Contents 25

ZBORNÍK SYMPÓZIA

O POČÍTAČOVEJ GEOMETRII

SCG´2005

Album

PROCEEDINGS OF SYMPOSIUM

ON COMPUTER GEOMETRY

SCG´2005

Volume 14

Slovak Society for Geometry and Graphics

Mechanical Engineering Faculty

Civil Engineering Faculty

Slovak University of Technology in Bratislava

September 2005, Kočovce, SR

ISBN 80-227-2278-2

Contents

J. Błahut: Fractals and Semigroups (2): A Proof of the Existence Theorem . . . . . . . . . . . .	5
M. Božek: Dense Placements of Monotone Polygons . . . . . . . . . . . . . . . . . . . . . . . . . . . . .	9
J. Černý, J. Vecková: A Note on a Generalization of Hacar Surfaces . . . . . . . . . . . . . . . . .	15
S. Cichacz, T. Sulima-Samujłło, Tomasz Wieja: Application of the Flash MX Program to Visualisation of Selected Metric Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .	21
J. Dietrich, T. Sulima-Samujłło, T. Wieja: Visualization of Principles of the Certain Geometrical Projections in the Cabri 3D Program . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .	25
R. Ďurikovič: Collection of Bubbles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .	29
M. Kmeťová: Line Conics and Dual Bézier Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .	31
M. Kočandrlová: Středy normálových křivostí elipsoidů v modelu bistatické altimetrie . . .	37
B. Koreňová: Výchova geometriou ku vnímavosti, rozpoznávaniu a kreativite . . . . . . . . .	42
S. Kudličková: Fair Cardinal Splines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .	50
D. Lizoňová: Geometria, história a umenie . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .	56
K. Maleček, D. Szarková: Pseudoparalelní přenos vektoru podél prostorové křivky . . . . .	60
I. Markechová: Cycloid, Coin Paradox and Bike Wheel Motion (Two Tesserae in Cycloid Beauty Mosaic) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .	66
L. Novák: Konstruktivní geometrie v studijním programu technologického studia UTB ve Zlíně. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .	70
D. Richtáriková, P. Kundis: Využitie fraktálovej dimenzie a entropie v analýze experimentálnych dát. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .	75
D. Velichová: Geometry Modules on Web . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .	81
E. Vranková: Niektoré vlastnosti množiny hustých rozmiestnení zhodných mnohouholníkov .	87
T. Wieja: Application of the Autocad Graphic Program to the Construction of the Computer Matrix of the Solar Line Geometrical Diagram. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .	93
R. A. Górska: Theory and Research on Spatial Abilities Teaching Methods and Graphics Related Courses. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .	99
L. Żakowska: Size - Interdisciplinary Approach to Research and Problem Solving in Engineering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .	100

Jerzy Błahut
Instytut Elektroniki Politechniki Śląskiej, ul. Akademicka 16, 44-100 Gliwice, Polska
e-mail: ewa.blahut@womkat.edu.pl

Miloš Božek
Comenius University, Faculty of Mathematics, Physics and Informatics, Mlynská dolina, 842 48 Bratislava, SR
e-mail: bozek@fmph.uniba.sk

Abstract. The aim of the present paper is to prove that the set D(M, N) of all dense placements of a polygon M to a polygon N in the case of monotone polygons with coincident monotony directions is a simple closed broken line, which is the boundary of a monotone polygon.
Keywords: Polygon, monotone polygon, translation, overlapping, densely placed polygons.

Jaroslav Černý¹, Jana Vecková²
FSv ČVUT Praha, Thákurova 7, 166 29 Praha 6, ČR
e-mail: ¹cerny@mat.fsv.cvut.cz

Abstract. In the paper there is defined a generalization of a Hacar surface. Several examples will be presented. The basic geometrical properties are studied in the selected types of surfaces.
Keywords: surface, Hacar surface, collineation

Silvia Cichacz¹, Tomasz Sulima-Samujłło², Tomasz Wieja³
AGH University of Science and Technology, Faculty of Applied Mathematics
30-059 Kraków, al.Mickiewicza 30, Poland,
e-mail: ¹cichacz@agh.edu.pl, ²sulsam@agh.edu.pl, ³twieja@poczta.fm

Abstract. In the article have been shown computer presentations of selected geometrical problems by means of the Flash MX program. The visualizations consist of two elements: axonometric drawing and construction in the Monge projection. Animation of the picture is controled by the user. Application of the Flash MX program allows to locate each of the visualizations as a web page.
Keywords: Flash MX, presentation, animation , visualization, metric problems, axonometry, Monge projection

