ZBORNÍK SYMPÓZIA
O POČÍTAČOVEJ GEOMETRII
SCG´2013
PROCEEDINGS OF SYMPOSIUM
ON COMPUTER GEOMETRY
SCG´2013
Volume 22
Slovak Society for Geometry and Graphics
Mechanical Engineering Faculty
Civil Engineering Faculty
Slovak University of Technology in Bratislava
October 2013, Kočovce, SR
ISBN 978-80-227-4041-8
Contents
| V. Bakurová: On Singularities of Pedal Curves in the Minkowski Plane
. . . . . . . . . . . . . . . . . . |
5 |
| V. Bálint, M. Kaukič, Š. Peško: Maximalizácia súčtu obsahov pomocou počítača. . . . . . . . . . . |
11 |
| M. Bátorová, P. Chalmovianský: Stability of Superelliptic Singularity Structure . . . . . . . . . . . . |
16 |
| M. Božek: On La Hire´s Theorem in the Minkowski Plane
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
23 |
| E. Capková, P. Krušinský, Z. Grúňová, M. Holešová: Harmónia a geometria vo
vybraných konštrukciách gotických krovov v regióne Turca . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
29 |
J. Čižmár: O určitej kubickej involúcii v Pn(k) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
|
35 |
| V. Čmelková: GeoGebra a LATEX v podpore výučby geometrie na Fakulte
prevádzky a ekonomiky dopravy a spojov Žilinskej univerzity
v Žiline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
50 |
A. Ferko, E. Šikudová: Predpoklady pre matematický model virtuálneho múzea. . . . . . . . . . . . . .
|
57 |
M. Gemeranová, P. Chalmovianský: Planar Reconstruction Using Multifocal
Lemniscates . . . . .
|
61 |
| Š. Gergelitsová, T. Holan: Gradované sady planimetrických úloh . . . . . . . . . . . . . . . . . . . . . . . . . |
67 |
| P. Chalmovianský: Deformations of Singularities and Their Description . . . . . . . . . . . . . . . . . . . . |
74 |
| S. Kudličková: Riadiace siete rozvinuteľných kubických Bézierovych záplat . . . . . . . . . . . . . . . . . |
80 |
| I. Markechová, H. Stúpalová: Synograph & Cylindrical Coordinates Approach in
Explicit Analytical Representation of Rotational Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
88 |
| E. Molnár, J. Szirmai: Densest Geodesic and Translation Ball Packings by Space
Groups pg21 in ~SL2R. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
94 |
| P. Novotný: Ukladanie kociek na pravouholník. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
96 |
| A. Nowak, W. Rokicki: Patterns of Nature Described by Mathematical Models in Shaping Contemporary Elevations of Architectural Objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
101 |
| D. Richtáriková: Aproximácia meraní funkciou dvoch premenných . . . . . . . . . . . . . . . . . . . . . . . . |
106 |
| W. Rokicki, E. Gawell: Algorithmic Modeling in the Development of Modern Architectural Forms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
109 |
| D. Rumanová, D. Vallo: Geometrické konštrukčné úlohy riešené v priestore s využitím CABRI 3D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
118 |
| M. Samuelčík: GeomForge - Framework for Geometric Modeling . . . . . . . . . . . . . . . . . . . . . . . . |
123 |
| I. Varhaníková: The Geometry of Various Kinds of Anamorphisms . . . . . . . . . . . . . . . . . . . . . . . . |
129 |
| D. Velichová: Minkowski Triples of Curve Segments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
137 |
| Symposium Programme . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
145 |
| List of Participants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
148 |
Abstracts
On Singularities of Pedal Curves in the Minkowski Plane
Viktória Bakurová
KAGDM FMFI UK, Mlynská dolina, 842 48 Bratislava, SR
1e-mail: viktoria.bakurova
fmph.uniba.sk
Abstract. We point out connection between singular points on pedal curve and inflexion points
on base curve in the Minkowski plane. We formulate hypotheses about the type of singular
points on pedal curve according to the position of pedal point and order of corresponding inflexion
points on base curve.
Keywords: Minkowski plane, pedal curve, singular point, inflexion point
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Lineárna kombinácia a rovinné krivky v programe GeoGebra
Vojtech Bálint1, Michal Kaukič2, Štefan Peško3
FPEDAS ŽU, Univerzitná 1, 010 26 Žilina, SR
1e-mail: balintfpedas.uniza.sk, 2e-mail: mikercatel.fri.uniza.sk, 3e-mail: stefan.peskori.uniza.sk
Abstrakt: Príspevok udáva maximálny súčet obsahov pre uloženie troch trojuholníkov do
kružnice bez prekrývania, ak konvexný obal tých troch trojuholníkov je šesťuholník.
