ZBORNÍK SYMPÓZIA
O POČÍTAČOVEJ GEOMETRII
SCG´2014
PROCEEDINGS OF SYMPOSIUM
ON COMPUTER GEOMETRY
SCG´2014
Volume 23
Slovak Society for Geometry and Graphics
Mechanical Engineering Faculty
Civil Engineering Faculty
Slovak University of Technology in Bratislava
October 2014, Bratislava, SR
ISBN 978-80-227-4256-6
Contents
Abstracts
Variabily vo výučbe geometrie
Martin Ambroz1, Božena Koreňová2
1SvF STU, Radlinského 11, 842 15 Bratislava, SR,
1e-mail: ambroz.martin.ml
gmail.com
1e-mail: bobakorengmail.com
Abstract. V prvej časti charakterizujeme výtvarné dielo Mariana Drugdu, v druhej
rozoberáme aspekty využitia variabilu vo výučbe. Tretia časť je venovaná modelovaniu
variabilu v GeoGebre.
Keywords: variabily, Euklidovské transformácie, GeoGebra, applet
Summary. This paper describes the geometric art of Marian Drugda, discusses aspects of using
variabils in education process and deals with modelling of variabils in GeoGebra.
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Prečo je triviálne vyzerajúci problém ťažký alias zlyhanie intuície
Vojtech Bálint
FPEDAS ŽU, Univerzitná 1, 010 26 Žilina, SR
1e-mail: balintfpedas.uniza.sk
Abstrakt: Príspevok ilustruje možné zdroje ťažkostí na dvoch rôznych problémoch.
Kľúčové slová: maximum, ukladanie gulí, ukladanie trojuholníkov
Summary. Contribution illustrates the possible sources of difficulties on two problems. While
looking for the maximum, but can only set the supremum resp. failure of intuition.
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Structure of Singularity of Cab Curve
Martina Bátorová
KAGDM FMFI UK, Mlynská dolina, 842 48 Bratislava, SR
e-mail: viktoria.batorovafmfi.uniba.sk
Abstract: This paper deals with singularities of Cab curves defined over the field of complex
numbers. We subject the defining polynomial of given curve to a series of parameterizations
and we describe the internal structure of its singularity placed at infinity.
Keywords: Cab curve, parameterization, deformation, singular point, structure
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On Singularities of Skew Pedal Curves
Miloš Božek
Comenius University, Faculty of Mathematics, Physics and Informatics
Mlynská dolina, 842 48 Bratislava, SR
e-mail: bozekfmph.uniba.sk
Abstract: We generalize concept of pedal curve in the Euclidean plane; we detect all singular
points of such curves and partially classify them.
Keywords: planar curve, curvature, Frenet formulas, inflexion point, singular point
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Kvadratická involúcia v Pn(k)
Ján Čižmár1, Monika Vojtašáková2
1Pedagogická fakulta Trnavskej univerzity,
Priemyselná 4, P.O. BOX 9, 918 43 Trnava, SR
2Fakulta matematiky, fyziky a informatiky Univerzity Komenského,
Mlynská dolina
842 48 Bratislava, SR
1e-mail: jan.cizmartruni.sk
2e-mail: monika.vojtasakova7gmail.com
Abstrakt: V článku je opísaná biracionálna korešpondencia v n-rozmernom projektívnom
priestore Pn(k) (n ? 2) nad algebricky uzavretým poľom k charakteristiky 0. Dvojica rôznych
regulárnych korešpondujúcich bodov inciduje s priamkou trsu priamok a je dvojicou bodov
harmonicky združených vzhľadom na priesečníky priamky trsu s regulárnou nadkvadrikou.
Sú charakterizované nasledovné množiny: fundamentálna varieta, množina všetkých
iregulárnych bodov, invariantná varieta a otvorená množina biregulárnych bodov.
Kľúčové slová: korešpondencia, kvadratická involúcia, fundamentálna varieta, množina
iregulárnych bodov, invariantná varieta, množina biregulárnych bodov
Summary. A birational correspondence in the n-dimensional projective space Pn(k) over an
algebraically closed field is investigated. Pairs of corresponding points are harmonically
conjugated with respect to pairs of intersection points of a hyperquadric with straight lines of
a bundle. The correspondence is a quadratic involution. Both fundamental and invariant
varieties are founded out as well as sets of irregular and biregular points.