Jerzy Dietrich¹, Tomasz Sulima-Samujłło², Tomasz Wieja³
AGH University of Science and Technology, Faculty of Applied Mathematics
30-059 Kraków, al.Mickiewicza 30, Poland,
e-mail: ¹jdietrich@wp.pl, ²sulsam@agh.edu.pl, ³twieja@poczta.fm

Abstract. An application of the graphic program CABRI 3D to visualization of the principles of selected geometrical projections has been presented in the article. In the beginning principles of the axonometric projection have been given. Then, basic information about the Monge method, mapping and central projections have been demonstrated, each of the methods being illustrated by means of the axonometric projection.
Keywords: visualization, projection, Monge projection, mapping projection, central projection, axonometric projection, basic elements of space

Roman Ďurikovič
Comenius University, Faculty of Mathematics, Physics and Informatics, Mlynská dolina, 842 48 Bratislava, SR
e-mail: durikovic@sccg.sk

Abstract. This paper deals with a complete computer simulation of soap bubbles from a dynamic perspective. We discuss the dynamic formation of irregular bubble clusters. The resulting model takes into account surface tension, film elasticity, and shape variations due to gravity and external wind forces.
Keywords: Physics-based animation, soap bubbles, spring model, dynamics

Abstrakt. V tomto článku popíšeme počítačovú simuláciu dynamiky mydlových bublín tému, ktorá je zaujímavá pre mnohých matematikov a fyzikov. Najskôr popíšeme jednoduchú geometriu malých zoskupení a navrhneme dynamicky systém tvorby zoskupení bublín. V našom modely berieme do úvahy povrchové napätie, elastické deformácie a vonkajšiu silu. Predkladáme tiež metódy na výpočet spoločnej plochy medzi bublinami, tvorbu bublín a ich deštrukciu.

Mária Kmeťová
FPV UKF Nitra, Tr. A. Hlinku 1, 949 74 Nitra, SR
e-mail: mkmetova@ukf.sk

Abstract. The paper deals with projective construction of envelope conic section, line conic and its corresponding point conic and generally, with dual Bézier curve and its corresponding point curve.
Keywords: projectivity, perspectivity, line conic, de Casteljau algorithm, dual Bézier curve

Summary: If p(t) =B₂₀ t₀ + B₂₁ t₁ + B₂₂ t₂ is a dual Bézier conic, then the corresponding tangent points form the conic T(t) = (1 - t)² V₀ + t(1 - t)V₁ + t²V₂, where V₀ = t₀ × t₁, V₁ = t₀ × t₂, V₂ = t₁ × t₂.
In general if p(t₀) = B_imp_i , i = 0,..., m is a dual Bézier curve of degree m, then the corresponding point curve has degree 2m - 2.
The dual line-based representation of planar curve segment was introduced by J. Hoschek. Their behaviour when used for design purposes is different. By using standard representation it is difficult to design cusps but quite easy to achieve inflections, whereas the dual representation tends to introduce cusps but very special conditions must be met to design an inflection [5].

Milada Kočandrlová
FSv ČVUT Praha, Thákurova 7, 166 29 Praha 6, ČR
e-mail: kocandrlova@mat.fsv.cvut.cz

Abstrakt. V bistatické altimetrii se ze znalosti polohy dvou satelitů a délky signálu vyslaného z jednoho na druhý satelit hledá bod odrazu signálu na zemském tělese. Při geodetickém výpočtu se využívá vhodný referenční elipsoid. Jednak s ohledem na praxi, k odlehlosti elipsoidu a geoidu, jednak vzhledem k přeurčenosti úlohy, nehledáme odrazný bod, ale bod o daných vlastnostech, který je nejblíže referenčnímu elipsoidu.
Klíčová slova: Bistatická altimetrie, referenční elipsoid, střed křivosti.

Summary. We know the state vectors of and , the delay, which span the first ellipsoid and the parameters of a reference ellipsoid of the Earth, and we solve for the minimum distance of both ellipsoids. The paper presents an iterative method for solving a minimum distance of two ellipsoids.

