Proceedings Conference MoNGeometrija 2010 9788679240408

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UNDER THE AUSPICES OF: Republic of Serbia – Ministry of Science and Technological Development Faculty of Architecture in Belgrade Faculty of Mechanical Engineering in Belgrade Faculty of Civil Engineering in Belgrade Faculty of Forestry in Belgrade Faculty of Transport and Traffic Engineering in Belgrade Faculty of Applied Arts in Belgrade

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25th National and 2nd International Scientific Conference moNGeometrija 2010 Title of Publication

Publisher | Izdavač Faculty of Architecture in Belgrade Arhitektonski fakultet u Beogradu

PROCEEDINGS | BILTEN Reviewers | Recezenti

Editor-in-Chief | Glavni urednik

Ph. D. Miodrag Nestorović Ph. D. Aleksandar Čučaković Ph. D. Marija Obradović Ph. D. Branislav Popkonstantinović M. Sc. Magdalena Dimitrijević M. Sc. Slobodan Mišić Co-Editor | Zamenik urednika

Ph.D. Miodrag Nestorović

M. Sc. Magdalena Dimitrijević

Serbian Society for Geometry and Graphics Srpsko udruženje za geometriju i grafiku SUGIG

Text formatting Miljan Radunović, Ph. D. Candidate

ISBN 978-86-7924-040-8 Numbers of copies printed | Tiraž: 100 Printing | Štampa: All rights reserved. No parts of this publication may be reprinted without either the prior written permission of the publisher. Izdavač zadržava sva prava. Reprodukcija pojedinih delova ili celine nije dozvoljena.

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Conference Organizers

Main Organizers Faculty of Architecture, Belgrade Faculty of Mechanical Engineering, Belgrade

Co-organizers Faculty of Civil Engineering, Belgrade Faculty of Forestry, Belgrade Faculty of Transport and Traffic Engineering, Belgrade Faculty of Applied Arts, Belgrade

Conference Committees

Executive Committee

Ph.D. Miodrag Nestorović, Full professor, Faculty of Architecture, Belgrade Ph.D. Branislav Popkonstantinović, Associate professor, Faculty of Mechanical Engineering, Belgrade Ph.D. Biserka Marković, Full professor, Faculty of Civil Engineering and Architecture, Niš Ph.D. Aleksandar Čučaković, Associate professor, Faculty of Civil Engineering, Belgrade Ph.D. Radovan Štulić, Full professor, Faculty of Technical Sciences, Novi Sad Ph.D. Ljubica Velimirović, Full professor, Faculty of Science and Mathematics, Niš

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Supervisory Committee

Ph.D. Miodrag Stoimenov, Associate professor, Faculty of Mechanical Engineering, Belgrade M.Sc. Zorana Jeli, Lecturer Assistant, Faculty of Mechanical Engineering, Belgrade Ph.D. Ratko Obradović, Associate professor, Faculty of Technical Sciences, Novi Sad Ph.D. Jelisava Kalezić, Associate professor, Faculty of Civil Engineering, Podgorica, Montenegro M.Sc. Marija Jevrić, Lecturer Assistant, Faculty of Civil Engineering, Podgorica, Montenegro Ph.D. Ljiljana Petruševski, Associate professor, Faculty of Architecture, Belgrade Ph.D. Dragan Petrović, Associate professor, Faculty of Mechanical Engineering, Belgrade Dr Mirjana Devetaković Radojević, Docent, Faculty of Architecture, Belgrade M.Sc. Gordana Đukanović, Lecturer Assistant, Faculty of Forestry, Belgrade Maja Petrović, architect, Faculty of Transport and Traffic Engineering, Belgrade

Organizational Committee

Ph.D. Vladimir Mako, Dean of the Faculty of Architecture, Belgrade Ph.D. Đorđe Vuksanović, Dean of the Faculty of Civil Engineering, Belgrade Ph.D. Milorad Milovančević, Dean of the Faculty of Mechanical Engineering, Belgrade Ph.D. Milan Medarević, Dean of the Faculty of Forestry, Belgrade Ph.D. Slobodan Gvozdenović, Dean of the Faculty of Transport and Traffic Engineering, Belgrade Ph.D. Vladimir Kostić, Dean of the Faculty of Applied Arts, Belgrade Ph.D. Aleksandar Čučaković, Associate professor, Faculty of Civil Engineering, Belgrade Ph.D. Branislav Popkonstantinović, Associate professor, Faculty of Mechanical Engineering, Belgrade Ph.D. Marija Obradović, Docent, Faculty of Civil Engineering, Belgrade M.Sc. Magdalena Dimitrijević, Lecturer assistant, Faculty of Civil Engineering, Belgrade M.Sc. Slobodan Mišić, Lecturer assistant, Faculty of Civil Engineering, Belgrade M.Sc. Biserka Nestorović, Lecturer assistant, Faculty of Forestry, Belgrade M.Sc. Biljana Jović, Lecturer assistant, Faculty of Forestry, Belgrade

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Scientific Review Committee

Professor Đorđe Zloković, Member of Serbian Academy of Sciences and Arts, Department of Technical Sciences, Belgrade Ph.D. Miodrag Nestorović, Full professor, Faculty of Architecture, Belgrade Ph.D. Hellmuth Stachel, University of Technology Vienna, Austria Ph.D. Gunter Weiss, Technical University of Dresden, Germany Ph.D. Emil Molnar, Technical University of Budapest, Hungary Ph.D. Aleksander Dvoretsky, Kiev National University of Building and Architecture, Ukraine Ph.D. Milena Stavrić, TU Graz, Institut für Architektur und Medien, Austria Ph.D. Aleksandar Veg, Full professor, Faculty of Mechanical Engineering, Belgrade Ph.D. Aleksandar Čučaković, Associate professor, Faculty of Civil Engineering, Belgrade Ph.D. Natasa Danilova, Faculty of Civil engineering, Architecture and Geodesy, Sofia, Bulgaria Ph.D. Carmen Marza, Technical University of Cluj-Napoca, Romania Ph.D. Sonja Gorjanc, Faculty of Civil Engineering, Zagreb, Croatia, Hrvatska Ph.D. Marija Obradović, docent, Faculty of Civil Engineering, Belgrade M.Sc. Branko Pavić, Full professor, Faculty of Architecture, Belgrade Ph.D. Ivana Marcikić, Full professor, Faculty of Applied Arts, Belgrade Ph.D. Hranislav Anđelković, Retired full professor, Faculty of Civil Engineering and Architecture, Niš Ph.D. Miroslav Marković, Retired full professor, Faculty of Civil Engineering and Architecture, Niš Ph.D. Lazar Dovniković, Retired full professor, Faculty of Technical Sciences, Novi Sad

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Topics Theoretical geometry, exposed by synthetical or analytical methodology: * Descriptive and constructive geometry * Projective geometry * Central projection, Perspective and Restitution * Cartography * Theory of Polyhedra * Fractal geometry Geometry and Graphics applied in Engineering and Architecture: * Engineering graphics * Computational geometry (algorithms, computer modeling of abstract geometrical objects, structures, procedures and operations) * Computer Aided Design and Drafting; Geometric and Solid Modeling; Product Modeling; Image Synthesis; Pattern Recognition; Digital Image Processing; Graphics Standards; Scientific and Technical Visualization * Kinematics Geometry and Mechanisms * Applications of Polyhedra theory * Fractals * Computational restitution * Stereoscopy and Stereography * Virtual reality Geometry applied in Visual Arts and Design: * Theory and application of Visual Aesthetics * Geometrical and mathematical criteria of Aesthetic values * Perception and meaning of colors * Geometrical forms applied in Visual Arts * Optical illusions and its applications History of Geometry: * Famous scientist and their contribution * Origin, derivation and development of particular geometrical branches * History of geometrical education Education and didactics: * Descriptive Geometry and Graphics Education, including the Reform of Education * Education Technology Research * Multimedia Educational Software Development * Virtual Reality Educational Systems * Educational Software Development Tools Research and so on

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DEVELOPING SPACE PERCEPTION BY MEANS OF 3D REPRESENTATIONS Dorin Bărbînţă1 Radu Dardai2 Raluca Nerişanu3

Abstract In the context of accelerated development of information technology, the use of the computer in education becomes a necessity. If traditional teaching methods have been verified by practice in time, modern methods make this verification with the help of the technical progress in the field of computers. This paper aims at presenting the intersection of two bodies and its solution by the traditional method and with the help of the modern computer. It is considered that the parallel manner of presentation of the two methods leads to a faster and easier understanding of the problem. Key words: intersection, prism, common solid, progress, engineering.

1

Dorin Bărbînţă, eng., PhD student, lecturer, Technical University of ClujNapoca, Romania, Faculty of Civil Engineering, Dep. of Geodesy and Engineering Graphics, e-mail: [email protected]. 2 Radu Dardai, eng., teaching assistant, Technical University of Cluj-Napoca, Romania, Faculty of Civil Engineering, Dep. of Geodesy and Engineering Graphics, e-mail: [email protected]. 3 Raluca Nerişanu, eng., PhD student, teaching assistant, Technical University of Cluj-Napoca Romania, Faculty of Civil Engineering, Department of Geodesy and Engineering Graphics, e-mail: [email protected].

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1. INTRODUCTION Descriptive geometry is one of the basic subjects for engineering as it contributes to establishing and developing space visual perception. Together with technical drawing, the two subjects aim at providing basic graphical engineering knowledge to the undergraduates and developing a universal graphical language of communication in engineering. The example taken by the authors refers to an intersection of two solids, first presented in and then with the help of the 3D representation made with computer software. 2. THE INTERSECTION OF TWO SOLIDS IN ORTHOGONAL PROJECTION A particular case was used for our discussion that is the intersection of two solid bodies. Figure 1 shows the intersection of two prisms placed in special, particular positions, one in perpendicular line to [W] and one in vertical position. The solution is made in the orthogonal projection on three planes of projection. To find the polygon of intersection, we used as auxiliary planes, frontal planes that are parallel to the edges of the two prisms. The representation of the common solid resulted from the intersection, in an orthogonal projection on two or three planes of projection is difficult to be understood by the undergraduates (Figure 2). It is for this reason that we have considered that a representation in axonometry can be faster and easier to understand. The training of a student to be able to solve problems requiring a good spatial perception requires common effort and a lot of energy on behalf of the teacher. In the classroom, a proper working rhythm can then be established for all the student groups who should be individually motivated and involved to actively take part in the classroom activities.

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Fig. 2 Representation of the common solid in an orthogonal projection on three planes of projection

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3. SOLVING THE INTERSECTION OF TWO SOLIDS WITH THE HELP OF THE CAD SOFTWARE We previously presented the intersection of two solids with the conventional or traditional method. Figure 3 presents the intersection with the help of a CAD software application. The final results are reached much faster, practically with some clicks. One can see the analysis of the two bodies intersected from any direction (Figure 4). The common solid is very easy extractable, as one can see in Figures 4 and 5.

Fig. 3 Representation of the intersection of two prisms with the help of CAD software

In the engineering field one cannot conceive education without computers today. The computer is the most suitable instrument for the engineers to express their way of thinking with respect to the filed they are working in. In general, people are reluctant to modern methods application and they think it is dangerous to abandon the study of the traditional disciplines or to suddenly replace them with subjects that are better adapted to today’s realities. [4]

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Fig. 4 Representation of the prisms, after the intersection, with the help of CAD software

Fig. 5 Representation of the common solid with the help of CAD software

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The computer is the invention of the genius, but at the disposal of the human and consequently the results depend upon the connection between the man and the instrument created for him. One should consider some conditions when thinking of reaching the purposes established: adapting the education process to the needs of the present-day situation and to those in perspective for the society, the continuous enriching and updating of the educational system and methods, achieving progress in the field of computers, information science, and communication technologies. [4] 4. CONCLUSIONS In all fields, solving a problem with the help of the computer and only with one and the same application, can lead to dependence on the software used. Such a fact can lead to inconveniences for the graduate entering the labour market. It is obvious that ideally the graduate leaving school should be able to work and used as many software programs as possible from those on the market and available for the field of study of the graduate.

Fig. 6 Examples of existing buildings

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Figure 6 presents an example of intersections of bodies used practically for the construction of buildings. The authors agree that the classical methods should not be entirely replaced by the very modern ones, both regarding to teaching and learning. It is rather recommendable to find a solution that mixes up the two methods mentioned. The CAD application used is useful, simplifies understanding and learning, but it cannot replace totally the classical part of descriptive geometry, it can only complete it. If we are able to blend the traditional and the modern in teaching, if we manage to make students take part in their own education, if the students we teach now are able to discover truths, then the methods we used in the educational process will be seen as the most suitable.

5. LITERATURE 1.

2.

3. 4.

Adir, V., Pascu, N.E., Graphic education as teaching method concerning the representation of pieces as views and sectional views, Proceedings of the 2nd International Conference of Engineering Graphics and Design, Dunarea de Jos University Galati, pp. 425-428, ISBN 978-973-667-252-1, Galati Romania, June 2007, Publisher Cermi, Iasi. Belea, G. Intersecţia poliedrelor în axonometrie. Volumul I - A VIIa ediţie a conferinţei naţionale cu participare internaţională, Craiova România, 2002. Zetea, E., Drăgan, D. Geometrie descriptivă–probleme. Editura Universităţii Tehnice, Cluj-Napoca, 1999. D. Bărbînţă, D. Drăgan, R. Nerişanu: Using alternative pedagogical methods in teaching graphics subjects, The 3rd International Conference on engeenering graphics and design, CLUJ 2009, Acta Tehnica Napocensis, 2009, ISSN 1221-5872, pag. 597.

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THE POTENTIAL OF GEOMERTICAL AND ARTISTIC CHARACTERISTICS OF THE HYPERBOLIC PARABOLOID IN APPLIED ARTS Petar Blagojević1 Filip Popović2 Tijana Tripković3

Résumé The aim of this paper is to research the hyperbolic paraboloid through applied geometry. There are three approaches to the theme of the hyperbolic paraboloid which have potentials in fine and applied arts as well as design: -The developing of the net of hyperbolic paraboloid, using cone segments. The aim is to show the efficiency of the developed net approximation method. -The pictorial projection on the net of the hyperbolic paraboloid. -Researching the rotation of the hyperbolic paraboloid, expecting a number of interesting forms. Keywords: hyperbolic transformation

paraboloid,

rotation,

projection,

geometric

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Petar Blagojević, 2nd year BA student of interior and furniture design, Faculty of Applied Arts, University of Arts in Belgrade, [email protected] 2 Filip Popović, 3rd year BA student of graphic design, Faculty of Applied Arts, University of Arts in Belgrade, [email protected] 3 Tijana Tripković , 2nd year BA student of interior and furniture design , Faculty of Applied Arts, University of Arts in Belgrade, [email protected]

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1. DEVELOPING THE NET OF THE HYPERBOLIC PARABOLOID

1.1. Introduction This section explores a method of developing the net of this non-developable surface. The examples of developing the net of non-developable surfaces which were found in available literature are mostly based on the use of polygons. One exception is Durer’s construction of the sphere net. He developed the sphere net using segments which were determined by sections of a circle as meridians. This approach is less complicated than the ones that use polygons, and also the net approximated in this way is more suitable for a sphere form, as it is constructed by use of the circle arches (and a sphere alone is formed by a rotation of a circle around its diameter). This Durer’s approximation was the direct inspiration for developing the net of hyperbolic paraoloid using the segments determined by curves which define it completely, and these are hyperbolae and parabolae. 1.2 Dividing the hyperbolic paraboloid into segments The xz cross-sections of hyperbolic paraboloid are translated copies of a common parabola P, and the yx crosssections are translated upside down copies of the same parabola P (Fig.1). This parabola will be used as a starting point for dividing the hyperbolic paraboloid into segments and developing its net.

