Contents 27

ZBORNÍK SYMPÓZIA

O POČÍTAČOVEJ GEOMETRII

SCG´2007

Album

PROCEEDINGS OF SYMPOSIUM

ON COMPUTER GEOMETRY

SCG´2007

Volume 16

Slovak Society for Geometry and Graphics

Mechanical Engineering Faculty

Civil Engineering Faculty

Slovak University of Technology in Bratislava

October 2007, Kočovce, SR

ISBN 978-80-227-2734-1

Contents

V. Bálint, V. Čmelková: Ako nájsť presné hodnoty počtu prostých priamok určených n bodmi v rovine? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..	5
J. Błahut: A Semigroup with No Finite Base (2) . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . .	12
M. Božek: Rolling of Broken Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . .	17
S. Cichacz, T. Wieja: Computer Aided Teaching of Central Projection in the Flash MX Program . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .	23
J. Dietrich, M. Gawron, T. Wieja: Application of 2D and 3D Computer Models to Construction and Presentation of Selected Geometrical Problems . . . . . . . . . . . . . . . . . . . .	27
R. Górska: Some Geometrical Constructions in Modern Edition (Flash MX) . . . . . . . . . . . . .	32
V. J. Havel, V. Sedlář: Testing Possible Central Projection Images . . . . . . . . . . . . . . . . . . . .	33
P. Chalmovianský: Visualisation of a Planar Algebraic Curve - Toplogy and Geometry . . . .	39
M. Kmeťová: On Generation of Ruled Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .	44
A. Koch, T. Wieja: Franciszek Sapalski "Geometria wykreślna z zastosowaniem do perspektywy, cieniów, kamieniarstwa, ciesiołki i innych konstrukcyy, wypracowana dla užytku szkoły woyskowey applikacyyney" - the First Descriptive Geometry Textbook in Poland - Ananlysis of the Time Methodology and Didactics of the Subject in the Context of the Contemporary Teaching Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .	48
S. Kudličková: Some Remarks on the Shape of Hermite Splines . . . . . . . . . . . . . . . . . . . . . .	53
D. Lizoňová: Geometrické tvary s mystickou symbolikou . . . . . . . . . . . . . . . . . . . . . . . . . . . .	59
I. Markechová: A Notelet to the Kakeya Needle Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . .	64
M. Nociar: Taxonómia prístupov k úrovniam detailu. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .	68
P. Novotný: Najhoršie pakovateľné štyri kocky. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .	78
P. Ňuňuková: Zhodné zobrazenia na ZŠ aplikovaním Cabri.. . . . . . . . . . . . . . . . . . . . . . . . .	82
D. Onačilová: E-learning for Computer Graphics: Urbánne rekonštrukcie . . .. . . . . . . . . . .	87
V. Palaj: Bézier Curves Intersection Using Monge Projection. . . . . . . . . . . . . . . . . . . . . . . .	90
M. Polan: Možnosti formátu SVG a jeho využitie v elektronickom publikovaní. . . . . . . . . . .	98
J. Procházková, J. Sedlík, D. Procházka: Direct B-spline Interpolation of CNC Path from Cloud of Points. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .	104
M. Smoleňová: Prehľad geometrických metód na modelovanie virtuálnej vegetácie. . . . . . . .	110
Z. Študencová, J. Zámožík: Faldy (Folding Surfaces). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .	115
E. Tercyńska, J. Tofil: Electronic Form of a Collection of Exercises in Descriptive Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .	121
J. Tofil: Know how? - Multimedia Lectures on Descriptive Geometry - Issues of Perspective.	125
M. Vajsáblová: Kužeľové projekcie . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . .	128
D. Velichová: Modelovanie sféroidov . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .	136
E. Vranková: Monotony of Rectangular Polygons and Their Dense Placement . . . . . . . . . . .	144
P. Zahradník: A Boundary of a Monotone Geometric Body in the Euclidean Plane . . . . .. . .	152
S. Bogacki: Application of 3d graphics in didactics and research
J. Poczęsna, M. Sroka-Bizoń: Adapting Requirements of Higer Education to The European Requirements. New Education Standards in Descriptive Geometry illustrated by the Course of Architecture and Urban Planning

Vojtech Bálint, Viera Čmelková
Katedra KMAHI fakulty PEDAS ŽU v Žiline, Univerzitná 1, 010 26 Žilina, SR
¹e-mail: balint

fpedas.utc.sk

Abstrakt. V práci je prehľad súčasného stavu najdôležitejších poznatkov o počte prostých priamok. Vety 3-5 poskytujú množstvo otvorených problémov.
Kľúčové slová: bod, priamka

Summary. The paper gives a short survey on the problem of ordinary lines. Theorems 3-5 offer many open problems.

