Contents 24

ZBORNÍK SYMPÓZIA

O POČÍTAČOVEJ GEOMETRII

SCG´2004

Album

PROCEEDINGS OF SYMPOSIUM

ON COMPUTER GEOMETRY

SCG´2004

Volume 13

Slovak Society for Geometry and Graphics

Mechanical Engineering Faculty

Civil Engineering Faculty

Slovak University of Technology in Bratislava

November 2004, Kočovce, SR

ISBN 80-227-2133-6

Contents

V. Bálint, V. Bálint ml., P. Novotný: Poznámka o p-číslach . . . . . . . . . . . . . . . . . . . . . . . . . .	5
J. Błahut: Contractive Semigroups and Their Minimal Sets: Some Properties. . . . . . . . . . .	13
M. Božek: On Drawings-up of Polygons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .	16
S. Cichacz, T. Sulima-Samujłło: Application of B-spline to Representation of Selected Lines and Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .	21
J. Černý, M. Kočandrlová: Trojuholníkové sítě na hyperbolickém paraboloidu . . . . . . . . .	26
J. Čižmár: Tradičné a aktuálne problémy slovenskej matematickej terminológie, špeciálne geometrickej . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .	32
M. Gawron, A. Koch, T. Sulima-Samujłło, T. Wieja: Visulization of Some Stereometric Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .	40
R. Górska: Construction of Some Lemniscate Curves in the P-projection . . . . . . . . . . . . .	45
V. Havel, V. Sedlář: K základní větě reálné afinní rovinné geometrie . . . . . . . . . . . . . . . . .	50
Z. Havranová: Izofóty na pravidelných mnohostenoch . . . . . . . . . . . . . . . . . . . . . . . . . . . . .	54
A. Hosťovecký: A Note on the Efficiency of Periodical Placements . . . . . . . . . . . . . . . . . .	58
M. Kmeťová: On Menelaus' Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .	64
S. Kudličková:. Working with Bézier Curves and Controlling Their Shape . . . . . . . . . . . . .	70
K. Maleček, D. Szarková: Paralelní a pseudoparalelní přenos vektoru . . . . . . . . . . . . . . . .	76
J. Minaroviech, M. Zimányi: Real-time Visualization of Urban Data Improved by Planar Horizon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .	81
K. Palac, T. Wieja: Application of the Cabri Computer Program to Geometrical Presentations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .	89
E. Pavelová: Kružnicový oblúk v dopravnom staviteľstve . . . . . . . . . . . . . . . . . . . . . . . . . .	97
P. Pech: Generalisations of Simson - Wallace Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . .	102
D. Richtáriková, Z. Zámožíková: Kružnicová inverzia v iteračnom systéme . . . . . . . . . . . . .	106
M. Samuelčík: Basic Subdivision Schemes on Triangular Meshes . . . . . . . . . . . . . . . . . . . .	113
D. Szarková, J. Zámožík: Priesečníky čiar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .	119
Z. Ungvarská: The Construction of Geometric Objects by Coons Method . . . . . . . . . . . . .	125
D. Velichová: Euler Curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .	131
E. Vranková: O stope mnohouholníka a jeho vnútra pozdĺž úsečky . . . . . . . . . . . . . . . . . . .	137

¹Vojtech Bálint, Vojtech Bálint ml., Pavol Novotný
Katedra matematiky, Fakulta prevádzky a ekonomiky dopravy a spojov,
Žilinská univerzita v Žiline, Dlhý diel, 01026 Žilina
e-mail: ¹balint@stuba.sk

Abstrakt. Práca pojednáva o možnosti rozdelenia konvexných mnohouholníkov na menšie homotetické kópie. Čiastočnú odpoveď na jednu z otázok položených v [1] dáva Veta 1, vo Vete 2 je dokázaný aj všeobecnejší výsledok.
Kľúčové slová: p-čísla, problém pakovania, homotetická kópia

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. In many industries we encounter the task of placement of plane figures over a plane domain when only translations of figures are allowed, see e.g. [4], [5], [6] or [8]. Very simple but useful tools for solving such problems in both automatic and interactive treatments are translations of a figure closely to another one in a given direction. The aim of our paper is to study such translations and present the computation of their lengths for polygons.
Keywords: Polygon, translation, overlapping, densely placed polygons.

