Contents 28

ZBORNÍK SYMPÓZIA

O POČÍTAČOVEJ GEOMETRII

SCG´2008

Album

PROCEEDINGS OF SYMPOSIUM

ON COMPUTER GEOMETRY

SCG´2008

Volume 17

Slovak Society for Geometry and Graphics

Mechanical Engineering Faculty

Slovak University of Technology in Bratislava

October 2008, Kočovce, SR

ISBN 978-80-227-2952-9

Contents

V. Bálint, V. Bálint jr.: Husté rozmiestnenia bodov v kockách . . . . . . . . . . . . . . . . . . . . . . .	5
S. Bogacki: Interactive 3D Graphics - Tools and Examples of Application . . . . . . . . . . . .	11
M. Božek: On Dense Placements Based on Arbitrary Rigid Motions in the Plane . . . . . . .	12
J. Čižmár, Z. Sklenáriková: Predvečer zrodu neeuklidovskej geometrie (neeuklidovské tendencie v elementárnej geometrii 18. storočia) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .	18
M. Dobelis: Geometry and Graphics as CAD Prerequisite in Engineering Education . . . .	26
M. Gawron, J. Dietrich, A. Lankosz, T. Wieja: Computer Visualization of the Polar-Stereographic Projection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .	32
M. Gawron, T. Wieja: Teaching Engineering Graphics Exemplified by Syllabus of Faculty of Physics and Computer Science of the AGH University of Science and Technology . . . .	34
R. Górska, A. Lankosz: Designing and Implementing New Courses at Engineering Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .	39
P. Chalmovianský: Classification of Simple Isolated Singular Points and Intersection Multiplicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .	44
M. Kmeťová: Construction of Ruled Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .	52
M. Kočandrlová, J. Vecková, S. Olivík: Internet Support of Constructive Geometry . . . . .	59
S. Kudličková: Designing with B-Spline Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .	63
A. Lankosz, R. Górska: Visualization and Some Properties of the Polar-Stereographic Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .	70
M. Nociar: Historický vývoj merania chyby pomocou kvadratických funkcionálov v procese zjednodušovania meshov. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .	75
L. Pletenac: Classification of Quartics, Deduced by Projective Transformation. . . . . . . . .	81
A. Sliepčević: Die Kegelschnitte und ihre verwandte Kurven.. . . . . . . . . . . . . . . . . . . . . . .	86
D. Stachová: Modelovanie jazdnej dráhy vozidla po vozovke s poškodeným povrchom . . .	87
M. Vajsáblová: Zobrazovacie rovnice pseudocylindrických priamkových zobrazení . . . . . .	91
D. Velichová: Minkowského súčet v geometrickom modelovaní . . . . . . . . . . . . . . . . . . . . .	96
M. Vojteková, M. Stacho: Grafická reprezentácia úlohy lineárneho programovania . . . . .	102
E. Vranková: Niektoré vlastnosti množiny hustých rozmiestnení ľubovoľných mnohouholníkov . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .	107
P. Zahradník: On Dense Placements of D-Monotone Geometric Bodies in the Euclidean Plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .	113
Anna Błach, Piotr Dudzik, Anita Pawlak: Rozwinięcia powierzchni stopnia drugiego w oparciu o miejsca geometryczne i makrokonstrukcje z zastosowaniem programu CABRI II PLUS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .	121

¹Vojtech Bálint, ²Vojtech Bálint jr.

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

fpedas.uniza.sk
²e-mail: vojtob

centrum.sk

Abstrakt. Práca podáva stručný prehľad výsledkov o hustých pakovaniach bodov do viacrozmernej kocky a poskytuje tak obrovské množstvo otvorených problémov.
Kľúčové slová: bod, sféra, kocka, hustota

Summary. The paper gives a brief survey of the results on the dense packings of points into the cube and offers a lot of unsolved problems.

