Dynamická učebnica deskriptívnej geometrie pre stavebné odbory

BEGANOVÁ Juliana, VAJSÁBLOVÁ Margita

Stavebná fakulta STU v Bratislave
Bratislava, SR
e-mail: juliana.beganova@stuba.sk, margita.vajsablova@stuba.sk

Dynamická učebnica deskriptívnej geometrie pre stavebné odbory
Digitálna forma učebných materiálov dáva študentovi efektívnejšiu možnosť osvojenia geometrických pojmov a konštrukcií. Vysokoškolská učebnica DESKRIPTÍVNA GEOMETRIA PRE STAVEBNÉ ODBORY bola publikovaná kolektívom autoriek v roku 2022 v elektronickej forme a je určená pre študentov študijných programov zameraných na staviteľstvo, architektúru, krajinné plánovanie, geodéziu a kartografiu. V príspevku ukážeme využitie rôznych foriem dynamických prezentácií geometrického učiva, ktoré zahrňujú teóriu a vizualizáciu pojmov a konštrukcií pomocou 2D obrázkov, stereoskopických obrázkov, animácií a videí s ohľadom na typ použitých softvérových prostredí a nástrojov. Uvedený prístup vo veľkej miere podporuje pochopenie a priestorovú predstavivosť študentov.

Dynamic university textbook of descriptive geometry for construction study programs

Contribution of a tangent of multiplicity one to the intersection multiplicity of two plane curves

BOSÁKOVÁ Adriana

Fakulta matematiky, fyziky a informatiky, Univerzita Komenského v Bratislave
Bratislava, SR
e-mail: adriana.bosakova@fmph.uniba.sk

Intersection multiplicity of two plane curves F and G at some point P has a well-known property of I_P (F,G)≥ mn, where m and n are the multiplicities of the point P on the curves F and G respectively. To each common tangent of F and G at P can be assigned a nonnegative integer, a number equal to its contribution to the intersection multiplicity. This can be done via local investigation methods. The sum of these contribution numbers of all common tangents is equal to the remainder R=I_P (F,G) − mn, We investigate the values of this contribution number for common tangents of the multiplicity 1.

Geometry of Schwarzschild spacetimes

BRAJERČÍK Ján

Katedra fyziky, matematiky a techniky
Fakulta humanitných a prírodných vied Prešovská univerzita v Prešove
Prešov, SR
e-mail: jan.brajercik@unipo.sk

The Schwarzschild spacetime is a basic model of the classical general relativity theory, describing the gravitational field outside a spherical mass. This contribution is devoted to its geometric theory. In a broader sense, a Schwarzschild spacetime is a smooth manifold, endowed with an action of the special orthogonal group SO(3) and a Schwarzschild metric, which is an SO(3)-invariant metric field, satisfying the Einstein equations. Explicit formulas of all Schwarzschild metrics on four-dimensional manifold Rx(R³\{(0,0,0)}) are introduced in a sense of spherical charts. To find these formulas we first classify all metrics on Rx(R³\{(0,0,0)}) invariant with respect to the action of SO(3), time-translations, and time-reflection.
The resulting family of Schwarzschild metrics is parametrized by a function and two real parameters, the integration constants. For any Schwarzschild metric, one of the parameters determines a submanifold, where the metric is not defined, the Schwarzschild sphere. In particular, the family admits a global metric whose Schwarzschild sphere is empty.
By the winding mapping of the real line R onto the circle S¹, these results are transferred to the manifold S¹x(R³\{(0,0,0)}) which topologically differs from Rx(R³\{(0,0,0)}). All our assertions are derived independently of the signature of the Schwarzschild metric; the signature can be chosen as an independent axiom.