Miloš BožekComenius University, Faculty of Mathematics, Physics and Informatics, Mlynská dolina, 842 48 Bratislava, SR
e-mail: bozek

fmph.uniba.sk

Abstract. The main aim of the paper is to discuss approximations of roulettes by broken lines induced by broken lines approximations of input curves.
Keywords: Roulette, rolling of a curve without slipping, simple broken line.

Silvia Cichacz¹, Tomasz Wieja²
AGH University of Science and Technology, Faculty of Applied Mathematics
30-059 Kraków, al.Mickiewicza 30, Poland,
¹e-mail: cichacz

agh.edu.pl
²e-mail: twieja

poczta.fm

Abstract. In the paper have been presented selected spatial problems performed in the Central Projection method and worked out in the Flash MX program in the form of computer animation. Separate elements of the constructions have been demonstrated on the visual axonometric diagram as well as in the Central Projection method.
Keywords: Flash, Monge projection, animation, visualization, vectorial picture, collineation.

Jerzy Dietrich¹, Marek Gawron², Tomasz Wieja³
AGH University of Science and Technology, Faculty of Applied Mathematics
30-059 Kraków, al.Mickiewicza 30, Poland,
¹e-mail: jdietrich

wp.pl²e-mail: mwgawron

poczta.fm
³e-mail: twieja

poczta.fm

Abstract. In the article we have presented classical constructions made by means of vectorial programs. The constructions have been shown in the form of 2D diagrams and 3D visualizations.
Keywords: computers model, construction, presentation, visualization, AutoCAD, CABRI 3D, CABRI 2D, CorelDRAW.

Renata Górska
CUT, Warszawska 24, 31-155 Kraków, Poland
e-mail: rgorska

pk.edu.pl

Abstract. The power of some geometrical planar constructions has been presented by means of using a modern tool, which is Micromedia Flash MX. The project has been addressed to pupils at higher classes from the ground school, to the students from high schools and to all fans of geometry. The beauty of many well forgotten constructions can be newly discovered through Internet presentations. Micromedia Flash is the most appropriate and useful software to create dynamic presentations. The student is able to run the ‘step-by-step’ constructions at the pace which is most effective and adequate personally for him or her. The powers of vector graphics together with the tool for exporting the final product into *.swf format enables application of the product for creation of a web page. A few examples of geometrical planar constructions will be presented as an example for geometrical e-learning.
Keywords: e-learning, Micromedia Flash, geometrical constructions

V. J. Havel¹, Vladimír Sedlář²
¹Department of Mathematics, Faculty of Electrical Engineering and Communications, Brno University of Technology, 61600 Brno,Technická 8, ČR,
e-mail: havel

feec.vutbr.cz
²Mathematical Institute of the Silesian University in Opava, Na Rybnicku 1, 746 01 Opava, Czech Republic,
e-mail: Vladimir.Sedlar

math.slu.cz

Abstract. This remark improves on some old contributions of Chetverukhin and Beskin to the classical criterion of Kruppa for a central axonometry to be a central projection. Several additional observations are attached.
Keywords: Desargues configurations, central projection, fundamental conic of a polarity.

Pavel Chalmovianský
Comenius University, Faculty of Mathematics, Physics and Informatics, Mlynská dolina, 842 48 Bratislava, SR
e-mail: Pavel.Chalmoviansky

fmph.uniba.sk

Abstract. We summarize existing methods for computing topology of a real planar algebraic curve. We consider symbolic as well as numeric approaches due to the different nature of the results and the speed of the algorithms we can obtain using them. The study of these methods serve as a preliminary study for the treating the similar problem in case of real curves in higher
dimensional spaces and in case of real surfaces in RP3, which singular points can form an algebraic curve.
Keywords: planar algebraic curves, approximation, singularities

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

ukf.sk

Abstract. Ruled surfaces are important objects in geometric modeling. In this article we would like to describe ruled surfaces as one parameter sets of lines on line congruencies using their Klein images.
Keywords: projective space, Plücker coordinates, Klein mapping, line complexes, line congruencies

Andrzej Koch¹, Tomasz Wieja²
AGH University of Science and Technology, Faculty of Applied Mathematics,
30-035 Kraków , al.Mickiewicza 30 , Poland
¹e-mail: makoch28

interia.pl
²e-mail: twieja

poczta.fm

Abstract. In the paper we present history and factual analysis of the first textbook on Descriptive Geometry printed on the Polish land in 1822 in Warsaw. The factual content of the reviewed textbook reflects well on the world – class standard of that time teaching Descriptive Geometry in Poland.
Keywords: Descriptive Geometry, history, teaching, construction, methodology, didactics

