BÍMOVÁ Daniela " Studijní materiály v GeoGebře pro výuku pravoúhlé axonometrie " Orthogonal axonometry is a kind of a parallel projection that is illustrative for displaying relatively small 3D objects. On the contrary, the principles of constructions in the orthogonal axonometry are more complicated in comparison with e. g. the Monge projection. There are some texts concerning the basic principles and properties of the orthogonal axonometry. These texts contain mostly the static black and white figures that are not enough helpful for the students which spatial imagination is not developed on sufficient level. Consequently, the study text, containing the basic principles, properties of the orthogonal axonometry, the coloured illustrative figures, and the corresponding figures in the version for anaglyphic glasses, was written. The coloured figures as well as the figures in the version for anaglyphic glasses were generated from the dynamic applets created in GeoGebra. Using the dynamic applets together with the study text seems to be very helpful for the students during their study. |
| BLAŽEK Jiří " On One Remarkable Curve " The article deals about properties of a third degree curve. The curve is defined as a locus of foci of conics tangent to given quadrilateral. The properties are derived in synthetic fashion. |
| BULANTOVÁ Jana " Topografické plochy v Geogebra " Nové výukové materiály pro předmět Konstruktivní geometrie na FAST VUT vytvořené v programu Geogebra. |
| GERGELITSOVÁ Šárka " Dělení příček a drobnosti v planimetrických konstrukcích " Příspěvek je určen pro workshop programu GeoGebra (www.geogebra.org), který je součástí konference. Na příkladech úloh (nejen) z planimetrie v něm ukážeme některé nástroje a funkce programu GeoGebra, především z jeho prostředí pro rovinnou geometrii, které usnadní přípravu appletů pro výuku. Zvolené úlohy budou ilustrovat hledání řešení a jeho demonstraci v dynamickém prostředí. |
| HAŠEK Roman " Řešení historických geometrických úloh pomocí počítače " Příspěvek je určen pro workshop programu GeoGebra (www.geogebra.org), který je součástí konference. Jeho cílem je představit vybrané nástroje a funkce programu GeoGebra, především z jeho prostředí rovinné a prostorové geometrie a počítačové algebry, při jejich využití vhodném pro výuku geometrie. Bude tak učiněno prostřednictvím řešení konkrétních úloh z historických pramenů, týkajících se například trisekce úhlu či vztahů prostorových útvarů. Nástroje programu GeoGebra budou použity k dynamickému modelování úloh a k nalezení jejich řešení, způsobem, který otevírá cestu k daným problémům již žákům na základních a středních školách. |
| CHALMOVIANSKÝ Pavel " Some Examples of Intersection Computed Using Schubert Calculus " We recall a construction of Schubert cells and show on tractable examples how to use it for computation of intersection problems in algebraic geometry which look difficult to solve. Basic properties of Grassmannians which are intimately connected to the Schubert calculus is mentioned. |
| KMEŤOVÁ Marika " Discovering Plane Curves by GeoGebra " The contribution aims to be a motivation in geometry teaching in prospective mathematics teacher education showing different constructions of plane curves as loci of points in dynamic geometry environment. |
| KOLÁŘOVÁ Dana " Tensegrity - studentské modely od základních geometrických struktur po mosty " Příspěvek seznamuje s tensegritními strukturami, jejich využitím ve výuce deskriptivní geometrie na Fakultě architektury ČVUT v Praze. Poukazuje na využití problematiky pro zlepšení mezipředmětových vztahů. |
| KOLCUN Alexej " Right-Triangulated Irregular Network Meshes " The RTIN meshes are widely used for the terreain representation. We introduce this approach for CT-scan based triangulations. 3D extension of this method is proposed. |
| KOSTELECKÁ Adéla " Konstrukce G1 spojitých ploch " V této práci se věnujeme algoritmu, který na sebe nerozeznatelně navazuje Bézierovy plochy. Po provedení algoritmu mají tyto plochy na hranicích společný tečný prostor. Tuto metodu nazvanou Chiyokura Kimura použijeme na čtyřúhelníkové a trojúhelníkové Bézierovy plochy. Dále se zabýváme navazováním více trojúhelníkových ploch pomocí nahrazení řídících bodů racionálními funkcemi. Vzniknou tak tzv. Gregory plochy. Pro obě metody předvádíme důkaz, že tyto plochy navazují G1 spojitě. Na závěr prezentujeme výsledky algoritmu na nepravidelném dvacetistěnu a dalších reálných geometrických objektech, jako je Standford Bunny. |
