Contents 29

ZBORNÍK SYMPÓZIA

O POČÍTAČOVEJ GEOMETRII

SCG´2009

Album

PROCEEDINGS OF SYMPOSIUM

ON COMPUTER GEOMETRY

SCG´2009

Volume 18

Slovak Society for Geometry and Graphics

Mechanical Engineering Faculty

Slovak University of Technology in Bratislava

October 2009, Kočovce, SR

ISBN 978-80-227-3141-6

Contents

M. Billich: Program WINGCLC vo vyučovaní kužeľosečiek . . . . . . . . . . . . . . . . . . . . . . . . . .	5
J. Błahut: Sequences of Selfsimilar Sets (2): the Trace of the Base . . . . . . . . . . . . . . . . . . . .	11
A. Błach, P. Dudzik, A. Pawlak: Macroconstructions in Programme Cabri II Plus Exemplified by Developments of Second Degree . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .	15
A. Błach, P. Dudzik, A. Pawlak: Surface Developments of Second Degree Based on Geometrical Places with the Aid of Cabri II Plus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .	19
A. Błach, A. Pawlak: Optimization of Assumptions Choice for Descriptive Geometry Tasks .	25
M. Božek: On Dense Placements of Star-Shaped Polygons . . . . . . . . . . . . . . . . . . . . . . . . . .	27
P. Csiba: Vlastnosti softvéru GeoGebra a jeho používanie . . . . . . . . . . . . . . . . . . . . . . . . . .	33
J. Čižmár: Aproximatívne geometrické konštrukcie . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .	39
V. Čmelková, V. Bálint ml., V. Bálint: Niekoľko novších výsledkov o p-číslach . . . . . . . . . . .	45
P. Chalmovianský: Singularity of an Algebraic Curve and Its Knot . . . . . . . . . . . . . . . . . . . .	51
M. Kmeťová: On Triangular Quadric Patches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .	58
S. Kudličková, M. Tisoň: Dynamický portál o rotačných plochách . . . . . . . . . . . . . . . . . . . . .	64
R. Merta, S. Sulwiński, K. T. Tytkowski: Descriptive Geometry Contrasting with Problem-Orientated Education or Return to Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .	70
M. Nagyová: Rôzne formy využitia programu GeoGebra vo vyučovaní matematiky . . . . . . .	75
P. Novotný: Zlepšenie dolnej hranice v nerovnosti G. Larchera a F. Pillichshammera . . . . .	82
B. Pabich: The Golden Ellipse´s Inventions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .	85
J. Poczęsna: Realization of Educational Standards in the Scope of Descriptive Geometry Exemplified by the Course of Environmental protection . . . . . . . . . . . . . . . . . . . . . . . . . . . .	91
E. Spernol, K. T. Tytkowski: Cycloidal Curves in Library of Flat Curves . . . . . . . . . . . . . . . .	97
D. Szarková: Tvorba výučbových materiálov a GeoGebra . . . . . . . . . . . . . . . . . . . . . . . . . . .	107
D. Velichová: Minkowského súčin v modelovaní plôch . . . . . . . . . . . . . . . . . . . . . . . . . . . .	113
P. Zahradník: Pathwise Monotone Geometric Bodies in the Plane and their Placements . . . .	120
S. Zachariáš: Stereografická projekce a kruhová inverze . . . . . . . . . . . . . . . . . . . . . . . . . . . .	128

Martin Billich

Pedagogická fakulta KU v Ružomberku
Námestie A. Hlinku 56/1, 034 01 Ružomberok, SR
e-mail: billich

fedu.ku.sk

Abstrakt. Dynamický geometrický softvér ponúka nové možnosti nielen na rýchle a presné rysovanie či znázorňovanie rôznych geometrických situácii v riešeniach klasických geometrických úloh (riešiteľných pomocou kružidla a pravítka), ale poskytuje študentom ideálne prostredie na skúmanie rôznych geometrických vzťahov, tvorbu hypotéz a ich overovanie. Tento príspevok sa zaoberá jedným z takýchto prostredí, konkrétne programom WinGCLC. Okrem základných informácii o programe v príspevku uvádzame možnosti jeho využitia pri štúdiu kužeľosečiek ako množín bodov daných vlastností.
Kľúčové slová: geometria, interaktívny softvér, WinGCLC, kužeľosečky.