Soňa Kudličková
FMFI UK, Mlynská dolina, 842 48 Bratislava, SR
e-mail: kudlickova

fmph.uniba.sk

Abstract. Hermite cubic segment and Hermite spline are modified by the tangent vectors. The shape of the Hermite spline is altered by a choice of the parametrization and desired shape is achieved with a construction of the fair spline.
Keywords: Hermite interpolation, continuity conditions, end conditions, circular arc

Denisa Lizoňová
KMDG TU Zvolen, Masarykova 24, 96053 Zvolen, SR
e-mail: lizonova

vsld.tuzvo.sk

Abstrakt: Používanie geometrických tvarov na vyjadrenie rôznych javov skutočnosti sa objavuje už v prehistorickom období a najväčší rozmach zaznamenáva v antike. Posolstvá a významy, ktoré sú takto symbolicky vyjadrené, sú aktuálne dodnes. Správna znalosť symbolu a jeho významov je jedným z krokov v úspešnej tvorivej práci dizajnéra. Využitie geometrickej symboliky pri navrhovaní produktov je jedným z mnohých faktorov, ktorými možno dosiahnuť pozitívne prijatie umeleckého predmetu verejnosťou.
Kľúčové slová: Platónske telesá, kvet života, zlatý rez, logaritmická špirála

Summary. Geometry plays a very important role in the field of human consciousness. We can see that all things throughout our universe seem to follow the same fundamental blueprint or geometric patterns. These geometrical archetypes reveal to us the nature of each form. They are also symbols of the underlying metaphysical principle of the inseparable relationship of the part to the whole. When geometry is used to explore that all life emerges from the same blueprint it becomes sacred. The value of geometry and its need for common life can be followed as early as the old time.

Iveta Markechová
Slovak University of Technology, Faculty of Materials Sciences and Technologies, Institute of Applied Informatics, Automation and Mathematics,
Department of Mathematics, Hajdóczyho 1, 917 24 Trnava, SR
e-mail: mark

mtf.stuba.sk

Abstract. The aim of this paper is partially cultural (to present some petty applications of famous Kakeya needle problem to traditional exercises on differential calculus, elementary geometry, kinematics, dynamics basic course at technical university) as well as partially provocative (to generate student’s interest in differential geometry, nonstandard reparametrization techniques, numerical parametrization, etc.).
Keywords: Kakeya needle problem, geometry, kinematics, parametric equations, reparame-trization

Michal Nociar
FMFI UK, Mlynská dolina, 842 48 Bratislava, SR
e-mail: nociar

sccg.sk

Abstrakt. Príspevok ponúka stručný prehľad metód na konštrukciu a udržiavanie rôznych úrovní detailov pre polygonálne aproximované objekty (najčastejšie siete trojuholníkov).
Kľúčové slová: Úrovne detailov, viacúrovňový model, zjednodušujúce operátory, simpliciálny komplex.

Summary. This short paper presents a survey on various methods for construction and maintenance of levels of detail for polygonal objects.

Pavel Novotný
F PEDaS, Žilinská univerzita, Univerzitná 1, 010 26 Žilina, SR
e-mail: Pavel.Novotny

fpedas.uniza.sk

Abstrakt: V práci je nájdené najmenšie číslo V₄ také, že každý systém štyroch kociek s celkovým objemom 1 sa dá uložiť do kvádra s objemom V₄.
Kľúčové slová: pakovanie kociek

Summary. Any system of four cubes of total volume 1 can be packed into a box of volume at most 1,5196303266.

Pavla Ňuňuková
FMFI UK Bratislava, Mlynská dolina, 842 48 Bratislava, SR
e-mail: nunukova

sccg.sk

Abstrakt. V dnešnej dobe sa na vyučovanie geometrie na základných a stredných školách využíva prevažne Cabri geometria. Je výhodné, ak učitelia majú predpripravené či už motivačné alebo iné príklady a úlohy. V tomto príspevku sa budeme niekoľkými typmi úloh zaoberať. Tiež sa pozrieme na možnosti iných softvérov, ktoré by bolo možné využiť pri vyučovaní. Porovnáme tieto softvéry navzájom, aby sme zistili ich prednosti a nedostatky.

Summary. This paper discusses about different types of software for teaching mathematics on elementary and secondary schools. Cabri geometry is used nowadays, and we try to find a better one because on our schools there is just its demo version. The second aim was to create new examples on how to teach translation and symmetries on elementary schools. The most important is motivation, so there are some examples how to use and create this in Cabri geometry.