Kľúčové slová: maximum, súčet obsahov
Summary. The paper gives the maximal sum of areas for the packing of three triangles into
a circle without overlapping in the case, if the convex hull of those three triangles is
a hexagon.
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Stability of Superelliptic Singularity Structure
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 treats the singularities of superelliptic curves over the field of complex
numbers. We subject these curves to a series of deformations and prove that the internal structure
of their singularities is independent of given deformations.
Keywords: superelliptic curve, parameterization, deformation, singular point
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On La Hire´s Theorem in the Minkowski Plane
Miloš Božek
Comenius University, Faculty of Mathematics, Physics and Informatics
Mlynská dolina, 842 48 Bratislava, SR
e-mail: bozekfmph.uniba.sk
Abstract: We prove a variant of La Hire’s theorem on orbits of a planar motion induced by
slipping of a line-segment with end-points constrained at two straight-lines in the Minkowski
plane.
Keywords: Minkowski plane, La Hire’s theorem, affine conic
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Harmónia a geometria vo
vybraných konštrukciách gotických krovov v regióne Turca
Eva Capková1, Peter Krušinský2, Zuzana Grúňová3, Michaela Holešová4
1Katedra mediamatiky a kultúrneho dedičstva, FHV, ŽU v Žiline, Univerzitná 8215/1,
010 26 Žilina, SR, e-mail: eva.capkovafhv.uniza.sk
2Katedra pozemného staviteľstva a urbanizmu, SvF, ŽU v Žiline, Univerzitná 8215/1,
010 26 Žilina, SR, e-mail: peter.krusinskyfstav.uniza.sk
3Katedra pozemného staviteľstva a urbanizmu, SvF, ŽU v Žiline, Univerzitná 8215/1,
010 26 Žilina, SR, e-mail: zuzana.grunovafstav.uniza.sk
4Katedra matematiky, FHV, ŽU v Žiline, Univerzitná 8215/1,
010 26 Žilina, SR, e-mail: michaela.holesovafhv.uniza.sk
Abstrakt: V nadväznosti na predchádzajúce stavebno-historické výskumy sa v rámci
projektu VEGA postupne realizuje geometrická analýza historických krovových konštrukcií z
územia SR, pričom prvým územím vhodným na skúmanie bol vybraný región Turca. V tomto
príspevku sa zaoberáme geometrickou analýzou troch gotických krovových konštrukcií z tejto
oblasti, konkrétne krovmi rímskokatolických kostolov: svätého Kozmu a Damiána
v Abramovej, svätého Petra v obci Turčiansky Peter a svätého Michala v obci Turčiansky
Michal. Geometrická analýza vychádza z princípov, ktoré boli v danom období používané pri
navrhovaní stavieb, ktorými okrem kvadratúry a triangulácie bolo používanie tzv. hudobných
pomerov a iracionálnych proporcií, ktoré odrážali zmysel pre dokonalosť vtedajších
architektov a staviteľov.
Kľúčové slová: geometrická analýza, historický krov, hudobné pomery
Summary. Following the previous building-historical research we provide a geometrical analysis of
historical truss constructions from the area of the Slovak republic. The first chosen territory
was the Turiec region. In this article, we deal with the geometrical analyses of three gothic
truss constructions from this area, specifically the Roman Catholic churches of: Saint Kozma
and Damian in Abramová, Saint Peter in Turčiansky Peter and Saint Michal in Turčiansky
Michal. The geometrical analysis is based on principles, which were used by the architects
and builders in the certain period of time for the design of the churches. These principles
included squaring and triangulation, as well as musical ratios, and irrational proportions and
reflected sense of harmony of their creators.
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O určitej kubickej involúcii v Pn(k)
Ján Čižmár
Pedagogická fakulta Trnavskej univerzity,
Priemyselná 4, P.O. BOX 9, 918 43 Trnava, SR
e-mail: jan.cizmartruni.sk
Abstrakt: V článku je opísaná biracionálna korešpondencia v n-rozmernom projektívnom
priestore Pn(k) nad algebricky uzavretým poľom k charakteristiky 0. Dvojicami
korešpondujúcich bodov – ak existujú – sú priesečníky priamok (n – 1)-rozmerného zväzku
priamok s nadkvadrikami zväzku nadkvadrík. Sú opísané množiny fundamentálnych,
iregulárnych a biregulárnych bodov korešpondencie.