Božena Koreňová
KMDG DF TU, T.G. Masaryka 24, 960 53 Zvolen, SR
e-mail: korenova@vsld.tuzvo.sk

Abstrakt. V článku rozoberáme význam a použitie geometrických štruktúr v dizajnérskej praxi a ich podiel na rozvoji intelektuálnych a tvorivých schopností. Uvádzame niekoľko návodov, ako vytvoriť geometrickú štruktúru, ako v nej hľadať zaujímavé krivky a tvary.
Kľúčové slová: grafický prejav, grafický dizajn, vývoj geometrických tvarov, foriem a kompozícií, zástavková konštrukcia hyperboly, sprievodičová konštrukcia kužeľosečiek, Archimedova a logaritmická špirála, obvodový uhol, zväzok priamok.

Summary. Presented article analyses the importance and application of the geometrical structures in design practice and their contribution to development of intellectual and creative abilities. It also gives some suggestions of how to create a geometrical structure and discover interesting curves and shapes in it.

Soňa Kudličková
FMFI UK, Mlynská dolina, 842 48 Bratislava, SR
e-mail: kudlickova@fmph.uniba.sk

Abstract. Cardinal splines as an application of Hermite blendig are investigated. The conditions of proportionality constant for getting fair - arclike, wavelike features - interpolating curves are proposed.
Keywords: Hermite interpolation, fair curve, cusp, loop, control polygon, continuity conditions, proportionality constant, tension, end conditions

Denisa Lizoňová
KMDG TU Zvolen, Masarykova 24, 96053 Zvolen, SR
e-mail: lizonova@vsld.tuzvo.sk

Abstrakt: Využitie geometrie pri navrhovaní produktov je jedným z mnohých faktorov, ktorými možno dosiahnuť presvedčivú mieru ich dokonalosti. Aplikácia geometrie a výtvarnej analýzy pri návrhu predstavuje pomocnú, ale nevyhnutnú zložku a jej význam a potrebu možno sledovať už v dávnej minulosti. História podáva jasné svedectvo o potrebe oprieť sa pri tvorbe o geometrické konštrukcie. Článok pojednáva o niektorých vybraných oblastiach a historických obdobiach.
Kľúčové slová: výtvarná geometria, geometrická harmónia, staroveká a stredoveká geometria

Summary. The exploitation of geometry in the process of designing is one of the many factors, witch can achieve the valid measure of aesthetic perfection. The value of geometry and its need for common life can be followed as early as old time. We can find and look through the value and the status of geometry in past especially in ancient Egypt and in Middle Ages.

Kamil Maleček¹, Dagmar Szarková²
¹FSv ČVUT Praha, Thákurova 7, 166 29 Praha 6, ČR, e-mail: kamil@mat.fsv.cvut.cz
²SjF STU Bratislava, Nám. slobody 17, 812 31 Bratislava, SR, e-mail: szarkova@sjf.stuba.sk

Abstrakt: V [4] jsme se zabývali pseudoparalelním přenosem vektoru podél křivky K na ploše

vzhledem k tečným rovinám plochy

v bodech křivky K. V tomto příspěvku se zabýváme pseudoparalelním přenosem vektoru podél prostorové křivky, která není spojena s plochou. Pomoci pohyblivé ortonormální báze zavádíme tři systémy rovin a odvozujeme podmínku pseudoparalelních přenosů vektoru podél křivky vzhledem k těmto systémům. V podmínce se ukáže význam invariantů – křivostí prostorové křivky.
Klíčová slova: přenos vektoru podél křivky, pseudoparalelní přenos vektoru podél křivky, křivosti křivky

Summary. In the paper we present the concept of the parallel displacement and the Levi-Civita parallelism of a vector along the space curve with respect to the given system of planes. There are defined three systems of planes by means of a moving orthonormal basis, and the condition of the Levi-Civita parallelism with respect to these systems. Special types of the Levi-Civita parallelism can be achieved for the Frenet trihedron of the curve taken as the moving basis. Curvatures of the curve appear in their condition. Example of the Levi-Civita parallelism along the cylindrical helix with respect to its osculating, rectification and normal planes is presented finally.

Abstract. The main aim of this paper is to present some simple ideas/questions generated by well-known coin paradox as well as by motion of a bike wheel. Looking for correct answers can make the atmosphere in classroom more joyful, creative and – especially – can help to make better understanding of cycloid, its various properties and its beauty.
Keywords: cycloid, coin paradox, bike wheel, recreational mathematics/physics

Ludvík Novák
AGH University of Science and Technology, Faculty of Applied Mathematics
30-059 Kraków, al.Mickiewicza 30, Poland,
e-mail: ¹kmbp@interia.pl, ²twieja@poczta.fm

