ZBORNÍK SYMPÓZIA
O POČÍTAČOVEJ GEOMETRII
SCG´2012
PROCEEDINGS OF SYMPOSIUM
ON COMPUTER GEOMETRY
SCG´2012
Volume 21
Slovak Society for Geometry and Graphics
Mechanical Engineering Faculty
Civil Engineering Faculty
Slovak University of Technology in Bratislava
October 2012, Kočovce, SR
ISBN 978-80-227-3798-2
Contents
Abstracts
On Singularities of Evolutes of Curves in the
Pseudo-Euclidean Plane
Viktória Bakurová
KAGDM FMFI UK, Mlynská dolina, 842 48 Bratislava, SR
1e-mail: martina.bakurova
fmph.uniba.sk
Abstract. The aim of the paper is to classify singular points of arbitrary order of evolutes of
curves in the pseudo-Euclidean plane. We point out a close connection of such singularities to
higher-order vertices of base curves.
Keywords: pseudo-Euclidean plane, curve, singular point, evolute
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Deformations of Hyperelliptic Curves of Genus 2
Martina Bátorová1, Pavel Chalmovianský2
KAGDM FMFI UK, Mlynská dolina, 842 48 Bratislava, SR
1e-mail: viktoria.batorovafmfi.uniba.sk, 2e-mail: pavel.chalmovianskyfmfi.uniba.sk
Abstract: This paper gives a basic overview of hyperelliptic curves of genus 2 over the field of
complex numbers. We subject these curves to a series of deformations and present some ideas
about the impact of these transformations on the process of resolution of their singularities.
Keywords: hyperelliptic curve, parametrization, deformation, singularity, resolution, blowup
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Lineárna kombinácia a rovinné krivky v programe GeoGebra
Martin Billich
Pedagogická fakulta KU v Ružomberku, Hrabovská cesta 1, 034 01 Ružomberok,
SR
e-mail: billichku.sk
Abstrakt: Veľmi rýchle tempo vývoja programu GeoGebra robia tento softvér dynamickej
geometrie komplexnejším a následne medzi jeho aktívnymi používateľmi aj čoraz viac
obľúbenejším. V tomto článku sa budeme zaoberať možnosťami vizualiácie lineárnej
kombinácie rovníc rovinných kriviek, ktoré poskytuje najnovšia verzia programu GeoGebra
Kľúčové slová: GeoGebra, lineárna kombinácia, rovinné krivky
Summary. Dynamic Geometric Software GeoGebra can be one of the most powerful tool for
teaching and demonstrating many topics in geometry, algebra and calculus. We have shown
some facts from the theory of linear combination of functions representing plane curves. In
this paper we used GeoGebra to visualize linear and quadratic plane curves, and their linear
combinations.
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On Singularities of Arbitrary Order
of Planar Pedal Curves
Miloš Božek1, Gabriel Foltán2
Comenius University, Faculty of Mathematics, Physics and Informatics
Mlynská dolina, 842 48 Bratislava, SR
1e-mail: bozekfmph.uniba.sk, 2e-mail: foltan.gfmph.uniba.sk
Abstract: We detect and partially classify all singular points of pedal curves in the Euclidean
plane.
Keywords: planar curve, curvature, Frenet formulas, inflexion point, singular point
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O niektorých klišé v riešení (nielen)
konštrukčných geometrických úloh
Ján Čižmár
Fakulta matematiky, fyziky a informatiky Univerzity Komenského,
Mlynská dolina, 842 48 Bratislava, SR
e-mail: jan.cizmarfmph.uniba.sk
Abstrakt: Článok analyzuje niekoľko klišé vyskytujúcich sa trvalo v metodických príručkách
o riešení konštrukčných geometrických úloh. Poukazuje na logickú neudržateľnosť niektorých
týchto klišé a kritizuje nekritické preberanie chybnej tradície. Taktiež skúma všetky štyri
kroky úplného riešenia konštrukčných geometrických úloh a vysvetľuje ich logický obsah.
Kľúčové slová: úplné riešenie konštrukčnej geometrickej úlohy, niektoré tradičné chybné
klišé, logický obsah štyroch krokov úplného riešenia
Summary. The paper analyses several clichés occurring steadily in methodological manuals
on solving of constructional geometrical problems. It shows the logical unsustainability of
some of them and critisises an uncritical adopting of erroneous tradition. It also investigates
all four steps of the complete solution of constructional geometrical problems and explains
their logical contents.
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Why is Geometry Beautiful?