Summary. Dynamic Geometry Software (DGS) offers new possibilities for the investigation of geometric loci. The common experience of DGS is that dynamic geometry tools help students to acquire knowledge about geometric objects. In this paper we present basic informations about WinGCLC – a tool for teaching and studying geometry and geometric constructions and for storing descriptions of geometrical figures. The system WinGCLC is based on the idea of formally describing figures, so it is close to mathematical way of thinking. We describe the possibility of using WinGCLC for the teaching of the conics and for the solving construction problems.

¹Anna Błach, ²Piotr Dudzik, ³Anita Pawlak

Geometry and Engineering Graphics Centre, Silesian University of Technology
ul. Krzywoustego 7, 44-100 Gliwice, Poland
¹e-mail: Anna.Blach

polsl.pl
²e-mail: Piotr.Dudzik

polsl.pl
³e-mail: Anita.Pawlak

polsl.pl¹

Abstract. The paper covers the domain which forms the continuation of the contents of paper “Surface developments of second degree based on geometrical places with the aid of CABRI II Plus’ presented at the Symposium on Computer Geometry in Kočovce in 2008.
Computer techniques incline to search for abilities of easy drawing of curves defined by a large number of points, as examples development line of intersections of second degree can be given.
Keywords: developments, geometric places, macroconstruction, CABRI II Plus.

¹Anna Błach, ²Piotr Dudzik, ³Anita Pawlak

Geometry and Engineering Graphics Centre, Silesian University of Technology
ul. Krzywoustego 7, 44-100 Gliwice, Poland
¹e-mail: Anna.Blach

polsl.pl
²e-mail: Piotr.Dudzik

polsl.pl
³e-mail: Anita.Pawlak

polsl.pl¹

Abstract. In the aspect of possibilities which computer techniques offer the aim has been set to determine algorithms which allow automatic drawing of curves developments of cylinder surfaces (of revolution and noncircular ones) and cone (of revolution and noncircular ones). For that purpose CABRI II Plus program has been used, and in particular, its options which allow generation of geometrical places.
Keywords: developments, geometric places, CABRI II Plus.

¹Anna Błach,²Anita Pawlak

Geometry and Engineering Graphics Centre, Silesian University of Technology
ul. Krzywoustego 7, 44-100 Gliwice, Poland
¹e-mail: Anna.Blach

polsl.pl
²e-mail: Anita.Pawlak

polsl.pl

Abstract. Computer techniques give ever bigger possibilities of improvements in the area of didactics. As an example Cabri II Plus program can be given, which offers possibilities of assumption modification and task solutions from descriptive geometry.
Keywords: Cabri II Plus

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

fmph.uniba.sk

Abstract. The aim of the presented paper is to prove that, under an additional technical assumption, the set D(M, N) of all dense placements of star-shaped polygons M and N is a simple closed broken line, which is the boundary of a star-shaped polygon.
Keywords: Stare-shaped set, strictly stare-shaped set, translation, overlapping of polygons, densely placed polygons.

Peter Csiba

Pedagogická fakulta, Univerzita J. Selyeho
Bratislavská 3322, 945 01 Komárno, SRe-mail: csiba.peter

selyeuni.sk

Abstrakt. GeoGebra je voľne dostupný matematický softvér, ktorý bol vyvinutý predovšetkým na to, aby sa pomocou neho dalo lepšie pochopiť a vyučovať matematiku. Cieľom príspevku je podať krátky (neúplný) pohľad na to, čo je to za softvér, ako sa používa a hlavne na čo sa dá použiť. Namiesto všeobecného opisu softvéru a jeho možností obsahuje niekoľko konkrétnych príkladov s ich popisom.
Kľúčové slová: GeoGebra, interaktívna konštrukcia, aplet.

Summary. GeoGebra is free and multi-platform dynamic mathematics software for learning and teaching. The aim of this paper was to present a short introduction and characterization of this software, from the point of view of practice. The paper contains some concrete examples with detailed description..

Ján Čižmár

Fakulta matematiky, fyziky a informatiky Univerzity Komenského
Mlynská dolina 842 48 Bratislava, SR
e-mail: cizmar

fmph.uniba.sk

Abstrakt. Článok sa zaoberá podstatou aproximatívnych geometrických metód riešenia konštrukčných geometrických úloh. Ako ukážky použitia takýchto metód sú uvedené niektoré riešenia troch klasických antických úloh – trisekcie uhla, zdvojnásobenia kocky a kvadratúry kruhu.
Kľúčové slová: aproximatívne geometrické konštrukcie, klasické úlohy antiky.