Abstrakt: Přestavba vysokoškolského studia podle zákona č. 111/1998 Sb na studium bakalářské a magisterské si vynutila v nových učebních plánech snížení rozsahu vyučovacích hodin z geometrie. Z obsahu vyučovaných předmětů technologického studia je ale zřejmé, že právě vědomosti z geometrie jsou nutným předpokladem úspěšného studia odborných disciplin vysokoškolského technického vzdělání.
Klíčová slova: geometrie v technickém studiu, geometrie jako základ studia odborných technických předmětů

Summary. Reconstruction university studies according to law No. 111/1998 Sb on study baccalaureate and dispensing chemist force in new instructional plans decrease range lessons from steering-swivel geometry. From content taught articles technological studies is but evident that the just knowledge from steering-swivel geometry are necessary presumption successful studies special discipline university technical education.

Daniela Richtáriková¹, Peter Kundis²
SjF STU, Nám. slobody 17, 812 31 Bratislava, SR
e-mail: ¹richtarikova@sjf.stuba.sk, ²kundis@sjf.stuba.sk

Abstrakt: V príspevku sa zaoberáme analýzou súborov reálne nameraných dát pomocou fraktálovej dimenzie, štatistickej entropie a Kolmogorov-Sinaiovej entropie.
Kľúčové slová: fraktálová dimenzia, Kolmogorov-Sinaiova entropia, reálny súbor dát

Summary. The paper deals with real data file analysis through fractal dimension and statistic entropy and Kolmogorov-Sinai entropy. The items were used in the level analysis and diagnostics research of performance and body stability of a shooter-gun system in discipline Air Rifle.

Daniela Velichová
SjF STU, Nám. slobody 17, 812 31 Bratislava, SR
e-mail: velichov@sjf.stuba.sk

Abstract. In the paper, there are presented some e-learning educational modules on Geometry designed in the scope of a European project within the Leonardo da Vinci scheme dMath. The aim of the project dMath is a creation of an open authorship system and database of electronic materials in mathematics utilising the brand new information and communication technologies, shared and applied throughout the European universities, technical particularly.
Keywords: e-learning, XML documents, MathML coding

Summary. V článku je prezentovaný e-learningový výučbový modul pre Geometriu, navrhnutý kolektívom medzinárodného projektu dMath v rámci európskeho programu Leonardo da Vinci. Cieľom projektu dMath je návrh otvoreného autorského systému a databázy elektronických učebných materiálov z matematiky pre virtuálne prostredie, vytvorených a využívajúcich najnovšie informačné a komunikačné technológie, ktoré budú voľne zdieľané na Internete a aplikovateľné na univerzitách v celej Európe, so zvláštnym zameraním na technické univerzity.

Edita Vranková
KMaI PdF TU, Priemyselná 4, 918 43 Trnava, SR
e-mail: evrankov@truni.sk

Abstrakt. Cieľom článku je štúdium niektorých vlastností množiny D(M,N) hustých rozmiestnení pre pravouhlé zhodné a súmerné mnohouholníky M, N.
Kľúčové slová: Pravouhlé, zhodné a súmerné mnohouholníky, množina D(M,N).

Summary. The aim of present paper is study some properties of the set D(M,N) dence placements for rectangular and mutually congruent symmetric polygons M, N.

Tomasz Wieja
AGH University of Science and Technology, Faculty of Applied Mathematics,
30-059 Kraków, al.Mickiewicza 30, Poland
e-mail : twieja@poczta.fm

Abstract. In the article have been presented the construction principles of the shadow projected from spatial objects of assumed parameters i.e. latitude, height of the Sun and determined hour. The constructions have been made by means of the AutoCAD graphic program. Resulting from it is a construction of the matrix of the solar line. Application of the designed matrix makes it possible to analyze the shadow projected from spatial objects depending on geographical location of the object.
Keywords: solar line, AutoCAD, azimuth, shadow, insolation, Monge projection

Renata Anna Górska
A9, CUT, Warszawska, 31-155 Kraków, Poland
e-mail: rgorska@pk.edu.pl

Abstract. The goal of this paper is to get insight into research on spatial ability development conducted individually and in the international co-operation over several years at technical universities in Poland, USA and Germany. The subjects tested were chosen among students of engineering majors, especially of those at the civil engineering, architecture, environmental and mechanical engineering. Various testing instruments and background questions were asked to the students. The tests used for measurement of spatial abilities are typically: the Mental Cutting test, the Mental Rotations Test, Purdue Spatial Visualisation Test: Rotations, Differential Aptitude Test. The results obtained in the test over several years were statistically analysed. Discussion on the role of perception and its influence on the enhancement of spatial skills will be also presented. Recent research conducted in co-operation with a colleague Zuzana Juščáková from the Technical University of Košice (Slovakia) who developed a new testing instrument TPS in frame of a VEGA project No 1/1407/04, which was granted by the Slovak Ministry of Education in 2004, gave a new results and evaluations. The author attempts to provide systematic analysis of the research done in Poland and to provide comparison to the results obtained in the international co-operation.

