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Tom 58(72), Fascicola 1, 2013 Visualization in teaching and learning mathematics

in elementary, secondary and higher education Branko J. Malešević1 Ivana V. Jovović2 Bojan D. Banjac

Abstract: In this paper we present our experience in using visualization in mathematics education. The experience with our university courses: “Computer tools in matematics” and “Symbolic algebra” provides the basis for mathematics teacher education program http://vizuelizacija.etf.rs/. The program is intended for elementary and high school teachers. The education program deals with modern techniques of visualization by using technologies such as GeoGegebra, JAVA and HTML. Keywords: GeoGebra, Gröbner’s bases, visualization in mathematics education, application of mathematics in engineering

1. INTRODUCTION

In the following sections we present our experiences in using visualization in mathematics education. In each section the concrete mathematical topics are selected and visualization techniques are briefly illustrated using GeoGebra and Java applets for different levels of education. 2. COMPUTER TOOLS IN MATHEMATICS Undergraduate studies in Electrical and Computer Engineering at School of Electrical Engineering, University of Belgrade, offer elective course “Computer tools in mathematics” (http://raum.etf.rs). Students can choose this course on Computer Science and Informatics module in the fifth semester. On the other modules there is, in the third semester, a similar accompanying course “Computer practicum in Mathematics 3” of the mandatory course “Mathematics 3” (consisting of Calculus 3 – Multiple Integrals, Complex Analysis, Laplace and Fourier transformation). “Computer practicum in Mathematics 3”, among the others software, uses free software Sage. The goal of the Sage project is to create a free open source alternative to Maple, Mathematica, and MatLAB, which are the most popular non-free closed source mathematical software systems.

Students are expected to produce a final project for the course “Computer tools in mathematics” using free software Sage and GeoGebra. Each of the projects deals with a specific topic presented as a website. GeoGebra enables students to make easily HTML pages with GeoGebra applets. Now, we will present one of the applets from the course. Applet is concerned with an application of mathematics in electrical engineering. We will introduce the ideal mass-spring-damper system with the amplitude gradually decreasing to zero and with mass m, spring constant k and damping coefficient b. For the oscillatory force it holds

( ),sF k y x= − (1)

and for the damping force we have

' ( ).dF b y x= − (2)

By Newton’s Law of motion the total force on the body is

''( ),tF m y x= (3)

where ''( )y x is the acceleration of the mass, '( )y x is the speed of the mass and ( )y x is the displacement of the mass relative to a fixed point of reference. Since

t s dF F F= + (4)

we obtain the second order homogenous differential equation with constant coefficients of the form

''( ) '( ) ( ) 0.m y x b y x k y x+ + = (5)

We are interested only in the case where the characteristic polynomial

1 University of Belgrade, School of Electrical Engineering, Department of Applied Mathematics, Bulevar kralja Aleksandra 73, 11000 Belgrade, Serbia, malesevic@etf.rs

2 University of Belgrade, School of Electrical Engineering, Department of Applied Mathematics, Bulevar kralja Aleksandra 73, 11000 Belgrade, Serbia, ivana@etf.rs

3 University of Belgrade, School of Electrical Engineering, Department of Applied Mathematics, Bulevar kralja Aleksandra 73, 11000 Belgrade, Serbia, bojan.b.mail@gmail.com 37

mailto:malesevic@etf.rsmailto:ivana@etf.rsmailto:bojan.b.mail@gmail.comhttp://raum.etf.rs/

2 0m b kλ λ+ + = (6)

has complex roots. Hence, the general solution of the motion of the mass-spring-damper system is given by

( )2 1 2( ) cos( ) sin( ) ,b xmy x e C x C xω ω

−= + (7)

where 24

2mk b

mω

−= . The solution can be rewritten

in the form

2( ) sin( ),b xmy x Ae xω ϕ

−= + (8)

where 2 21 2A C C= + and 1

2

arctanCC

ϕ = . So the

solution ( )y x oscillate between the two curves

2( )b xmy x A e

−= and 2( )

b xmy x A e= . If we consider the

differential equation (5) with initial conditions

0

1

(0) ,'(0) ,

y yy y

=

= (9)

we obtain

2 20 1 1

2

20

0 1

( )2 ,

4

4arctan .