Jenő Horváth1, Ágota H. Temesvári2
1Budapest, Hungary
e-mail: hta49t-online.hu
2University Pécs, H ungary
e-mail: temesvarigamma.ttk.pte.hu
Abstract: In this paper we would like to show how beautiful elementary geometry is. We
deal with different solutions for exercises. You can find exercises also in non-Euclidean
geometry.
Keywords: axioms, transformations, vectors, non-Euclidean geometry
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Multifocal Lemniscates - A Short Introduction
Pavel Chalmoviamský
Comenius University, Faculty of Mathematics, Physics and Informatics
Mlynská dolina, 842 48 Bratislava, SR
e-mail: Pavel.Chalmovianskyfmph.uniba.sk
Abstract: We recall the curves called multifocal lemniscates occurring in many areas such as
CAGD, complex analysis, root estimation, convergence of interpolation techniques, approximation
of domain boundary by a finite set of Jordan curves and others. The class of the curves
contains certain famous curves such as Cassini’s ovals and using logarithming one gets a broad
class of quasi-lemniscates, certain conic sections among them. The curves appeared recently as
a new possible modeling tool due to flexibility and arbitrary topology. The interesting challenges
are hidden inside this class which are to be revealed. The question of convexity, singularities
of the dual curves and complex extensions of these curves are still opened.
Keywords: algebraic curves, multifocal lemniscate
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Colored Marching Tetrahedra: a Surface Reconstruction Method for Multiple Interacting Liquids
Michal Chládek1, Juraj Onderik2, Roman Ďurikovič3
Department of Applied Informatics, FMFI UK, Mlynská dolina, 842 48 Bratislava, SR
1e-mail: michal.chladekfmfi.uniba.sk
2e-mail: juraj.ondrejikfmfi.uniba.sk
3e-mail: roman.durikovicfmfi.uniba.sk
Abstract. Reconstructing surface of a multi-fluid immiscible flow is still an open problem. In
this paper we propose an approach to reconstruct surface between multiple immiscible fluids
using tetrahedralization of fluid particles. A multi-fluid flow can be generated from a single
fluid flow by generating particles of the second fluid. In this way we get a surface moving with
the fluid and thus convert the Eulerian surfacing problem to a Lagrangian problem.
Keywords: surface reconstruction, terahedralization, multi-fluid flowy
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Kubická funkcia a jej graf
Dominika Koštová
Fakulta matematiky, fyziky a informatiky Univerzity Komenského, Mlynská dolina, 842 48 Bratislava, SR
e-mail: dominikakostovagmail.com
Abstrakt: Zámerom tohto článku je objasniť ako sa mení kubická funkcia a jej graf
v závislosti od jej koeficientov.
Kľúčové slová: polynóm, kubická funkcia, graf funkcie, koeficient funkcie
Summary. The aim of the paper is to illustrate how cubic function and its graph changes
depending on its coefficients.
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Konštrukcia rozvinuteľných plôch
Soňa Kudličková
FMFI UK, Mlynská dolina, 842 48 Bratislava, SR
e-mail: kudlickovafmph.uniba.sk
Abstrakt. Konštrukcia rozvinuteľných Bézierovych plôch s okrajovými Bézierovymi
krivkami 2° je opísaná pomocou podmienok rozvinuteľnosti, ktoré sú vyjadrené pomocou
riadiacich bodov okrajových kriviek a medzivrcholov získaných v Casteljauvom algoritme.
Charakterizované sú štyri typy rozvinuteľných plôch v náväznosti na štyri konfigurácie
určené medzivrcholmi.
Kľúčové slová: rozvinuteľné priamkové plochy, Bézierove plochy, Casteljau algoritmus.
Summary. The paper deals with geometric design of the developable Bézier surfaces B12(u,v).
The conditions for developability are derived from the Casteljau algorithm and expressed as
a set of equations that must be fulfilled by the Bézier control points Vij, i = 0,1, j = 0,1,2.
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Výtvarná charakteristika a aplikácie priestorových plôch
Denisa Lizoňová
KMDG, Drevárska fakulta, TU Zvolen,
T.G. Masaryka 24, 96053 Zvolen, SR
e-mail: denisa.lizonovatuzvo.sk
Abstrakt: Príspevok ukazuje na prepojenie medzi geometriou, výtvarnou geometriou
a dizajnom. Práca s tvarom vyžaduje nielen ovládanie potrebných konštrukcií, ale aj dobrú
znalosť výtvarného a psychologického pôsobenia. Len v tejto kombinácii je možné
plnohodnotne využiť potenciál, ktorý prepojenie geometrie a dizajnu poskytuje. Po tvarovovýtvarnej
stránke sa článok orientuje na vybrané plochy technickej praxe, ktoré predstavujú
širokú základňu pre praktické využitie pri tvorbe nových produktov.