Daniela Onačilová
FMFI UK, Mlynská Dolina, 842 48 Bratislava, SR
e-mail: dankao

zmail.sk

Abstrakt: Táto práca sa venuje vyučovaniu počítačovej grafiky. Konkrétne v oblasti urbánnych rekonštrukcií. Jej výsledkom je stránka zverejnená na internete, ktorej obsahom sú tutoriály pre autorské nástroje (hugin, Blender, TrueSpace).
Kľúčové slová: urbánne rekonštrukcie, e-learning, počítačová grafika

Vladimír Palaj
KAGDM FMFI UK, Mlynská dolina, 842 48 Bratislava, SR
e-mail: palaj

fmph.uniba.sk

Abstract. Intersection of spatial curves became general problem in the sphere of geometric modeling and computational geometry. To find a set of points of the curve/curve intersection could be performed by several algorithmic techniques. At the present time there exist several different approaches to this problem but the endeavor is to avoid difficulties in calculation which are mainly results of polynomial representation of higher degree curves. We stay indeed at the original polynomial approach, e.g. Bézier clipping method, established by Nishita et al. in [1990]. Presented paper describes the mathematical background of this method that solves existing problem of two curves intersection on class plane Bézier curves only.
Keywords: Bézier curve, orthogonal projections, Bézier clipping, Monge projection

Michal Polan
KAGDM FMFI UK, Mlynská dolina, 842 48 Bratislava, SR
e-mail: michal

polan.sk

Abstrakt: V príspevku sa zaoberáme základnými vlastnosťami formátu SVG a jeho praktickým použitím pri publikovaní elektronických dokumentov.
Kľúčové slova: SVG, grafický formát, vektorová grafika, e-learning, obrázok, publikovanie

Jana Procházková¹, Josef Sedlák², David Procházka³
^1,2FSI VUT, Technická 2896/2, 616 69 Brno, CZ
e-mail: jprochazkova

gmail.com
³MZLU v Brně, Zemědělská 1, 613 00 Brno, CZ
e-mail: xproch17

pef.mendelu.cz

Abstract: Reverse Engineering (RE) is technology used in CAD, CAM, CAE systems for shorter time for development, construction and production of components. The gist of RE is reproduction of some existing thing – circuit, bone, etc. New area of interest in RE in mechanical engineering is interested in classifying massive points. Our intension here is to describe the method for scanning of components (using MicroScribe) and principle of interpolation of surfaces based on B-spline curves. This method is used in health services for the joint replacements.
Keywords: reverse engineering, B-spline interpolation, joint replacement, cloud of points

Katarína Smoleňová
FMFI UK, Mlynská dolina, 842 48 Bratislava, SR
e-mail: smolenova

fmph.uniba.sk

Abstrakt: V práci je predstavená klasifikácia geometrických modelov stromov založená na rôznych stupňoch komplexnosti. Vzhľadom na požadovanú úroveň detailov môžeme hovoriť o detailnej, globálnej alebo viacúrovňovej reprezentácii. Dôraz je kladený na detailnú reprezentáciu a geometrické metódy na modelovanie vetviacich štruktúr stromov.
Kľúčové slová: modelovanie stromov, geometrická reprezentácia stromov, modulárna reprezentácia, vetviace štruktúry, realistické zobrazenie.

Summary. There are several geometric methods for modeling of branching stuctures of trees. The aim is a realistic representation of trunk and branches. This paper is concerned with classification of geometric models based on the approach of creation of continuous model from a discrete set of geometric primitives. The main difficulty is to define exact representation of branch junctions, which is actual for detail representation of trees.

Zuzana Študencová¹, Jozef Zámožík²
¹SvF STU, Radlinského 11, 813 68 Bratislava, SR
e-mail: studenc

math.sk
²MtF STU, Paulínska 16, 917 24 Trnava, SR
e-mail: zamozik.j

seznam.cz

Abstrakt: S útvarmi typu fald (folding surface) sa stretávame v bežnom živote. V príspevku venujeme predovšetkým pozornosť geometrii jedného typu faldov.
Kľúčové slová: folding surfaces, fald, faldový element, fázová poloha

Summary. This paper describes geometric figure, surfaces – folding surfaces. We know this surfaces from live more as from geometry. Our focus is on one of more folding surfaces.