Sławomir Bogacki

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

polsl.pl

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 paper is to discuss possibilities of extending the concept of dense placements of plane figures with respect to translations to the corresponding concept based on all rigid motions in the Euclidean plane.
Keywords: Geometric figure, dense placement, rigid motion in the Euclidean plane.

¹Ján Čižmár, ²Zita Sklenáriková

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

fmph.uniba.sk
²e-mail: sklenarikova

fmph.uniba.sk

Abstrakt. Článok približuje prínos matematikov 18. storočia G. G. Saccheriho a J. H. Lamberta k tzv. teórii rovnobežiek. V prácach venovaných problematike elementárnej geometrie súvisiacej s piatym Euklidovým postulátom dospeli k celému radu výsledkov vyplývajúcich z alternatívy postulátu. Tieto dôsledky sa v 19. storočí stali súčasťou systematickej teórie hyperbolickej geometrie N. I. Lobačevského a J. Bolyaia.
Kľúčové slová: Chajjámov-Saccheriho štvoruholník, Hajthamov-Lambertov štvoruholník, hypotéza ostrého uhla a jej dôsledky.

Summary. The paper depicts a contribution of two mathematicians of the 18th century – G. G. Saccheri and J. H. Lambert – to the theory of parallel lines. They have achieved several results of non-Euclidean character in the works on the alternative to the 5th Euclid´s postulate. These results have become an important part of the theory of the non-Euclidean hyperbolic geometry elaborated by N. I. Lobachewsky and J. Bolyai in 20th and 30th of the 19th century.

Modris Dobelis

Department of Computer Aided Engineering Graphics, RTU
Azenes iela 16/20-439, Riga, LV-1048, Latvia
e-mail: Modris.Dobelis

rtu.lv

Abstract. The present time can be characterized as an era of digital revolution which has resulted in a creativity burst in different design applications. Rapid advancements in Information Technology (IT) are still under way and are widely used to communicate design projects practically in all areas of engineering. The paper deals with an exploration of some elements of IT – how they can be integrated into university curricula. In academic courses contemporary Computer Aided Design (CAD) knowledge basically is about the development of both computer aided design and problem solving skills in the context with fast changing CAD media. It becomes more difficult to teach the common principles behind individual software which is general and will most likely survive ever-changing software versions. Limited university class contact hours for the subjects do not allow development of detailed courses for the training of particular software. This study presents the evaluation of the capability of the architectural and mechanical engineering students for their problem solving skills. Statistical analysis of the data revealed strong linear correlation between the time to complete particular exercises and preliminary CAD competence level among the students, thus clearly demonstrating the significance of CAD training in effective usage of software.
Keywords: 3D Modeling, Computer Aided Design, Engineering Education

¹Marek Gawron, ²Jerzy Dietrich, ³Adam Lankosz, ⁴Tomasz Wieja

AGH University of Science and Technology, Faculty of Applied Mathematics
30-059 Kraków, al.Mickiewicza 30, Poland
¹e-mail: mwgawron

posta.fm²e-mail: jdietrich

wp.pl
³e-mail: awip

interia.pl
⁴e-mail: twieja

poczta.fm

Abstract. In the article has been shown a computer visualization of a certain projection of polar stereographic type in which some objects belonging to projective space R3 are represented on a projective plane R2. The visualization has been carried out by means of the modeling in the AutoCAD program and with the use of the principle of polarity in relation to a curvilinear quadric of revolution (paraboloid), and of the properties of the quadrics stereographic projections. It has been proved in empirical way that to any point of space R3 corresponds to one polar plane with respect to the paraboloid and intersecting the latter in a conic (ellipse). The projection of the conic from a point at infinity belonging to the paraboloid, on the plane R2 (tangent to the paraboloid at its vertex) is a circle.
Keywords: Polar-Stereographic Projection, visualization, AutoCAD, paraboloid