Kľúčové slová: korešpondencia, biracionálna transformácia, involúcia 3. stupňa,
fundamentálny bod, iregulárny bod, biregulárny bod
Summary. A cubic involution in the n-dimensional projective space Pn(k) over an algebraically
closed field of characteristic 0 is described. Both fundamental and irregular varieties are
found and the set of biregular points is specified as well.
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GeoGebra a LATEX v podpore výučby geometrie na Fakulte
prevádzky a ekonomiky dopravy a spojov Žilinskej univerzity
v Žiline
Viera Čmelková
FPEDAS ŽU, Univerzitná 1, 010 26 Žilina, SR
e-mail: viera.cmelkovafpedas.uniza.sk
Abstrakt: V príspevku autorka približuje niekoľko riešených príkladov z geometrie
vytvorených pomocou dynamického matematického softvéru GeoGebra a typografického
systému LaTeX pre študentov prvého ročníka bakalárskeho štúdia Fakulty PEDAS Žilinskej
univerzity slúžiacich ako pomôcka pri opakovaní stredoškolského učiva.
Kľúčové slová: deskriptívna geometria, GeoGebra, posunutie, otočenie, stredová súmernosť,
osová súmernosť
Summary. In this paper author brings few tasks from geometry, that have been developed
with help of dynamic mathematical software GeoGebra and typographic system LaTeX
for students in the first year of bachelor’s study at the Faculty PEDAS at the University
of Žilina, as an aid helping students to repeat secondary knowledge.
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Predpoklady matenatického modelu virtuálneho múzea
Andrej Ferko1, Elena Šikudová
Comenius University, Faculty of Mathematics, Physics and Informatics
Mlynská dolina, 842 48 Bratislava, SR
1e-mail: ferkofmph.uniba.sk
Summary: We witness many implementations of virtual museums and exhibitions. However, the
mathematical model, connecting the physical world with computer representation and
implementation, requires certain completion. We collect the assumptions and unavoidable
definitions for the conceptual framework for such a future formal model.
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Planar Reconstruction Using Multifocal
Lemniscates
Mária Gemeranová1, Pavel Chalmovianký2
Comenius University, Faculty of Mathematics, Physics and Informatics
Mlynská dolina, 842 48 Bratislava, SR
1e-mail: Maria.Gemeranovafmph.uniba.sk
2e-mail: Pavel.Chalmovianskyfmph.uniba.sk
Abstract: Multifocal lemniscate in E2, defined via multipolar coordinates, is the set of points
in E2, whose product of distances to a finite set of fixed points is a constant. In our work, we
approximate an input set of points in E2 using lemniscates, i. e. we look for an appropriate
set of foci and a radius value to get a sufficiently small approximation error. Our algorithm
computes a position of one focus and an appropriate radius value, duplicates focus and looks
for optimal positions of the achieved foci. The process of duplication and optimization of the
foci is repeated until the achieved error is under chosen threshold.
Keywords: multipolar coordinates, multifocal lemniscates, approximation, reconstruction, Newton’s
descant method in optimization
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Gradované sady planimetrických úloh
Šárka Gergelitsová1, Tomáš Holan2
MFF UK v Praze, Malostranské nám. 25, 118 00 Praha 1, ČR
1e-mail: sarkagbm.cz
2e-mail: Tomas.Holanmtf.cumi.cz
Abstrakt. V příspěvku se zamýšlíme nad možnostmi, jak žákům usnadnit nácvik základních
dovedností a způsobů myšlení potřebných k řešení problémů v planimetrii. Popisujeme, jak
k tomu může přispět systém GeoTest, který funguje již více než dva roky, a navrhujeme
příklady tzv. gradovaných řetězců (sad) sestavených z planimetrických úloh v systému
dostupných.
Klíčová slova: planimetrie, gradované sady úloh, online, GeoTest
Summary. The paper reports the observation of what kinds of tasks teachers assign to their pupils.
Although students should solve a wide range of different problems, some problems in the
prepared set of tasks in GEOTEST database stay omitted. As a way how to draw attention to
these problems we propose and describe four sets of tasks arranged by their difficulty.
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Deformations of Singularities and Their Description
Pavel Chalmovianký
Comenius University, Faculty of Mathematics, Physics and Informatics
Mlynská dolina, 842 48 Bratislava, SR
e-mail: Pavel.Chalmovianskyfmph.uniba.sk
Abstract: We are preparing for the notion of an infinitesimal deformation in the context of
local singularities. We give an example and illustrate several notions leading to the module of
the first order deformations and its interpretation in terms of normal sheaf. We illustrate some
of the notions on ADE singularities.
