CSGG 2022

Participants

Abstracts information

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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.

Jan Šafařík

Fakulta stavební, VUT v Brně