VTOL propeller development in start-up environment

CIBULKA Jaroslav

České vysoké učení technické v Praze
Fakulta strojní, Ústav technické matematiky
Praha 2, ČR
e-mail: jaroslav.cibulka@fs.cvut.cz

City air mobility transport with vertical take-off and landing capabilities challenges engineers across development spectra. Fast prototyping of parametric propeller geometry will be introduced as well as start-up approach to new geometry requirements. Capabilities and deficiencies of open propeller geometry software will be compared to classical licensed software approach.
SLA 3D printer used as a suitable tool for basic geometry evaluation and aero-acoustic testing will be described.

Single-parameter mesh primitive counting formulas for subdivision surfaces

ČAVARGA Martin

Katedra Algebry a Geometrie FMFI UK
Bratislava, SR
e-mail: mcavarga@gmail.com

Methodology
Our approach begins with the formal definitions of a mesh, tessellation-changing operations, and subdivisions. Building on this foundation, we derive a new counting formula for Loop 4:1 subdivision in the context of triangle meshes. We extend this methodology to other subdivision techniques like Catmull-Clark.
                               

1. Formal Definitions: Explanation of core terms to set the foundation for the methodology.
2. Counting Formula for Loop 4:1: We derive a formula, supported by rigorous proof, for estimating vertex density in Loop 4:1 subdivisions of triangle meshes.
3. Catmull-Clark & Other Subdivisions: Extension of the methodology to other common subdivision techniques.

Results and Performance
We present numerical and empirical results to demonstrate the efficacy of our proposed method in estimating vertex density and mesh primitive counts. Our approach is crucial in problems like Lagrangian shrink-wrapping. We also evaluate the possibility of efficient memory pre-allocation.

Sliceforms in education of architecture students

ČEČÁKOVÁ Stanislava, KOLÁŘOVÁ Dana

Ústav nosných konstrukcí FA ČVUT v Praze
Praha 6, ČR
e-mail: cecaksta@fa.cvut.cz, kolarova@fa.cvut.cz

„Geometry is a part of mathematics that does not belong to mathematicians. It plays a role in all cultures in defining aesthetics from patterns through architecture to other aspects of day to day design. “
John Sharp

The presentation focuses on the use of sliceforms in the education of architecture students at CTU.

It shows not only historical models created via this method, but also the work of current students who used the latest technologies for their creation (CAD modelling, GeoGebra, laser cutting, 3D printing).

Sliceforms are useful for improving cross-subject relationships, most notably showing a close relationship between mathematics and descriptive geometry.

Exhibition of student’s models:
https://media.cvut.cz/cs/foto/20230517-slice-forms-fa

On multicriteria triangulations

DAS Manaswita

Comenius University
Bratislava, SR
e-mail: manaswita.das@fmph.uniba.sk

We survey the research of ´Multicriteria Optimized Triangulation´. This approach extends the classical Delaunay triangulation by incorporating optimization objectives such as edge lengths, angles, and even user-defined constraints. Multiple authors preferred stochastic methods mainly containing genetic optimization and edge flip. This approach proves useful in various applications, including computer graphics, mesh generation, and geographic information systems, where different factors need to be considered simultaneously for constructing accurate and adaptable geometric structures. Hence we will dive into the explorations done and results achieved in successive experiments afterwards, and propose the classification of various criteria. We present a unifying notation and discuss recent findings within the broader context of subgraphs of multicriteria triangulations of planar pointsets.

Ukázky konstrukcí oválů v ostravské architektuře 19. století

FERDIÁNOVÁ Věra, HOLEŠOVÁ Michaela

Katedra matematiky, Přírodovědecká fakulta, Ostravská univerzita
Ostrava, ČR
e-mail: vera.ferdianova@gmail.com

Stavebná fakulta Žilinskej univerzity v Žiline
Žilina, SR
e-mail: michaela.holesova@uniza.sk

Výuka geometrie, zejména planimetrie na středních školách je zaměřena na základní konstrukce a využití shodností a podobností v konstrukčních úlohách. V nepovinně volitelných seminářích nebo na technicky orientovaných školách se představují konstrukční techniky pro kuželosečky, ale konstrukce oválů bývá na pozadí. Studenti se s problematikou setkávají povětšinou až na vysoké škole. Příspěvek bude ilustrovat různé konstrukce oválů na původní technické dokumentaci budov v Moravské Ostravě 19. století a přiblížení této problematiky studentům.