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Fig. 1 - xz and yz cross sections of hyperbolic paraboloid

An example of dividing the hyperbolic paraboloid into segments (Fig.2) shows that the net will be constructed from cone segments. A cone of each segment has a different apex angle (β), where the generatrix of the referring cone matches the tangent of the central parabola of the hyperbolic paraboloid. In the middle of each segment there is a parabola of the hyperbolic paraboloid which is placed between two parabolae of the cone from which the segment is cut off. These two parabolae do not belong to the hyperbolic paraboloid.

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Fig. 2 – Dividing the hyperbolic paraboloid into segments

1.3. The process of positioning the observed parabola on the specific cone In order to locate the wanted segments on a cone, it is necessary to determine the position of the observed parabola on the cone. The orthogonal projection of a right circular cone with its parabola section represents the system of parabolae that have mutual focus, which matches the orthogonal projection of the cone apex (Fig. 3).

Fig. 3 – Orthogonal projection of a cone with its parabolas

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In order to locate the given parabola of the hyperbolic paraboloid on the cone it is necessary to geometrically transform the orthogonal projection of this parabola into the plane parallel to the cone base. The example of this process is given in the Figure 4. The transformation angle α is equal to the angle between the generatrix and the basis of the cone.

Fig. 4 – Geometrical transformation of the given parabola

Figure 5 shows the example of locating the parabola on the cone using the orthogonal projection of the cone.

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Fig. 5 – The location of the parabola on the cone

1.4. The construction of the net segments Once the parabola is located on the cone, the construction of the cone layout and the wanted segment on it can be done. The development of the cone layout with parabolae is shown in an example of developing the first segment of the hyperbolic paraboloid net. All of the of cone apexes are turned downwards (Fig. 6).

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Fig. 6 – The position of the cone apex, where one half of hyperbolic paraboloid is observed Fig. 7 – The position of the cone segmen

Figure 7 shows the location of the segment where the central parabola of hyperbolic paraboloid is found on the cone using the explained method, and the second parabola is constructed by placing the segment into the upper projection of the cone. The cone layout is constructed by a 15° angled rectification of the circle arch referring to CUSANUS and SNELL. Then the real distances of the parabolas intersection with the sides of the cone from the apex are transferred on the layout.4 Second parabola is used as the first parabola on the next segment, and the whole process is repeated for other segments (Fig. 4-7). Figure 8 presents the approximated net of the hyperbolic paraboloid constructed using the explained method. The main flaw of this approximation is the geometric deviation of the net edges from the edges of the hyperbolic paraboloid. The segment edges are concave towards the center of the hyperbolic paraboloid. This deviation comes as a result of locating the 4

Dr. phil. Karl Strubecker, VORLESUNGEN ÜBER DARSTELLENDE GEOMETRIE, Göttingen, 1967., p.142.

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segment on the first cone projection, and then transferring it on the second projection. It is clear that the alignment of the net edges and the edges of the hyperbolic paraboloid in one projection would cause their misalignment in the other. Figure 9 shows the net constructed from the segments where all of the cone apices are turned upwards (Fig. 2). This is opposite to the net from the Figure 8. This net has significantly minor misalignments from the edges of the hyperbolic paraboloid, than the one mentioned above. Here the edges of the net are convex towards the center of the hyperbolic paraboloid.

Fig. 8 – The net approximation Fig. 9 – The net approximationBoth of these two nets have one mutual

flaw. Both nets have the central parabola which belongs to hyperbolic paraboloid, but this parabola doesn’t exist on other segments. This is because the second parabola of one segment is used as the first parabola of the next segment every time. The parabolas differ as the segments approach the apex of cone, or get farther from it, depending on the method. In Figure 10 is an example of the net which is constructed by combination of these two methods, so the cone apices of two neighbor segments are opposed. The cone apices of segments s1, s3 and s4 are upwards, and the cone apices of the segments s2 and s5 are directed downwards. This enables every segment,

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except segment s2 to have one parabola which is also the parabola of hyperbolic paraboloid. Never the less these parabolae are not positioned in the middle of segments, but are slightly dislocated which is shown in Figure 11.

Fig .10 - The net approximation Fig.11 – The dislocation of parabolae

2. IMAGE PROJECTION ON THE HYPERBOLIC PARABOLOID

The idea to apply an image on the surface of a hyperbolic paraboloid, beside the opportunity to develop its net, is also to project the image directly to the surface. It is possible to do that using methods with parallel rays or from one point.

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Fig. 12 Projection of the image directly to the surface using methods with parallel radiances (perspective view)

Figure 13 shows the image projection (in this caselogo) that is applied on the surface of the hyperbolic paraboloid which is inscribed in a cube. The orthogonal projection of the hyperbolic paraboloid is a square grid, so the image projected on it, is an illusion of a flat plane. This stands only for this point of view.

Fig. 13 Orthogonal projections

All edges and essential points of logo were marked from the top view, and projected on a grid (so it could be easier

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and more precise to route the image). Then, with parallel rays those points are projected to the grid from the side view. 3. ROTATION OF THE HYPERBOLIC PARABOLOID The criteria for the axes of rotation were to take the elements that define the cube. Once this is determined the rotation is done in x, y, or z coordinate. Examples: - Rotating around the center of the cube – R ((Fig. 14), the axes of rotation through this point are the y (Fig. 15) and z (Fig. 16) coordinates.

Fig. 14 Cube centre

Fig. 15 Y rotation

Fig. 16 Z rotation

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- Rotating around an edge of the cube G-C, with its point R (Fig. 17). The axes of rotation through this point are coordinates z (Fig. 18) and y (Fig. 19).

Fig. 17 Cube edge

Fig. 18 Z rotation

Fig. 19 Y rotation

- Rotating at the points R1, R2, R3, R4 and R5 which define a parabola (Fig. 20). The axe of rotation is the y coordinate. Fig. 21 and Fig. 22 are cross sections of these rotations.

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Fig. 20 Parabola points

Fig.21 Y rotation

Fig. 22 Cross section of the y rotation

-The axe of rotation -y- goes through the intersection (R1, R2, R3, R4 and R5) of the symmetry axe of the cube and the chords of the parabola (Fig. 23). The y axe is normal on the axe of symmetry of the cube. Examples are in Fig. 24.

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Fig. 23 Parabola chords

Fig. 24 Y rotation

-Rotation with multiple axes (Fig. 25): 1. Rotation from the centre of the cube, around both y and z axe at the same time. 2. Rotation from the centre of the cube, around both y ,x and z axe at the same time. 3. Rotation from the diagonally of the side of a cube, around both y and z axe at the same time. 4. Rotation from the diagonally of the side of a cube, y, x and z axe at the same time. 5. Rotation from the diagonally of the side of a cube, x and z axe at the same time. 6. A point outside of the cube, y rotation, 7. A point outside of the cube, x, y and z axe at the same time 8. Rotation around the edge of the cube, y and z axe at the same time 9. Rotation around the edge of the cube, x and z axe at the same time

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Fig. 25 Rotation with multiple axes

CONCLUSION The aim of this work was to apply geometry in art. This was accomplished by separate approaches, which had the hyperbolic paraboloid as subject. The results of this work are applicable in both art and design, confirming the advantages of applied geometry. LITERATURE 1. Đurović Vinko, Nacrtna goemetrija, „Naučna knjiga“, Beograd, 1977. 2. Dr. phil. Karl Strubecker, Vorlesungen über darstellende geometrie, Göttingen, 1967. 3. Zloković Đorđe, Koordinirani sistem konstrukcija, „Izdavačko preduzeće Građevinska knjiga“, Beograd, 1969.

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GEOMETRY OF TRANSITIONAL DEVELOPMENT SURFACES Vladimir Calic5 Branislav Popkonstantinovic 6 Zorana Jeli 7

Resume The technique has often the problem of transition from one shape and the size of the cross section of water to another. The continuity of the transition changes to pieces easily accomplished when He made this casting. However, at low pressure lines (ventilation, pneumatic transport) that allow thin-wall lines, transitional pieces are made of thin sheet metal or welding summer. Technology and need continuous change of the cross section requires that surfaces transitional pieces to be developing that could make the cutting and bending sheets without their creasing or stretching. We will consider cases of transition from polygon to polygon and polygons on a flat curve. Tool use SolidWorks 3D software that has the ability to work with planes and cubes. The basic concept of work based on the application of geometry in the industry making parts from sheet metal deformation, cutting and welding. Key words: surface, plane, polygon, sheet metal

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V. Calic, student of graduate year, Faculty of Mechanical Engineering, Belgrade 6 B. Popkonstantinović, Ph.D., Assoc. Prof., Faculty of Mechanical Engineering, Belgrade 7 Z. Jeli, M.Sc., Assistant, Faculty of Mechanical Engineering, Belgrade

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1.

INTRODUCTION

The technique has often the problem of transition from one shape and the size of the cross section of water to another. The continuity of the transition changes to pieces easily accomplished when He made this casting. However, at low pressure lines (ventilation, pneumatic transport) that allow thin-wall lines, transitional pieces are made of thin sheet metal or welding summer. Technology and need continuous change of the cross section requires that surfaces transitional pieces to be developing that could make the cutting and bending sheets without their creasing or stretching. We will consider cases of transition from polygon to polygon and polygons on a flat curve. Tool use SolidWorks 3D software that has the ability to work with planes and cubes. The basic concept of work based on the application of geometry in the industry making parts from sheet metal deformation, cutting and welding. Method of making depends on the size of overall measures of the part, so we consider two solutions, cutting and wrapping the whole separation layer on the main parts of which consists, flat triangles and triangular parts of the envelope cone slant. The main problem that arises when bending sheet metal edges are sharp polygon problem that must be rounded, depending on the angle of bending, the type of material, method of bending. In the case of our examples fillet radius must be greater than one thick sheet. When cutting the layer should take into account the direction of the sheet metal rolling, bending should be normal on the fibers to avoid shooting in the area of sheet metal bending. 2.

THE TRANSITION FORM ONE POLYGON TO ANOTHER

When the polygons of the same kind of perspective and a collinear position, interim development surface layer seems truncated pyramid or prism (as it is associated with the polygon that is cut. Lie in the same plane. As soon as the two long passing loop their development area will consist of two triangular plane (the triangle is determined by a longer and longer as the final point of the other, there are, therefore, two solutions).

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Figure 1.

Figure 2. In the figures 1 and 2 is shown in axonometry crossing the quadrangle and the pentagon shown in pairs orthogonal projection case, but when the same polygons lie in parallel planes. Polygons are placed so that the 2-3 and BC lie on the same plane, so the developed layer transition pieces consist of the real size of a trapezoid, triangular and 1-4 C, because of symmetry, three pairs of equal triangles.

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Figure 3 For proper triangutation coupling surface is best to imagine that the two polygons in space rests flat on the outside. Then, for example, even set up through the side AB pentagon, turning around and straight, square touch the outside of the crown 2, which means that it will even set aside 1-2 in the square outside the first touch the Pentagon in crown A. This means that the transition from AB to 1-2 do triangles AB-2 and 1-2-A. (Fig. 3). 3.

SOLUTION OF THE PROBLEM USING 3D SOFTWARE

In two parallel planes create drawings that present a view from below and from the top, between the two drawings drawn line represents the path of movement of one drawing to another (fig. 4, 5 and 6).

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Figure 4

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Figure 5.

Figure 6

4.

CROSSING FROM A FLAT CURVE TO THE POLYGON

The shell of the development areas for the transition from straight to curve polygon composed of flat triangles and triangular parts of slanted cone layer (of each other the same number of pages as polygon vertices). For the surface smoothness of development it is essential that the plane of the triangle is duo tangent plane of two adjacent cone, that is. tangent to the curve is directly in line with the corresponding side polygons. Now the triangle is tangent to cone is generating and the transition from flat to conical surface smooth plane.

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Figure 7 The figure 8 shows in orthogonal projections, the transition from circular pieces of square shape, when the plane circle and a square cut in the finality of the (left) and when they intersect in infinity (right). (Note the difference in the position of point D, in the second case, due to double symmetry, is sufficient to determine the right size of a flat triangle and develop a layer of a conical part.).

Figure 8

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5.

SOLUTION OF THE PROBLEM USING 3D SOFTWARE

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Figure 9 Figure 9 shows the method of creation, in two different planes construct circular and square shape. Tangent line joining the two planes, tangent line represents the line of keeping the transition from circular to square shape. The distance between the casings used for development. With options in SolidWorks LOFTED BENDS mark in the circular and square form, (in our case we select the thickness of sheet). In Figure 10 we open the envelope to receive choosing Flatten.

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Figure 10

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6.

FINAL REMARKS

This work was carried out presentations, problem-solving crossing polygons of the polygon and the transition from plane curve in the polygon, to solve problems that result from bending sheet metal. All solutions are received exclusively using 3D software. 7. 1. 2.

LITERATURE

Dovniković L. Nacrtna geometrija, Matica Srpska, Novi Sad Gagić Lj., Nacrtna geometrija, Naučna knjiga, Beograd

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THE PLACE OF GEOMETRY IN TRANSILVANIAN SCHOOLS AT THE BEGINNING OF XX CENTURY Crăciun Florina1 Drăgan Florin2 Drăgan Delia3

Resume The study is “a look” over almost two hundred years of education in Romanian schools, insisting on geometry. The reason of choosing this subject is the less information we have found about geometry (separated from the mathematical study) as one of the discipline considered important in education. In hard conditions, geometry, like other disciplines continued for centuries to have an important role in education. The two influences upon educational system, (German and Austrian, on one hand, and French on the other hand) have been dissolution into one Romanian educational system. Representatives of Romanian science and culture of the XIX and XX century succeeded in difficult conditions to diminish the educational postponement comparative with the developed European nations. Key words: geometry, technical drawing, Transylvania, education

1

8

INTRODUCTION

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Florina Crăciun, Ph.D.Eng., associate professor, Technical University of Cluj-Napoca, Faculty of Civil Engineering, Department of Geodesy and Engineering Graphics, E-mail: [email protected], Phone: 0264 - 401842 2

Florin Drăgan, eng., PhD, professor, Technical University of Cluj-Napoca Romania, Faculty of Electrical Engineering, Department of Electrical Measurements, e-mail: [email protected]. Office Phone: 0040 264 401520. 3 Delia Drăgan, eng., PhD, professor, Technical University of Cluj-Napoca Romania, Faculty of Civil Engineering, Department of Geodesy and Engineering Graphics, E-mail: [email protected]. Office Phone: 0040 264 401842.

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This study is an occasion for “a look” over almost two hundred years of education in Transylvanian schools, insisting on geometry (descriptive geometry, geometrical and technical drawing.) The reason of choosing this subject is the less information we have found about geometry (separated from the mathematical study) as one of the discipline considered important in education, for about two hundred years. On the other hand, there is less information about Romanian education in Transylvania province, before it joined to Romanian kingdom. Transylvania has always been nearer to the central European influence that might be seen in structure and curriculum of its first schools. We want to underline the receptivity of Romanian scholars at the European different domains newness. The paper is also “a humble” dedication to those representatives of Romanian science and culture of the XIX and XX century who succeeded in difficult conditions to diminish the educational postponement comparative with the developed European nations. 2. SOME ASPECTS ABOUT THE DISCIPLINE Drawings representing building plans had been found together with mathematical calculus, on ancient clay tablets from Orient, much before the invention of paper [3]. From antiquity, people had to measure and build and have alwais needed projects before. Old Greeks gave much importance to mathematics and geometry. Platon said: “Everything is ordered in Shape”. The three dimensional drawing reached the highest level in Renaissance. Centuries after Vitruvius wrote the first Book of Architecture (the I-st century B.C.) which contains the Greek conceptions about proportions based on mathematical relations, Luca Pacioli in his “De Divina Proportione”, recommended it to his generation. Artists and architects have since been interested in perspective and projections. At the beginning of the XV-th century, Alberti, Brunelleschi and Leonardo da Vinci developed mids to represent building plans and also mechanical contrivances [3]. The apparition of Descriptive Geometry is associated with numerous problems faced with fortification constructions in the XVIII-th century, in France. Gaspard Monge, Comte de Peluse (1746 - 1818) was a French mathematician, recognized as the inventor of Descriptive Geometry [3]. Like Analytic Geometry, Descriptive Geometry found a relationship between points from space and their projections on plan.