¹Marek Gawron, ²Wieja Tomasz

AGH University of Science and Technology, Faculty of Applied Mathematics
30-059 Kraków, al.Mickiewicza 30, Poland
¹e-mail: mwgawron

posta.fm
²e-mail: twieja

poczta.fm

Abstract. In the article have been presented the principles of the Engineering Graphics teaching program in the context of the teaching standards of the Bologna System. In Poland change of the teaching programs in 2007 caused a change in the number of lecture and practice hours. In the Division of Descriptive Geometry and Engineering Graphics of AGH the students are offered classes on both Descriptive Geometry and Engineering Graphics. In the paper we present an original teaching program (syllabus) of Engineering Graphics for the students of Faculty of Physics and Computer Science. In the presentation elements of computer graphics application (the Cabri Geometre II and AutoCAD programs) to Engineering Graphics have been included.The demonstrated program is based on original contents of classes worked out in the Division of Descriptive Geometry and Engineering Graphics. Presentation of separate assignments and teaching methods while practical solving of some given problems will make it possible to assess and substantially discuss the conception elaborated by the team members of the Division of Descriptive Geometry and Engineering Graphics of AGH.
Keywords: Education, engineering graphics, technical drawing, AutoCAD, Cabri Geometre.

¹Renata Górska, ²Adam Lankosz

¹CUT, Warszawska 24, 31-155 Kraków, Poland
e-mail: rgorska

pk.edu.pl
²AGH University of Science and Technology, Faculty of Applied Mathematics
30-059 Kraków, al.Mickiewicza 30, Poland
e-mail: awip

interia.pl

Abstract. It has been assumed that the generic engineering skills will be reinforced by means of 3D design and modeling in the new engineering curriculum. Thus the most modern and up-to-date software has been introduced within the proposed courses. In the paper the outline of the new courses titles and contents will be described. Some examples of students works will be presented.
Keywords: New courses design, Engineering education, Bologna Declaration

Pavel Chalmovianský

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

fmph.uniba.sk

Abstract. The goal of this paper is to present a fan of methods used to classify singularities of algebraic varieties and critical points of the corresponding functions. We focus on isolated singularities of low degree. These can be classified via Morse theory and give the so called ADE-singularities. We focus also on the unfoldings of the varieties. The stability of the class or certain invariants of the singularity is important when numerically detect the singular points. We also consider the intersection multiplicity for planar curves. The geometric interpretation in presence of singular points is given. Most of the methods were studied from [5].
Keywords: algebraic varieties, critical point, singularities

Mária Kmeťová

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

ukf.sk

Abstract. The paper presents one type of construction of ruled surfaces from several possibilities [3], [6]. The construction is based on the theory of determine ruled surface as an intersection of three line complexes in general position. Figures are drawn by program Maple.
Keywords: line complex, line congruence, ruled surface.

¹Milada Kočandrlová, ²Jana Vecková, ³Stanislav Olivík

Faculty of Civil Engineering CTU in Prague
Thákurova 7, 166 29 Prague 6, Czech Republic
¹e-mail: kocandrlova

mat.fsi.cvut.cz
²e-mail: veckova

mat.fsi.cvut.cz
³e-mail: olivik

mat.fsi.cvut.cz

Abstract. During teaching the subject Constructive Geometry in our faculty, we are often in situation when we aim to support the explained subject matter by practical utilization. We sought for creating web pages that would be closely linked with the lecture notes [4] that are used during the teaching and that were compiled exactly for the subjects KOG, KOGA, KG2. We can offer presentations used in lectures, solutions of examples from the lecture notes, supplementary figures (plane as well as spatial), interesting internet links or free chat with students’ questions and teachers’ responses.
Keywords: tuition, geometry, internet support.

Soňa Kudličková

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

fmph.uniba.sk

Abstract. A classical problem of the construction of the B-spline curves is considered. B-spline curves are typically specified in terms of a set of control points, a knot vector and an order. The shape modifications of the B-spline curves are realized on the knot vectors or on the control points. To understand each modification of B-spline curve means to demonstrate the influence of the associated B-spline functions on the shape of the curve in the same figure.
Keywords: B-spline functions, parametrization, continuity conditions, B-spline curve, shape modification.