Silvia Cichacz¹, Tomasz Sulima-Samujłło²
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

Abstract. The spline function takes advantage of vertices (de Boor points) as the steering points of the curve. The command SPLINE of AutoCAD generates a curve interpolating it between given points. In the article has been presented a possibility of application the curves of the spline type to representation of contour lines belonging to topographic surface originally determined by sets of points. They have also been applied to approximate representation of a slope line as well as of the contour lines of the slope surfaces passing through this line.
Keywords: interpolate curve, spline function, B-spline, de Boor points, spline, layer, topographic surface, slope line, slope surface

Jaroslav Černý¹, Milada Kočandrlová²
FSv ČVUT Praha, Thákurova 7, 166 29 Praha 6, ČR
e-mail: ¹cerny@mat.fsv.cvut.cz, ²kocandrlova@mat.fsv.cvut.cz

Abstrakt. Skořepinové konstrukce se často nahrazují prutovými konstrukcemi. Na hyperbolickém paraboloidu lze síť konstruovat s využitím jeho struktury přímkové i translační plochy. V příspěvku jsou studovány vlastnosti trojúhelníkových sítí na dvou typech segmentů hyperbolického paraboloidu, nejčastěji používaných v praxi.
Klíčová slova: Hyperbolický paraboloid, trojúhelníková síť, prutová konstrukce.

Summary. The static impact of the spatial bar shaped system is similar to the influence of the corresponding shell. The above quotation inspired this paper which is concerned with the covering of the area of the hyperbolic paraboloid with plane triangles. Basic geometric properties of two types of triangle grids are studied.

Ján Čižmár
FMFI UK, Katedra geometrie, Mlynská dolina, 842 48 Bratislava, SR;
e-mail: cizmar@fmph.uniba.sk
PdF UK, Katedra matematiky, Račianska 59, 813 34 Bratislava, SR;
e-mail: cizmar@fedu.uniba.sk

Abstrakt. Článok sa zaoberá niektorými tradičnými a aktuálnymi problémami slovenskej matematickej terminológie. Rozoberá niektoré chybné názory na zásady tvorby odborných termínov. Uvádza niekoľko príkladov nesprávnych, problematických alebo diskutabilných termínov a dáva odporúčania na opravy a zdokonalenie. Načrtáva niekoľko hlbších problémov, ktoré by sa mali vyriešiť v najbližšej budúcnosti.
Kľúčové slová: odborné termíny, zásady tvorby odborných termínov, tradičné a súčasné problémy a spôsoby ich riešenia

Summary. This paper is concerned with several traditional and contemporary problems in the Slovak mathematical terminology. It analyses some erroneous views at the principles in the formation of technical terms. It presents several examples of incorrect or problematic or contentious terms and brings recommendations of correction and improvement. It sketches some deeper problems, which are to be solved in the near future.

Marek Gawron¹, Andrzej Koch², Tomasz Sulima-Samujłło³, Tomasz Wieja⁴,
AGH University of Science and Technology, Faculty of Applied Mathematics
30-059 Kraków, al.Mickiewicza 30, Poland,
e-mail: ¹mgawron@wms.mat.agh.edu.pl, ² koch@agh.edu.pl, ³sulsam@agh.edu.pl, ⁴twieja@poczta.fm

Abstract. In the article several stereometric problems have been solved. After presenting their spatial solution the AutoCAD program has been applied to show the solutions by means of its methods along with dynamic visualization in 3D space. All the problems have finally been visualized by means of the Corel Draw program
Keywords: visualization, stereometric problems, Auto CAD, Corel DRAW

Renata Górska
CUT, Faculty of Architecture, A-9, Warszawska 24, 31-155 Kraków, PL
e-mail: rgorska@pk.edu.pl

Abstract. Construction of some lemniscate curves will be the point of interest of this paper. Specifically the curves, which are images of either a circle or an ellipse in the P-projection, which has been widely described in works [2], [3], together with their visualization with the use of a GEONE_XT software will be presented.
Keywords: Projective geometry, theory of transformations, Booth lemniscates, GEONE_XT

