SVIBOR - Project code: 1-01-266

MINISTRY OF SCIENCE AND TECHNOLOGY

Strossmayerov trg 4, HR - 10000 ZAGREB
tel.: +385 1 459 44 44, fax: +385 1 459 44 69
E-mail: ured@znanost.hr

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Project code: 1-01-266


GEOMETRICAL STRUCTURES AND THEIR APPLICATION IN TECHNICS


Main researcher: ŠČURIĆ, VLASTA (46205)


Assistants
Type of research: basic
Duration from: 10/07/91. to 12/31/95.
Papers on project (total): 27
Institution name: Geodetski fakultet, Zagreb (7)
Department/Institute: Institute for Higher Geodesy
Address: Ulica fra Andrije Kačića Miošića 26
City: 10000 - Zagreb, Croatia
Communication
Phone: 385 (0)1 4561-222
Fax: 385 (0)1 445-410
Summary: The pencils of the cone section types IV, VI, and VII in isotropic plane have been completely classified. The equations of these pencils have been given in their normal form, as well as the interpretation of all geometric invariants. The diagrams are presented by means of computer and plotter. It has been proved that the special hyperbola of the isotropic plane is isotropically analogous to the equilateral hyperbola of Euclidean plane. The classification of the quadrics in the flag space has also been given, as well as some metric relations. With the quadrics bundle arranged in quadrics pencils, the rays of Niče's oriented complex have been determined. The ray congruences of this complex have been researched, as well as the surfaces made by the points adjecent to these rays. M-model of the hyperbolic space in Moebius plane has been made. The graphic projecting in the hyperbolic space has been given. The Monges method of projections in H3-space is interpreted in M-model. The round surfaces in construction engineering have been applied at the level of the projects. By means of isogonal transformation the focal curve of the range of conics has been brought into the connection with the polar curve of the stretchable four-bar linkage in the cinematics. The transformation of the projective space, called quartic inversion, has been defined and its properties researched. Quartic inversion was connected with pedal deducing of surfaces, and pedal surfaces of some (1,2) congruences was constructed by computer. The library of classes (object-oriented programming) corresponding with the basic geometric elements and transformations has been formed.

Keywords: two- and three dimensional space: real projective, affine, isotropic, Euclidean, hyperbolic; flag space, Moebius plane, surface, curve, pencil and bundle of surfaces of second order, pencil of conic sections, classification, normal form, curve of the centers, curve of isotropic focal points, transformation, ray complex, ray congruence, design, theory of mechanisms, object-oriented programming

Research goals: The goal of the project is to classify still unresearched conesection pencils in the isotropic plane, as the basis for the additional research of quadrics pencils in the isotropic spaces; to research as many isotropic spaces analogous with those in Euclidean space, as well as among the isotropic spaces themselves, as possible. To establish the hyperbolic perspective and there presentation of technical objcts with an aim to achieve the posibility of having the methods of macro-projecting at disposal in the future (big dimensions, high speeds). To prepare focal properties of higher circular curves and surfaces in the real projective space for the processing by meansof computer graphics, which makes it possible to expect the wider usage of round surfaces in the construction engineering. To derive pedal curves of cone sections in the isotropic plane. To show the connection between the pedal derivation and square inversion through polar reciprocal curve, and to examine the properties of the derived curves of 4th order. To derive the pedal surfaces of the (1,2) congruence by means of quartic inversion. Linear programming of the material for teaching geometrical subjects and for making models.

COOPERATION - INSTITUTIONS


  1. Name of institution: Montanuniversitaet, Institut fuer Mathematic und angewandte Geometrie
    Type of institution: University/Faculty
    Type of cooperation: Joint publishing of scientific papers
    City: Leoben, Austrija
Other information about the project.
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Last update: 10/12/95
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