¹Adam Lankosz,²Renata Górska

¹AGH University of Science and Technology, Faculty of Applied Mathematics
30-059 Kraków, al.Mickiewicza 30, Poland
e-mail: awip

interia.pl
²CUT, Warszawska 24, 31-155 Kraków, Poland
e-mail: rgorska

pk.edu.pl

Abstract: The goal of the paper is to present visualization and some properties of a special case of a polar-stereographic transformation, which base consists of the sphere. Some special cases will be discussed and visualized both in 2D and 3D spaces.
Keywords: Projective and Descriptive Geometry, Spatial Transformations, AutoCAD, Visualization

Michal Nociar

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

sccg.sk

Abstrakt. V príspevku je uvedený historický vývoj metód, ktoré riadia proces zjednodušovania meshov prostredníctvom vyčísľovania chýb kódovaných pomocou kvadratických funkcionálov.
Kľúčové slová: Úroveň detailov, zjednodušovanie, kódovanie chyby kvadratickými funkcionálmi, simpliciálny komplex (mesh).

Summary. Garland and Heckbert published first paper on QEM based simplification of polygonal meshes in 1997. Since then many researchers have used QEM because it offers a good trade-off between speed and quality of simplification process. During the next decade many publications have appeared which were devoted to applications or modifications of QEM, but only several of these represent substantial theoretical improvements and generalizations to original QEM. This short paper presents most notable ones.

Lidija Pletenac

Građevinski Fakultet, Sveučilišta Rijeka
V. C. Emina 5. 51 000 Rijeka, Croatia
e-mail: pletenac

gradri.hr

Abstract: In this paper there are classified projective curves of 4^th order in the projective plane, which can be deduced using a special projective transformation: a pencil of polarities.
Keywords: quartic curves, projective plane, deduction, projective transformation, polarity.

Ana Sliepčević

Građevinski Fakultet, Sveučilišta Zagreb, Croatia
e-mail: anas

grad.hr

Abstract. In der Ebene sind ein beliebiger Kegelschnitt c und ein Punkt P als der Pol gegeben. Dem Paar (c, P) ordnet man verschiedene verwandte Kurven zu. Es sind: polar reziproke Kurve, Inversionskurve, negative Fußpunktkurve, Äquidistantkurve usw. Solche Kurven wurden konstruiert und eine zur anderen in Zusammenhang gebracht. Aus der Konstruktion schließt man dass die Äquidistantkurve mit einer anderen Trabantkurve zusammenfällt. Man konstruiert weiter die Hüllkurve der Kegelschnittnormalen und beweißt, dass diese mit der Kurve aller Krümmungszentren des gegebenen Kegelschnittes koinzidiert.
Keywords: Kegelschnitt, Fußpunktkurve, negative Fußpunktkurve, Dualkurve, polar-reziproke Kurve, Homothetie.

Darina Stachová

FPV ŽU, Hurbanova 15, 010 26 Žilina, SR
e-mail: darina.stachova

fpv.uniza.sk

Abstrakt: Interakcia v systéme vozidlo jazdná dráha predstavuje aktuálny problém, ktorému venujú pozornosť inžinieri na celom svete. Predkladaný príspevok je venovaný matematickému opisu funkcie P(x) určujúcej trajektóriu náboja na kolese a funkcie nahradzujúcej skutočný tvar nerovnosti v jazdnej dráhe f(x), ak túto nerovnosť predstavuje určitá časť konvexnej krivky.
Kľúčové slová: trajektória, konvexná krivka, singulárne body.

Summary. In order to ensure safety of vehicles driving on uneven roads, search and removal of singular points is considered. This is because existence of these points causes the trajectory of the vehicle to not match the shape of the road in places, where the road is damaged. The results that were obtained from numerical and experimental analysis will serve automobile designers, road designers, and road administration agencies in relation to roads’ lifespan and their maintenance.