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Pohľad na začiatky programu GeoGebra
Viera Čmelková
FPEDAS ŽU, Univerzitná 1, 010 26 Žilina, SR, e-mail: viera.cmelkova@fpedas.uniza.sk
e-mail: viera.cmelkovafpedas.uniza.sk
Abstrakt: V príspevku je podaný krátky pohľad do histórie dynamického matematického
softwéru GeoGebra. Sú priblížené verzie GeoGebra 1.0, GeoGebra 2.0, GeoGebra 3.0
a GeoGebra 3.2.
Kľúčové slová: GeoGebra, geometria, výučbový program
Summary. The paper brings a brief review of the history of dynamic mathematical software
GeoGebra. A closer insight to versions GeoGebra 1.0, GeoGebra 2.0, GeoGebra 3.0
and GeoGebra 3.2 is presented here.
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GeoGebra vo vyučovaní stredoškolskej matematiky
Lilla Koreňová
FMFI, Mlynská dolina, 842 48 Bratislava, SR
e-mail: lillakorenovafmph.uniba.sk
Abstrakt: Pre študentov je digitálny svet prirodzenou súčasťou ich každodenného života.
Vybavenie škôl, ako sú interaktívne tabule, hlasovacie zariadenia, notebooky a tablety, ale aj
free softvér, ako napríklad GeoGebra, nastoľujú otázky ohľadom ich efektívneho využitia vo
vyučovacom procese. Klasické a inovatívne vyučovacie metódy a formy majú v digitálnom
prostredí nový rozmer, no vznikajú aj nové metódy priamo súvisiace s využívaním
digitálnych technológií. V príspevku poukážeme na niektoré aspekty vyučovania matematiky
pomocou softvéru GeoGebra a uvedieme niekoľko námetov na takéto inovatívne vyučovanie.
Kľúčové slová: GeoGebra, metóda riadeného skúmania, digitálne technológie vo vyučovaní
Summary. For students, the digital world is a natural part of their everyday lives. Nowadays,
schools are equipped with interactive whiteboards, voting devices, notebooks and tablets, as
well as with free software such as GeoGebra, etc. This raises the question of their effective
use in the educational process. Classical and innovative teaching methods and forms gain
a whole new dimension in a digital environment. There are also new, rising methods tightly
connected to and focused on digital technologies. In this contribution, we'd like to point out
some aspects of teaching mathematics with the GeoGebra software and we'll provide a few
scenarios for such innovative teaching.
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Isogonal and Isotomic Transformations Revisited
Péter Körtesi
University of Miskolc, 3515 Miskolc, Egyetem út 17 POB 10, HU
e-mail: pkortesigmail.com
Abstract: The symmedian lines and the symmedian point of a given triangle present
interesting properties. Part of these properties can be formulated in a more general context for
isogonals. In a triangle the isogonal of a line passing through one of the vertices of the
triangle is a line symmetric to the bisector of the given angle. In a similar way, the isotomial
of such a line can be introduced as the line intersecting the opposite side in a symmetric point
to the middle of that side. It can be proven that the three isogonals resp. isotomials of three
concurrent lines which are passing through the three vertices of the triangle, are
concurrent.This property serves as definition for the isogonal resp. isotomial transformation,
the image of a given point in this transformation will be the intersection point of the three
isogonals, resp. isotomials of the three lines that are passing through the given point and the
vertices of the triangle. The paper is aimed to present some of the properties of this
transformations, and to visualise them, using GeoGebra.
Keywords: Cevian, trilinear coordinate, isogonal conjugate, isotomic conjugate
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Geometrický dizajn rozvinuteľných kvadratických
Bézierovych plôch
Soňa Kudličková
Fakulta matematiky, fyziky a informatiky Univerzity Komenského, Mlynská dolina, 842 48 Bratislava, SR
e-mail: kudlickovafmph.uniba.sk
Abstrakt: Článok sa zaoberá ďalšími možnosťami konštrukcie vrcholov riadiacej siete
rozvinuteľnej kvadratickej Bézierovej plochy, ktoré závisia na výbere piatich stupňov
voľnosti a následnom modelovaní vybraných rozvinuteľných plôch pomocou škálovacích
faktorov.
Kľúčové slová: rozvinuteľné priamkové plochy, Bézierove plochy, Casteljau algoritmus,
komplanárnosť medzivrcholov, stupne voľnosti, škálovacie faktory
Summary. This paper investigates the geometric design of developable quadratic Bézier surfaces.
The developability condition has been combined with Casteljau algorithm, resulting in a set of
constraint equations imposed on the control points. The concrete examples, using partial
scaling factors, have been presented for design of developable quadratic Bézier surfaces.