Examples of ovals in 19th-century architecture in Ostrava

The teaching of geometry, especially planimetry, in high schools focuses on basic constructions and the use of congruences and similarities in construction problems. In optional seminars or in technically oriented schools, construction techniques for conics are introduced, but the construction of ovals tends to be in the background. Students are usually not exposed to the problem until they are at university. The paper will illustrate various constructions of ovals on the original technical documentation (blueprints) of buildings in 19^th century Moravian Ostrava, and it will show how the issue can be introduced to students.

Osobnosti slovenskej matematiky-inšpirácia a životné vzory pre budúce generácie

GUNČAGA Ján, TKAČIK Štefan

Pedagogická fakulta
Katedra didaktiky prírodovedných predmetov v primárnom vzdelávaní
UK v Bratislave
Bratislava, SR
e-mail: guncaga@fedu.uniba.sk

Pedagogická fakulta
Katolícka univerzita v Ružomberku
Ružomberok, SR
e-mail: stefan.tkacik@ku.sk

V rámci projektu KEGA č. 004KU-4/2022 "Osobnosti slovenskej matematiky II" pokračujeme v úspešnej edícií Osobností slovenskej matematiky. Projekt je zameraný na zachytenie života a diela významných osobností slovenskej matematiky v 20. storočí ich vlastnými slovami. V rámci príspevku chceme prezentovať spracované materiály o doc. RNDr. Daniele Velichovej, CSc. Je významnou osobnosťou v oblasti geometrie a geometrického modelovania útvarov. Venuje sa aj počítačovej grafike, e-learningu vo vyučovaní matematických disciplín. Jej príklad môže slúžiť ako inšpirácia pre mladých a začínajúcich vedcov a ukazuje, že aj matematika môže byť zaujímavou a nádhernou disciplínou prístupnou pre mladých.

On the use of Bézier bicubic surface for shape reconstruction

HLAVOVÁ Marta

Ústav technické matematiky FS ČVUT v Praze
Praha 2, ČR
e-mail: marta.hlavova@fs.cvut.cz

How to exactly interpolate quadrilateral grid of points? In this article, some possibilities of using a Bézier bicubic patch for interpolation surface modelling are presented.

Ames´s illusions

KOKTAVÁ Marie

Mendelova univerzita v Brně
Brno, ČR
e-mail: provazni@mendelu.cz

Last year, as usual, we participated in the Czech European Researchers´ Night at Mendelu. Since the theme was “With all the senses”, as geometers we focused on vision. We created two of Adelbert Ames Jr.´s most famous illusions, namely the Ames Room and the Ames Window. In this paper we will describe how these illusions work and how we recreated them.

Are natural fractals really fractals?

KOLCUN Alexej

Ostravská univerzita PřF, KIP
Ostrava, ČR
e-mail: alexej.kolcun@osu.cz

Self-similar structures attract our attention for a very long time. In the contribution, we will focus on constructions that are inspired by the Koch curve. We will point out the problematic nature of the BCM (Box Counting Method).

Triangulation of implicit surfaces with singularities

KORECOVÁ Kristína, CHALMOVIANSKÝ Pavel

Comenius University
Bratislava, SR
e-mail: pavel.chalmoviansky@fmph.uniba.sk

A zero set of a suitable function is visualized using marching triangle technique with adaptation by Gauss curvature. Such an approach cannot be used in presence of singularities at the surface. We consider a description of ADE classes of singularities on a surface. Using suitable parameters, a neighborhood of a singularity is triangulated and the result is glued to the triangulation of the regular part of the surface. The algorithm is able to deal with bounding polyhedra as well. Examples of the approach are shown in detail.