2

om ky by y myAmk b

y mk bby my

ϕ

+ +=

−

− = + (10)

Fig. 1. Mass-spring-damper system

Now, we will give the detailed explanation of the GeoGebra applet presented on the Fig. 1. Firstly, we create four sliders for mass m, spring constant k, damping coefficient b and time t. Slider for time (not visible in Fig. 1.) is necessary for the animation. Insert Buttons “Start” and “Stop” are used to begin and

terminate animation. These tools are used to execute StartAnimation[t] and StartAnimation[false] GeoGebra script commands with a single click on buttons. The Input Boxes “ (0)y = ” and “ '(0)y = ” are used to change interactively initial conditions and they are connected to object “ 0y ” and “ 0 'y ”. Numbers

c and β are defined as 24c mk b= − and 2bm

β = − .

We need these numbers for the written text at the bottom of the applet on Fig. 1.

Fig. 2. Algebra view

Parametric Curve “ ( )eo p ” is used to visualize the

oscillating string. The Curves “ h( )p ”,“ h max( )p ”

and “ h min( )p ” present the solution of differential equation and the curves between that solution oscillates. The rest of the applet deals with printing text with interactive objects. 3. SYMBOLIC ALGEBRA Master studies in Electrical and Computer Engineering at School of Electrical Engineering, University of Belgrade, offer elective course “Symbolic algebra” (http://simba.etf.rs) on Applied Mathematics module. The course is dedicated to the study of the theory of Gröbner bases and their applications in the various fields of electrical engineering. Here we present two examples of the wide range of applications (in Robotic, Cryptography, Coding theory, ...) of Buchberger’s algorithm. The algorithm is a method of transforming a given set of generators for a polynomial ideal into a Gröbner basis with respect to some monomial order. Gröbner basis of the ideal obtained in this manner is not unique. To obtain unique reduced Gröbner basis we use algorithm Tab. 3. Application has been developed in Java and it was a part of the master thesis of the Phd student Bojan Banjac, who works as a student assistant at the Department of Applied Mathematics. Talented

38

http://simba.etf.rs/http://en.wikipedia.org/wiki/Ring_idealhttp://en.wikipedia.org/wiki/Gr%C3%B6bner_basishttp://en.wikipedia.org/wiki/Monomial_order

students who choose course “Symbolic algebra” are encouraged and challenged to participate with new applications of the algorithm in conferences in this field. In the past two years, three groups of students have successfully presented their papers at the Symposium “Mathematics and Applications”, Faculty of Mathematics, University of Belgrade, Serbia.

Tab. 1. Multivariate division algorithm

Tab. 2. Buchberger’s algorithm

Tab. 3. Algorithm for obtaining a reduced Gröbner basis

One of application created by students under the supervision of professor Malešević is “Vrba”. The idea was to show expansion of the monomial ideal using Buchberger’s algorithm and to describe in detail the way it works. Two and three dimensional graphs are used to demonstrate functionality of the algorithm for polynomials in two or three unknowns. Programming language Java was chosen, so the application can be available for all students independently from a computer they access. A combination of standard libraries for graphic user interface which are part of java execute environment and JOGL library are used for this purpose. JOGL library is created as wrapper around the OpenGL library written in program language C. Therefore, the application is not portable in the way that the most Java applications are. But this problem is solved by skilfully written files for running the applet.

Fig. 3. Main screen of application “Vrba”

SymPy library is used for the calculation of Gröbner basis. This library requires interpreter for programming language Python. Although various Java libraries make this possible, their size combined with the low speed make application slower for launching and execution from predicted one. In order to eliminate these complications, in the later versions of the application, new specialized library for calculation is written and implemented.

Fig. 4. Main screen of application “Robohan”

Here we will present one more application named “Robohan”. It is an application of Buchberger’s algorithm in Inverse Kinematics. Problem of robotic arm movement can essentially be reduced to solving system of polynomial equations using the theory of Gröbner bases. The idea for multimedia project

Multivariate division algorithm [1] Input: f 1, . . . , f k, f Output: a1, . . . , a k, r a1 := 0, . . . , a k := 0, r := 0 p := f WHILE p ≠ 0 DO i := 1 divisionoccured := false WHILE i ≤ k and divisionoccured = false DO IF LT(f i) divides LT(p) THEN a i:= a i + LT(p)/LT(f i) p := p − LT(p)/LT(f i) · f i divisionoccured := true ELSE i := i + 1 IF divisionoccured = false THEN r := r + LT(p) p := p − LT(p) (STOP)

Buchberger's algorithm [2] Input: ),...,( 1 kffF = generating set for ideal I Output: ),...,( 1 sggG = Gröbner basis for ideal I