Václav Havel¹, Vladimír Sedlář²
¹Ústav matematiky fakulty elektrotechniky a informatiky, VUT, Technická 8, 616 00 Brno, ČR, e-mail: havel@feec.vutbr.cz
²Matematický ústav, Slezská univerzita, Na Rybníčku 1, 746 01 Opava, ČR, e-mail: Vladimir.Sedlar@math.slu.cz

Abstrakt. Je podán důkaz věty zmíněné v nadpisu bez prostředků projektivní geometrie.
Klíčová slova: vektorový prostor, regulární lineární transformace, afinní rovina, afinní zobrazení

Summary. There is investigated the theorem saying that every automorphism (affinity) of a real affine plane induces a nonsingular linear transformation of the underlying vector space of translations. The present Contribution contains a proof which as far as possible get out of the means of Projective geometry.

Zuzana Havranová
SvF STU, Radlinského 11, 812 31 Bratislava, SR
e-mail: havranova@sjf.stuba.sk

Andrej Hosťovecký
FMFI UK, Mlynská dolina, 842 48 Bratislava, SR
e-mail: hostovecky@fmph.uniba.sk

Abstract. The fundamental inequality e ? 1 for the efficiency of periodical layout of geometrical figures is proved in details, under some natural assumption on the figure.
Keywords: Geometrical figure, periodical layout, two-dimensional Lebesgue measure.

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

Abstract. The paper deals with Menelaus` Theorem in elementary plane geometry, their generalizations to n-sided polygons and with applications of CAGD version of Menelaus` Theorem in theory of Bézier curves.
Keywords: Menelaus` Theorem, Ceva`s Theorem, Hoehn`s Theorem, area principle, de Casteljau algorithm, blossoming

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

Abstrakt. Interactive curve design for the spatial Bézier curves in the space E3 is presented. The tools for shape modifications of the Bézier curves are introduced by the operations on the control vertices.
Keywords: control polygon, Bézier curve, Bernstein polynomials, shape modifications

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 příspěvku je uvedený pojem paralelní a pseudoparalelní přenos vektoru podél křivky K. v rovině a na obecné ploše. V [2] a [3] se k zavedení pseudoparalelního přenosu užívá absolutní derivace. My jsme užili integrálních invariantů křivky, které jsou uvedeny v [1]. Zde je také velmi stručně popsán námi uvedený způsob zavedení pseudoparalelního přenosu.
Klíčová slova: paralelní přenos, pseudoparalelní přenos, křivost, geodetická křivost.

Summary. The concept of the Levi-Civita parallelism of a vector along the plane curve K. is introduced on the base of the curve integral invariant [1], as an alternative approach to the one inserted in [2] and [3] by the absolute derivative.

Ján Minaroviech¹, Marek Zimányi²
Department of Computer Graphics and Image Processing
Comenius University, 842 48 Bratislava, SR
e-mail: ¹jmi@realtime.sk, ²zimanyi@fmph.uniba.sk

Abstract. This paper describes several alternatives of planar horizon creation and also different speed-ups of this process. It also gives a guide for simple creation of effective set of large occluders from this horizon. Paper gives a guide how to use the horizon or occluders generated from horizon with chosen existing algorithms.
Keywords: Occlusion culling, Visibility culling, Planar Horizon, Occluder fusion, Urban Walkthroughs

Pałac Krzysztof¹, Wieja Tomasz²
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

Keywords: CABRI, visualization, animation, mapping projection, Monge projection, axonometry, central projection

Eva Pavelová
SvF TU, Vysokoškolská 4, 042 00 Košice, SR
e-mail: pavelova@post.sk

Abstrakt: V technickej praxi musia projektanti a konštruktéri pri navrhovaní cestnej, vodnej alebo železničnej komunikácie často prepojiť dve priame komunikácie rôzneho smeru, priamu cestu spojiť s kruhovým oblúkom alebo prepojiť dva kruhové oblúky. Navrhované riešenie musí vyhovovať predpísaným normám a nesmie veľmi narušiť okolitú prírodu. Dané prepojenie je navrhované buď ako kružnicový oblúk alebo prechodnicami. Tento článok popisuje matematické riešenie prepojenia komunikácií s využitím kružnicového oblúka.
Kľúčové slová: smerové oblúky, prechodnice, výškový oblúk, kružnicový oblúk

Summary. A constructor must often connect a line road with a circular curve. This paper describes the mathematical method of solving the given problem.