Margita Vajsáblová

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

stuba.sk

Abstrakt: V príspevku uvádzame charakteristiku triedy pseudocylindrických priamkových zobrazení, konkrétne ekvidištančných a Eckertových zobrazení - ekvivalentného a kompenzačného. Hlavným cieľom je formulácia zobrazovacích rovníc týchto zobrazení, ktoré sme odvodili za účelom ich použitia na vykreslenie obrazu prvkov referenčnej sféry v prostredí Mathematica 6.0.
Kľúčové slová: referenčná sféra, pseudocylindrické zobrazenie, ekvidištančné, ekvivalentné zobrazenie.

Summary. In this paper we give an outline of characteristics of pseudocylindrical linear cartographic projections, equidistant projections, Eckert equal-area projection and Eckert compensated projection. The main aim is formulation of their mapping equations, which we derived for projection of parallels, meridians and boundaries of continents by our programs in system Mathematica version 6.0.

Daniela Velichová

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

stuba.sk

Abstrakt: Cieľom článku je predstaviť niektoré základné poznatky o Minkowského súčte bodových množín a uviesť pár príkladov ilustrujúcich výpočet súčtu konvexných množín, o.i. mnohouholníkov. Uvedených je niekoľko príkladov aplikácií Minkowského súčtu pri modelovaní zonotopov a listov plôch euklidovského priestoru ako Minkowských súčtov bodových množín určených vektorovými rovnicami.
Kľúčové slová: Minkowského súčet, zonotop, modelovanie plôch.

Summary. Aim of this paper is to introduce some basic properties of the Minkowski sum of point sets and to present few examples illustrating calculation of the Minkowski sum of convex sets, e.g. polygons. There are presented some examples on application of Minkowski sum in modelling zonotops, surface patches and solid cells in the Euclidean space as Minkowski sums of the point sets determined by vector equations.
Key Words: Minkowski sum, zonotop, modeling of surfaces.

¹Mária Vojteková, ²Milan Stacho

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

fpedas.uniza.sk
²e-mail: milan.stacho

fpedas.uniza.sk

Abstrakt: Úlohu lineárneho programovania s dvoma premennými možno riešiť algebricky ale aj graficky. V článku je uvedený ilustračný príklad grafického riešenia úlohy lineárneho programovania s tromi premennými, ktorý sa obvykle rieši iba algebrickou Simplexovou metódou.
Kľúčové slová: lineárne programovanie, grafické riešenie, konvexný polyéder

Summary. It is possible to solve a problem of linear programming with two variables in algebraic way and graphic way, too. The aim of our article is to represent graphic solution of problem containing three variables. Usually this problem is solved only by algebraic Simplex method.

Edita Vranková

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

truni.sk

Abstrakt. Cieľom článku je ukázať, že pri splnení určitých geometrických podmienok má stredová, resp. osová súmernosť ľubovoľných mnohouholníkov M, N za následok stredovú, resp. osovú súmernosť množiny D(M,N) hustých rozmiestnení.
Kľúčové slová: stredovo a osovo súmerné mnohouholníky M, N, množina D(M,N) hustých rozmiestnení..

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 show that the main result of [2] may be extended to arbitrary D-monotone geometric bodies. If M and N are two D-monotone geometric bodies we prove that the set I(M, N) of their intersections is a D-monotone geometric body, too. Moreover, the set D(M, N) of dense placements of bodies M and N is a boundary of I(M, N).
Keywords: geometric body, D-monotonicity, set of dense placements.

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. W aspekcie możliwości jakie dają techniki komputerowe postawiono sobie za cel określenie algorytmów pozwalających na automatyczne rysowanie rozwinięć krzywych przekroju powierzchni walcowych (obrotowych i nieobrotowych) i stożkowych (obrotowych i nieobrotowych). W tym celu wykorzystano program CABRI II Plus, a szczególnie jego opcje pozwalające na generowanie miejsc geometrycznych oraz macrokonstrukcji.
Keywords: rozwinięcia, miejsce geometryczne, macrokonstrukcja, CABRI II PLUS.