FG =: },|),{(: jijiji ffGffffM =∧∈=

WHILE 0M ≠ DO (p,q):= a ordered pair from M )}.{(\: qpMM =

),(: qpSS =

h:=normal(S,G)

IF 0h ≠ THEN : {( , ) | }M M g h g G= ∪ ∈ : { }G G h= ∪ (STOP)

Algorithm for obtaining a reduced Gröbner basis[3] Input: },...,{ 1 sggG = Output: }ˆ,...,ˆ{ˆ 1 kggG =

GG =:ˆ FOR ALL Gg ˆ∈ DO IF )(|)()|ˆ( gltltgG µµµ ≠∈∃ THEN }{\ˆ:ˆ gGG = ELSE }){\ˆ,(: gGgremg = FOR ALL Gg ˆ∈ DO

)(

:glc

gg =

(STOP)

39

“Robohan” is to present in an appealing and attractive way usage of the previous algorithms. Knowledge of Computer Graphics, theory of Gröbner bases and of generating sounds was needed. As with the previous application, programming language Java was chosen, so application can be placed on the course site. Standard libraries for graphic user interface were used. Some of calculations were carried before applications starts, and some were implemented during its execution. This decision was made because the execution time was very long for some calculations. The sort algorithm was used for sorting boxes by robotic arm. One of idea was that the pre-recorded sound accompanied application. Problem was in the delay of sound reproduction. To avoid this, project participants used JSyn library to form sound waves that follows execution of application. 4. PROGRAM OF EDUCATION FOR TEACHERS OF MATHEMATICS Department of Applied Mathematics, at School of Electrical Engineering, University of Belgrade, has started program of education "The visual representation of some mathematical content using computers" (http://vizuelizacija.etf.rs/) in 2011/12. Institute for the promotion and teaching resources Serbia (http://www.zuov.gov.rs/) accredited this program for 2013/14. school year. The program is designed for mathematics teachers in elementary and secondary schools. Program is conducted by members of Department of Applied Mathematics in the premises of the Computer Center of School of Electrical Engineering and in Regional Centers for Professional Development of Education Personnel.

Fig. 5. Picture from the seminar

Program lasts for two days, total time is 16 hours. The training has two parts: plenary lectures and workshops. Plenary lectures focus on some interesting topics and parts of program GeoGebra related to visualization of some mathematical content using

computers. During workshops, educators and participants work together to design simple applets in GeoGebra. Participants engage in an iterative process of applet creation that can be part of their teaching techniques. Here we provide a simple applet for visual representation of translation.

Tab. 6. Example of translation

REFERENCES [1] D. Cox, J. Little, D. O’Shea, Ideals, Varieties, and Algorithms, Third Edition, Springer, 2007; [2] K. Forsman, Hitchhiker guide to Gröbner bases, Research Institute for Symbolic Computation, Linz, Tehnical Report 0374, 1992; [3] P., Grayson, Robotic Motion Planning, MIT Undergraduate Journal of Mathematics, Tom I, 2001, pp. 57-68; [4] A., Heck, Bird's-eye view of Gröbner Bases,Nucl. Instrum. Meth. A, Tom 389, 1997, Fasc. 1-2, pp. 16-21; [5] J. Hill, J. Brannen, Briljantno HTML5 i CSS3, CET Computer Eqipment and trade, 2011; [6] J. Hohenwarter ,M. Hohenwarter, Introduction to GeoGebra, 2011; [7] P., Grayson, Robotic Motion Planning, MIT Undergraduate Journal of Mathematics, Tom I, 2001, pp. 57-68; [8] K., Kendricks, Solving the Inverse Kinematic Robotics Problem: A Comparison Study of the Denavit-Hartenberg Matrix and Groebner Basis Theory, 2007; [9] F., Winkler, Gröbner bases at work: Inverse Kinematics in Robotics, Computer Algebra, 2010; [10] W. Yan, H. Lu-bin, Y. Ting-li, Inverse Kinematics Analysis of General 6R Serial Robot Mechanism Based on Gröbner Base, Higher Education Press and Springer-Verlag, 2006.