| KREJČÍŘ Miroslav " Kompoziční geometrie gotiky " Architektonická díla byla v období gotiky navrhována pomocí geometrických rámců, které vycházely z praxí ověřených poměrů délek (šířek) a výšek stavebních konstrukcí. Proporce stavby, které se na realizovaných návrzích po statické a estetické stránce osvědčily, byly aplikovány i na návrhy pozdější, čímž se vymezovalo aktivní kompoziční tvarosloví. Při geometrizace se tak setkáváme s opakujícími se pravidly, vyplývajícími z principu čtverce (ad quadratum), rovnostranného trojúhelníka (ad triangulum) či výjimečně kružnice (ad circulum). |
| KUDLIČKOVÁ, Soňa, BÁTOROVÁ Martina, MACKOVOVÁ Alžbeta " Logické prvky v didaktickej výstavbe a praxi " V príspevku sa sústredíme na úlohu logiky vo výučbe deskriptívnej geometrie a na jednotlivé techniky dokazovania tvrdení. Stručne uvádzame základné typy dôkazov a ilustrujeme ich na ukážkach z planimetrie a stereometrie. Upozorňujeme na metodické a technické úskalia induktívnych a deduktívnych metód dôkazov, ponúkame niekoľko náhľadov do pojmotvorného procesu v deskriptívnej geometrii a prezentujeme viacero ilustračných príkladov vhodných na priame zaradenie do vyučovacieho procesu. |
| LÁVIČKA Miroslav " Approximate Reconstructions of Rational Cubics with Inexact Input " The talk is devoted to one selected problem from geometric modelling and related applications when exact symbolic computations are sometimes used also on objects given inexactly, i.e., when it is not adequately respected that numerical or input errors may significantly influence fundamental properties of considered algebraic varieties. We present a simple algorithm for an approximation of a non-rational planar cubic which is assumed to be a perturbation of some unknown rational planar cubic. The input curve is given by a perturbed polynomial or by perturbed points sampled from the original curve. The designed method will be presented on several commented examples. |
| MACKOVOVÁ Alžbeta " Voronoi Diagram in Hyperbolic Geometry " The construction and properties of the Voronoi diagram in Euclidean geometry have been studied and are known since P. Dirichlet. In this paper, we discuss the construction and properties of the Voronoi diagram in hyperbolic geometry which are studied recently. We work with the Poincaré disk model and focus on determining and llustrating the conditions that must be met for a Voronoi diagram for a given set of points to be non-degenerate, both in hyperbolic and Euclidean notions. |
| MAKOVNÍK Marcel " Refinement of Unorganized Point Sets via Quadric Fitting " In this paper we propose a refinement scheme working on sets consisting of points with corresponding normal vectors. Each point of the input set and its local neighbourhood is fitted by a quadric surface. Then, a set of points is selected from the surface and included in the refined set. Particularly, we focus on the construction of such sets with respect to the uniformity of the sampling of the refined point set. |
| MENCÁKOVÁ Kristýna, ŠAFAŘÍK Jan " Grafické znázornění řešení soustav lineárních diskrétních rovnic " Příspěvek obsahuje několik vybraných příkladů soustav lineárních diskrétních rovnic se zpožděním a grafické znázornění jejich řešení. Jedná se jednak o statické obrázky těchto řešení použité například pro články do časopisů. Dále jsou někerá řešení zobrazena v pohyblivých appletech vytvořených pomocí programu GeoGebra. |
| MOLNÁR Emil, PROK István, SZIRMAI Jenő " E5 › E2 Animation of Regular and Other Nice Solids with Visibility " In previous works (see [1], [2], [3]) the authors extended the method of central projection to higher dimensions, namely, for E4 › E2 projection from a one dimensional centre figure, together with a natural visibility algorithm. All these are presented in the linear algebraic machinery of real projective sphere PS4 or space P4(V5, V5, ~). In this presentation we further develop this method for E5 › E2 projection by the exterior (Grassmann – Clifford type) algebra (with scalar product) and implement on computer with other effects of illumination, e.g. for (regular and maybe other nice) polytopes on the base of the homepage http: //www.math.bme.hu/~prok The machinery is applicable for any d-dimensional projective space Pd and p-dimensional image. References [1] KATONA, J., MOLNÁR, E. Visibility of the higher-dimensional central projection into the projective sphere. Acta Math. Hungar. 123(3) (2009) 291–309. [2] KATONA, J., MOLNÁR, E., PROK, I. Visibility of the 4-dimensional regular solids, moving on the computer screen. Proc. 13th ICGG (Dresden, Germany, 2008). [3] KATONA, J., MOLNÁR, E., PROK, I., SZIRMAI, J. Higher-dimensional central projection into 2-plane with visibility and applications. Kragujevac Journal of Mathematics Volume 35 Number 2 (2011), 249–263. |