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Monge helped to found the first European polytechnic school, Ecole Polytechnique, in 1794 and was Professor of Descriptive Geometry there for more than ten years. In 1798, he published the first book of Descriptive Geometry. After 1795 Descriptive Geometry became an important discipline in technical schools, as well as in the traditional ones, not only in France, but also in Austria, Germany, Russia and USA.

3. HISTORIAN MILIEU OF THE TRANSYLVANIAN EDUCATION DEVELOPMENT The development of the Romanian education was tightly connected with the political and cultural events from the center and southeast Europe and its history is identical with that of the Romanian Provinces. Before referring to the period mentioned in the title of the paper, it is necessary to bring to mind some important moments from the preceding century. Under the influence of the renascence cultural stream in Vest and Central Europe, the Hungarian king, Romanian by father, Matia Corvin, concentrate around him humanist scholars and established the Universities in Bratislawa and Buda. The next century, Nicolaus Olahus, archbishop and regent of Hungary, friend with Erasmus from Rotterdam, systematized the education and established in 1566, with the help of Jesuits, the University of Târnava (Sâmbăta). Another Jesuit University (with three faculties: theology, philosophy and juridical) functioned in Cluj between 1581-1603. Nicolae Pătraşcu, son of Michael the Brave was student there, in 1585. [2] After the Congress from Carlowits, in 1699, the War between The Holy League and Turkish Empire, Transylvania was subordinated to the Austrian Emperor. The Claudiopolitanian Academy from Cluj, reestablished in 1693, became in 1773, College, and in 1786 Regal High School. Among the romanian personalities who studied at this college, the most important are: Gheorghe Şincai, Petru Maior, Gheorghe Lazăr, Ioan Piuariu-Molnar, Aron Pumnul, Al. Papiu Ilarian, Avram Iancu, George Bariţiu. [1] The confessional education was transferred to Alba-Iulia, beside the roman-catholic episcopate, and among the medicine and philosophy studies, technical education courses were organized too. Unfortunately, in 1790, the Regimen from Cluj abolished the reforms in education, as punishment to those who were considered guilty for the Popular Inssurection guided by Horia, Cloşca and Crişan.

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The great ideological and cultural movement of the Romanian enlightenment, known as “Şcoala Ardeleană”, was affirmed in these circumstances. The grek-catholic bishop in Blaj Inochentie Micu, established the guidelines of Romanian spirit emancipation. The representatives of “Şcoala Ardeleană” movement elaborated “Supplex Libellus Valachorum” charter and brought a great number of scientific works based on historical, philological, cultural, demographycal, argumentations for backing their demands [1] The book “Ansichten von Siebenbürgen” published in 1816 in Sibiu offers an approximate image of the Transylvanian education from the beginning of the XIX century. Alexis Bethlen, the author, said in his book that greek-catholic Romanians had good schools in Blaj and Năsăud and orthodox Romanian in Sibiu and Brasov [1]. In reality there was another ordinary school for orthodox Romanians in Arad (from 1812) and a catholic one in Oradea. A statistical of Transylvanian schools in 1841 showed this: in 2840 villages there were 1628 schools but only 298 Romanian (285 for the orthodox people in Sălişte, Răşinari, Sadu, Avrig, etc. and 14 for the greek-catholic people in Orlat, Năsăud, etc) [1] In 1842, the Regimen from Cluj passed a law, proposed by the Hungarian aristocracy, according to which, “in all the schools of the Province – except the saxon ones - the teaching language must be Hungarian language” (even if the schools in Blaj were established and sustained by Romanians) It is no wonder that among the demands of Revolution in 1848 was the access of Romanian people to culture, the establish national schools with useful disciplines in curriculum and a Romanian University. An official statistic from 1851 [1] indicated the following status about education in Austrian Empire: 10 universities, 5 law academies, 9 surgical schools, 4 mountain institutes, 1 agricultural institute, 3 technical institutes, 38 theoretical schools, 262 high schools. From 74613 students, 16385 were Germans, 20054 Slavons, 11052 Hungarians, 21732 Italians, 3196 Jews, 1108 Romanians and 85 others (figure 1).

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figure 1. An official statistic from 1851 indicating the status about education in Austrian Empire [1]

Referring to this situation, George Bariţiu underlined the necessity of establishing schools, “with our own resources, by all sacrifice” [1]. In 1855 the German language was impose as the teaching language untill 1867, when Austria was defeated at Königgrätz and Sadova and Transylvania was given to Hungary. This event caused disastrous consequences in education too [1]. In those difficult historical conditions, the Romanian education was organized by the two confessions (the greek-catholic from Blaj and orthodox from Sibiu). 4. THE STUDY OF GEOMETRY IN THE CURRICULUM OF THE SUPERIOR GYMNASIUM FROM BLAJ, BETWEN 1880-1915 In 1799 the popular school from Blaj became Gymnasium, and its curriculum is the same as all others from the Empire, especialy „Santa Barbara” from Wien [1]. In 1829 Timotei Cipariu started to teach in Romanian language the German philosophy, offering to Romanian pupils an alternative to the latin culture. In 1831 the Gymnasium became College, with Simion Bărnuţiu as chairman. He tought philosophy, algebra, geometry and universal history (even this

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was not aloud by authorities). Bărnuţiu was substitute by Aron Pumnul in 1846, for teaching philosophy. He was a good teacher with great influence upon his pupils, therefore he was obliged to leave Blaj and went to Cernăuţi [1]. We studied the yearbooks of the greek-catholic education in Blaj, between years 1880 – 1915 [5] and observed that “geometria desemnativă”(as it was mentioned in archives), were all the years one of the main discipline. In the first class of the high school, pupils learned to represent points, lines, and angles and had to solve problems about angle measurements and their relationship. In the second class, they studied three-dimensional drawing; solids with plan figures or curve surfaces as borders, symmetry and surfaces and volumes calculus. Figures having lines as borders, perimeters calculus, areas calculus and topography elements were subjects to be study in the third class. In the last class the study of geometry consisted in circle, ellipse, parabola and spirals representation and problems. Because the curricula was almost the same as in other schools all over the Empire, the main manual used for this discipline was first a translation made by Iosif Hossu, of a german manual (by Wagner and Fodru, 1887). The interest in the discipline lead to the publishing in 1888 of another book, “Theoretical and Practical Guide for the Geometry Study” by Borgovan [5]. In the The Year-Book of the greek-catholic education from Balázsfalva (Blaj), in 1914 – 1915, was mentioned that the study of geometry in the fourth form of the high school, was based on the manual wrote by F.C. Domşa [5]. Looking for the authors mentioned before, we found Iosif Hossu in 1912 as joint author of the project for the high education reorganize, presented in “Proposals and Decisions of Iaşi University Council “ (28.02. 1912). The Technical Industrial School was established in 1884 in Cluj, sustained by The Commerce and Industry Chamber and Commerce and Industry Ministry, being connected in 1887 with The Drawing School and Technical Museum. Beside the teoretical education, The Technical Industrial School ensured practical disciplines, displayed in particular workshops and factories.

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FIGURE 2. TECHNICAL INDUSTRIAL SCHOOL BUILDING, NOW THE FACULTY OF ELECTRONICS AND ELECTRO-MECHANICS, CLUJ-NAPOCA

5. SOME IMPORTANT MOMENTS IN THE DEVELOPMENT OF GEOMETRY AND DESCRIPTIVE GEOMETRY IN ROMANIA In 1895 was published in Iaşi the book “Arithmetical elements in a real representation” written by Amfilohie Hotiniul, after a book of Alessandro Conti. It was the first mathematical book written in Romanian, which contained geometrical notions. Gheorghe Asachi in 1812 introduced descriptive Geometry among the other disciplines at the Engineering Class from Princely School from Iaşi. The study of geometry is in curricula of Saint Sava School in Bucharest (established by Gheorghe Lazăr) from 1776, and Descriptive Geometry since 1818 [4]. In 1836, Carol Valstein, painter and teacher at Saint Sava College, wrote “Drawing and Architecture Elements”, the first didactic manual of this kind in Romanian language. In 1851, Alexandru Orăscu elaborated his own Descriptive Geometry for High school curricula [2]. Alexandru Costinescu was the first Romanian engineer and architect (graduated at Wien Politechnic Faculty) who became

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professor at Iaşi, then director and analitical geometry teacher at „The Bridge and Roads” School from Bucharest. 6. THE EDUCATION STATUS IN TRANSYLVANIA, AFTER THE GREAT UNION Among other positive effects, which appeared soon after The Union of Transylvania with Romania, in 1918, was the increasing of the literate’s number. In comparison with 1910, when 49% were literates, in 1930, their number increased to 64%. Those who were educated after 1918 represented 41% from the total number of literates in 1930 and 54% from those in 1910. That means that the education progress in 12 years after Union was almost equivalent with halph a century before [4]. The Universities represented the High education from Cluj and Timişoara. Transilvanian teachers and personalities who went in Romania were invited to come back. The same curricula have been established for all schools and universities from all provinces. Wellknown personalities composed the teaching staff In the new University of Cluj-Napoca: Victor Babeş, Ioan Ursu, Emil Racoviţă, Sextil Puşcariu, Theodor Capidan, Vasile Pârvan, Dimitrie Gusti, etc. The Inauguration of the University of Cluj took place in 1 February 1920, with 171 members of the teaching staff and 2034 students, besides members of regal family and representatives from all over the world [2]. The festivity was the occasion for fixing connections with professors from other universities. Sextil Puşcariu, the rector of the University of Cluj was nearly to get assistance from the French administration for establishing a High School for Mining and Metallurgy. This event does not finalized, because a Politechnical School was allready established in Timişoara. Anyhow, in Cluj was inaugurated in the same day a School of Art and Crafts, changed in Medium Technical School. The admission to this school depended on the results at tests in algebra, geometry and drawing. The school became in 1922 The Technical Conductor School, the single one having an electric profile. Descriptive Geometry and drawing were basic disciplines in curriculum of the Bridge and Roads School established also in 1920 in Cluj (in the same time with similar schools in Bucharest, Chişinău and Cernăuţi) [2]. Starting with year 1930-1931, The Technical Conductor School and Bridge and Roads School became a single one, having as

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centre the buiding from Bariţiu Street (Faculty of Electronics and Electromechanics). After the Viena Dictate (30.08.1940), the North Transylvania was given to Hungary and The University, with Comercial Academy, National Theatre, Philharmonic, The Ethnographic Museum and Romanian printing houses had to leave Cluj. The University of Cluj went in Sibiu, while The Electromechanical School, The Faculty of Science and Agricultural Academy moved in Timişoara, until September 1945. 5440 Romanian teachers and educators, 550 engineers, 953 lawyers, 744 physicians and 24336 pupils and students had to leave Transylvania between september 1940 until march, 1944 [4]. While being in Timişoara, because of refuge and war, the vacancy chairs had to be occupied. Engineer Octavian Costea was hired for teaching Descriptive Geometry. He continued his activity when the Mechanical Institute came back in Cluj in 1945. In 1953 The Mechanical Institute became The Polytechnical Institute [4]. Although the disciplines from curriculum varied in time, Descriptive Geometry and Industrial drawing continued to have an important place in engineering educational activity.

7. THE ROLE OF GEOMETRY, DESCRIPTIVE GEOMETRY AND ENGINEERING GRAPHIC IN THE DEVELOPMENT OF A CREATIVE MIND Because it works with representations, Geometry may be said that operates with symbols. A symbol is much more than a sign, it is the essence of a thing, and image process and it presume an interpretation. Symbolic image is capable to induce a spiritual activity and symbol is a variable, which does not change the value of the relation, while getting a new connotation. On the other hand, engineering graphics, based on certain rules, is a universal language. Standardization continue to be a civilize factor [6]. Knowing and respecting standards used in graphical representations means the posibility of international collaboration in designing products. Draftsman’s hand work may be substitut by computer, but no operator can renounce to geometrical knowledge.

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8. CONCLUSION Human society from XX-th century has evolved. New disciplines and scientific brances apeared and are to be learned and computing techniques have been most developed. It is natural that geometry and drawing do not occupy the same important place in educational system, but taking them off from curricula may have negativ results in perception and spatial sight, in developing a creative thinking. In hard conditions, as mentioned before, geometry, like other disciplines continued for centuries to has an important role in education. After Union, Romania made great efforts to developed educational system. In the same time, between Romanian scholars there were tight relationship, that helped in establishing common curriculum all over the country. In present there are tendencies to reduce drawing activities, but this may not be a good decision.

9. LITERATURE 1. Albu N., Istoria şcolilor româneşti din Transilvania între 1800 – 1867, Didactis-Pedagocical Printing House, Bucharest, 1971. 2. Giurescu Dinu C., Istoria ilustrată a românilor, Sports-Tourism Printing House, Bucharest, 1981. 3. Giesecke, Mitchel, Engineering Graphics, Macmillan Publ., Boston, Massachusetts, USA. 4. Nistor, I.S., Istoria Învăţământului tehnic din Cluj-Napoca, UTPRES, ClujNapoca, 2004. 5. *** yearbooks of the greek-catholic education in Blaj, between years 1880 – 1915, “Octavian Goga” County Library, Cluj-Napoca 6. *** Funcţiune şi formă, Meridians Printing House,1989.

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DEFINING THE PRINCIPAL AXES OF THE QUADRIC CONE GENERAL CASE WITH ELLIPTIC BASE SECTION CURVE Aleksandar Čučaković9 Magdalena Dimitrijević10 RESUME This paper presents an constructive procedure of determining three mutually orthogonal principal axes (three planes of symmetry) of the quadric cone, the general case with elliptical base section curve.The constructive procedure is based on establishing correlative corespondance between the base curve plane (points and lines) and bundle of lines and planes at the vertex of the cone.At the base curve plane, two pairs of collocal, corelatively associated planes are set. After overlapping two of them, the other two collocal planes become collinear. Three double points in two generally collinear planes are intersection points between three principal axes and base plain of the cone. Key words: The general case of cone; the main axes of cone; corellative transformation; polarity; auto polar tetrahedron

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PhD Associate proffessor, Faculty of Civil Engineering, Belgrade, [email protected] 10

MSc Teaching Assistant, Faculty of Civil Engineering, Belgrade, [email protected]

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1. THE INTRODUCTION It is well known that each cone has three mutually orthogonal axes, so called – the principal axes of cone. They define three orthogonal planes i.e. three planes of symmetry of cone. The position of the principal axes of an right circular cone are not metter of interest. In case of an quadric cone (general case), with elliptic, parabolic or hyperbolic base curve, with one axis or plain of symmetry specified, a simple geometric construction is available to determine the other two axes, or the other two planes of symmetry. In case of quadric cone (elliptical, parabolic or hyperbolic type of base section curve), with no specified elements, constructive procedure for determining three principal orthogonal axes i.e. three symmetry planes, is complex geometric problem. 1.1 The correlative bundles of lines and planes in space The cone τ is set with base curve section – ellipse k, center point K, in the horizontal plane H, and vertex V above. The minimal distance from vertex V to the base plane is determined by radius of circle k1. Center point K1 is orthogonal projection V’ of point V on the base curve plane H. The points and lines in base plane H, with vertex V of cone τ, form the bundle of lines and planes. They are mutuallly in correlative correspondance. Each point in the base plane H, related to a base curve - ellipse k of cone τ, induces on the correspondent polar line the involutory sequence of points [2;10]. Also, each involutory mapped pair of points on the polar line, with correspondent polar point of plain H form an auto polar triangle [4;74]. Connection line of vrtex V and corresponding polar point in the base plane H is carrier for an involutory pencil of planes respectively to the cone τ.This pencil of planes induces in corresponding polar plane an pencil of lines. Each pair of corresponding lines from involutory pencil of planes (in the polar plane), with corresponding polar line forms an auto polar tetrahedron of cone [2;11]. Each plane, from pencil of planes, intersects cone in two generatricess, and polar plane in one straight line. This three lines are in dual harmonic relation to a polar line. In the involutory pencil of lines there is an orthogonal pair of lines. Regarding this, it is necessary to determine polar line, which is orthogonal to a polar plane.

