Blagojevic Popovic Tripkovicthe Potential of Geomertical and Artistic Characteristics of the Hype

8/8/2019 Blagojevic Popovic Tripkovicthe Potential of Geomertical and Artistic Characteristics of the Hype

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THE POTENTIAL OFGEOMERTICALAND ARTISTICCHARACTERISTICS OF THE HYPERBOLIC PARABOLOID IN

APPLIED ARTS

Petar Blagojevi1

Filip Popovi2

Tijana Tripkovi3

Rsum

The aim of this paper is to research the hyperbolic paraboloidthrough applied geometry. There are three approaches to the themeof the hyperbolic paraboloid which have potentials in fine and appliedarts as well as design:-The developing of the net of hyperbolic paraboloid, using conesegments. The aim is to show the efficiency of the developed netapproximation 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 paraboloid, rotation, projection, geometric

transformation

1 Petar Blagojevi, 2nd year BA student of interior and furniture design,

Faculty of Applied Arts, University of Arts in Belgrade, [email protected] Popovi, 3rd year BA student of graphic design, Faculty of Applied

Arts, University of Arts in Belgrade, [email protected]

Tijana Tripkovi , 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 aremostly based on the use of polygons. One exception is Durersconstruction of the sphere net. He developed the sphere net usingsegments which were determined by sections of a circle as meridians.This approach is less complicated than the ones that use polygons, andalso the net approximated in this way is more suitable for a sphereform, 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 Durers approximation was the direct inspiration for developingthe net of hyperbolic paraoloid using the segments determined bycurves which define it completely, and these are hyperbolae andparabolae.

1.2 Dividing the hyperbolic paraboloid into segments

The xz cross-sections of hyperbolic paraboloid are translatedcopies of a common parabola P, and the yx cross-sections aretranslated upside down copies of the same parabola P (Fig.1). Thisparabola will be used as a starting point for dividing the hyperbolicparaboloid 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. Acone of each segment has a different apex angle (), where thegeneratrix of the referring cone matches the tangent of the centralparabola of the hyperbolic paraboloid. In the middle of each segmentthere is a parabola of the hyperbolic paraboloid which is placedbetween 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 thespecific cone

In order to locate the wanted segments on a cone, it is necessaryto determine the position of the observed parabola on the cone. Theorthogonal projection of a right circular cone with its parabola sectionrepresents the system of parabolae that have mutual focus, whichmatches 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 onthe cone it is necessary to geometrically transform the orthogonalprojection of this parabola into the plane parallel to the cone base.

The example of this process is given in the Figure 4. Thetransformation angle is equal to the angle between the generatrixand the basis of the cone.

Fig. 4 Geometrical transformation of the given parabola

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

Fig. 5 The location of the parabola on the cone


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1.4. The construction of the net segments

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

Fig. 6 The position of the coneapex, where one half of hyperbolic

paraboloid is observed

Fig. 7 The position of the conesegmen

Figure 7 shows the location of the segment where the centralparabola of hyperbolic paraboloid is found on the cone using theexplained method, and the second parabola is constructed by placingthe segment into the upper projection of the cone. The cone layout isconstructed by a 15 angled rectification of the circle arch referring toCUSANUS and SNELL. Then the real distances of the parabolasintersection with the sides of the cone from the apex are transferredon the layout.4 Second parabola is used as the first parabola on the

4

Dr. phil. Karl Strubecker, VORLESUNGEN BER DARSTELLENDE GEOMETRIE,Gttingen, 1967., p.142.


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next segment, and the whole process is repeated for other segments(Fig. 4-7).

Figure 8 presents the approximated net of the hyperbolicparaboloid constructed using the explained method. The main flaw ofthis approximation is the geometric deviation of the net edges fromthe edges of the hyperbolic paraboloid. The segment edges areconcave towards the center of the hyperbolic paraboloid. Thisdeviation comes as a result of locating the segment on the first coneprojection, and then transferring it on the second projection. It isclear that the alignment of the net edges and the edges of thehyperbolic paraboloid in one projection would cause theirmisalignment in the other.

Figure 9 shows the net constructed from the segments where allof the cone apices are turned upwards (Fig. 2). This is opposite to thenet from the Figure 8. This net has significantly minor misalignmentsfrom the edges of the hyperbolic paraboloid, than the one mentionedabove. Here the edges of the net are convex towards the center of thehyperbolic paraboloid.

Fig. 8 The net approximation Fig. 9 The net approximation

Both of these two nets have one mutual flaw. Both nets have thecentral parabola which belongs to hyperbolic paraboloid, but thisparabola doesnt exist on other segments. This is because the secondparabola of one segment is used as the first parabola of the nextsegment 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


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segments are opposed. The cone apices of segments s1, s3 and s4 areupwards, and the cone apices of the segments s2 and s5 are directeddownwards. This enables every segment, except segment s2 to haveone parabola which is also the parabola of hyperbolic paraboloid.Never the less these parabolae are not positioned in the middle ofsegments, but are slightly dislocated which is shown in Figure 11.

Fig .10- The net approximation Fig.11 The dislocation ofparabolae

2. IMAGE PROJECTION ON THE HYPERBOLIC PARABOLOID

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


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

Figure 13 shows the image projection (in this case- logo) thatis applied on the surface of the hyperbolic paraboloid which isinscribed in a cube. The orthogonal projection of the hyperbolicparaboloid is a square grid, so the image projected on it, is anillusion 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 thetop view, and projected on a grid (so it could be easier and moreprecise to route the image). Then, with parallel rays those pointsare projected to the grid from the side view.


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3. ROTATION OF THE HYPERBOLIC PARABOLOID

The criteria for the axes of rotation were to take the elementsthat 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. 14Cube 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. 17Cube edge

Fig. 18 Z rotation Fig. 19 Y rotation

- Rotating at the points R1, R2, R3, R4 and R5 which define aparabola (Fig. 20). The axe of rotation is the y coordinate. Fig. 21 andFig. 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


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parabola (Fig. 23). The y axe is normal on the axe of symmetry of thecube. Examples are in Fig. 24.

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 atthe same time.2. Rotation from the centre of the cube, around both y ,x and z axe atthe same time.3. Rotation from the diagonally of the side of a cube, around both yand z axe at the same time.4. Rotation from the diagonally of the side of a cube, y, x and z axe atthe same time.5. Rotation from the diagonally of the side of a cube, x and z axe atthe 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 time8. 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. urovi Vinko, Nacrtna goemetrija, Nauna knjiga, Beograd, 1977.2. Dr. phil. Karl Strubecker, Vorlesungen ber darstellende geometrie,

Gttingen, 1967.

3. Zlokovi ore, Koordinirani sistem konstrukcija, Izdavakopreduzee Graevinska knjiga, Beograd, 1969.

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Blagojevic Popovic Tripkovicthe Potential of Geomertical and Artistic Characteristics of the Hype