Parametrizace vybraných šroubových ploch a jejich zobrazení v programech GeoGebra a Maple

KRÁLOVÁ Alice

Lesnická a dřevařská fakulta, Mendelova univerzita v Brně
Ústav matematiky
Brno, ČR
e-mail: alice.kralova@mendelu.cz

V příspěvku ukáži odvození parametrických rovnic pro některé přímkové a cyklické šroubové plochy. Tyto parametrizace umožňují vytvořit vizualizace těchto ploch v programech GeoGebra a Maple prostřednictvím statických modelů nebo animací. Oba programy jsou porovnány z hlediska funkčnosti pro zobrazení ploch, náročnosti jejich použití a kvality grafických výstupů.

Geometrické modelovanie: Variačný princíp

KRUPKA Demeter

Lepage Reasearch Institute
Prešov, SR
e-mail: demeter.krupka@lepageri.eu;

Medzi aktuálne problémy súčasného výskumu v diferenciálnej geometrii patria jej aplikácie v teórii riadenia (regulácie) mechanických systémov. Prednáška je zameraná na úvod do geometrickej teórie riadenia (geometrická robotika).
Zahrnuje metódu modelovania matematických a fyzikálnych procesov a problémov stability v technických vedách pomocou variačných princípov. Sú tiež diskutované nové možnosti konštrukcie variačného popisu (pohybových) rovníc a možnosti jeho rozšírenia v geometrickej robotike.
Prednáška nevyžaduje špeciálne matematické znalosti.

Geometric modeling: Variational principle

The current research in differential geometry includes applications of geometric methods in control of the motion of mechanical systems. In this presentation we discuss elementary ideas of the geometric control theory (geometric robotics) and introduce a method of mathematical modeling of
processes and stability problems in physics and engineering using variational principles. New possibilities of introducing the control parameters and extensions of the variational description of equations of the motion are also discussed. The lecture does not require special mathematical knowledge.

Descriptive geometry in Slovenia, decline or oportunity

KUŠAR Domen

Faculty of Architecture
Ljubljana, Slovenia
e-mail: Domen.Kusar@fa.uni-lj.si

Descriptive geometry is a science that has been indispensable in all areas of technical education for many years.In addition to the way of representing spatial elements and their relationships on a two-dimensional medium, it also developed a spatial representation.We have been following the downward trend of spatial ability among students of the Faculty of Architecture of the University of Ljubljana since 1999.
With the advent of computers, especially computer graphics, the attitude towards descriptive geometry also changed.Since computers took over the presentation of space, it was no longer necessary. Therefore, the number of hours was reduced, and in many faculties, it was simply abolished or combined with other similar subjects.In Slovenia, this happened at many technical faculties.Part of the blame is also on the side of educators who did not understand the situation and the role that descriptive geometry can play and adapted the material to current needs or even upgraded it.
At the Faculty of Architecture of the University of Ljubljana, beside classical knowledge we included into the course computer programs and new teaching methods.Flexibility helped us to carry out the subject remotely without major problems during Covid.With the project of pilot updating the course using information and communication technology, we wanted to bring the course even closer to students and make it accessible at any time.However, it turned out that digitization also has its limitations, as pedagogical work also requires a personal approach and communication between the pedagogue/assistant and students.This means finding a balance between the use of modern techniques and "classical" pedagogical approaches with the aim of achieving the best possible knowledge and the greatest possible spatial representation that future architects will need in their work.

Euclidean symmetries of implicit surfaces in 3D

LÁVIČKA Miroslav

University of West Bohemia, Pilsen
Plzeň, ČR
e-mail: lavicka@kma.zcu.cz

We present a complete algorithm to compute the rotational, axial, reflectional and central symmetries of an algebraic surface defined by means of its implicit equation. The functionality of the proposed algorithm is illustrated with several examples.