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The observed cone τ defines correlative bundle of lines and planes { }. The bundle { } consists of polar axes and associated polar planes, respectively to the cone τ. A new right circular cone τ1 is set within the same vertex (V). It’s axis is perpendicular to a circular base of cone. Diameter of base circle k1 is equal to perpendicular distance from vertex V to the plane H (all the generators make the angle of 45o to the plane H). This circular right cone determines new correlative bundle of lines and planes { 1}.There are only three mutually orthogonal axes mapped to adequate polar planes in correlative bundle { }, respectively to cone τ, while in the correlative bundle { 1} respectively to cone τ1, all polar axes and corresponding planes are mutually orthogonal [2;25].The double lines/double planes of those two corelative bundles are principal axes/planes of summetry for the quadric cone τ. 2. COLLOCAL CORRELATIVE MAPPED PLANES Plane H, with two base section curves: of cone τ and cone τ1, intersect two collocal correlative bundles { } and { 1} in two pairs of collocal corelative corresponding planes (α,β) and (α1,β2). After overlaping planes β and β1, the other two, α and α1 become collinear.

Fig.1 Two cones with base section curves in generally collinear planes

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Two collocal, generally collinear planes α and α1 are defined with pair of mapped points K and K1 and vanishing lines r and i1. Vanishing line r is polar line of pole - center point K1, respectively to ellipse k.Vanishing line i1 is antipolar line of pole K (or polar line of inversed pole Ki), respectively to circle k1. (fig.1) The inversion came out of the fact that cone τ1 is real representative of the imaginary cone defined with vertex V and absolute conic curve. In two collinear mapped planes α and α1, centar point K of ellipse k, corresponds to center point K1 of circle k1. Both are in correlative correspondence to the infinite line t∞≡ t1∞ in base curve plane β≡β1. Conjugated diameters of elipse k1 form an involutory pencil of lines, while conugated diameters of circle k form circular involutory pencil of lines. Pairs of parallel lines in planes β≡β1 (linking lines between points A∞, B∞ and C∞ on infinite line t∞≡ t1∞ and center points K/K1) have their corresponding lines – conjugated diameters of ellipse k (a, b, c) and circle k1 (a1,b1,c1) in planes α and α 1.(fig.2)

Fig.2 Two involutory pencils of lines respectively to ellipse k and circle k1

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2.1 Hyperbola of Apollonius as a result of two projective mapped pencils of lines Conjugated diameters a, b, c and a1,b1,c1, in planes α and α1 form two projective mapped pencils of lines in vertexes K and K1 (center points of ellipse k and circle k1). They produce an conic curve of 2nd order, in this case, hyperbola h1 with orthogonal asimptotes – hyperbola of Apollonius. (fig.3) Two projective mapped pencils of lines were translated to one overlaping point K≡K1 in collocal “position”. Using Steiner’s construction procedure over sequence of points on 2nd order curve (Steiner’s circle), for two perspective mapped pencils of lines (with vertexes S and S1), the axis of perspectivity (double lines d1 and d2) determined directions of asymptotes of hyperbola.

Fig.3 Steiner’s construction (left top corner) Two perspective mapped pencils of lines defining asymptotes of hyperbola

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The other two projective mapped pencils of parallel lines (R) and (R1), with infinite vertexes R and R1, determined position of asymptotes as1 and as2 and center point of hyperbola. Pencils of lines (R) and (R1) were intersected with two lines a and a1, producing two perspective mapped sequences of points. The vanishing points G and G1 on two perspective mapped sequences of points, on lines a and a1, are referent points for position of asymptotes as1 and as2.(fig.3) 2.2 Projective mapped pencils of lines with vertexes in focal points of two collinear planes Double points D1, D2 and D3 for two generally collinear planes α and α1 are intersecting points between two conic curves (any type) derived from two pairs of projective mapped pencils of lines. One conic curve is hyperbola of Apollonius h1, with orthogonal asymptotes. The other appropriate choice of conic curve is circle, because of accuracy of intersection (between circle and hyperbola). In order to get a circle, as a result of mapping, it is necessary to set vertexes of two projective mapped pencils of lines in focal points of planes α and α1. In process of constructing focal points, the first step is determination of vanishing points P and O1 on vanishing lines r and i1 in planes α and α1. Those are corresponding points to infinite points O∞ and P1∞ of directions nr and p1i1.The main perpendiculars gn , gn1 of planes α and α1 pass through points P and O1. According to a known rule: „ Some lines in plane α ,from pencil of parallel lines set through infinite vertexes, have their corresponding lines in plane α1 passing through corresponding vanishing points and focal point L1“ (and vice versa), the focal points L and L1 were determined [1]. The important fact is that the angle which those corresponding lines make with the adequate vanishing lines has the same value. Following the rule [1], in plane α1 , the direction w1 (in vertex K1 ) and infinite point W1∞ was addopted. Corresponding line w in plane α (in vertex K) intersects vanishing line r in vanishing point W. Eventualy, there is a line in plane α, set through point W, which make the angle of 69o with vanishing line r (the same angle is between direction w1 and vanishing line i1, in plane α1) and intersecting the main perpendicular gn in focal point L.The same procedure followed for the focal point L1, using arbitrary direction q and point Q∞ in plane α.(fig. 4)

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The other pair of focal points M and M1 is symetrical to L and L1 respectively to vanishing lines r and i1. Two projective mapped pencils of lines with vertexes in focal points L and L1, will produce a circle l as a conic curve. The circle l is defined with three points: focal points L and L1 and point of intersection of two main perpendiculars gn and gn1. There are four points of intersection between circle l and hyperbola h1.Three of them are double points D1, D2 and D3 of two collocal mapped collinear planes α and α1.

Fig.4 The focal points within planes α and α1 Steiner’s supplementary construction (right corner down)

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The new circle m was adopted using three points: focal points M and M1 and intersection point between main perpendiculars gn and gn1.Two projective mapped pencils of lines in points M and M1 produce new hyperbola h2, also with orthogonal asymptotes (hyperbola of Apollonius). Constructive procedure: in two projective pencils of lines M (x, y, z) and M1o(x0 , y0, zo) which produce a circle m, one pencil , in vertex M1o, was symmetrically transformed around axis gn1 into pencil M1 (x1 , y1, z1), in order to produce hyperbola h2. (fig.5)

Fig.5 Two projective mapped pencils of lines for hyperbola h2

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As the result of intersection of three curves: hyperbola h1, circle l, hyperbola h2, three double points D1, D2 and D3 of an auto polar triangle, in two generally collinear planes α and α1,appear. Their linking lines o 1, o2 and o3 to a vertex V are three principal axes of cone τ, forming an auto polar tetrahedron (in space) with vertexes D1, D2, D3 and V. (fig.6) These axes form 3 mutually orthogonal planes of symmetry of cone τ. (fig.7)

Fig.6 Three principal axes of quadric cone τ

3. CONCLUSION The determination of principal axes of the quadric cone τ, with an elliptic base section curve and vertex V, is based on determination of vertexes (double points) D1, D2 and D3 of an auto polar tetrahedron in generally collinear planes α and α1. Those are intersection points between two conic curves: hyperbola h1 and circle l, or circle l and hyperbola h2, as results of mapping of two pairs of projective pencils of lines in focal points. If determined focal points L, L1 and M, M1 for two planes α and α1, the easiest way to determine an auto polar tetrahedron i.e. axes of symmetry for the quadric cone τ, is to make

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one more projective transformation in focal points M, M1 deriving hyperbola of Apollonius (h2).

Fig.7 Three planes of symmetry of quadric cone τ

4. LITERATURE

1. A. Zdravković Jovanović, Metric and alignment invariants of collinear spaces, PhD thesis, Faculty of Architecture, Belgrade 1985. 2. R. Janićijević, Spatial disposition of circle sections on cone surfaces preseka (constructive approach), MSc thesis, Faculty of Architecture , Belgrade 1989. 3. V. Niče, Introduction to synthetic geometry, Školska knjiga, Zagreb, 1956 4. V. Sbutega, Synthetic geometry I, Script for postgraduate studies, Faculty of Architecture , Belgrade, 1975.

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OPTIONAL COURSE ENGINEERING GRAPHICS ON DEPARTMENT FOR LANDSCAPING ARCHITECTURE AT THE FACULTY OF FORESTRY, UNIVERSITY OF BELGRADE Aleksandar Čučaković11, PhD Biljana Jović12, MSc This paper is dedicated to our dear colleague Jelena Maksic†,PhD professor who establish subject Engineering graphics at the Faculty Forestry, University of Belgrade. Resume The paper gives a brief overview of the course content, aim and outcomes of the optional course Engineering graphics, in the school 2009/10.year, the Department of Landscape Architecture at the Faculty of Forestry in Belgrade. A good example of what should be done on the other technical faculties, how to preserve the basic course Descriptive geometry and how to establishment new optional courses as well as how to position course on the corresponding higher year of study. In addition this paper points out the importance of preserving the existence of basic course Descriptive Geometry at all technical faculties, with the possibility for curriculum supplement the theoretical foundations of computer graphics. Key words: Descriptive geometry, Computer graphics, Engineering graphics, Optional courses

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Faculty of Civil Engineering, Belgrade University, Bulevar Kralja Aleksandra 73/1, Belgrade, Serbia, [email protected] 12 Faculty of Forestry, Belgrade University, Kneza Viseslava 1, Belgrade, Serbia, [email protected]

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1.

INTRODUCTION

It is known that Descriptive Geometry provides a base for the development of spatial perception, imagination and intuition of the students. It is necessary to preserve its role, also in contemporary conditions, theoretical principles and procedures need to be extended by the basic theoretical principles and procedures that are used in computer graphics. Consequently course of descriptive geometry should remain as a separate obligatory course in the first year of studies,where basic procedures and principles of descriptive geometry and computer graphics would be studied. All areas of geometry which go beyond the basic areas, should be left and offered for advanced training and studing on optionaly courses that should be studied at higher academic years. Since basic descriptive geometry course lasts only 3 months it is unacceptable to put all the knowledge from this discipline in the basic course, especially from pedagogical reasons. 2. DESCRIPTIVE GEOMETRY AND ENGINEERING GRAPHICS

Reform of education at the Faculty of Forestry University of Belgrade changed names in many courses. It happened with the subject of Descriptive Geometry at the Department for the wood processing. The new name of the previous course Descriptive geometry is not only lost “Descriptive” but lost the title “Geometry”, and a new name following course “Applied engineering graphics”, which is inadequate scientific name of an area that is taught on this subject. This title is not only inadequate, but degrades Descriptive geometry as a discipline. At the Departments of environmental engineering and landscape architecture name Descriptive geometry course has remained unchanged. Also there is a special optional course

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Engineering graphics where the basic principles and methods of work in the AutoCAD graphics program is learned. Optional course Engineering Graphics is a kind of substitute for the subject of “Descriptive geometry with technical drawing” that existed under the old program. Descriptive geometry remains the same and part of the course Technical drawing is substituted with course Engineering graphics. Course in Descriptive geometry should remain as a separate obligatory course in the first year of studies where basic procedures and principles of descriptive geometry and computer graphics are studied. That was done in two departments on the Faculty of Forestry. The course Engineering graphics is located on the second year of study as an elective course. This is a good example of how the other technical faculties should be positioned courses Descriptive geometry and Engineering graphics or other optional courses related to this discipline. The teaching content of the course of Descriptive Geometry need to amend the relevant fields of computer graphics. Students should learn a basic course descriptive geometry developed by considering the perception ability of 3D space and got all the necessary skills in computer graphics, so that during the course and after its completion, students could work in any existing graphical program constructively to solve problems in Descriptive geometry or professional courses. 3. COURSE ENGINEERING GRAPHICS

The aim of the course is learning about the ways and possibilities and principles of constructing and shaping in geometric 2D and 3D space, using computer software AutoCAD. Developing skills for effective modeling and composition of geometric forms with opportunities for their final design and presentation of the corresponding projections.

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Fig1 3D geometric modeling

Through this course the necessary knowledge for 3D geometric modeling of space in a computer graphics program AutoCAD is aquaired (Fig.1). Knowledge could be applied in solving concrete problems in the professional subjects, and later in practical or investigative work. 2.1 Content of the course The contents of the course are the theoretical classes and laboratory exercises. In the theoretical classes the procedures and method of constructing the basic geometric elements and shapes in the plane and 3D space are learned as well as ways of setting and use of commands. Control drawings using layers. Ways and commands to construct polygonal lines and curves in the plane and 3D space. Graphic constructing in 3D space. Determining the point of view on the 3D geometric space and the types of views. Spatial coordinate system. Global and local rectangular coordinate system, absolute and relative coordinates of the 3D space and the way to set coodinate of the points. Types of 3D modeling, generation ways, command, and opportunities. Wireframe modeling, surface modeling, and solid modeling of geometric forms (Fig2).

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Fig. 2 Wireframe, surface and solid modeling

Commands for modifications and transformations of the objects in the plane and 3D space (Fig.3).

Fig. 3 Transformation and modification of the objects

Cutting and intersection of the regular and irregular solids (Fig.4).

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Fig. 4 Cutting and intersection of regular solids

Final processing of 3D objects. Laboratory exercises are conducted in a computer classroom to produce graphic work on specific tasks folowing theory processed during the lectures. Each student has the opportunity to do on a separate computer given exercise (Fig.5).

Fig.5 Computer lab.

4. GEOMETRIC MODELS

Computer graphics has already developed into a powerful tool that it used to be experts in solving very complex problems. It serves the research and presentation of extremely large and extremely small objects. With the help of interactive graphics for computer engineers solve spatial problems, but directly in the three-dimensional virtual space. Results are displayed graphically at the same time in all the desired projections.

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Models are used to display real or abstract objects not only with the aim of creating images, but also to show their structure or properties. The model is a simplified description of the object. It may be a mathematical model, geometric or experimental. With the help of computer modeling replaced many traditional techniques, such as a model for testing and similar. We can say that the computer model is electronic version of the objects. Computer modeling is a much wider area, but we can now look only at models whose main goal is a graphic interpretation (Fig.6).

Fig.6 Geometric model

Geometric model shows objects whose geometric properties of natural looking have graphical representation. It consists of geometric shapes and geometric transformations. If we are very interested in visualization, we want to achieve the impression of real three-dimensional space. The first step, and the entire foundation of this work is defining the geometric properties and shapes of all objects in the scene. The next step is animation. AutoCAD is a professional CAD program that appeared for personal computers. It is a software package for technical drawing, powerful in all fields of technology, but still has its limitations, especially in 3D (Fig.7).