40

http://vizuelizacija.etf.rs/http://www.zuov.gov.rs/

CoverConference presentationConference sectionsOrganizing comitteeScientific conference comitteeConference chairpersons

Index of papers Index of authorsADIR, G.Fsup 165

ADIR, V.Fsup 165

ALB, C.Fsup 268

ANATII, S.Fsup 233

ANDO, N.F1 67Fsup 79

ANGHEL, A.A.Fsup 21F2 109

AVRAMESCU, A.M.F1 95Fsup 51

BANJAC, B.D.F1 37

BARABAS, I.F2 37

BARGLAZAN, M.F1 123

BAROIU, N.Fsup 31

BARSAN, A.Fsup 177Fsup 181

BARSAN, L.Fsup 177Fsup 181

BATOG, I.Fsup 125

BALCAU, M.F2 37Fsup 13Fsup 17

BARBANTA, D.F2 45

BELEA, G.F1 105

BERBINSCHI, S.Fsup 31Fsup 261

BICA, S.M.F1 101

BJELIC, I.F1 41F1 87Fsup 41

BLASCU, L.Fsup 233

BODEA, S.F1 91F2 19Fsup 13Fsup 17

BOLOS, C.Fsup 221

BRANKO, J.F1 37

BUDISAN, T.F2 61F1 115

BLAGOJEVIC, Z.F1 53

CAPATANA, I,Fsup 235

CARTOFEANU, C.Fsup 233

CAPRINCIU, G.Fsup 157

CHERNIKOV, A.Fsup 161

CHISCOP, F.Fsup 219

CIOTEA, M.Fsup 221

CISMARU, C.F1 128

CLINCIU, M.R.Fsup 197

CLINCIU, R.Fsup 193

CORSIUC, G.F2 53F1 21

CRACIUN, F.F1 63

CRISAN, N.F2 19Fsup 13Fsup 95

CRISAN, A.T.F2 45Fsup 25

DARDAI, R.Fsup 268

DANILA, V.L.Fsup 21F2 109F2 41

DIACONESCU, O.Fsup 71

DJORDJEVIC, D.F1 75

DJUKANOVIC, G.F1 105F1 75F1 110

DINTU, S.Fsup 235Fsup 233

DOBRE, D.F1 57Fsup 209

DOBRE, A.F1 57Fsup 209

DRAGAN, D.F2 45Fsup 25

DRAGAN, F.F1 63

DUCA, E.F2 53

DUCA, M.Fsup 205Fsup 253

DUMITRESCU, C.F2 23F2 91Fsup 45

DUTA, A.Fsup 55Fsup 137

DANAILA, V.L.F2 41F2 111

ELES, G.F1 5

ENACHE, I.C.F1 95Fsup 51Fsup 219

FARCAS, A.F1 21

GANEA, D.Fsup 83Fsup 145Fsup 149

GHERGHINA, G.Fsup 129Fsup 133Fsup 137

GHIOLTEAN, L.F2 49Fsup 67

GIUREA, D.F2 23Fsup 265

GOANTA, A.M.Fsup 99Fsup 103

GRADINARU, R.Fsup 38

GROZA, O.F2 79

GRUJOVIC, D.F1 105F1 75

HARAGA, G.F1 45

HERBAN, S.I.F1 119

IGIC, M.F1 53

IONITA, E.Fsup 59

IONITA, S.F2 11F1 9Fsup 63

IOSIF, A.F1 5

JANIC, M.F1 105F1 75

JIANU, S.F1 123

JOVANOVIC, N.F1 49

JOVOVIC, I.F1 37

KISS, C.F2 105

KRASIC, S.F1 87F1 71Fsup 115

KRSTIC, H.Fsup 111Fsup 115

LAZOVIC, G.F1 79

MACHIDON, O.F1 128

MAIER, A.F2 91

MALESEVIC, B.F1 110

MAN, E.T.F1 128

MARIN, D.F2 11F2 15Fsup 63Fsup 257

MARKOVIC, B.F1 83

MATIC, V.F1 75

MEREUTA, E.F2 87Fsup 87Fsup 91Fsup 107Fsup 185Fsup 189

MEREUTA, C.Fsup 83Fsup 145Fsup 149

MIREA, M.F1 13Fsup 121

MISIC, S.F1 79

MORARESCU, A.Fsup 35

MOSKALENKO, A.Fsup 161

MUSCA, I.A.F1 17

MARZA, C.F2 53F1 21

MUSAT, C.C.F1 119

NEAGOE, M.Fsup 181

NERISANU, C.Fsup 268

NERISANU, R.Fsup 25

NECHITA, M.Fsup 201Fsup 261

NICOLAU, V.Fsup 201

NICOLIC, V.F1 83

NICOLIC, O.F1 83

OBRADOVIC, M.F1 110F1 79

OPREA, G.Fsup 9Fsup 205Fsup 253

ORBAN, M.F2 33Fsup 5

PASCU, N.E.Fsup 165

PAVEL, C.F2 5

PECICAN, A.H.Fsup 241Fsup 247

PEJIC, P.F1 87F1 71F1 41

PETRE, I.F2 29

PETRESCU, L.F2 15Fsup 257

PETROVIC, M.F1 110

PITRU, A.Fsup 137

POMAZAN, V.Fsup 153Fsup 157

POPA, C.F2 29

POPA, C.L.Fsup 215

POPA, V.Fsup 215

POPA, D.Fsup 129Fsup 133Fsup 137

POPESCU, D.I.F1 17Fsup 67

POPKONSTANTINOVIC, B.F1 79

POVIAN, C.Fsup 45

PRICOP, C.F1 128

PRUNA, L.F1 27F1 31Fsup 229Fsup 237

PADURARU, C.F1 128

RACOLTA, A.Fsup 265

RADU, I.V.F2 11F1 9

RAICU, L.F2 83

RANDELOVIC, D.Fsup 111Fsup 115

RUGESCU, A.Fsup 9

RUS, M.A.F2 87Fsup 87Fsup 91Fsup 107Fsup 185Fsup 189

RUSU, A.M.F2 65

SAFTENCUFsup 241Fsup 247

SASS, L.Fsup 55

SCURTU, L.I.Fsup 95

SHIBATA, A.F1 67Fsup 79

SILVASAN, C.F2 97F2 101

SIMION, I.F1 95Fsup 51Fsup 75

SLONOVSCHI. A.F1 27F1 31Fsup 229Fsup 237

STIRBU, I.Fsup 235

STOIANOF, C.M.Fsup 38

STANCIU, V.F2 5

STANILA, A.F2 71F2 75

STANILA, O.F2 71F2 75

SULETEA, A.Fsup 233Fsup 235

SZITAR, MF1 101

TARAS, I.P.Fsup 173

TEODOR, R.F.Fsup 201

TOCARIU, L.F2 57Fsup 169

TOMA, A.F2 71F2 75

TODORUT, A.F2 37

TRIPA, M.S.F2 49

TUDORAN, M.S.Fsup 83Fsup 145Fsup 149

TUTUNEA, D.Fsup 129Fsup 133Fsup 137

UNGUREANU, D.F2 29

URDEA, M.Fsup 141