Parametric CAD model of axial blood pumps components

LINKEOVÁ Ivana

České vysoké učení technické v Praze
Fakulta strojní Ústav technické matematiky
Praha 2, ČR
e-mail: ivana.linkeova@fs.cvut.cz

The paper presents the development of a fully parametric CAD model of axial blood pumps components in Grasshopper, a graphical algorithm editor integrated in Rhinoceros. Since scientific papers report very little (if any) geometric information about the pumps, a special reverse engineering process was used to develop the parametric CAD model.

Conchoids and subdivision

MAKOVNÍK Marcel

Katedra algebry a geometrie
Fakulta matematiky, fyziky a informatiky Univerzita Komenského
Bratislava, SR
e-mail: marcel.makovnik@fmph.uniba.sk

We discuss the refinement of a planar polyline using conchoids. For each segment of the polyline, we construct a conchoid, which interpolates its endpoints. This is achieved by choosing a feasible coordinate system, scaled by the input global parameter. Then, we choose new points from the interpolating conchoid, symmetrically to the horizontal axis. Afterwards, we transform the new pair of points to the original coordinate system. The refined polyline is obtained in the "corner-cutting" fashion, i. e. by joining the subsequent pairs of new points. The process of refinement may be applied repeatedly to achieve the desired level of detail. The proposed refinement scheme is approximating and non-linear. We provide several examples that demonstrate the behaviour of the refinement. Also, we inspect on the influence of the value of the global parameter. For the specific value, we obtain the well-known Chaikin´ s algorithm.

The Kárteszi point of a triangle by the absolute sine theorem of János Bolyai. To the Memory of my Mentor Professor Ferenc Kárteszi (1907-1989)

MOLNÁR Emil

Department of Geometry, Institute of Mathematics Budapest
University of Technology and Economics, HU
email: emolnar@math.bme.hu

Professor Ferenc Kárteszi was a charismatic personality of the Hungarian mathematics education and science (finite, non-Euclidean and descriptive geometry). He made presentation trips also in Slovakia. His didactical credo on the role of problem solving in mathematics education will be illustrated and reported here: Starting with an elementary geometrical problem how to get the solution, generalization for theory, application and further on.

Impact of surface continuity on 3D model

MÜLLER Jakub

ČVUT v Praze, Strojní fakulta
Ústav technické matematiky
Praha 6, ČR
e-mail: jakub.muller@fs.cvut.cz

In an engineering environment, continuity is one of the important concepts. Not only does it have a significant impact on the aesthetic aspect of products, which is an essential parameter for commercial success, but it is equally a very crucial parameter from a technological point of view.
The continuity of the geometry of 3D CAD models determines how smooth the transition between two adjacent surfaces of the model is. C2 continuity, i.e. continuity up to the second derivative, has become the standard for the vast majority of CAD software today.
This work focuses on increasing the degree of contiguous surfaces to C3 continuity using CAD software Rhinoceros 7 to improve the quality of the model for CNC processing.

Variations on Fregier´s Theorem

ODEHNAL Boris

University of Applied Arts Vienna
Vienna, Austria
e-mail: boris.odehnal@uni-ak.ac.at

Fregier´s theorem states: Choose a conic c and a point P on c. The hypothenuses of all right triangles with the right angle at P pass through a single point F, the Fregier point. The set of all Fregier points of a conic is called the Fregier conic. In fact, this is a theorem that belongs to projective geometry. We shall see that there is a close relation to generalized offsets of conics. Futher, we give some generalizations of Fregier´s result.