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Fig. 7 Geometric modeling

For 3D modeling in the space you need proper assessment and interpretation models. You also need to set or adjust view of the model.Auto-CAD, have two sets of coordinate systems. Global (world) coordinate system (WCS) set the computer program and user coordinate system (UCS), which binds to any point in the drawing. The basic elements of graphics and the point and line. They have made all the signs, graphics primitives and transformations that is used for performing a model (Fig.8).

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Fig.8 Performing a model

5. CONCLUSION

Contents volume of descriptive geometry and computer graphics is constantly expanding. It is absurd or nonsense to put a large amount of material in a basic course, which lasts only three months. Material that exceeds the theoretical foundations of descriptive geometry and computer graphics should be stored in the optional courses that would be positioned in the higher years of study and at higher levels of study. The forms of realization of teaching the subject Descriptive Geometry must be preserved. It is unacceptable to replace it or supplement it with inadequate forms such as writing seminary papers. Work on seminar tasks takes time for students instead of working on concrete geometry exsersizes. Students in this way introduced into the research work unnecessarily and

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before they are qualified to doing research in an unknown area for them. Exercise in which students solve concrete geometry problems, by convetional means or any computer program are the most important in the process of studuing and unreplaceable form of learning. Student learn the material in this discipline succesfully by drawing required number of examples. Working with students on the exercises on the subject Descriptive Geometry is a very demanding and hard part of the job for the person who performed the exercise.To facilitate that hard part of the job (try not to work with students) teaching assistants attempt to change or suplement the form of teaching. It is fatal error for the discipline also for students, in this way they are discourage and deviate from the right direction and target training. This is the same as when you entered the car school to learn to drive a car, and the school has the opportunity to give you a chance to try to drive a car, but they want from you to do a written paper on how to drive a car. A teacher who does not want to work with students, shall declared a students as antitaleted. Talent is required for artistic singing, painting, dancing, composing, acting,etc. but again the great work is necessery! For the basics knowledge of geometry, mathematics, chemistry, physics, biology, history, geography, computer science, etc. not need to be talented. Any student who is entered on technical faculties, came to learn basics of Descriptive Geometry, as taught driving skills, playing football, cooking, make-up or knitting. The paper gives an overview on the risk (harm) for the course because of arbitrary changing the basic course Descriptive Geometry in contemporary conditions, also the negative consequences with the introduction of new (alternative forms of teaching) or new forms of inadequate training (either as an alternative or supplement to existing forms of teaching) in primary Descriptive geometry course. Department of Landscape Architecture at the Faculty of Forestry, University of Belgrade, is a good example as it should

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be at the other technical faculties, to preserve the basic course Descriptive geometry and the establishment of new optional courses and their positioning (placement) to the appropriate higher years of study. 6. LITERATURE 1. Cocchiarella L.: Geometry and graphics in spatial invention: amongmind, hand and digital means, Proceedings of 12th international conference on geometry and graphics, Salvador, Brazil, (2006). 2. Čućaković A., et al.: Opšti i posebni nastavni sadržaji u edukaciji u nacrtnoj geometriji i inženjerskoj grafici, XXIII konferencija za nacrtnu geometriju i inženjersku grafiku, Zbornik radova, Fakultet tehničkih nauka, Novi Sad, (2006.) 3. Maksić J.,et al: Primena novih metoda u nastavi nacrtne geometrije usklađenih sa Bolonjskom konvencijom i njihov značaj za razvoj prostorne vizualizacije, XXIII konferencija za nacrtnugeometriju i inženjersku grafiku, Zbornik radova, Fakultet tehničkihnauka, Novi Sad, (2006). 4. Masanori, Y., et al. Interactive Education Of Descriptive Geometry Using Auto CAD LT, Proceedings of 10th ICGG, Kyiv, Ukraine, (2002). 5. Masatsune, I. Introducing 3-D CAD Software Into Graphic Education For Freshmen In Engineering Course, Proceedings of 10th ICGG, Kyiv, Ukraine, (2002). 6. Mitić, M., et al. Predlozi za unapređenje primene nacrtne geometrije u nastavnom procesu na odseku za pejzažnu arhitekturu, Šumarskog fakulteta, Univerziteta u Beogradu , XXIV konferencija za nacrtnu geometriju i inženjersku grafiku, Zbornik radova, Vrnjačka banja, (2008). 7. Kenjiro, S., Application Of Descriptive Geometry Procedures In Solving Spatial Problems With 3D-CAD, Proceedings of 10th ICGG, Kyiv, Ukraine, (2002). 8. Stachel H.: The status of todays descriptive geometry related education (CAD/CG/DG) in Europe, 40th anniversary of the Japan Society for Graphic Science, Tokio, Japan, (2007). 9. Suzuki K., et al.: Development of graphics literacy education implementation of comercial 3D-CAD/CG software into graphic science course, Proceedings of 12th internationalconference on geometry and graphics, Salvador, Brazil, (2006).

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FUNDAMENTAL MATRIX IN EPIPOLAR GEOMETRY Natasha Kirilova Danailova13 Venzislav Dakov Radulov14 (Abstract) The information and communications in our society are performed by computer and visual technics, in which the images representation is of great importance. That’s the reason for the increasing interest to the multiple view geometry during the last years. It analyzes the relations between the object views by changing of the camera’s position The epipolar geometry covers the projective geometry properties between two perspective views. An arbitrary projective space coordinate system is considered. A correlation is established between the points in the first image plane and a pencil of lines in the second image plane. The algebraic representation of the epipolar geometry is the fundamental matrix F. Some representations of the matrix F are derived: 1. F as a function of the two projective matrixes 2. F as a function of the homography between the two pencils of epipolar lines. Key words: epipolar geometry, epipoles, homography, projection matrix, fundamental matrix

13

Natasha Kirilova Danailova, Assoc. Prof.,University of Architecture, Civil Engineering and Geodesy, blv. H. Smirnensky 1, Sofia 14

Venzislav Dakov Radulov, Ass. Prof., University of Architecture, Civil Engineering and Geodesy, blv. H. Smirnensky 1, Sofia

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I. Introduction. The planar image creation of the three dimensional world is investigated lately in the framework of the projective, affine or Euclidian geometry. By special choice of Decart coordinate system in the space with origin in the projection center (camera coordinate system) a simple matrix equation is obtained between the homogeneous space coordinates of a point and its central projection [1], [2]. Some correspondences can be established between two or three and more images, which properties can be applied. In the general case the projection plane is determined by pixels and the coordinate origin doesn’t coincide with the camera. We introduce in [6] projective coordinate space system as well as projective homogeneous and no homogeneous coordinates, which are the most common case of coordinates. The projective coordinates have a nice property - they don’t change after central projection. The projective coordinate system К(U, V, W, O, E) is defined by 5 points, called basic points, so that no four of them lie on a plane. We call U, V, W – excluded points, О – zero point and Е –unique point. It is shown in [6], that if A(u A , v A , w A , t A ) is an arbitrary point with projective homogeneous coordinates in projective coordinate system К(U, V, W, O, E), and A ' (u 'A , v 'A , w A' ) is the central projection of А, so that we project on arbitrary projection plane with projection center O, and the projective homogenous coordinates of A' are in K ' (U ' ,V ' ,W ' , E ' ) – the central projections of the points U, V, W and E, then the following relations are true: 1   0 0 0  1   PA  A' , P0 0 0 ,  A . E  0  1 0 0    

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The matrix P is called projection matrix of size (3 x 4) and rang (P)= 3 In case of two projection centers and two projection planes we denote the corresponding projection matrixes by P and Q. The mapping is induced between the both images, called homography, defined by the matrix H. Projective mapping is established between the images of the one plane and the so called epipolar lines of the other plane, which is given by functional matrix F of rang 2. There exists a relation between the matrix of homography H and the functional matrix F (see [6]). The homography and the functional matrix are necessary for description of the space coordinates of a point, if its two images are known. This is the problem for reconstruction of the object. The projective, affine and Euclidian reconstruction can be investigated by adding an additional information for the camera. A part of the known results in the epipolar and trifocal geometry are described in the monographies [1] and [2]. II. Projective mapping between the pencils К and К’. Let consider the double central projection by given projection system (С,ω) with projection matrix Р and projection system (С',ω') with projection matrix Q. The points K  CC ' and K '  CC ' ' are called Kern points (epipoles). They are resp. the images of С' by (С,ω) and of С by (С',ω'). Each line k in ω, passing through К , is called Kern (epipolar) line for ω . By analogy each line k' in ω', passing through К, is called Kern (epipolar) line for ω. If m is an arbitrary point in ω, then m  K  k is its corresponding Kern line. Let consider the pencils of Kern lines with centers К and К'. Let к is an arbitrary Kern line in ω and denote by К* its intersection point with the line t     ' Follows, that the line к' = К*К' is a Kern line in ω'. Thus each Kern line in ω is compared

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to a kern line in ω' and intersects ω' in the corresponding line к'.

+Let denote the Kern lines and their images: к1, к2, к3, к4 and к'1, к'2, к'3, к'4 resp.. The double ratio is preserved: (к1, к2, к 3, к4) = (К1*,К2*,К3*,К4*) =( к' 1, к' 2, к'3, к'4). The correspondence between the pencils К and К' preserves the double ratio and it is the projective one. It can be represented by nonsingular square matrix. III. Fundamental matrix-properties and representations Let m be an arbitrary point in ω, which is different of К, and m  K  k is its corresponding Kern line.The lines СС' and к define a plane, which intersects ω' at the line к'. The line к' evidently is a Kern line in ω'. There exists a correspondence, which compares each point m in ω to the line к' in ω'. The fundamental matrix F is such that Fm  k ' .

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Properties: 1.

FK  0, K ' F T  0, m'T Fm  0 .

2. F '  F T . Proof: Let F :    ' is a fundamental matrix such that Fm  k ' . Let F':  '   is the fundamental matrix such that F ' m '  k . Follows that m T F ' m'  0 . But

m'T Fm  0 therefore 3.

m T F T m'  0 . or

F  [ K ' ] QP 

Proof: k '  K '  m'  [ K ' ] QM  [ K ' ] QP  m. k '  [ K ' ] QP  m. 4.

F'  F T .

We

have

F  Q  T P T [ K ] .

Proof: In [6] it is shown that if g is an arbitrary line in ω , then P T g   – is a plane, passing through C, and Q T   g ' - is a line in ω'. It follows, that Fm  k '  Q  T   Q  T P T k  Q  T P T ( K  m)  Q  T P T [ K ] m. 5.

F  [ K ' ] H .

Proof: From Hm  m' we get K ' Hm  K 'm' . Follows that [ K ' ] Hm  k '  Fm. . 6. H T F  [ K ] . Proof: [ K ] m  K  m  k  H T k '  H T Fm. . 7.

H T F  F T H  0.

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Proof: From H T F  [K ] , which is asymmetrical matrix, we get consequently [ K ]T  [ K ]  ( H T F ) T  F T H . 8. F is asymmetrical matrix with rang 2. 9. F  [ K ' ] ( X ' , Y ' )h( X , Y )[ K ] , where: - [K ' ] transforms each point m' in ω' to its Kern line : k '  K 'm' - if X ' ( x1' , x 2' , x3' ) and Y ' ( y1' , y 2' , y 3' ) are arbitrary points, then the matrix  0  ' ' ( X ' , Y ' )   x1 y 2  x2' y1'  ' ' ' '  x1 y 3  x3 y1

 x1' y 2'  x 2' y1' 0 x2' y 3'  x3' y 2'

 x1' y3'  x3' y1'    x 2' y3'  x3' y 2'   0 

compares each point B (b1 , b2 , b3 ) to-the point B ' (b1' , b2' , b3' , ) , for which

b1'  (b2 y 2'  b3 y3' ) x1'  (b2 x2'  b3 x3' ) y1' b2'  ( b1 y1'  b3 y3' ) x 2'  (b1 x1'  b3 x3' ) y 2'

b3'  (b1 y1'  b2 y 2' ) x3'  (b1 x1'  b2 x 2' ) y 3' One can be easily checked, that B' lies on the line X'Y' i.e. B'  X '  Y ' -

h is the projective transformation between the pencils of Kern lines with centers К and К'

-

if

X ( x1 , x2 , x3 )

and

Y ( y1 , y 2 , y 3 ) are arbitrary

points, then the matrix

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0   ( X , Y )   x1 y 2  x2 y1 x y  x y 3 1  1 3

 x1 y 2  x2 y1 0 x 2 y 3  x3 y 2

 x1 y 3  x3 y1    x2 y 3  x3 y 2   0 

compares each Kern line к to its intersection point A  k  XY . - [K ] -transforms each point m in ω to its Kern line k  Km. Proof: Let X and Y be arbitrary points in ω, such that the line s = XY doesn’t pass through К. By analogy X' and Y' are arbitrary points in ω', such that the line s' = X'Y' doesn’t pass through К'. Let М be an arbitrary point and CM    m, C ' M   '  m' are its images by (С, ω) and by (С', ω'), resp. Let denote by k  m  K and k '  m' K ' the pair of the kern lines, which are corresponding by the projective transformation h. If A  k  s and A'  k ' s' , then A' k '  0 . We get

A'  k 's '  (m'K ' )  ( X 'Y ' )  m' [ K ' ] x ( X ' , Y ' ). On the other side, k '  hk  h( A  m)  h( s  k  m)  h( X  Y )  k )m   h( X , Y )  k  m  h( X , Y )  ( K  m)  m  h( X , Y )  [ K ] m. From

A' k '  0

and m'T Fm  0 we get the desired equation.

IV Algebraic representation of the fundamental matrix Let B1 be a base with center С, and take arbitrary X  , Y and Z   C ' By analogy B2 is a base with center С', and let take X '  X  , Y'  Y and Z '  C . We have the following coordinates: In В1

In В2

C(0; 0; 0; 1)

C(0; 0; 1; 0)

X  (1; 0; 0; 0)

X  (1; 0; 0; 0)

Y ( 0; 1; 0; 0)

Y ( 0; 1; 0; 0)

Z   C ' (0; 0; 1; 0)

Z   C ' (0; 0; 0; 1)

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One can easily checked, that the transformation from В1 to В2 can be realized by the matrix 1  0 A 0  0 

0 0 0  1 0 0 . 0 0 1  0 1 0 

Evidently, that A  AT  A 1 . If we have for the Kern points K(0; 0; k; 1) and K' (0; 0; k'; 1) in the base В1, then in the base В2 we get that K(0; 0; 1; k) and K' (0; 0; 1; k). Let b1 be a base in ω with center К, x   CX    , y   CY  . By analogy b2 is the base in ω' with center К, x'   C ' X    ' , y '   C ' Y   '. Let М be an arbitrary point and CM    m, C ' M   '  m ' are their images by (С, ω) and by (С', ω') resp. Then PM  m,

where

is the projection matrix for (С, ω). It is

is valid P  m  M . We get

known [6] , that for

consequently: P  m  M (in В1), AP  m  AM  M (in В 2),

m' (in b 2)  Qm(in B 2) Then m'  QM  QAP  m m'  Hm

In [6] a matrix Н is defined, for which

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Thus we get

=

Conclusion: The epipolar geometry is used in all projects dealing with binocular stereo. It san be described by the Functional matrix. The properties and representations of this matrix prepare the ground for its estimation and recovery. The algebraic representation is suitable for computer introduction and applications.

LITERATURE 1. Faugeras O. , Luoung Q. T., ed. T. Papadopoulo, The Geometry of Multiple Images, MIT Press, 2001. 2. Hartley R.,. Zisserman A., Multiple View geometry in Computer Vision (Second edition), Cambridge University Press , 2003. 3. Смирнов С. А., Стереоперспектива в фотограметрии, Москва, “Недра”,1982г. 4. Матеев А., Проективна геометрия, „Наука и изкуство”, София,1972г. 5. Радулов В., Двойно централно проектиране, Доклади на Юбилейна конференция на УАСГ, София, 2007 г. 6.Georgiev G., Danailova N., Radulov V., Geometry of two cameras, Proceedings of International Scientific Conference in UACEG , Sofia, okt.2009.