URSU-FISCHER, N.F1 17

VALTER, N.Fsup 9Fsup 205Fsup 221Fsup 253

VASILESCU, I.F2 83

VERESIU, S.F2 87Fsup 87Fsup 91Fsup 107Fsup 185Fsup 189

VILCEANU, C.B.F1 119

VIRCHENKO, V.G.Fsup 173

PapersFascicola 1ContentSIMULATION OF THE TWO-DIMENSIONAL LIQUID MOTION WHEN PASSING THROUGH THE BARS OF A RAKING SCREEN Iosif Anton, Eleş GabrielWATER DISINFECTION WITH UV EQUIPMENT Ionuţ-Valentin RADU Stela IONIŢĂAUTOCAD SOLUTION FOR A ROOF Monica MIREADetermination of Equal Sound Level Curves Behind theNoise BarriersIoana Alexandra MUSCĂ, Nicolae URSU-FISCHER, Diana Ioana POPESCUMeans of Representing the Plumbing Installations Studied to Building Services Engineering Program, Carmen Mârza, Anagabriela Farcas, Georgiana CorsiucChoosePanels a computer program for panels design in civil engineering Liviu PRUNĂ Andrei SLONOVSCHIDetermination of Some Dimensioning Characteristicsin the Computer Assisted Drawings Casein the Civil EngineeringAndrei SLONOVSCHI, Liviu PRUNĂVisualization in teaching and learning mathematicsin elementary, secondary and higher educationBranko J. Malešević, Ivana V. Jovović, Bojan D. BanjacGeometric Patterns in the Floor of the Church of the Holy Archangels near Prizren, Igor BJELIĆ, Petar PEJIĆModeling Eco-Friendly Products in CAD SystemsGeorgeta HARAGASpecific Features of Modeling a Structure ContainingSlices of Paraboloid in Different Modeling SoftwareNenad JOVANOVIĆLogo as basis for architectural shaping, Milica Igic, Zeljko BlagojevicGeometric principles for the study of repeated patternsin one and two dimensionsDaniel Dobre, Adrian DobreThe Role of Information in Engineerign DesignCrăciun Florina, Drăgan FlorinStudy about the expression of void spacecomposed by sequence of subdivided planesAkihiro SHIBATA, Naomi ANDOAnalysis of Mapping of Hyperbolical Paraboloid with theAid of Absolute Conic in General Collinear SpacesSonja Krasic, Petar PejicPencils of Curves of the 4th and 3rdOrder Obtained asHarmonic Equivalents of Hyperbolic-Elliptic Pencil ofConicsGordana DJUKANOVIC, Djordje DJORDJEVIC, Milorad JANIC,Dragomir GRUJOVIC, Vjaceslava MATICGenerating a Type of Concave Cupolae of Fourth SortSlobodan MIŠIĆ, Marija OBRADOVIĆ, Goran LAZOVIĆ,Branislav POPKONSTANTINOVIĆPossibilities of Application of Transcendent Conesin the Form of Architectural StructuresVladan NIKOLIĆ, Olivera NIKOLIĆ, Biserka MARKOVIĆPossibilities of Application of Transcendent Conesin the Form of Architectural StructuresVladan NIKOLIĆ, Olivera NIKOLIĆ, Biserka MARKOVIĆComparative Study on Photogrammetric Methods for 3DModeling of Building HeritagePetar PEJIC, Sonja KRASIC, Igor BJELICWays of evaluating the knowledge of descriptive geometryBodea SandaCreating Realistic Images in a 3D Scene UsingIlluminations and Renderings to Create an AnimationAna-Maria AVRAMESCU, Ionel SIMION, Ioana-Catalina ENACHEIndustrial design and graphic design in architecturaleducationSmaranda-Maria BICA, Mirela-Adriana SZITARDigital Surveying Map as a Basis for Landscape DesignIllustrated by the Example of the Botanical Garden inKragujevac , Milorad JANIC, Dragomir GRUJOVIC, Gordana DJUKANOVICGenerating Curves of Higher Order Using theGeneralisation of Hügelschäffer’s Egg Curve ConstructionMarija OBRADOVIC, Branko MALESEVIC, Maja PETROVIC3Gordana DJUKANOVICThe hyposthases of symmetry Budişan TiberiuModern geodetic methods for monitoring landslides –Case study “The approach road between Orşovamunicipality and Topleţ plateau, Dranic peak, MehedinţiCounty”Clara – Beatrice Vîlceanu, Sorin Ioan Herban, Cosmin Constantin MuşatContribution to dimensioning of ditch with wallSergiu JIANU, Mircea BĂRGLĂZANMathematical modeling of advection- diffusive transportprocesses of certain pollutants from point sources in thePodu Iloaiei reservoir- Case StudyCristian Păduraru, Corneliu Cismaru, Eugen Teodor Man, Claudiu Pricop,Ovidiu Machidon