Ewa Tercyńska¹, Jolanta Tofil²
Geometry and Engineering Graphics Centre, Silesian University of Technology
ul. Krzywoustego 7, 44-100 Gliwice, Poland
¹e-mail: ewa.terczynska

polsl.pl
²e-mail: jolanta.tofil

polsl.pl

Abstract. Short summary of an electronic form of a set of exercises for the first year students of the Faculty of Civil Engineering of the Silesian University of Technology.
Keywords: descriptive geometry, a set of exercises, repetitions, Power Point, Cabri

Jolanta Tofil
Geometry and Engineering Graphics Centre, Silesian University of Technology
ul. Krzywoustego 7, 44-100 Gliwice, Poland
e-mail: jolanta.tofil

polsl.pl

Keywords: descriptive geometry, perspective, multimedia presentation, didactic solutions.

Margita Vajsáblová
SvF STU, Radlinského 11, 813 68 Bratislava, SR
e-mail: margita.vajsablova

stuba.sk

Abstrakt: Obsahom príspevku je ukázať jednotlivé typy kužeľových projekcií referenčnej plochy zeme, ich geometrické vlastnosti s ohľadom na použitie v kartografii. Hlavným cieľom je sformulovať zobrazovacie rovnice kužeľových zobrazení s pohyblivým stredom v rovine rovníka, príp. v rovine rovnobežkovej kružnice a podmienky pre zobrazenie bodov guľovej plochy v závislosti od voľby kužeľovej plochy a polohy stredu premietania.
Kľúčové slová: referenčná plocha, zemepisné súradnice, kužeľové projekcie

Summary. The theme of this paper is to show all types of the conic perspective projections of reference surface of Earth and their geometric properties considering their application in cartography. The cardinal aim is formulation of transformation equations of conic perspective projection with rotation of the projection centre located on the equator plane or on the latitude circle plane. Conditions of the conic projections of the sphere are determining in dependence of the conic surface location and the projection centre position.

Daniela Velichová
¹SjF STU Bratislava, Nám. slobody 17, 812 31 Bratislava, SR
e-mail: daniela.velichova

stuba.sk

Abstrakt: V článku sú opísané niektoré vlastnosti a formy plôch nazvaných sféroidy, ktoré sú špeciálnou podskupinou zovšeobecnených cyklických dvojosových rotačných plôch sférického typu. Na ilustráciu je pripojených niekoľko ukážok sféroidov modelovaných v programe Maple.
Kľúčové slová: dvojosová rotácia, cyklické dvojosové rotačné plochy sférického typu, sféroid, modelovanie plôch

Summary. Some of the properties and forms of surfaces called spheroids are described in the paper. Spheroids form special subgroup of cyclical two-axial surfaces of revolution of the spherical type. Several samples of surfaces created in the programe Maple are included.

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

truni.sk

Abstract. The placement of mutually congruent polygons belongs to the easiest placing tasks. The presented paper fasten on paper [7]. There were showed some geometrical properties of the set D(M,N) of congruent rectangular polygons M, N symmetrical about some point or line. Now, the aim is to show some geometric properties of D-monotone rectangular polygons and the monotony influence on the geometric structure of D(M,N).
Keywords: Monotone polygon, rectangular polygon, the set D(M,N).

Peter Zahradník
FMFI UK, Mlynská dolina, 842 48 Bratislava, SR
e-mail: zahradnik

fmph.uniba.sk

Abstract. The aim of the present paper is to define the term of a monotone geometric body in the Euclidean plane with respect to a given direction. In a case that the geometric body is bounded and has a connected interior we prove the theorem by which it is easier to determine whether the geometric body is monotone with respect to a given direction.
Keywords: Geometric body, boundary, monotony, connectivity

Sławomir Bogacki
Geometry and Engineering Graphics Centre, Silesian University of Technology
ul. Krzywoustego 7, 44-100 Gliwice, Poland
e-mail: slawomir.bogacki

polsl.pl

Abstract. In the presentation the examples of application 3D graphics in didactics and research in the selected fields will be shown. The animation and modeling software will be also introduced in the presentation.
Keywords: visualization, computer animation, didactic solutions. multimedia presentation

Janina Poczęsna, Monika Sroka-Bizoń
Geometry and Engineering Graphics Centre, Silesian University of Technology
ul. Krzywoustego 7, 44-100 Gliwice, Poland
e-mail: rjm4

polsl.pl

Abstract. Up till now education within the scope of architecture and urban planning has been realized at the majority of Architecture Departments of public technical universities as fulltime MSc studies. Descriptive geometry education was realized as 90 hours of classes. New education standards with two level system of studies will be realized in the new academic year 2007/2008. Gathered experience connected with realization of didactic classes will let us form first assessment of new education system.
Keywords: European system of higher education, education standards, desciptive geometry