The family of uniform polyhedra

PABICH Bronek

XX LICEUM OGÓLNOKSZTŁĄCĄ KRAKÓW
Wyższa Szkoła Nauk Pedagogicznych
Wieliczka, Polska
e-mail: bronek.pabich@gmail.com

Few mathematicians in geometry deal with homogeneous polyhedra. Many of my friends, serious mathematicians find these polyhedrons difficult to create and do not see their use in other fields of knowledge. Hence, little interest in this issue.
However, these polyhedrons are so captivating and beautiful, they require so much hard thought on how to create cardboard models or 3D prints of them.
They were mostly created in the 20th century. They are somewhat analogous to what Archimedean polyhedra are to Platonic polyhedra, with the difference that we now allow faces to be non-convex polygons. There are 54 of them in total, but the last ones were discovered in the 1970s and are very complicated.
In my presentation, on the example of one of the uniform polyhedra, I will show the principle of their construction and I will show the ones whose models I made with my students.

3D printing as a teaching tool in mathematics teaching

Konstrukce racionálních čísel pomocí origami

PEŠKA Patrik, VITKOVÁ Lenka

Katedra algebry a geometrie
Přírodovědecká fakulta Univerzita Palackého v Olomouci
Olomouc, ČR
e-mail: patrik_peska@seznam.cz, lenka.vitkova@upol.cz

Potřeba rozdělit papír je snad nejzákladnějším problémem, se který se setkáváme při jakémkoli skládání. Větší nesnáze však přicházejí, má-li být výsledný produkt origami symetrický a vyžaduje tudíž dělení papíru na stejné části. Je nasnadě, že „půlit“ papír je velice jednoduché, ale setkáváme se i s problémem, jak zkonstruovat takové dělení, kdy jmenovatel výsledného hledaného zlomku není mocninou čísla 2. V prezentaci ukážeme, jak toho lze dosáhnout pomocí jednoduchých algoritmů.

[1]  Lang, J. Robert. Origami and Geometric Constructions [online]. 1996. Dostupné z:       ttps://langorigami.com/wp-content/uploads/2015/09/origami_constructions.pdf
[2]  Hort D., Rachůnek, J. (2003). Algebra1. VUP Olomouc.
[3]  Sekanina M. (1986). Geometrie I. SPN Praha.
[4]  Jukl M. (2006). Lineární algebra. Univerzita Palackého Olomouc.

Priamkové plochy - tlačené 3D modely vytvorené v prostredí Rhinoceros

RÜCKSCHLOSSOVÁ Tatiana

Stavebná fakulta STU v Bratislave
Bratislava, SR
e-mail: tatiana.ruckschlossova@stuba.sk

Priamkové plochy sú v stavebníctve a architektúre často využívané pre ich matematickú jednoduchosť a prispôsobivosť. Sú efektívne z hľadiska použiteľného materiálu a pevnosti konštrukcie. Poskytujú možnosť vytvárať jedinečné architektonické prvky, ktoré sú estetické a moderné. Študenti architektúry sú nielen s touto skupinou plôch oboznámení v rámci predmetu Deskriptívna geometria II. Vyučovací proces sme vhodne doplnili skupinou tlačených 3D modelov. V nadväznosti na predchádzajúce skúsenosti pri tvorbe takýchto modelov, naďalej testujeme možnosti ich vytvárania pomocou rôznych vhodných softvérov. V tomto prípade sme modelovanie priamok na týchto plochách reprezentovali pomocou nástroja “pipes“ v prostredí Rhinoceros v kombinácii s nadstavbou Grasshopper.

Selected geometric constructions and proofs

ŘADA Jakub

Mathematical Institute
Faculty of Mathematics and Physics
Charles University, Prague
Praha, ČR
e-mail: rada@karlin.mff.cuni.cz

Geometry has been put on the back burner in many countries, and calculus is used much more. We present some geometric proofs and constructions instead of using calculus. It will be focused on one of the many geometrical proofs of Pappus-Pascal´s theorem and the construction of an osculating circle of an ellipse.