Supplementary FascicolaContentOn Tangency in Apparent Double Points of an IntersectionMagdalena ORBANThe Graphic Determination of the Specific Elements of theConic SectionsGabriel OPREA, Ana RUGESCU, Narcisa VALTERParticular Aspects of Rotation MethodSanda BODEA, Nour-I. CRIŞAN, Monica BĂLCĂUDetermining the real size of a triangular board –comparative methodsMonica BĂLCĂU, Sanda BODEAConsideration on the Multiple Threads RepresentationAlina Angelica ANGHEL, Vanda Ligia DĂNĂILĂStudy On The Various Representations Of The Viviani Conoid Surface Delia DRĂGAN, Adrian Tudoreanu – CRIŞAN, Raluca NERIŞANUCAD Modeling of Helical Cylindrical Surfaces withApplications for Helical DrillsNicuşor BAROIU, Silviu BERBINSCHIDesign of Profiles a Spiral Propulsion SystemAna MORARESCUWhy Realistic Graphic Looks Bad into an Interactive Enviroment Claudiu-Marian STOIANOF, Roman GRADINARUThe Geometry of Architectural Drawings of Arches on theSarcophagus of Patriarch Saba the IVthIgor BJELICThe Networks and the Geometrical PlanesEquipartitionsCristina POVIAN, Cristian DUMITRESCUModeling and simulation of manufacturing architecturesIoana - Catalina ENACHE, Ionel SIMION, Ana - Maria AVRAMESCUPlatonic SolidsLudmila SASS, Alina DUTA3D Modeling Of Regular, Semi-Regular And Star-ShapedPolyhedraElena IONITAStudies regarding to the DifferentialGearbox DesignDumitru MARIN, Stela IONIŢĂDesign Configuration under Parametric ControlDiana Ioana POPESCU, Lucia GHIOLŢEANThe Influence of the Wing’s Position on an Airplane’sFlightOlivian DIACONESCUGeometric Modeling in Engineering CurriculumIonel SIMIONFabrication of Furniture by Using Algorithmic DesignNaomi Ando, Akihiro ShibataEstablishing the Active Space of theHuman’s Lower Limb Usinga Java Programming SequenceDaniel GANEA, Marian Sorin TUDORAN, Claudiu MEREUŢĂGrapho-Analytical Method for Solving the Equation ofMotionElena MEREUTA, Madalina RUS, Silvia VERESIUModeling a Planar Mechanism using the Features Provided by Working Model Elena MEREUTA, Silvia VERESIU, Madalina RUSApplication Programing Interface to Generate HelicalCompression SpringsLiviu-Iacob SCURTU, Nour CRISANConsiderations on ComputerAided Design of MachineElementsAdrian Mihai GOANTAComparative Study of the Controls Generating ExternalThreads with Different Design SoftwareAdrian Mihai GOANTAThe Monitorisation of the Movement of a Series of GearsHaving Fixed Axes According to a Know LawMadalina RUS, Elena MEREUTA, Silvia VERESIUThe Influence of Different Geometrical Shapes and Colorson the Human Psyche Through Design in ArchitectureHristina KRSTIĆ, Dušan RANĐELOVIĆDrawing of Complex Geometric Surfaces Using SoftwarePackages on the Example of Hyperboloid, HyperbolicParaboloid and ParaboloidDušan RANĐELOVIĆ, Hristina KRSTIĆ, Sonja KRASIĆPlane Sections in Cilyndric-Conic Bodies with theAUTOCAD ProgramMonica MIREAAutomated Drawing – a More Productive WorkIonel BATOGConsideration about the Design of Automotive MufflersDragos TUTUNEA, George GHERGHINA, Dragos POPAAbout Element of Advanced Part ModellingGeorge GHERGHINA, DragosTUTUNEA, Dragos POPAAbout the Simulation Environment for Dental ImplantStudiesDragos POPA, Alina DUŢĂ, Allma PITRU,Dragoş TUTUNEA, George GHERGHINAStandardized Features and Database