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GRAPHIC TRANSFORMATION OF HYPERBOLIC PENCILS OF CIRCLES INTO PENCILS OF CONICS AND THESE INTO PENCILS OF CURVES OF THE 3rd OR 4th ORDER Gordana Djukanovic15 Vjaceslava Matic16

Summary This research deals with transformation of hyperbolic pencils of circles and their specific features into pencils of curves of the 3rd and 4 th order and their specific features and vice versa. The basic transformation will be inversion, which will be interpreted in two ways: as quadratic transformation in classical projective geometry and as pure symmetry in relativistic geometry, with constant parallels between two geometric systems and possibilities of their explication and generalization. In this paper, hyperbolic pencils of circles map into pencils of conics by homology, which then map into pencils of conics. Finally pencils of conics map into pencils of curves of the 4th or 3rd order (this mapping entails mapping of singular conics which are broken into corresponding pairs of straight lines). Key words: Pencils of circles, pencils of conics, pencils of curves of the 3rd or 4th order, homology, inversion.

1

Gordana Djukanović, M.Sc., Asssistant lecturer, Faculty of Forestry, Belgrade University 16 Vjaceslava Matić, Ph.D., Full professor, Faculty of Forestry, Belgrade University

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1. INTRODUCTION There are several different ways to obtain pencils of conics: by interesecting quadric with a conic of plane, by cutting a pencil of quadric with a plane or by stereographic projection of conics of circles from a sphere to a plane, etc. Harmonic symmetry is a completely bijective transformation, which means that a pencil of curves of the 3rd or 4th order has the same number of cutting points as a pencil of conic. In relativistic geometry, ellipse, hyperbola and parabola are interpreted as spatial curves of the 4th order which are obtained by intersecting a sphere and an elliptic cylinder. An elliptic cylinder which with a sphere makes an ellipse meets the sphere in an isolated dual point i.e. in the viewer’s antipodal point. If the viewer moves to another point on the sphere, he won’t see the curve as an ellipse, but as a curve the 3rd or 4th order. All the curves which are obtained by harmonic symmetry of an ellipse for different viewer’s positions also have a mutual harmonic symmetry. An elliptic cylinder which with a sphere makes a hyperbola meets the sphere, but it also cuts it on the curve which has a self-cutting dual point in the viewer’s antipode. Finally, an elliptic cylinder which with a sphere makes a parabola meets and cuts the sphere only on one side, so that the intersected curve of the sphere and cylinder has a spire in the antipode. In this paper, hyperbolic pencils of circles by homology map into pencils of conics, which then by harmonic inversion (symmetry) map into corresponding pencils of curves of the 3rd or 4th order. 2. CURVES OF THE 4th ORDER AS HARMONIC EQUIVALENTS THE PENCILS OF CONICS Figure 1 shows a hyperbolic pencil of circles which by homology maps onto elliptic-elliptic pencil of conic. The pencil of circles is positioned to the point at infinity v1 in such a way that homology maps it onto a pencil of conics which contains a hyperbola (circle 1 intersects the point at infinity), a parabola (circle 2 meets the point at infinity), ellipse (circle 3 doesn’t intersect the point at infinity). The pencil of curves of the 4th order contains curves of hyperbolic, parabolic and elliptic type. The ellipse maps into the curve of the 4th order which has one circular and one linear axis of autosymmetry. The axes aE= aH= aP, which pass as rays through the centre S, map into themselves. Thus, the curve of the hyperbolic type also has one

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circular and one linear axis of autosymmetry. The parabolic curve of the 4th order has only a linear axis of autosymmetry, while the other axis of symmetry is contracted in the antipode. Osculatory circles of hyperbola map into osculatory circles of the curve of the 4th order, while asymptote of the hyperbola map into osculatory circles in the node. Two mutual pairs of points of this EE pencil of conics are conjugate imaginary. Typical representatives of these conjugate imaginary points are the points at which complementary conics intersect (Fig.2). There is a complementary ellipse for the obtained hyperbola h1 and a complementary parabola for the parabola p2 . Disintegrated pencil of conics is composed of two conjugate imaginary straight lines which are the linking points for the two pairs of conjugate imaginary points. Typical representatives of these straight lines are the parallel lines which pass through two pairs of fundamental points of the conjugate pencil of conics. Figure 3 shows the enlarged pencil of conic and its corresponding pencil of curves of the 4th order. In Figure 4 the centre S is positioned at the point on the conic h1. Figure 5 shows a hyperbola with axes aH i bH , which by harmonic symmetry (S,s) maps into the curve of the 3rd order with circular axes of symmetry a H i b H through which the curve inverts into itself. Since the hyperbola has two axes of symmetry a H i bH , the mapped curve of the third order also has two circular axes of symmetry a H i b H. The asymptotes of the hyperbola, as osculatory `straight lines` in the antipodal node of the hyperbola map into osculatory circles in the node of the curve of the third order. The osculatory circle of the hyperbola Os map into asymptote of the curve of the third order. The centre of the circle is determined through the centre of the hyperbola C1. Tangent lines in the node of the curve of the third order are parallel to the asymptotes of the hyperbola. The angle of the selfintersection is an invariant of the hyperbola harmonic symmetry with the curve of the third order. These tangent lines are rays of inversion and they map into themselves. They meet and cut the hyperbola in the node S , whereas they meet and cut the curve of the third order in the node S. In Figure 6 the centre of the inversion S is outside of the ellipse which means that by applying harmonic symmetry we obtain a curve of the fourth order – an oval. Due to its irregular shape one might conclude it is an asymmetric curve. However, it has a harmonic symmetry both with the ellipse and with all the curves which are harmonically

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symmetrical with the ellipse. This `oval ` has two circular axes of selfsymmetry ( a E i b E) and it is centrally symmetrical in respect to the points C 1 i C 2=S. The point S is an isolated point of the curve of fourth order, while the point S is an isolated antipodal point of ellipse. The osculatory circles S are drawn into the apexes of ellipse 1, 2, 3 and 4. These circles map into osculatory circles of

the the the the

curve of the fourth order at points 1 , 2 , 3 i 4 . Circle 2 which meets the point at infinity by a homology maps into parabola. In Figure 7 the centre of inversion S is outside of the parabola, so by a harmonic symmetry we obtain a curve of the fourth order with one singular point, a cusp of the first order at the point S. The obtained curve has one axis of harmonic symmetry a P. The other axis of symmetry is contracted at the point of the curve`s cusp at which three apexes of the curve are squeezed as well, which means that the curve of the fourth order has only one apix. Parabola is a border curve between a number of hyperbolas and ellipses, which means that the group of curves of the fourth order with a parabolic point is a border between hyperbolic and parabolic curves of the fourth order. Hyperbolas differ in respect to their angle of selfintersection – while hyperbola has an asymptote as an angle of selfintersection, ellipse has an angle between its conjugate imaginary asymptotes (or the angle where diagonals of tangent rectangle intersect). Parabola’s asymptotes coincide with the axis and the angle of self-intersection equals zero. In the antipode, in the cusp two branches of the curve meet, all the parabolas are similar, i.e. all parabolic curves with a cusp have mutual harmonic symmetry. We can obtain all parabolic curves from one of them. 3. CONCLUSION In relativistic geometry, inversion is a completely bijective transformation, which means that the pencil of curves of the 3rd and 4th order intersect at the same number of points as a pencil of conics. This paper deals with only one type of pencils of conics: ellipticelliptic type and its harmonic equivalents. Previous papers dealt with the following types: parabolic-parabolic, parabolic-elliptic, parabolichyperbolic pencil of conics. The future papers will deal with hyperbolic-hyperbolic, hyperbolic-elliptic pencil of conics and inverted pencil’s of the curves of the third and fourth order.

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4.LITERATURE 1.Dovnikovi} L..:Harmonija sfera, Matica srpska, Novi Sad, 1999

2.Stavri} M.: Harmonijska sinteza i konstruktivna obrada povr{i vi{ih redova, Doktorska disertacija, 2002.

1 T1k1

u1

S1 v1 v2 T1h u2

T1e T2e T2k3

T2k1

2

T2p T2h

CK3

bH

as 1 T1k3

v1 p2

bE k3

h1

bE p2

a= a= aP H E Tp

S e3

k2

k1

s e3

h1

h1

as 1

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Fig.1 Harmonic equivalent of EE pencil of conic, for the centre S on the axis of symmetry

as 1 p2

p2

1 3

4 2

h1

h1

as 1 Fig.2 Complementary pencil of EE pencil of conic is HH pencil of conic

Fig.3 Harmonic equivalent of EE pencil of conic – enlarged

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Fig.4 EE pencil of conic and its harmonic equivalent - EE pencil of the curves of the third and fourth order

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Fig.5 Mutual harmonic symmetry of the hyperbola from EE pencil of conic and the curve of the third order with two circular axes of symmetry

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Fig.6 Mutual harmonic symmetry of the ellipse from EE pencil of conic and the curve of the fourth order with two circular axes of symmetry

Fig.7 Mutual harmonic symmetry of the parabola from EE pencil of conic and the curve of the fourth order with one axis of symmetry

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VISUALISATION AND ANIMATION OF GEOMETRIC TOPICS Gordana Djukanovic17 Milena Stavric 2 Resume This paper presents the process of teaching in which Euklid DynaGeo, Cabri3d, Great Stella, Sketch Up and AutoCAD software packages have been implemented. A great number of examples and models have made it much easier for the students to understand spatial relations because the use of these software packages improves the coordination between the two brain hemispheres, which is essential for understanding spatial relations and for creative expression of future engineers. The significance of this paper lies in the presentation of experience gained by implementing various software packages and spatial models which have enhanced the quality of education. Key words: Applied Engineering Graphics, 3d modelling, software packages for drawing and modelling

1

Gordana Djukanović, M.Sc., Assistant lecturer, Faculty of Forestry, Belgrade University 2 Milena Stavrić, PhD, Faculty of Architecture, Graz University of Technology, Austria

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1. INTRODUCTION

The introduction of computer technologies into teaching led to the inevitable changes in it. Teaching materials changed as well as names of certain subjects and new subjects which support these reforms were introduced. Hence, in the academic year 2007/08, at the Faculty of Forestry in Belgrade, at the department of Wood Processing, the subject Descriptive geometry got the new name - Applied engineering graphics. The aim of this subject is that by studying basic principles of geometry a student can develop the skills of spatial organization and creative solving and modelling of furniture and presentation and performance of practical tasks. One of the basic tasks of engineers in furniture design and defining of relationships between the elements of design is the understanding of space. Various means and methods are used in teaching to facilitate the understanding of space, shaping and modelling of spatial forms. Beside the classical way of teaching (sketching, drawing on a sheet of paper) several programme packages for graphic presentation, animation of three-dimensional objects and task practice are being applied. This paper presents five different softwares which are being used in the subject Engineering graphics at the Faculty of Forestry in Belgrade. The way in which specific softwares were implemented in teaching will be presented with practical examples of the elaboration of certain thematic units. The inspiration for such use of certain softwares was found in numerous examples which have been used at the Graz University of Technology for years. 2. SOFTWARES FOR THE VISUALIZATION OF GEOMETRIC TOPICS In today's era of computerization a large number of CAD packages are being used in different fields of engineering. A large number of these packages are predefined for professional use, while a pretty small number of them are suitable for use in education. Among them, a small number of programmes which are oriented towards solving of certain geometric problems, and which can be effectively used in the preparation and performance of geometry teaching. Based

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on the experience in the field of CAD, the authors of this paper have chosen several softwares to use in the elaboration of several geometric topics. The aim of such elaboration is to present certain topics in a more dynamic way and to enable easier understanding and mastering of spatial relations to students. 2.1 Cabri 3D Cabri 3D is a program intended for education in geometry. The main feature of this program is that it is based on the fundamentals of descriptive geometry: fundamental spatial elements (point, line and plane), fundamental spatial transformations (translation and rotation) and correlation of elements (parallelity and normality). Familiarity with these notions is a base for successful and creative utilization of this software. On the basis of the above mentioned fundamental elements and functions of this software, very complex geometric topics can be generated and animated. The proof for that is the fact that this program won the prestigious 2007 BETT Award in the Digital Content: Secondary (Core Subjects) category. Spanish researchers carried out a study over 6 years of 15,000 high school students and 400 teachers. They proved that students have better understanding of spatial relations, and that they are better at solving mathematical problems by 30% thanks to spatial models. Spatial relations become more easily understandable using graphic interpretation of mathematical (geometric) problems, which is the main task of descriptive geometry, in the first place. Our professors were making spatial sketches of tasks which were still twodimensional and more difficult to understand than the 3d models which can be made using the program Cabri 3d. It is simple and understandable software which can be efficiently applied in the preparation of teaching and direct work with students. In working with students the following can simply be done: a) Construct 2-D and 3-D figures From the simplest to the most complex, by combining fundamental geometric objects such as points, angles, segments, circles, planes, solids and transformations. b) Create expressions Using fundamental algebraic concepts, such as numbers, variables and operations. c) Connect geometry and algebra

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By measuring lengths, angles, areas and volumes and than making a record of those numeric values on figures and using them in calculations and algebraic expressions. An example of the use of this software in teaching is shown in Figure 1 The example. The example elaborates on the topic of rotation and finding of the actual size of geometric shapes. Figure 1 the shows rotation of a polygon in the first auxiliary plane. By using animation it is possible to rotate the whole polygon plane to the frontal plane and read the actual size of the polygon in one of the projections. Separate rotating circles and the axis of rotation are inscribed in animation, which enables easier comprehension of this program unit. We can say that with such kinds of animation, certain topic units of spatial geometry can be simplified and explained into detail.

Figure 1 Model of plane rotation in Cabri 3d program-actual hexagon size

Figure 2 elaborates on the topic unit of plane rotation in an arbitrary position around the first trace and positions of a cube whose face is on the given arbitrary plane. The animation of the rotating plane and the circle of rotation in the point Tz are shown in three segmental pictures. By using this example it is possible it is possible to analyze all positions of the cube in this plane such as: edges of the cube are parallel to projection planes, apex of the cube is on the projection plane,... All conditionalities of the position are visible on

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the model which makes the introspection of geometric relations in projections easier. Cabri3D has very good visualization tools. The generated elements can be seen from different sides, separate sheets with precisely defined projections and the kind of projection (isometry, axonometry and perspective) can be generated and everything can be prepared for good quality printing. In the use of these sheets attention should be paid to the positions of the axes, which are not marked in the same way as in the standard literature in this country, and which can be baffling for the students. Unfortunately, vector models cannot be obtained as the output so that this program is not compatible with other CAD softwares. Geometric models can be exported in the HTML format which enables direct use of animated models in WEB environment. This software is not cheap and the educational license can be bought at the price of 227.24€. It can be summed up, that according to our experience the application of this software in the preparation of teaching gave excellent results whereas the use of this software in practice exercises should be limited to simpler examples which can be realized in a limited time during exercise classes. An important advantage of this program is the export into WEB environment, where certain topics can be animated and be at students' disposal.