Keywords: singularities, deformations, infinitesimal deformation space
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Riadiace siete rozvinuteľných kubických Bézierovych záplat
Soňa Kudličková
Fakulta matematiky, fyziky a informatiky Univerzity Komenského, Mlynská dolina, 842 48 Bratislava, SR
e-mail: kudlickovafmph.uniba.sk
Abstrakt: Podmienky rozvinuteľnosti resp. komplanárnosti sú určené pomocou riadiacich
bodov okrajových Bézierovych kriviek 3° a medzivrcholov získaných v Casteljauvom
algoritme a následne sú aplikované pri konštrukcii vrcholov riadiacej siete rozvinuteľnej
kubickej Bézierovej záplaty.
Kľúčové slová: rozvinuteľné priamkové plochy, Bézierove plochy, Casteljau algoritmus,
konvexné štvoruholníky
Summary. Geometric design of the cubic developable Bézier patches from two boundary curves
is presented. 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. We study
the four classes A, B, C and D of the cubic developable Bézier patches.
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Synograph & Cylindrical Coordinates Approach in
Explicit Analytical Representation of Rotational Surfaces
Iveta Markechová1, Hana Stúpalová2
Institute of Applied Informatics, Automation and Mathematics,
Faculty of Material Sciences and Technology in Trnava
Slovak University of Technology in Bratislava, Hajdóczyho 1, 917 00 Trnava, SlR
1e-mail: iveta.markechovastuba.sk, 1e-mail: hana.stupalovastuba.sk
Abstract. Analytical expression of a geometrical object – line, surface, massif – via
parametrical equations is standardly used in elementary computer graphics course at technical
university. Nevertheless, basic mathematics course themes operate with analytical
representations from wider spectrum of the forms: there is an explicit version, implicit one,
the set of parametrical equations, etc. The aim of this note is to offer an alternative that is on
the authors` best known unused in this framework, based on cylindrical coordinates plus
synograph approach, leading to technically relatively simple and in light of mental
visualization easy explicit analytical expression of certain type of rotational surfaces.
Keywords: surface of revolution, cylindrical coordinate system, governing line/directrix,
parametrical equations, WinPlot, synograph
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Densest Geodesic and Translation Ball Packings by Space
Groups pg21 in ~SL2R
Emil Molnár1, Jenő Szirmai2
Institute of Mathematics, Department of Geometry,
Budapest University of Technology, Hungary
Egry József u. 1. H. II. 22, H – 1521 Budapest XI
1e-mail: emolnarmath.bme.hu
2e-mail: szirmaimath.bme.hu
Abstract. Brief information on the newest results in the classical ball packing problem
applied to the ~SL2R space.
Keywords: Thurston geometries, discrete group in ~SL2R, density of ball packing
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Ukladanie kociek na pravouholníki
Pavel Novotný
KKMAHI, FPEDaS, Žilinská univerzita,
Univerzitná 1, 010 26 Žilina, SR
e-mail: pavel.novotnyfpedas.uniza.sk
Abstrakt: V práci je nájdené najmenšie číslo A4 také, že každý systém štyroch kociek
s celkovým objemom 1 sa dá uložiť na pravouholník s obsahom A4.
Kľúčové slová: ukladanie kociek
Summary. Any system of four cubes of total volume 1 can be packed onto a rectangle of area at
most 1,8587123.
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Patterns of Nature Described by Mathematical Models in Shaping Contemporary Elevations of Architectural Objects
Anna Nowak1, Wiesław Rokicki2
Warsaw University of Technology, Faculty of Architecture, Structural Design Department,
Koszykowa 55, 00 659 Warsaw, PL
1e-mail: anna.patrycja.nowakgmail.com, 2e-mail: wrokickipoczta.fm
Abstract. The inspiration with nature constitutes an essential aspect in the architectural design.
The development of digital tools used currently in architecture made it possible to transfer the
possibilities found in nature into the methods of shaping the space and the architectural
surface. Consequently, the process of design using mathematical models describing standards
of the nature is becoming one of the most interesting trends in the shaping of contemporary
architecture.
Keywords: surfaces, structure, architecture, nature, mathematics.
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Aproximácia meraní funkciou dvoch premenných
Daniela Richtáriková
SjF STU Bratislava, Nám. slobody 17, 812 31 Bratislava, SR
e-mail: daniela.richtarikovastuba.sk
Abstrakt: Článok pojednáva o aproximácii trojrozmerných dát mocninovou funkciou dvoch
premenných, pričom sa aplikujú poznatky zo základných kurzov matematiky na technických
vysokých školách.
Kľúčové slová: experimentálne dáta, aproximácia mocninovou funkciou dvoch premenných
Summary. The paper deals with fitting of 3-dimmensional experimental data set by power
function in two variables.