Abstract. Inthis contribution we show generalizations of the well known Simson-Wallace theorem. To discover and to prove statements we use both classical methods and the methods of computer algebra which are based on Groebner basis computations, see [1]. This method enables us to find and prove such statements which are often very difficult to solve using techniques of synthetic geometry especially in space. All computations were done by computer algebra system CoCoA.
Keywords: Simson-Wallace theorem, Automatic theorem proving.

Daniela Richtáriková¹, Zuzana Zámožíková²
¹SjF STU, Nám. slobody 17, 812 31 Bratislava, SR, e-mail: richtarikova@sjf.stuba.sk
²KM, PdF UK, Račianska 59, 813 34 Bratislava, SR, yuyana@excite.com

Abstrakt: V článku predkladáme tvorbu limitných útvarov použitím nelineárnych zobrazení v iteračnom systéme transformácií. Jadro generujúcej transformácie tvorí kružnicová inverzia.
Kľúčové slová: kružnicová inverzia, IFS, limitný útvar

Summary. The paper deals with iterated system limit set modelling similar to Barnsley’s Chaos Game using circle inversion composed with additional non-linear transformation.

Martin Samuelčík
FMFI UK, Mlynská dolina, 842 48 Bratislava, SR
e-mail: samuelcik@fmph.uniba.sk

Abstract. Subdivision curves and surfaces are very useful parts of current geometric modeling. In this paper we present two basic subdivision schemes for triangular meshes, Loop scheme and modified butterfly scheme. Main part of this work is data structure for storing triangular mesh that saves memory and subdivision schemes can be performed fast using this structure. We also give steps how to perform described schemes on that structure.
Keywords: subdivision, triangular mesh, Loop scheme, modified butterfly scheme

Dagmar Szarková¹, Jozef Zámožík²
¹SjF STU, Nám. slobody 17, 812 31 Bratislava, SR, e-mail: szarkova@sjf.stuba.sk
²MtF STU Bratislava, Paulínska 16, 917 24 Trnava, e-mail: zamo@zoznam.sk

Abstrakt. V príspevku sú uvedené dve metódy riešenia úlohy o priesečníkoch rovinných čiar. Predložené metódy sú použiteľné aj na hľadanie reálnych koreňov pre širokú triedu rovníc.
Kľúčové slová: čiara daná parametricky a implicitne, graf funkcie, priesečník čiar, uzlový bod

Summary. There are two methods for solving the problem of the intersection of plane curves introduced in this paper. Presented methods are useful also for searching real roots for a wide range of equations. The size of the area which is chosen can have an effect on the amount of solutions obtained, as seen in the figures in the examples.

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

Abstrakt. Thi Coons construction is used for the derivation of the surface subdivision rules for the special kind of quadrilaterial surfaces, which boundary curves are made by the method of subdivision. The spline properties of the Coons patches are utilized here.
Keywords: B-spline patch, Coons patch, Subdivision.

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

Abstract. The Euler curve is introduced in the paper as the trajectory of the general rotational movement determined by three consecutive rotations about three different axes located to the co-ordinate axes in the space. Different forms of the trajectory curve are presented and discussed with respect to the three defined Euler angles of rotations about the separate axes.
Keywords: general rotation, Euler angles, Euler rotation formulae, Euler curve

Abstrakt. V práci je uvedená Eulerova krivka ako trajektória všeobecnej rotácie určenej zložením troch postupných rotácií okolo troch rôznych osí umiestnených do súradnicových osí priestoru. Prezentované sú rôzne formy krivky ako dráhy pohybu bodu podrobeného otáčaniam okolo jednotlivých osí o definované Eulerove uhly.

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

Abstrakt. Cieľom príspevku je ukázať topologický rozdiel medzi stopou mnohouholníka a stopou vnútra mnohouholníka pozdĺž úsečky. Vo všeobecnosti, uvedomiť si rozdiel medzi stopou uzavretej a otvorenej množiny (pozdĺž úsečky).
Kľúčové slová: stopa množiny pozdĺž inej množiny, mnohouholník, vnútro mnohouholníka, stopa mnohouholníka pozdĺž úsečky

Summary. The aim of the presented paper is to show the difference between trace of the polygon and trace of polygon's interior along a line segment. In general, to realize a topological difference between trace of closed and open set (along a line segment).