in CAD ApplicationsMihaela URDEAGraphical Method for Setting Up the Limits of theActive Space of Human Upper LimbMarian-Sorin TUDORAN, Daniel GANEA, Claudiu MEREUTAModelling the Bones of the Human Upper LimbMarian-Sorin TUDORAN, Daniel GANEA, Claudiu MEREUTAAssembly Top Down Design in Parametric CADValentina POMAZANAutomation of the Differential MechanismParametric DesignValentina POMAZAN, George CAPRINCIUModeling of Operation Modes of Front-End Loaderby the Autodesk Inventor PackageAlexander CHERNIKOV, Alina MOSKALENKOAirport Signage DesignVictor ADIR, George ADIR, Nicoleta Elisabeta PASCUThe Shape as a Means of Visual Communication in DesignLiliana TOCARIUComputer combinatorial-variation geometric modeling ofengineering objectsTARAS I.P., VIRCHENKO V.G.Methodology for Reducing the Product EnvironmentalImpact from the Early Phases of the Design ProcessLucian BARSAN, Anca BARSANThe Objective of Materials Recycling in the PackageEco-Design ProcessLucian BARSAN, Anca BARSAN, Mircea NEAGOEThe Geometrical Synthesis of the Cam MechanismsSilvia VEREŞIU, Elena MEREUŢĂ, Mădălina Alice RUSThe Geometrical Synthesis of the Slider – CrankMechanismsSilvia VEREŞIU, Mădălina Alice RUS, Elena MEREUŢĂApplications Developed for Illustrating the TechnologicalDimensioning Method. Case of a Threaded PartRamona CLINCIUApplications Developed for Representation of the Flangesin Technical DrawingMihaela Rodica CLINCIUMetallic plate behaviour under impact loadsManuela NECHITA, Viorel NICOLAU, Renardo-Florin TEODORBlack and white binomial psychological effects of itsdesign and ergonomicsMioara DUCA, Narcisa VALTER, Gabriel OPREAAspects regarding the dynamic behavior of elasticcouplings with metallic flexible membranesDaniel DOBRE, Adrian DOBREAddiction of compression process in the screwcompressors with the transverse profile rotorsCamelia Lǎcrǎmioara POPA, Viorel POPASolutions to improve production capacityIoana Catalina ENACHE, Narcisa VALTER, Florina CHISCOPIssues on representation of dual worm-face gear driveMargareta Ciotea, Codruţa BoloşPowerPoint use in building self assessment tests forgeometry descriptive disciplineLiviu PRUNĂ, Andrei SLONOVSCHICreating interactive materials forvirtual educationSergiu DINTU, Angela ŞULETEA, Lilia BLAŞCU, Silvia ANATII,Cristian CARTOFEANUThe use of different types of animation for the goal ofincrease of accessibility of electronic textbooksSergiu DINTU, Angela ŞULETEA, Iurie CĂPĂŢÂNĂ, Ion ŞTIRBUSoftware to study the streight lineAndrei SLONOVSCHI, Liviu PRUNĂTechnical graphics project – where, when, how and why?Doina SAFTENCU, Arseniu Hanzi PECICANIndividual work or teamwork?Arseniu Hanzi PECICAN, Doina SAFTENCUTechnical and economic arguments to use infograficrepresentations during the design modular control devicesNarcisa VALTER (POPESCU), Mioara DUCA, Gabriel OPREAA useful direction in Graphical Sciences teachingLigia PETRESCU, Dumitru MARINEvaluation of Computer Aided Design Skills inEngineering EducationManuela NECHITA, Silviu BERBINSCHIProgressive Bimodular PartitionsAndrei RACOLTA, Diana GIUREAComparative Study of Results Got After Applying MCTand MRT Tests with Male and Female Undergraduates inCivil EngineeringClaudia ALB, Raluca NERIŞANU, Radu DARDAI