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Algorithmic Modeling in the Development of Modern Architectural Forms
Wiesław Rokicki1, Ewelina Gawell2
Faculty of Architecture at the Warsaw University of Technology (WAPW),
Structural Design Department,
ul. Koszykowa 55, p.216, Warsaw, Poland
1e-mail: wrokickipoczta.fm,
2e-mail: ewelina.wysokinskagmail.com
Abstract: New trends and directions of research in architecture maintain the creation of
structural forms using mathematical and geometric solutions that result in chaos theory. The
application of generative design methods allows for a deeper understanding and creating
forms, such as chaotic fractals, turbulence, folding, thus giving new creative opportunities for
architects and necessary optimization tools. In recent years the impact of digital technology
on the new language of architecture has increased enormously, and the pursuit of an optimal
design is an objective measure of value and quality of architecture and engineering solutions.
This article includes an analysis of some algorithmic trends in creating structural forms that
coincide with the idea of Pythagoras that “the whole thing is a number”. It is an important
element of understanding the nature of building materials in the surrounding natural world.
Keywords: generative design methods, generative modeling, algorithmic code
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Geometrické konštrukčné úlohy riešené v priestore s využitím CABRI 3D
Lucia Rumanová1, Dušan Vallo2
FPV UKF v Nitre,
Tr. A. Hlinku 1, 949 74 Nitra, SR
1e-mail: lrumanovaukf.sk, 2e-mail: dvalloukf.sk
Abstrakt: V príspevku sa venujeme konštrukčným geometrickým úlohám a metódam ich
riešenia, pričom netradične, vzhľadom k obsahu výučby stereometrie na našich školách, sa
zameriavame na úlohy stereometrické. Pri vizualizácii riešení týchto úloh používame ako
dôležitú pomôcku softvér Cabri 3D, ktorý umožňuje virtuálnu manipuláciu s útvarmi v priestore,
a teda je vhodný hlavne na použitie vo vybraných fázach riešenia úlohy.
Kľúčové slová: konštrukčné úlohy, geometria, riešenie úloh, priestor, Cabri 3D
Summary. In this article we present the solutions of construction geometric tasks. We are
concerned with solid geometry variants of standard construction tasks that are taught in our
schools. The solutions are represented by using software Cabri 3D which allows virtual
manipulation with 3D objects. Some didactically important comments related to problems in
educational process are attached, too.
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GPU Visualization of Cubic Bézier Volumes
Martin Samuelčík
1FMFI UK Bratislava, Mlynská dolina, 842 48 Bratislava, SR
2Vis Gravis, Jána Stanislava 29, 841 05 Bratislava, SR
1e-mail: samuelcikfmph.uniba.sk, 2e-mail: samuelcikvisgravis.sk
Abstract: As one part of the geometric modeling research at our faculty, we started with the
developement of the complex system and framework that can be easily used for the
implementation of the new modeling and visualization techniques. Therefore we have created
programming library that contains all algorithms needed to achieve this goal and we also have
created application that uses this library and is used for practical modeling work. Whole
framework is called GeomForge. Presented paper gives an overview of our system, showing
its most important and most useful capabilities.
Keywords: geometric modeling, framework, API, application
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The Geometry of Various Kinds of Anamorphisms
Ivana Varhaníková
KAGDM, FMFI UK, Mlynská dolina, 842 48 Bratislava, SR
e-mail: varhanikovasccg.sk
Abstract. Anamorphic illusions with their ability to impress the observer interest as well the
artist as the mathematicians. The beauty of anamorphism that could be seen only from special
point of view or in particular auxiliary object combines the geometry and art. In this paper
we tried to summarize the known transformations of several chosen anamorphism and briefly
sketch the properties of another three dimensional ones.
Keywords:anamorphism, transformation equations, planar anamorphism, anamorphic statues,
luminary anamorphism, optical illusions
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Minkowski Triples of Curve Segments
Daniela Velichová
SjF STU Bratislava, Nám. slobody 17, 812 31 Bratislava, SR
e-mail: daniela.velichovastuba.sk
Abstract: Some ideas about modelling surfaces by means of Minkowski combinations of
three curve segments are presented in the paper. There are three different possible forms of
Minkowski triples of point sets, Minkowski sum triple, Minkowski product triple and
Minkowski mixed triple, all based on Minkowski operations of two point sets in the
Euclidean space En. Examples of Minkowski triples of three circles are presented.
Keywords: Minkowski sum and product, Minkowski linear, matrix and arithmetic
combination of point sets, Minkowski triples
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