A variety of graphics related courses are used to enhance spatial abilities of engineering students: descriptive geometry (DG), engineering graphics (EG), computer graphics (CG) and/or 3D Visualisation course. The research revealed that the students who had completed a course with descriptive geometry elements included in the content performed on the spatial tests much better than those with pure computer graphics. It was seen in the highly significant gains between pre- and post-tests at the CUT, MTU and UKL. However, the analysis of the gain scores for the spatial tests at MTU, where there was a great deal of "hands-on" problem solving and hand-made sketching led the authors to conclusion that the courses that stress hands-on sketching and drawing tend also to significantly improve spatial skills and significantly more than Computer Aided Design Methods. This was summarized by means of the following (Suzuki, 2004):
Descriptive geometry > EG/ED (hand drawing) > EG (3D-CAD).
Various teaching methods are implemented in the mentioned above courses. Drawing on a blackboard or on an overhead projector accompanied by the overhead presentations of the real-life architectural constructions is the methodology use at CUT and UKL. The lecture in the classroom is usually supported with a manual (Leopold, 1999) or with the handouts given out to the students. Some materials are provided to the students through the internet site (Górska). At MTU a Multimedia software and a book has been developed to accompany a 3D Visualisation course. Recent research conducted by Sorby (2005) revealed the fact that if the students use the workbook together with a Multi-media software their performance on spatial tests is better than if they use only a workbook either only a software. It means that the type of instruction plays important role in spatial abilities development.
Keywords: Spatial Visualization Abilities, Graphics education, Gender differences

Lidia Żakowska
A9, CUT, Warszawska, 31-155 Kraków, Poland
email: lzakowsk@pk.edu.pl

Abstract. European policy regarding the elderly aims at maintaining their mobility. This is the central element of their integration with society. Senior citizens want to stay autonomous and independent as far as possible. This work presents some results of the on-going research project SIZE (Life quality of senior citizens in relation to mobility conditions), which is founded by the European Commission within the 5th Framework Programme, RTD Programme “Quality of life and living resources”, key action “The ageing population and disabilities”, contract No QLK6-CT-2002-02399. The general objectives of SIZE are:
- to explain and describe the present mobility and transport situation, the problems, needs and wishes of different groups of senior citizens from their own perspective compared with experts’ points of view (“experts” being sociologists, psychologists, traffic experts, experts on gerontology, architects and urban designers, urban planners, politicians, policy makers, experts of other related EU projects, etc.);
- to motivate action by the authorities and other relevant groups in society who are, or feel, responsible in this area, among others by making discrepancies in problem identification transparent;
- to identify relevant solutions for existing problems and to provide guidance for setting up and implementing policies aimed at “keeping the elderly mobile”.
The problem of senior citizens’ mobility is interdisciplinary in nature. In consequence, studying the problem demands both: specific research methodology, which would allow tapping this diversity and complexity, as well as multidimensional solutions. The methodology used in SIZE project, including a combination of focus group interviews, in-depth interviews and questionnaires, conducted with seniors and experts target groups, proved to be successful.
Road and traffic infrastructure, as technical aspects of moving around are very important factor determining senior citizens’ mobility. This statement is strongly supported by data gathered in all countries participating in SIZE project.
Research SIZE Project partners:
University of Vienna

Institute of Sociology

Anton Amann

Austria
FACTUM OHG

Ralf Risser

Austria
University of Erlangen – Nuremberg

Institute for Psychogerontology

H. J. Kaiser

Niemcy
National University of Ireland - Traffic Research Unit

Donncha O´Cinneide

Irlandia
Universitá degli Studi Roma Tre Di.P.S.A.

Lucia Martincigh

Włochy
Lund University – Department of Technology and Society

Agneta Stahl

Szwecja
Politechnika Krakowska (CUT) – Wydział Architekury

Lidia Żakowska

Polska
Centrum Dopravního Výzkumu (CDV) – Transport Research Centre

Karel Schmeidler

Czechy
Psychonomy Research Unit

Universitat de Valencia

Hector Monterde i Bort

Hiszpania
Keywords: Life quality, mobility, age, European research