Figure 2 Rotation and different positions of the cube in relation to the projection planes

2.2 Euklid DynaGeo Euklid DynaGeo is one more useful Windows program which uses fundamental geometric elements and principles for the generation of

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complex geometric compositions. That is a program for solving geometric problems in two dimensions. It is possible to carry out all geometric transformations with it and draw all elements, like on a two-dimensional drawing paper. The main difference from drawing on a sheet of paper is that if certain geometric interdependencies are established, each dynamic change of certain elements is preconditioned by the already defined interdependencies. So, for example, perpendicular bisector will always stay a perpendicular bisector, a line segment joining the points A and B will always run from A to B, a parallel line will always stay parallel, if you have correctly constructed the circle through the three points of a triangle, then this circle will always run through these 3 points, no matter how you drag the edges of the triangle ... It enables the creation of dynamic drawings which allows that some points be moved, without the loss of interdependence of geometric figures established during drawing. This is also a very important characteristic of all dynamic and object oriented CAD softwares. All functions are placed into five subgroups: construction, mapping, form and colours, measuring and calculating, and animation. Here are some of the basic options provided by this software: 1. Create all the standard geometric objects: points, lines, segments, circles, polygons... 2. Use the built-in standard constructions: perpendicular, bisector, perpendicular bisector, ... 3. Use the built-in standard mappings: reflection, translation, rotation, 4. Measure distances and angle widths 5. Calculate and watch mathematic expressions 6. Colour the objects in your drawings 7. Create macros for animation 8. Use functions and their dependences in solving geometric problems 9. Draw upper row curves as animated point trajectories 10.Export your dynamic drawings into web pages In pictures 3 and 4 there is an example from the field of perspective. In these examples a vertical projection plane was chosen.

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Figure 3 Perspective of a cube and its shade on a horizontal plane The lower parts of the drawing show the object in the first projection, whereas the upper part of the drawing shows the same object in perspective. The proportion of projective planes is the same as the proportion of the object in the base. In these examples it is interesting that they simulate the process of generation of perspective figures which occur in all CAD programs i.e. that the construction of perspective picture is performed through the penetration of visual rays through the plane of the projection plane. In these examples it is possible to move the position of the viewer, the distancing of the viewer in relation to the projection plane or the object, change the height of the horizon, change the position of the object and its size. As a result of animation we obtain the change in the perspective picture depending on the above chosen changes. In the pictures fundamental elements of the perspective sketch such as vanishing points, distant points etc. are given in different colours. Picture 4 shows construction of complex architectural building and its shadow for a parallel light source. As for the perspective it is possible

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to change the light source and see the change of the shade in real time.

Figure 4 Perspective layout of an object and its shade on a horizontal plane

One more thing which is interesting in these examples is that one two-dimensional program is used for the explanation of perspective and that three-dimensional space is simulated in it. The advantage of this approach is that construction of the perspective picture is reached synthetically and analytically. With this software the complete genesis of construction can be revised and learned by students in a more efficient way. Such animations are very well accepted by students during teaching and they enable more efficient lessons to teachers. From this program it is possible to export only Html format in which in web environment the positions of selected points can be dynamically changed. 2.3 Great Stella A very useful program in studying regular polyhedrons is Great Stella. This program can be used by pupils, students as well as other lovers of three-dimensional polyhedrons. Figure 5 gives a layout of the screen of this program. On the right side of the screen 3D model of polyhedron can be seen, and on the left side the net of this polyhedron. The model can be moved with a left click of the mouse

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and it can be rotated so that it can be seen from all sides. Nets can be printed for the production of paper models.

Figure 5 Polyhedron model and its net

Small Stella, Great Stella and Stella4D are various types of this program. Small Stella has a fixed list of built-in models and can print out nets for over 300 polyhedra. Great Stella is more advanced, with many more built-in models, and tools for building literally trillions of new ones. Stella4D includes everything from Great Stella, but adds support for 4D models. 4D models can be projected into 3D, and you can see their 3D cross-sections and nets. The programs are fun and easy to use. Some of the jargon in the descriptions below may be a bit baffling at first, but you don't need to know anything about maths or geometry to be able to use these programs and create amazing polyhedra. They should suit everyone from the amateur enthusiast to the seasoned expert. The lessons at the Faculty of Forestry involve the study of Platonic solids (cube, tetrahedron, octahedron, icosahedron and dodecahedron). So, by turning on the info option the students can learn the following for each polyhedron: whether the solid is one whole or a compound of two or more solids, number of faces, number of edges, number of apexes, its dual solid, whether there is the

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stellation option for it and they can print the net of the solid. In order to become more familiar with the geometry of these solids they had to make regular cardboard polyhedra for homework. During lessons students were shown the possibilities of this program and making of other polyhedra as compounds of more regular polyhedrons or making of new polyhedrons with the stellation option.

Figure 6 Cube model and its model faceting

As students of the Faculty of Forestry also use our program in furniture design we tried to make a direct connection with the principles of furniture design. A very important fact is that most furniture is made from panels, so that decomposition of furniture of complex design into separate panels is one of the topics in our program. That field is polyhedra as solids with high regularity of inner geometry, which makes them important for design and economical furniture production. The principle of construction of complex design from “one prototype“ in which it is possible to process a number of similar pieces of furniture is present in the work of our students with the program Great Stella. The performances of this program are shown on the example of a cube. Figure 6 shows model faceting whose apices coincide with the apices of a cube. Figure 7 shows the model of the compound of 3 and 4 cubes. Dual polyhedron for the cube is octahedron; hence figure 9 shows the model of five dodecahedra and one 4D model.

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Figure 7 Model of compounds of 3 and 4 cubes

Figure 8 Model of compounds of a cube and an octahedron, 5 dodecahedra, and one 4d model

Models with the stellation option are very interesting. Figure 9 shows the stellation model for dodecahedron. Figures 6,7,8 and 9 were taken from the site http://www.software3d.com/.

Figure 9 Model of stellation for dodecahedron and making of a paper model

2.4 Google SketCh-Up Google SketchUp is a vector program which has recently become one of favourite tools for the modelling of 3D models due to its simplicity, intuitive usage and the possibility of animation. It is free software being used by a large number of people. Pro version is not free of charge and the only difference from the basic version is the possibility of exporting of different vector formats such as DWG, 3Ds, DXF, OBJ etc., and other formats. People around the world exchange

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what they made using Google 3D Warehouse. It is a huge browser of produced models which can further be used without paying attention to author’s rights. Sketch-Up has a small number of tools which can be enriched with additional functions by using different taster combinations. Since the Sketch-up 7 version it is possible to generate parametric models and perform parametric deformations of geometric objects. This part is one of the tools of future, which can be effectively used in teaching and it a focus of these authors’ future research.

Figure 10 Sketch-Up program interface and the model of penetration of line through plane This program is not intended for solving of geometric problems (as it was the case with the previously mentioned softwares), but it has several tools which can be perfectly well used for the animation of geometric problems. Actually, geometry generated in the sketch-up can be animated with the possibility of the change of attributes of the object. The following example (Figure 10) shows a picture from the animation of the principle of understanding of the penetration of line through plane.

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Figure 11 Animation sequences: positioning of the auxiliary plane, intersection of the auxiliary plane and the arbitrary plane and finding of the point of penetration It is well known in geometry that for the penetration of line through plane, an auxiliary plane is positioned through the given line, followed by searching for its intersection with the given plane and further from the intersection of two lines we obtain the point of penetration of the line through plane. This way of getting the penetration of line through plane is explained in the above mentioned way but with animation and taking the auxiliary plane to be the same as the light plane this geometric topic can be elaborated in a more suitable and clearer way (Figure 11). 2.5 Auto Cad Auto CAD is a comprehensive program which is widely being used both for drawing in 2d and modelling in 3d. Powerful tools enable material allocation and rendering of 3d models. Here is an example of the angular perspective and a simple model drawn in 2D and later in 3D. With the use of a model the students obtain better understanding of the position of a projection plane and vanishing points of directions a and b and the image of the object on the projection plane. Figure 12 shows the angular perspective of the given object. Spatial model of the angular perspective is shown in Figure 13.

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Figure 12 Angular perspective in 2d and the object model in 3d

Figure 13 - Angular perspectives- 3d model of the object and the projection plane

3. CONCLUSION This paper is a presentation of the process of teaching using program packages Cabri3d, Euclid Dynageo, AutoCADA-a, Great Stella and Google Sketch Up. Students' results were improved with the use of different softwares and the combination of spatial sketches and models modelled using them. The objective of the subject- to understand what the object looks like and draw it on a twodimensional sheet of paper or two-dimensional monitor, or make its model in one of the programs, was achieved. After a large number of models and examples it is much easier for students to understand spatial relations because the use of these softwares creates a more intensive connection between the two brain hemispheres (L and R mode) which is necessary for understanding spatial relations and creative expression of future engineers. 4. LITERATURE 1. Leopold C. Geometrische Grundlagen der Architekturdarstellung, Kohlkammer, 2009. 2. Obradovic M.: Racunarska geometrija-skripta, Izvod sa predavanja za studente geodetskog odseka, 2009.

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3. Omura G.: Introducing AutoCAD 2008, Wiley Publishing Inc. Indianopolis, Indiana, 2007. 4. Rajković P. : Kompjuterska Geometrija – Skripta za studente I godine, Mašinski fakultet, Niš 2007.) 5. Štulić R., Segedinac T.: Kompjuterski podržana nastava Nacrtne geometrije: Matematički model za određivanje i prezentaciju međusobnih preseka paraboličkih kvadrika, Zbornik XXII Savetovanja moNGeometrija 2004, Beograd, 2004, 171-183. 6. Stavric, M., Wiltsche, A., Schimek, H.: 2005, New Dimension in Geometrical Education, KoG 9, pp. 45 – 54; 7. Stavric, M., Schimek, H., Wiltsche, A.:2007, Didactical Integration of analogue and digital tools into architectural education, in: CAADFutures2007, Sydney, pp. 61 – 70; 8. Stavric, M., Schimek, H., Wiltsche, A.: 2008, Cybertecture, the use of digital and analogue media in architectural education, in: Digital Thinking in architecture, Civil, Engineering, Archaeology, Urban Planning and Design: Finding the ways, EuropIA, Montreal, Springer Publishing, pp. 39 – 50; 9. http://www.cabri.com/ 10. www.iam.tugraz.at 11. http://www.software3d.com/Stella.php#great#great 12. http://sketchup.google.com/product/gsu.html

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GENDER BASED IQ AND SPACE PERCEPTION RELATIONSHIP Delia Drăgan1 Florin Drăgan 2 Raluca Nerişanu 3

Abstract This paper analyses the results of tests administered to first year students of the Faculty of Civil Engineering of Cluj-Napoca. In the first test, the objective consisted in determining the intelligence coefficient of the students, while in the second test, we assessed the spatial perception level reached by undergraduates, after learning the descriptive geometry and technical drawing modules. Finally, data were interpreted considering the subjects gender, correlations were made and conclusions were drawn. Key words: IQ, MCT (Mental Cutting Test), spatial perception, students’ gender

1

Delia Drăgan, eng., PhD, professor, Technical University of Cluj-Napoca Romania, Faculty of Civil Engineering, Department of Geodesy and Engineering Graphics, E-mail: [email protected]. Office Phone: 0040 264 401842. 2 Florin Drăgan, eng., PhD, professor, Technical University of Cluj-Napoca Romania, Faculty of Electrical Engineering, Department of Electrical Measurements, e-mail: [email protected]. Office Phone: 0040 264 401520. 3 Raluca Nerişanu, eng., PhD student, teaching assistant, Technical University of Cluj-Napoca Romania, Faculty of Civil Engineering, Department of Geodesy and Engineering Graphics, E-mail: [email protected]. Office Phone: 0040 264 401976.

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1. INTRODUCTION Graphical subjects, Descriptive Geometry and Technical Drawing, have a fundamental role for the training of the undergraduates in the Faculty of Civil Engineering. The training in graphical subjects for the students of Land Measurements and Cadastre covers four semesters. In the first academic year, semesters one and two, are dedicated to Descriptive Geometry. The study of Descriptive Geometry helps students learn object representation rules for various projection systems: double orthogonal projection, axonometry, projection with elevations. Students become familiar at the same time with geometric reasoning and abstract level that are mandatory in all engineering representations. In Descriptive Geometry, during the first semester, the study refers to: the double orthogonal projection of the point, straight line, plane, methods of transforming projections, representation of polyhedrons, of cylindrical-conical surfaces, notions of orthogonal and oblique axonometry. During the second semester, the projection with elevations and more in-depth study related to special surfaces used in civil engineering, such as conoids, cylindroids, hyperbolic paraboloids are taught. Based on the knowledge acquired during the first two semesters in Descriptive Geometry, in the second year, during semester three in the subject of Technical Drawing students learn how to layout projections and sections of solids and how to dimension them. They also learn the basic rules related to constructions representations. In the last semester, semester four", the study continues with topographic and cartographic drawing. The syllabus is organised so that at the end of the four semesters the undergraduates master sufficient knowledge to be able to produce or “read” situational plans, topographic plans, elevations, sections in buildings, and other types of civil engineering drawings. The spatial view is to be strengthened during further studies and students manage to simply imagine the situations on the ground by reading plans. 2. TEST DEVELOPMENT METHODS In the test, 96 first year undergraduates were included 44 male students and 52 female students. The test was administered at the beginning of the second semester of the academic year 2009-2010,

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after the first module of Descriptive Geometry was covered. First, the students had the test for establishing the intelligence coefficient, secondly the test that appreciates their ability of spatial perception, i.e. Mental Cutting Test. 2.1. TEST TO DETERMINE STUDENTS’ INTELLIGENCE COEFFICIENT (IQ) The test to determine students’ IQ was a classical intelligence coefficient test. The students had to answer 60 questions. As an example, Table 1 presents the first 3 questions for IQ determination purpose. To each correct answer, a point was allotted. Table 1 Example – the first 3 questions for IQ determination

1. Choose from the five letter marked shapes (illustrations) bellow, the one that logically matches the matter in discussion: it is for

that

is it for?

(A) B) (C) (D) (E) 2. Which one of five shapes below does not identify itself with the others? (A) (B) (C) (D) (E) 3. Which one of five diagrams below does not identify itself with the others? (A)

(B)

(C)

(D)

(E)

Depending on score and age, the students were included in a model table (Table 2) and the intelligence coefficient was calculated accordingly. Up to 80 points, the coefficient is very small, between 81 and 100 –average, between 101-110 - above the average, between 111120 - high, between 121-130 - very high, over 131 - we speak about exceptional capabilities.

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Table 2 Model table – IQ calculation

Age Score IQ Age Score IQ Age Score IQ

22 80

23 82

24 84

25 86

26 88

35 36 37 38 39 106 108 110 112 114 48 49 50 51 52 132 134 136 138 140

16-25 28 29 92 94 16-25 40 41 42 116 118 120 16-25 53 54 55 142 146 150 27 90

30 96

31 32 33 34 98 100 102 104

43 44 45 46 47 122 124 126 128 130 56 57 58 155 160 161

2.2. TEST TO DETERMINE STUDENTS’ SPATIAL PERCEPTION The test to determine spatial perception consisted in asking students to mentally cut 26 solids. For each problem, 5 solutions were given, with only one correct. The MCT (Mental Cutting Test) started by giving two examples of how to solve the problems so that students could better understand the task. Table 3 shows the two examples and five of the 26 solids in question. The students who solved correctly between 23 and 26 solids received the score for exceptional, for 18-22 right answers, the score was very well, for 12-17 right answers the score was well, for 5-11 right answers- sufficient and under 5 right answers meant unsatisfactory answer. Table 3 MCT –2 examples and 3 solids in question

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3. RESULTS The results of the IQ test were the following: Out of 44 boys tested, 2 subjects exhibited special qualities (4,55%), 24 subjects –a very high IQ (54,55%), 14 subjects (31,80%) –a high IQ and 4 subjects (9,10%) an IQ level above the average. In the case of female students, 2 girls (3,85%) exhibited special qualities, 26 girls (50%) had a very high IQ, 18 girls (34,60%) a high IQ, 4 girls (7,7%) an IQ level above the average and 2 (3,85%) an average IQ level (Fig. 1, 2).