| PECH Pavel " Factorization of Some Locus Equations and Related Problems " By investigation of locus equations we sometimes encounter problems with factorization of resulting polynomials. In a few examples we will show how to behave in such cases. |
| PIRKLOVÁ Petra " Procvičování Mongeova promítání v programu GeoGebra " Monge projection, part of descriptive geometry, isn´t popular among students. There are a lot of topics for learning, on the contrary, time for learning is very short. Students very often complain for low number of exercises for practising the Monge projection. Above all most of the tasks are without solutions. But solution without construction is useless and knowledge of the construction of the solution of a problem is very important. Therefore, we created worksheets of with 133 problems concerning the Monge projection. The problems start at the basic level and end with the complex problems. Then gradual construction process (using a slider) and verbal or symbolic description of the construction were created for each problem in the dynamic 3D software GeoGebra. GeoGebra book containing solutions for all 133 problems of the worksheets was built |
| SROKA-BIZOŃ Monika " DIAD-Tools Project - Presentation of Didactic Materials to Support the Teaching of Construction Drawing " The international project under the Erasmus+ Programme “Development of Interactive and Animated Drawing Teaching Tools” - DIAD-tools No2017-1-LT01-KA202-035177 is realized by partners from Estonia, Latvia, Lithuania, Slovakia and Poland. The project implementation was started on October 1 2017 and will last until March 30, 2020. The main goal of the DIAD-tools project is to create interactive tools support the learning of technical drawing. After completing the project tools will be available on the online platform available for university´s students, college´s students, school´s and universities´ teachers from different countries. The project group from Silesian University of Technology has been elaborated didactic materials in the field of construction drawing. These materials have been divided in five parts: 1. Architectural and construction drawings - general principles, 2. Graphic designations of the building materials, 3. Dimensioning at civil engineering drawings, 4. Construction elements, 5. Construction drawings - examination of a construction project, reading of the drawings practical assignments. In the report authors will present problems in realization of the universal didactic materials in such field as construction drawing. |
| STACHEL Hellmuth " Movement of conics on quadrics " There are well-known examples of conics which can be moved on quadrics. Apart from the trivial case of circles on a sphere, paraboloids are surfaces of translation, even with a continuum of translational nets of parabolas. On quadrics of revolution, each planar section can be moved. Conversely, under rotation about an axis a, each conic c symmetric with respect to a plane through a, sweeps a portion of a quadric. What's about general quadrics Q? There is a three-parametric set of cutting planes, but the size of an ellipse or hyperbola depends only on its two semiaxes. This parameter count reveals that on each quadric there exist ellipses or hyperbolas with a one-parameter set of congruent copies on Q. We present parametrizations for such movements on ellipsoids and hyperboloids. There is a close connection beetween the movement of ellipses and hyperbolas on central quadrics and the theory of confocal quadrics. For studying the movement of parabolas, another strategy is necessary. Finally, we discuss infinitesimal motions of conics. |
| SURYNKOVÁ Petra " Removing Duplications from the List of Quadrilateral Meshes " Quadrilateral meshes are widely used in various practical applications such as computer graphics, numerical simulations, production industries and many more. We present the process of removing duplications from the list of quadrilateral meshes of a certain class. The list of quadrilateral meshes is obtained using our enumeration framework. We show the sketch of filtering the lists using unique numbering and demonstrate the process on concrete examples. |