Fascicola 2ContentAbout the IN-YOLOGICAL Interpretation of aComplex PlaneVirgil STANCIU, Cristina PAVELTheoretical and Applied Aspects ofAxonometric RepresentationsStela IONIŢĂ, Dumitru MARIN, Ionuţ –Valentin RADUGraapho-analytical studies of a conical surfaceand its applications in industrial designDumitru MARIN, Ligia PETRESCUMethods to Determining the Tangent Plane to a SphereNour CRISAN, Sanda BODEAComparative study of planar representations of spatialforms for architecture studentsDiana Giurea, Cristian DumitrescuMethods for Obtaining the Intersection Curves for theCylindrical SurfacesCarmen POPA, Ivona PETRE, Dan UNGUREANUConsiderations of Curvature Determination of SomeSpecific Plane CurvesMagdalena ORBANParticular Cases when Constructing a SphereMonica BĂLCĂU, Istvan BARABAS, Adrian TODORUŢMethod to Develop a Bend Consisting of Conical ElementsVanda – Ligia DănăilăStudy Concerning Various Representations of the KüperConoid SurfaceDelia DRĂGAN, Dorin BĂRBÎNŢĂ, Adrian Tudoreanu – CRIŞANIssues on Ruled Conoid-Type SurfacesLucia Margareta GHIOLŢEAN, Mihai Sorin TRIPAConsiderations Regarding the Geometrical Constructionof Distributors/Collectors Equipment Used inVentilation/Air Conditioning.Carmen MÂRZA, Eugenia DUCA, Georgiana CORSIUCShadow Representation in Central Perspective, aDescriptive Geometry ApplicationLiliana TOCARIUDesigning a Quasicircular Pool Using DescriptiveGeometryTiberiu BUDIŞANStructural form in Architecture of the 20st and 21stCentury Based on Warped SurfaceAna-Maria RUSUCable Mesh StructuresAneta STANILA, Ana-Maria TOMA, Oana STANILAThe V-Shaped ColumnsAna-Maria TOMA, Aneta STANILA, Oana STANILAGeodesic StructuresOthilia GROZASections and Intersections of the Surfaces in theHistory of GeometryLucian RAICU, Ileana VASILESCUThe Intersection of Polyhedrons – Teaching - LearningAspectsMadalina RUS, Silvia VEREȘIU, Elena MEREUȚĂRuled surface formsAlexandra MAIER, Cristian DUMITRESCUSections in Cylinders by the Projection on InclinedPlanes.Tangent to the Sectioning Curve in Particular Points.Tangent Generatrix Method.Claudiu SILVĂŞAN3-D Curves Intersection between cylindres:Projection on curved surfacesTangent to the curve in particular pointsClaudiu SILVĂŞANIntersection between two surfaces with AutoCad- the descriptive geometry method Gheorghe BELEA Cantemir KISSMethod to develop a bend consisting of cylindricalelements with axes laying in different planesVanda – Ligia Dănăilă, Alina – Angelica Anghel

Timisoara in images