IQ - MALE 60,00% 54,55%

50,00%

PERCENTS

40,00%

31,80% 30,00%

20,00%

9,10%

10,00% 4,55%

0,00% IQ

Fig.1 IQ results – male

For the Spatial Perception text, the results found were: In the case of boys: very good – 4 subjects (9,10%), good – 18 subjects (40,90%), satisfactory – 18 subjects (40,90%) and unsatisfactory – 4 subjects (9,10%). The girls had the results as follows: 6 girls (11,54 %) reached level very well, 22 girls (42,30 %) received level - well, 20 girls (38,46 %) – satisfactory and 4 girls (7,70 %) obtained only level unsatisfactory. The results are graphically presented in the diagrams in Fig.3, 4.

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IQ - FEMALE 60,00%

50,00% 50,00%

40,00%

PERCENTS

34,60%

30,00%

20,00%

10,00%

7,70% 3,85%

3,85%

0,00% IQ

Fig.2 IQ results - female

MENTAL CUTTING TEST - MALE 45,00% 40,90%

40,90%

40,00%

35,00%

PERCENTS

30,00%

25,00%

20,00%

15,00%

10,00%

9,10%

9,10%

5,00%

0,00% MCT

Fig.3 MCT results – male

In the correlation between the intelligence index and „spatial intelligence” index, the findings were as follows: 100% of the boy and

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girls that exhibited special qualities with respect to IQ, also obtained level „very well” in the Mental Cutting Test (MCT). MENTAL CUTTING TEST - FEMALE 45,00% 42,30% 40,00%

38,46%

35,00%

PERCENTS

30,00%

25,00%

20,00%

15,00% 11,54% 10,00%

7,70%

5,00%

0,00% MCT

Fig.4 MCT results - female

Of 24 boys with a very high IQ, 2 (8,34%) had the score „very well”, 16 (66,66%) reached score „ well”, 4 (16,66%) obtained the level „satisfactory” and 2 (8,34%) got „unsatisfactory” – Fig. 5. Girls presented the following situation: from the 26 girls with a very high IQ, 4 (15, 38%) had the score „very well”, 14 (53, 85%) reached score „ well”, 6 (23, 07%) got the „satisfactory” level and 2 (7, 70%) got the „unsatisfactory” level – Fig. 6. Boys with a high IQ presented in MCT (Fig. 7) the following results: 0% got „very well”, 2 (14,29%) got score „ well”, 10 (71,42%) received score „satisfactory” and 2 (14,29%) received level „unsatisfactory” . In the case of girls having a high IQ, the MCT findings were (Fig. 8): 0% received score „very well”, 8 (44,44%) got level „ well”, 8 (44,44%) reached level „satisfactory” and 2 (11,12%) were given score „unsatisfactory”.

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IQ VERY HIGH - MALE 70,00%

well (16) 66,66%

60,00%

PERCENTS

50,00%

40,00%

30,00%

20,00%

10,00%

sufficient (4) 16,66%

very well (2) 8,34%

unsatisfactory (2) 8,34%

0,00% MCT

Fig. 5 Relationship between IQ and space perception

IQ VERY HIGH - FEMALE 60,00% well (14) 53,85% 50,00%

PERCENTS

40,00%

30,00% sufficient (6) 23,07% 20,00% very well (4) 15,38%

10,00%

unsatisfactory (2) 7,70%

0,00% MCT

Fig.6 Relationship between IQ and space perception

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IQ HIGH - MALE 80,00% sufficient(10) 71,42% 70,00%

60,00%

PERCENTS

50,00%

40,00%

30,00%

20,00% well (2) 14,29%

unsatisfactory (2) 14,29%

10,00% very well (0) 0,00% 0,00% MCT

Fig. 7 Relationship between IQ and space perception

IQ HIGH - FEMALE 50,00% well (8) 44,44%

45,00%

sufficient (8) 44,44%

40,00%

35,00%

PERCENTS

30,00%

25,00%

20,00%

15,00% unsatisfactory (2) 11,12% 10,00%

5,00% very well (0) 0,00% 0,00% MCT

Fig.8 Relationship between IQ and space perception

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As for those with an IQ above the average level, 100% of the 4 boys and 100% of the 4 girls had a MCT „satisfactory”. At the same time 100% of the two girls with an average IQ had a „satisfactory” score in MCT. We should mention that one of the mistakes made during the test was that of not making a break between the two types of tests (IQ and MCT). After having finished the IQ level test, some of the students were tired or lacked patience and consequently they treated the second test with more superficial attitude. That is why, mainly boys with average high IQ were found with relatively poor results in the MCT test. WE should also mention that the undergraduates tested had not been taught Technical Drawing yet, the only „support points” they had to solve the MCT consisted in the chapter of „Intersection of polyhedrons” from the course of Descriptive Geometry. All the students tested who underwent the course in Land Measurements and Cadastre had entered the first year of study after an entrance examination whose average score consists in a written test in The Faculty admission competition of the undergraduates tested was the highest in the Faculty of Civil Engineering. 6. LITERATURE 5.

6.

7.

8.

Bărbînţă D., Dardai R., Drăgan D.: Study concerning the development of the spatial perception of the undergraduates, Proceedings on the International Conference on Engineering Graphics and Design, Galaţi 2007, Editura Tehnică, Ştiinţifică şi Didactică IAŞI, ISBN 978-973-667252-1, pag. 433. Drăgan D.: Results of the MCT (Mental Cutting Test) used with the undergraduates of the Agricultural Cadastre College, Al 3-lea Simpozion Internaţional al USAMV Cluj, Cluj-Napoca, 18-23 oct.2004, Buletinul USAMV-CN nr. 61/2004, ISSN 1454-2382, Editura AcademicPres®, pag.470. Drăgan D., Bărbînţă D.: The assessment of spatial perception for the civil engineering’s students, Proceedings on the international conference on engeenering graphics and design, 2005, Bren Prod s.r.l., Bucureşti, ISBN 973-648-471-8, pag. 459. Drăgan D., Drăgan F., Nerişanu R.: Study on the relationship between the spatial perception and the IQ, International Conference on Engineering Graphics and Design, Cluj (Romania)12-13 June 2009, pag.621.

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STUDY OF SURFACES GENERATED BY ANTISYMMETRIC PARABOLA BRANCHES Delia Drăgan1 Raluca Nerişanu2 Claudia Alb3 Florin Drăgan4

Abstract The paper aims at studying some translation surfaces with plane director generated by antisymmetric parabola branches with parameters. The control of parabolic points from the surface situated on a given curve permits the determination of the introduced parameters. Key words: translation surfaces, parabolic points, directrix plane, free form.

1

Delia Drăgan, eng., PhD, professor, Technical University of Cluj-Napoca Romania, Faculty of Civil Engineering, Department of Geodesy and Engineering Graphics, E-mail: [email protected]. Office Phone: 0040 264 401842. 2 Raluca Nerişanu, eng., PhD student, teaching assistant, Technical University of Cluj-Napoca Romania, Faculty of Civil Engineering, Department of Geodesy and Engineering Graphics, E-mail: [email protected]. Office Phone: 0040 264 401976. 3 Claudia Alb, eng., PhD student, teaching assistant, Technical University of Cluj-Napoca Romania, Faculty of Civil Engineering, Department of Geodesy and Engineering Graphics, E-mail: [email protected]. Office Phone: 0040 264 401971. 4 Florin Drăgan, eng., PhD, professor, Technical University of Cluj-Napoca Romania, Faculty of Electrical Engineering, Department of Electrical Measurements, e-mail: [email protected]. Office Phone: 0040 264 401520.

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1. INTRODUCTION It is known that by correlating two (function and strength) of the three basic factors, that is function-structure-shape, in the ”art of building” the outcome shall be a structural form, and that by adding the aesthetic factor to them, an architectural form is obtained. The term of free form introduced by Curt Siegel [4] refers to releasing form from geometrical constraints and maintaining the contact with the structural order. Free forms are, in general, full of originality and for this reason they single out. For their study it is necessary to set up computerbased mathematical models, low scale models and prototypes. In general, the process of creation of architectural shapes contains the following stages: The stage of passing from the imaginary to the spatial form is made up of a series of images conceived by each designer (form creator) that makes use of the codes and symbols belonging to the language of the form to express his or her own vision. In this stage that is strongly affected by emotional factors, the architect has the possibility of showing his/her talent and phantasy. This is the most beautiful period of creation as phantasy becomes reality by generating, analysing and selecting among spatial forms while looking for the architectural form. In Figure 1, one can see the design stage image of the building named Walt Disney Concert Hall that was designed by the American architect Frank Gehry. The stage of passing from the spatial form to structure is the one when the architect –the creator of the form – and the engineer the creator of rational structures- have the optimal relationship of cooperation. During this stage, the geometry of the form, the structural strength, the load decomposition and transmission take place. The stage of passing from structure to architectural form analyses the balance between the shape reached and the structure conferring it strength and stability. During this stage, the following aspects are considered: the manner in which the requirements of the design topic have been observed, the relationship between the architectural ensemble and the environment and last but not least, economic factors.

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Fig. 1 Walt Disney Concert Hall, architect. Frank Gehry - sketch

In the process of spatial forms generation, the architect, respectively the designer can use combinations (intersections) of simple geometrical volumes and can also resort to complex volumes containing surfaces generated with the help of the graphical problems. A spectacular demonstration of the of what to apply surfaces generated with the help of the computer means in architecture lies in the design of the building presented previously, that is Walt Disney – The Concert Hall from Los Angeles (California ) – fig. 2.

Fig.2 Walt Disney Concert Hall – axonometric projection

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The sculptural form of this building erected between 1999 and 2003, Figure 3, as well as that of other similar designs produced by the same architect (such as the Gugenheim Museum from Bilbao - Spain), constitutes one of the strongest and obvious contrasts with the rational forms that have dominated architecture for centuries on end.

Fig.3 Walt Disney Concert Hall - Los Angeles (SUA)

In the previously mentioned cases, for the conversion of the physical models (prototypes initially made of cardboard) into digital models the software CATIA (Computer Aided Three Dimensional Interactive Application) was used. This software oriented more towards the study of the surfaces than that of three dimensional models had been previously used in the aero spatial industry. The computer-based surface generation, with the means of mathematical relationships, has the advantage that it enables a relatively easy exploration of a wide range of forms. These statements can be supported by papers [1] and [2] where the mixed type of interpolation methods such as (blending) - (Coons) or of type Hermite and Birkhoff, families of free surfaces to be used in constructions were generated. The surfaces presented in this paper are generated by asymmetrical branches of parabolas with a director and controlled parabolic points and are formed of elliptic and hyperbolic type surfaces separated by the line of parabolic points. This line can be a straight line or a plane curve, dependent upon the right choice of parameters.

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2. GENERATION OF A SURFACE WITH A DIRECTOR PLANE AND CONTROLLED PARABOLIC POINTS In the plane YOZ, one considers two half parabolas with the vertex in point M0 (0, y0, z0 ), y0>0, z0>0, (Figure 4) of the equations:

(1)

z  z0    y  y0  , if y  y0 2

respectively (2)

z  z 0    y  y0  , if y  y0 . 2

If we set the conditions that these half parabolas pass through points A(0,0,h), h>0, respectively B(0,f,0), 00 and it is given by technical conditions, can be written: K:

x2  y2 

R K2

h  z 0 2

 ( z  z 0 ),

(2)

The intersection curve between the cone and the frontohorizontal cylinder, fields as it follows:

 2 R K2 2 x  y  (z  z0 )  (K∩Cf0)  h  z 0 2 y 2  z 2  R2 . f0 

(3)

In Fig. 2, representing only the vertical projection, a particular case for the previous situation is presented, as an application of Monge’s theory: Two second order surfaces circumscribed to the same quadric intersect in two plane curves. Thus, Fig. 2 shows the connection of two pipes of varying diameters (C 1 and C2) through a frustum of a cone, part of cone (S). In this case, cone (S) and cylinder C1 are circumscribed to a sphere so that the intersection curve unwraps in ellipses (a’a1’) and (b’b1’), where from branches a’e’ and b’e’ are useful for the joining. Point e’ corresponds to the joint chord of the two ellipses, perpendicular to the plane of the axes.

346

'

a'

b'

'

e'

t' b1'

t1' a1'

s'

Fig. 2.Particular case

3.CONNECTING PIPELINES USING TRANSITION PIECES 3.1. The graphical solution This paragraph will be dedicated to the connection of large diameter cylinder shaped pipelines, often met in outside sewerage works. This provides the transit between two cylinders: a frontalhorizontal cylinder with the base contained in the lateral plane [W] and a vertical cylinder with the base situated in a horizontal plane. In Fig. 3a, the representation on three planes of projection of the piece is shown. The representation of the lower frontal –horizontal semicylinder was skipped because it does not present interest from the point of view of the part graphical solution. The reduction is made from two oblique cylindrical nappes and two triangle faces. A solution for a flow with no excessive turbulence consists in the one where triangles 1C2, respectively 3D4 – contained in the frontal plane – are isosceles, with the unequal side (12 ≡ 34) equal to the diameter of the vertical cylinder. The intersection curve between the connecting oblique cylinders and the frontal-horizontal cylinder is determined with the help of the frontal auxiliary planes F1...F5, drawn through important points. Let us take plane [F4] as an example. This plane intersects the frontal- horizontal cylinder along the generatrices 7” ≡ 8”, while the oblique cylinders follow generatrices M-7, N-8, and at their

347

intersection, points 7(7, 7’), respectively 8(8, 8’) are obtained. In this way, curves 1-7-5-9-3 and 2-8-6-10-4 are obtained. In Fig.3b, half of the development was represented (with the part taken symmetrical). This contains triangle Do3o4o with its true size in vertical projection (as it is contained in a frontal plane) and the development of a cylindrical nappe. In order to find the latter, it is necessary to have a section with a vertical projecting plane [P] perpendicular to the generatrices of the joining frontal cylinder. The section is an ellipse arc βδαεγ, yielding directly with lines of recall and whose true size is determined using the coincidence with the vertical projection plane. One observes that the section is obtained with a vertical projecting plane perpendicular to the generatrices. In this manner the arc of ellipse is turned into a straight line of length βoδ oαoεoγo, called stretchout line. Perpendicular lines are drawn from the points of the normal section and segments 1=’1’, C =’c’, α5 = α’5’, αA = α’a’ etc. are measured, as the generatrices of the joining frontal cylinder. In Fig. 4, this piece was represented in oblique frontal isometric axonometry. F1''

c'=d'= n'=n1' m'=m1'

a'

c''

b'

F4 ''

F3''

F5''

a''=b''=

F2''

d''

P'

5' 7'=9'

1'=3'

F1

F3

2 n

a

9

3

9''=10''

3''=4''

8

6

b

n1

m1

F5 F2

c m

5

1''=2''

7''=8''

2'=4'

1 7

F4

5''=6''

6' 8'=10'

4

10

d

Fig. 3.a.Transition piece in orthographic representation

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Fig. 3.b.The development of the piece

Fig. 4. 3D representation of the piece

3.2. The analytical solution Analytically, the curve of intersection (Γ) between the transition piece and the fronto-horizontal cylinder is composed by a succession of warping curves and straight lines, namely:

    153  34  462  21. (4) One uses the next notations: Rfo = the radius of the fronto-horizontal cylinder;

349

Rv = the radius of the vertical cylinder; α = the angle between the generatices of the oblique cylinder with the horizontal plane of projection; m= ctg α; t = parameter; t  [-a, a], respectively 0