Summary. This paper deals with the substance of approximative geometric methods of solving constructional geometric problems. In the role of examples of applying such methods are presented several solutions of three classical ancient problems – angle trisection, cube duplication and circle squaring.

¹Vojtech Čmelková, ²Vojtech Bálint ml., ³Vojtech Bálint

Katedra KMAHI fakulty PEDAS ŽU v Žiline
Univerzitná 1, 010 26 Žilina, SR
¹e-mail: viera.cmelkova

fpedas.uniza.sk
²e-mail: vojtob

centrum.sk
³e-mail: balint

fpedas.uniza.sk

Abstrakt. Okrem stručného prehľadu známych výsledkov poskytuje článok nové výsledky (Lema 3, Veta 4) o p-číslach.
Kľúčové slová: p-číslo, parketovanie, trojuholník, štvorec, n-uholník.

Summary. Besides the short survey of the known results about Partridge numbers this paper gives also some new results in the Lemma 3 and Theorem 4.

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 present a construction of a knot corresponding to an isolated singular point of an algebraic curve in the complex plane. The local structure of the singular point is transformed to the structure of the knot, for which we have several well known topological characterizations such Alexander polynomial and other invariants.
On the other hand, certain types of knots can be transformed or characterized via planar algebraic curves and their singular points (however not all).
We give an example of computation of such characteristics using Puiseux series and visualize the whole process using carousel diagram based on a sequence of approximations for the parametrization.
Keywords: algebraic curves, singular points, knot.

Mária Kmeťová

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

ukf.sk

Abstract. The paper contents overview of various constructions of (mainly quadratic) triangular patches. Examples are illustrated by program Maple.
Keywords: quadratic triangular Bézier patches, WRD construction, Gordon-Coons patch.

¹Soňa Kudličková, ²Miroslav Tisoň

Comenius University, Faculty of Mathematics, Physics and Informatics
Mlynská dolina, 842 48 Bratislava, SR
¹e-mail: sona.kudlickova

fmph.uniba.sk
²e-mail: miroslav.tison

fmph.uniba.sk

Abstrakt. V príspevku predstavíme internetový portál o rotačných plochách vytvorený v systéme CMS Joomla!. Navrhnutý portál je diferencovaný vzhľadom na stupne vzdelávania (základná, stredná, vysoká škola) a podľa etáp vyučovacieho procesu (učebné texty, riešené príklady, zbierka úloh).
Kľúčové slová: rotačné telesá, rotačné plochy, výučbové materiály, on-line, informačno-komunikačné technológie.

Summary. The paper deals with description of a new web portal Rotacneplochy.sk. The e-learning forms of a topic - surfaces of revolution – for different levels of education are presented.

¹Ryszard Merta, ²Stanisław Sulwiński, ³Krzysztof T. Tytkowski

¹Faculty of Automatics, Electronics and Informatics, Silesian University of Technology
ul. Akademicka 13, 44-100 Gliwice, Poland
^2,3Geometry and Engineering Graphics Centre, Silesian University of Technology
ul. Krzywoustego 7, 44-100 Gliwice, Poland
²e-mail: stanislaw.sulwinsky

polsl.pl
³e-mail: krzysztof.tytkowskil

polsl.pl

Abstract: In this paper attention is paid to the renaissance of geometry following from problem-orientated education. Complex industrial design processes cause CADD programs to be formed as if, by accident, and latter to develop such fields as: modeling geometry, constructive geometry and calculation geometry. So, in the process of the teaching of descriptive geometry, we should not forget about problem-orientated education and the new fields of geometry e.g. fractal geometry, fundamentals of metric topology or linear space (vector notation, matrix form of calculation).
Keywords: descriptive geometry, problem-orientated education.

Magdaléna Nagyová

Fakulta Matematiky, Fyziky a Informatiky UK
Mlynská Dolina 842 48, Bratislava, SR
e-mail: nagyova

gmail.com

Pavel Novotný

FPEDaS, Žilinská univerzita
Univerzitná 1, 010 26 Žilina, SR
e-mail: pavel.novotny

fpedas.uniza.sk

Abstrakt: V práci je zlepšený dolný odhad pre súčet druhých mocnín vzdialeností medzi vrcholmi konvexného n-uholníka s jednotkovým obvodom.
Kľúčové slová: vzdialenosť, konvexný n-uholník

Summary. We improve the lower bound in the inequality of G. Larcher and F. Pillichshammer for the sum of squared distances between vertices of a convex polygon with unit perimeter.