Seven versions of de Casteljau algorithm

ŠÍR Zbyněk

Univerzita Karlova, MFF
Praha, ČR
e-mail: zbynek.sir@mff.cuni.cz

In memory of Paul de Casteljau (1930-2022) we will present various aspects of his famous algorithm. First, we will outline the original affine version, including a short presentation of the original French paper (de Casteljau 1959). Then we will analyze the projective version (Farin 1983) and will present two interpretations within the classical projective geometry. We will also study alternative versions of the rational algorithm based on the factorization of the curve denominator (Šír and Jüttler 2015). Finally, we will show the generalizations of the de Casteljau algorithm to the case of spline curves, complex curves a various surfaces.

Sedm tváři de Casteljau algoritmu

Inspirováni vzpomínkou na Paula de Casteljau (1930-2022) se budeme věnovat různým aspektům jeho slavného algoritmu. Nejprve připomeneme vlastnosti původní afinní verze včetně náhledu na původní článek (de Casteljau 1959).
Poté přestavíme některé z velkého množství zobecnění tohoto algoritmu.
Podrobně vysvětlíme jeho racionální verzi (Farin 1983) a vysvětlíme dva pohledy jak chápat jeho projektivní aspekty.
Představíme i jeho alternativní verze založené na rozkladu jmenovatele (Šír and Jüttler 2015). Dále ukážeme jeho zobecnění pro splajnové křivky, křivky v komplexním oboru a různé typy ploch.

Application of Euler path

VORÁČOVÁ Šárka

Fakulta dopravní ČVUT v Praze
Praha, ČR
e-mail: voracova@fd.cvut.cz

Graph theory is a valuable mathematical modeling tool with a wide variety of geometric connections. It deserves more attention in all types of education. Starting with Euler paths, graph coloring, and Hamilton icosian game pupils can learn the terms and methods important for future programming. This concept can be very useful in real-life applications, such as how to solve transportation problems. In the contribution, the Hamiltonian paths and Eulerian cycles are presented together with their connection to the vehicle routing problems.

On computation of symmetries of algebraic objects

VRŠEK Jan

Západočeská univerzita v Plzni
Plzeň, ČR
e-mail: vrsekjan@kma.zcu.cz

We will discuss the problem of computation of symmetries of algebraic objects. In many cases the direct approach can be simplified via a reduction to an easier problem. We will demponstate this on the case of symmetries of planar polynomial vector fields.

3D-Versions of theorems related to Miquel´s Theorem

WEISS Gunter

TU Dresden & TU Vienna
Vienna, Austria
e-mail: weissgunter@gmx.at

The elementary geometric Miquel theorem concerns a triangle ABC and points R,S,T on its sides, and it states that the circles Օ-ART, Օ-BRS, Օ-CST have a common point M, the Miquel point to these givens. For M there exists a two-parametric set of possibilities, such that there exists a one-parameter set of point triplets R,S.T to a given point M. Choosing R,S,T in special ways one receives the so-called “Beermat theorem”, the Brocard theorems, and the Simson-Wallace theorem as special cases of Miquel’s theorem. Remaining within Euclidean geometry we deal with 3D modifications of these theorems. It turns out that, while the Miquel theorem can be generalized to n-simplices, 3D versions of the Brocard theorems need some modifications. The 3D Simson-Wallace theorem based on Miquel’s theorem is different from the standard generalization (see e.g. P. Pech: Generalization of Simson–Wallace theorem: planar and spatial formulation, JGG 2023), but it connects properties of the 2D case with Brocard’s modifications.

Calculating on a parabola

ZAMBOJ Michal

Faculty of Education, Charles University
Praha 1, ČR
e-mail: michal.zamboj@pedf.cuni.cz

K. G. Ch. von Staudt described simple geometric constructions of arithmetic operations in his Beiträge zur Geometrie der Lage. We discuss a special case of a parabola in particular. Elementary and derived constructions of addition and multiplication are presented synthetically and analytically, and straightforward algebraic observations are interconnected with the deeper geometric properties of a parabola. We focus on constructions of arithmetic, geometric, and harmonic mean. Von Staudt´s constructions are also discussed in relation to the Matiyasevich-Stechkin parabola and Möbius´ parabolic nomogram.