Klúčové slová: Výtvarná geometria, dizajn, tvar, priestorové plochy.
Summary. Art geometry builds a link between geometry as science and geometry as art, which develops
great possibilities for its use within creative process of building art and design work. The
harmonious link between knowledge and innate sense is a key moment for arising objects of
the top aesthetic quality, which is the main intention of each designer. Theoretical knowledge
and gained experience can be applied in art and geometric analyses of selected surfaces and
while creating compositions.
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The Regular 4-Solids Move in the Computer 2-Screen With Visibility and Shading of 2-Faces
Emil Molnár1, István Prok2
Budapest University of Technology and Economics, Institute of Mathematics,
Department of Geometry,
Egry József u. 1. H. II. 22, H – 1521 Budapest XI
Hungary
1e-mail: emolnarmath.bme.hu
2e-mail: prokmath.bme.hu
Abstract. 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, ~) over a real vector
space V5 for points and its dual V5 for hyperplanes up to the usual equivalence ~ (expressed
by multiplication by positive real numbers or non-zeros, respectively).
In this presentation we attempt to further develop the exterior (Grassmann) algebra method
(with scalar product) by computer to other effects of illumination, e.g. for (regular) polytopes
on the base of the homepage
http: //www.math.bme.hu/~prok
Below we present the computer figures of the six regular 4-polytopes. We plan further
develop the method in the above homepage
Keywords: regular 4-polytopes, animation in the computer screen with visibility and shading
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Pokrývanie pravouholníkov štvorcami
Pavel Novotný
KKMAHI, FPEDaS, Žilinská univerzita,
Univerzitná 1, 010 26 Žilina, SR
e-mail: pavel.novotnyfpedas.uniza.sk
Abstrakt: V práci sa hľadá najväčšie číslo An také, že každým systémom n štvorcov s celkovým obsahom 1 je možné pokryť nejaký pravouholník s obsahom An .
Kľúčové slová: pokrývanie štvorcami
Summary. We search the largest number An such that any system of squares of total area 1 permits
a covering of some rectangle of area n A .
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The Discovery of the Golden Elleipse
Bronislaw Pabich
Poland
e-mail: pabichfm
Abstract. The lecture shows how during the regular math class can lead to the discovery of a
particular ellipse. The author of this discovery called her golden ellipse because its properties
apply to the golden ratio. The lecture is presented discovered theorem and its proof.
Keywords: Cabri II PLUS , construction a square inside triangle, didactic of math, golden
ellipse
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Planar Cubic Space-Like Bézier Curves in Three Dimensional Minkowski Space
Barbora Pokorná1, Pavel Chalmovianský2
KAGDM, FMFI UK, Mlynská dolina, 842 48 Bratislava, SR
1e-mail: barbora.pokornafmph.uniba.sk
2e-mail: pavel.chalmovianskyfmph.uniba.sk
Abstract. This paper considers planar cubic Bézier curves in three-dimensional Minkowski
space. We shall consider the conditions for the control points A;C;D;B of the Bézier curve
such that the Bézier segment is pointwise space-like. For the control point C, we shall give a
geometrical interpretation of the feasibility condition provided the rest of the control points are
fixed.
Keywords: cubic Bézier curve, space-like curve, conic section
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Geometria kompozície: Gustav Klimt
Daniela Richtáriková
SjF STU Bratislava, Nám. slobody 17, 812 31 Bratislava, SR
e-mail: daniela.richtarikovastuba.sk
Abstrakt: Príspevok pojednáva o geometrických prvkoch v kompozícii vybratých diel
Gustava Klimta pri príležitosti 150-teho výročia narodenia.
Kľúčové slová: matematika a umenie, geometria, kompozícia, Gustav Klimt
Summary. The paper focuses on geometrical aspects in composition of three Gustav Klimt’s
paitings, Music II, 1989, Portrait of Margareth Stonborough-Wittgenstein, 1905 and Knight,
Stoclet Frieze, 1905-09.
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GPU Visualization of Cubic Bézier Volumes
Martin Samuelčík
FMFI UK Bratislava, Mlynská dolina, 842 48 Bratislava, SR
e-mail: samuelcikfmph.uniba.sk
Abstract: The aim of this work is to increase speed of rendering for trivariate Bézier
volumes of maximal degree 3. These volumes are natural extension of parametric Bézier
curves and surfaces. The rendering at interactive rates is necessary in many areas, for examle
for fast overview of object shape or for real-time editation of such volumes. We will use programmable functionality of modern graphics hardware to migrate computation of
volume’s points to processor at graphics card. In this work, we investigate the use of vertex
shaders and tessellation shaders for computation of object points using cubic polynomials. We
will also present implementation of proposed migration and show, that usage of shaders
brings high speedup to rendering rates. It is also shown, that on testing hardware with current
drivers, algorithm is faster in the case of vertex shaders than in the case of tesselation shaders.