Abstract. In the lecture I present how, starting from a known and simple tasks with planimetry can lead to the discovery of the structure where the ellipse parameters are precisely related to the golden ratio.
Keywords: golden ratio, golden ellipse, dynamic geometry, software CABRI, square inscribed in a triangle

Janina Poczęsna

Geometry and Engineering Graphics Centre The Silesian University of Technology, Gliwice, Poland

Abstract: The aim of his paper is to show the role of educational standards in creation of European Space of Higher Education after conference in Glasgow. Realization of standards in the scope of the subject engineering graphics at the course Environmental Protection at the Silesian University of Technology has been presented.
Keywords: European system of higher education, educational standards, descriptive geometry.

¹Elżbieta Spernol, ²Krzysztof T. Tytkowski

Geometry and Engineering Graphics Centre, Silesian University of Technology
ul. Krzywoustego 7, 44-100 Gliwice, Poland
¹e-mail:elzbieta.spernol

polsl.pl
²e-mail: krzysztof.tytkowskil

polsl.pl

Abstract: In this paper the authors have presented the historical aspect of each curve which occurs in the library. The purpose of the prepared set is popularization of the curious curves and presentation of the history of research over them together with their special cases.
Keywords: flat curves, history of geometry, cycloidal curves.

Dagmar Szarková

SjF STU, Námestie slobody 17, 812 31 Bratislava, SR
e-mail: dagmar.szarkova

stuba.sk

Abstrakt: Cieľom príspevku je prezentácia rôznych typov apletov – výučbových materiálov vytvorených v dynamickom matematickom softvéri GeoGebra voľne prístupnom na internete a o možnostiach ich vytvárania. GeoGebra slúži predovšetkým pre účely vyučovania matematiky a geometrie.
Kľúčové slová: výučbový materiál, aplet, softvér GeoGebra.

Summary. The paper deals with the development of different types of applets - instructional materials created in GeoGebra, dynamic mathematical software freely accessible on the Internet, which is used primarily for the purpose of teaching and learning mathematics and geometry. Not all applets, especially the dynamic ones, could be presented in the paper, so the above described various types can be found in the learning modules cykloidy.html, which is located in the Slovak database portal of the European Virtual Laboratory of Mathematics, at the address http://slovak.evlm.stuba.sk/elearning/ and entitled "Cyklické krivky”.

Daniela Velichová

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

stuba.sk

Abstrakt: V článku je zavedený pojem Minkowského súčinu bodových množín euklidovského priestoru a uvedené sú niektoré jeho vlastnosti. Minkowského súčin bodových množín sa dá aplikovať ako generujúci princíp pri modelovaní kriviek a plôch zaujímavých tvarov a vlastností. Uvedených je niekoľko príkladov jeho využitia pri modelovaní oblúkov kriviek, listov plôch a buniek masívov ako Minkowských súčinov bodových množín určených vektorovými rovnicami.
Kľúčové slová: Minkowského súčin, modelovanie kriviek, plôch a masívov.

Summary. Minkowski product of the point sets in the Euclidean space is introduced in the paper, and some of its basic properties are presented. Minkowski product of point sets can be applied as the generating principle in modelling curves and surfaces of interesting forms and properties. Several examples of its usage for modelling of curve segments and surface patches are included, determined as Minkowski products of point sets defined by vector representations.

Peter Zahradník

Comenius University, Faculty of Mathematics, Physics and Informatics
Mlynská dolina, 842 48 Bratislava, SR
e-mail: zahradnik

fmph.uniba.sk

Abstract. The aim of the present paper is to introduce a geometrically demonstrative definition of a monotone geometric body in the plane based on joining of points of the body by a certain curve lying in it. Moreover, if there are given two such monotone bodies in the plane we show that the set of intersections of them is also a monotone body and the set of their dense placements is its boundary.
Keywords: Geometric body, pathwise monotonicity, vertical line, set of intersections and dense placements of two bodies.

Abstrakt. V příspěvku jsou vykresleny dráhy těžiště a ortocentra trojúhelníka a středů kružnice vepsané i opsané. Tento trojúhelník je tvořen ručičkami na hodinovém ciferníku (nejen dvanácticiferného, ale i pro obecnější případ). V úvodu je ukázána souvislost použité metody kruhové inverze a stereografické projekce.
Klíčové slova: Stereografická projekce, kruhová inverze, GeoGebra