| ŠÍR Zbyněk " Rational Curves of Given Direction " We describe all rational curves with a given tangent indicatrix. We analyze their degree, singular points and cusps. We also discuss various applications. |
| VÁGOVÁ Renáta " GeoGebra - prepojenie digitálneho a materiálneho sveta " Využívanie technológií sa v posledných desaťročiach vyvíjalo veľmi rýchlo a v súčasnosti ponúkajú nové príležitosti pre výučbu matematiky. Najmä integrácia dynamických geometrických softvérov s 3D tlačou nám ponúka možnosť transformovať digitálne reprezentácie trojrozmerných objektov do ich materiálnej fyzickej podoby. V tomto príspevku uvedieme stručný prehľad a ukážku nami navrhovaných pracovných listov pre žiakov v rámci tematického celku Rez kocky. Naším cieľom je vytvoriť materiály a navrhnúť také vyučovanie, ktoré vzájomne spája digitálne (virtuálne) a fyzické (reálne) zdroje vyučovania a učenia sa priestorovej geometrie. |
| VELICHOVÁ Daniela " Výukové vídeá - produkty projektu DIAD-Tools " V prezentácii uvedieme informácie o projekte DIAD-Tools programu Erazmus+ zameranom na vytvorenie výukových video-materiálov určených pre podporu predmetov Technické kreslenie, Deskriptívna geometria, Konštrukčná geometria, a pod. na stredných a vysokých školách technického zamerania. |
| VORÁČOVÁ Šárka " Slovník těles v Geogebře " Prezentace má za cíl představit nový soubor interaktivních appletů k výuce těles na základní a střední škole a podělit se s kolegy o skušenosti s jeho používáním. Náš digitální vzdělávací zdroj provází žáka stereometrií od úvodního seznámení s typy těles a základními pojmy až po procvičování aplikací výpočtu povrchu a objemu. Našim cílem bylo vytvořit materiály podporující kreativitu žáků a tím snad i přispět k cestě učitelů za heuristickou metodou výuky. |
| VRÁBLÍKOVÁ Jana " Discrete Connections on Triangular Meshes " In this talk we discuss constructing parallel tangent vector fields on discrete surfaces. We introduce analogies of notions from differential geometry for discrete surfaces, which we represent by triangular meshes, and we show how to use these concepts when constructing tangent vector fields that are parallel at the whole surface. At the end we describe algorithm for constructing these vector fields and show some examples. |
| WEISS Gunter " Similarfaced and Equifaced Polyhedrons " Properties of polyhedrons or polytopes are either of projective geometric or of affine nature or their proper place of action is a metric space. An example for a rather projective geometric property is a pair of two perspective k-simplices, which reveal additional “remarkable points”, c.f. (Weiss & Ebisui, 2017). Such perspective simplices can be interpreted as perspective projections of cross-polytopes in a higher-dimensional projective space. Thereby the coordinate field can be chosen arbitrarily with the only restriction that its characteristic is unequal 2, see (Weiss & Ebisui, 2017). Examples of affine geometric properties are e.g. affine regularity and the centroid of a convex polytope (understood as the set of vertices or as a solid), while the centroids of lower dimensional “skins” of a polytope, e.g. its edges or 2-faces, are metric properties. The concept “equifaced” is a matter of the considered metric. It might mean the equal content of d-faces of an n-polytope as well as congruent d-faced polytopes, (1?d Another (Euclidean or non-Euclidean) metric property of a polytope worthy to be mentioned are their different sets of altitudes: E.g. the set of “vertex altitudes” of a tetrahedron consists, in general, of four generators of a special regulus, see (Havlicek & Weiss, 2002), while the set of “edge altidudes” of a tetrahedron consists of three concurrent segments, see (Weiss & Havlicek, 2002). |
| ZAMBOJ Michal " 1-2-3-Spheres in the 4-Space " Recently, we have studied visualizations of the 4-space in the double orthogonal projection onto two mutually perpendicular 3-spaces. A studied object is projected onto its two conjugated images in the modeling 3-space – 3D graphics software in the computer screen. In this contribution, we use this method of projection to construct spherical, circle and point intersections of a 3-sphere with a 3-space, plane and line. The provided step-by-step constructions are created in the interactive software GeoGebra 3D and the participants may follow them online at https://www.geogebra.org/m/a8zxntdh |