Keywords: Bezier volumes, OpenGL, visualization, shaders
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Dynamická podpora KoGe
Dagmar Szarková
SjF STU Bratislava, Nám. slobody 17, 812 31 Bratislava, SR
e-mail: dagmar.szarkovastuba.sk
Abstrakt. Cieľom príspevku je poukázať na možnosti využitia rôznych typov apletov –
dynamických výučbových materiálov vytvorených v softvéri GeoGebra voľne prístupnom na
internete a softvéri Maple, ktoré uľahčujú pochopenie výkladu látky v predmete Konštrukčná
geometria (v skratke KoGe), resp. sú dobrou vizualizačnou podporou pre samoukov pri štúdiu
konštrukčnej geometrie.
Kľúčové slová: dynamický výučbový materiál, aplet v softvéri GeoGebra a Maple
Summary. Nowadays, almost each student has a personal computer at his disposal, therefore it is
necessary to inform about possibilities to use dynamic support for the study of Constructive
geometry. Different types of applets are available, which are dynamic instructional learning
materials developed in software GeoGebra that is provided free on Internet, and in software
Maple. Dynamic support enables better understanding of the basic concepts presented in the
subject Constructive geometry. It is a good visualisation support for self-learners in their
study of space geometry, enhancement of better spatial abilities and imagination.
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Geometry in Teachers´Traning - Plans
Ágota H. Temesvári
Pécs University, Pécs, H
e-mail: temesvarigamma.ttk.pte.hu
Abstract. The Bologna System was introduced also in Teachers’ Training in 2006. After six
years the government has accepted the professional arguments against the situation. It is
planned that teacher’ training will be consolidated. The decision regarding professional
knowledge and teaching competence to be acquired was the duty of a national committee.
Keywords: Teacher’ Training, geometry
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Vizualizácia mnoholistých funkcií komplexnej premennej pomocou Riemannovej plochy
Miroslava Valíková1, Pavel Chalmovianský2
KAGDM, FMFI UK, Mlynská dolina, 842 48 Bratislava, SR
1e-mail: miroslava.valikovafmph.uniba.sk
2e-mail: pavel.chalmovianskyfmph.uniba.sk
Abstrakt. Na vizualizáciu niektorých komplexných funkcií sa ˇcasto používa ofarbovanie defini
ˇcného oboru. Táto metóda zlyhá pri kreslení grafu mnoholistej funkcie. Preto sa pri niektorých
takýchto funkciách používajú Riemannove plochy. Na takejto ploche sa mnoholistá funkcia
správa ako jednolistá. V našej publikácii ukážeme, ako sa dá vytvorit’ Riemannova plocha pre
racionálne komplexné odmocniny polynomických funkcií a ich súˇcin. Riemannovu plochu vytvárame
nad Riemannovou sférou, ˇco nám zabezpeˇcuje vizualizáciu celého definiˇcného oboru
danej funkcie vrátane nekoneˇcna. Kl’úˇcové body zobrazenia – vetviace singulárne body lokálne
zvýrazˇnujeme špirálovým obrazom malej kružnice so stredom v singulárnom bode.
Kl’účové slová: mnoholisté komplexné funkcie, Riemannova plocha, komplexné odmocniny
Summary.
We often use the domain coloring method for the visualization of complex functions.
This method does not give a correct image, when a multi-valued function is visualized. Using
Riemann surface associated with the function leads to an improvement. On such a surface the
multi-valued function behaves like a single-valued.
The aim of this work is to describe a technique for visualizations of the multi-valued
complex functions on the Riemann surfaces. We create the Riemann surface over the Riemann
sphere of the product of rational power. Each branch point is highlighted by a spiral image of a
small enclosing circle.
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Minkowski Combinations of Point Sets
Daniela Velichová
SjF STU Bratislava, Nám. slobody 17, 812 31 Bratislava, SR
e-mail: daniela.velichovastuba.sk
Abstract: Paper brings few ideas about a concept of possible Minkowski combinations of
point sets, which can be defined as generalisation of set operations Minkowski sum and
Minkowski product of two point sets in the Euclidean space En. Minkowski sum linear and
matrix combination, Minkowski product linear and matrix combination, and Minkowski
arithmetic combination of point sets are introduced and illustrated on examples.
Keywords: Minkowski sum and product, Minkowski linear, matrix and arithmetic
combination of point sets
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