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Projective Metrics and their Roles in Visualization
Emil Molnár
Institute of Mathematics, Department of Geometry,
Budapest University of Technology, Hungary
Egry József u. 1. H. II. 22, H – 1521 Budapest XI
e-mail: emolnarmath.bme.hu
Keywords: Projective metric sphere and plane: Euclidean, spherical, hyperbolic, Minkowski
and Galilei plane, higher-dimensional generalization by Grassmann-Clifford exterior algebra,
visualization
Maximálny súčet objemov dvoch kužeľov
Pavel Novotný
KKMAHI, FPEDaS, Žilinská univerzita,
Univerzitná 1, 010 26 Žilina, SR
e-mail: pavel.novotnyfpedas.uniza.sk
Abstrakt: V práci sa rieši problém, ako rozdeliť kruhový výsek na dva menšie výseky tak,
aby bol súčet objemov kužeľov, ktorých plášte sú vytvorené z týchto výsekov, maximálny.
Kľúčové slová: kužeľ, objem, maximum
Summary. We deal with the following problem: How to divide a circular sector into two smaller
sectors so that the sum of the volumes of the cones whose lateral surfaces are created from
this sectors is maximal.
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The Simson-Wallace Theorem Extension on Skew Quadrilaterals
Pavel Pech
Faculty of Education, University of South Bohemia
Jeronýmova 10,
371 15 České Budějovice, CZ
e-mail: pechpf.jcu.cz
Abstract. The well-known Simson–Wallace theorem reads: If P is a point in the circumcircle
of a triangle ABC then orthogonal projections of P onto the sides of ABC are collinear. The generalization on skew quadrilaterals in 3D space is as follows: Let
K, L, M, N be orthogonal projections of a point P onto the sides AB, BC, CD, AD of
a skew quadrilateral ABCD respectively. Then the locus of P such that K,L,M,N are
coplanar is a cubic surface H = 0. In the paper some relations between a skew quadrilateral
ABCD and its associate surface H are studied.
Keywords: Simson–Wallace theorem, cubic surfaces, loci of points
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Learning Using New Technology
Ivanka Stipančić Klaić
Faculty of Civil Engineering, Osijek, HR
e-mail: istipangfos.hr
Abstract: In which direction the learning process is heading today and how adults can use
new devices when they learn mathematics.
Key words: Pad, laptop, learning mathematics, cooperative behaviour in learning
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Symbolical Computations on B-Spline Curves and
Surfaces
Márta Szilvási-Nagy1, Szilvia Béla2
Dept. of Geometry, BUTE, Egry József u. 1, H-1111, Budapest, HU
1e-mail: szilvasimath.bme.hu,
2e-mail: belusmath.bme.hu
Abstract: In this short survey algorithms for interpolating or approximating one or more
curve segments are presented by using uniform B-spline curves of degree four. The
computations are carried out symbolically so that the resulting formulas give the control
points of the required B-spline curve expressed by the input data. By substituting the given
numerical data into the expressions of these pre-computed control points the B-spline curve is
generated directly by its equation. We show also the extension of this method to interpolation
or approximation of surface patches by B-spline surfaces. The computations have been made
by the symbolical algebraic program package Wolfram Mathematica.
Keywords: B-spline curves, B-spline surfaces, interpolation, approximation
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Basics of Videogrammetry
Daniela Velichová
SjF STU Bratislava, Nám. slobody 17, 812 31 Bratislava, SR
e-mail: daniela.velichovastuba.sk
Abstract. Basic concepts and algorithms for reconstruction of real dimensions of a 3
dimensional moving object from its video records are introduced. Basic formulas for
algorithms of point positioning and calibration calculation are explained.
Keywords: photogrammetry, videogrammetry, point cloud reconstruction, camera
calibration, bundle method, interior image orientation, exterior orientation
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Minkowski Circular Triplets
Daniela Velichová
SjF STU Bratislava, Nám. slobody 17, 812 31 Bratislava, SR
e-mail: daniela.velichovastuba.sk
Abstract: Special forms of a surface denoted as Minkowski circular triplet are presented in
the paper generated as Minkowski mixed triples of three circles, each positioned in one of the
three coordinate planes and concentrically with two others sharing common centre at origin.
Differential characteristics and intrinsic properties of both surface forms are derived and
several views of surface forms are visualised.
Keywords: Minkowski sum, Minkowski product, Minkowski combination of point sets, Minkowski tripless
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