Konstruktivna obrada geometrijskih površi i njihova primena u arhitekturi

The basics of architecture is complex process permeated with experiments, innovations and imagination. Architecture and geometry are inseparably connected, and architecture without geometry cannot exist. The history of geometry is in the same time also a part of history of architecture and vice versa. The foundation of every successful architecture is skillful and rational application of geometrical surfaces in shaping complex space structures, with reaching aesthetically quality forms. Current researches of form and visual conceptions in architecture demand one comprehensive knowledge, overview and constructive treatment of geometrical surfaces and are based upon the application of new computer technologies. Classification of surfaces is elaborated in the context of the possibility of their application in architecture and design. Shown division is clear and usable as a starting basis for further analyses and successful planning of form of architectonic objects, as well as other space structures. In practice more frequent are used ruled geometrical surfaces because of the simplicity of projecting, determination of basic parameters, as well as simplicity of practical works. In this doctoral thesis are shown the surfaces that represent basic gathering of the used forms in architecture and design. There have been analyzed the surfaces of the second degree and their application in architecture. The starting hypothesis is proved that constructive treatment of the surfaces of the second degree it is possible to do with the introduction of general colineation of the space where infinitely distant plane of space is put to finality. With space colineation one can determine the picture of absolute conic of the space in infinite plane. After that, the common elements of the picture of absolute conic of the space and the picture of infinite distant conic of quadric, are determined, which allows the determination of the axis, center, circular dots and circular sections with the quadrics. For constructive treatment of quadrics it is necessary to determine the common elements of the conics pair, no matter for reality. This procedure is based only on geometrical methods and it is possible to use to any other conics pair. Determination of common elements of conics pair is solved with the establishment of collinear relation between two collocal fields determined with those two conics. With finding focuses of the fileds and the application of the procedure for determination of direct connection between one real double straight line and focus of general conlienar fields, the sides of common auto-polar triangle are determined and after that also the cutting - section points of conics pair. There were shown and in details explained the examples of determination of the three conic sections such as pair of real conics, pair of imaginary conics, i.e. real and imaginary conics. Ruled surfaces are analyzed of the higher order and transcendent productive surfaces. Criteria are also set that the analyzed surfaces should fulfill, so that they could be justified and successfully applied in projecting and shaping of various space structures. These are first of all universality, simplicity of projecting and practical use, visual qualities and characteristics, rationality and economy. There have been suggested the innovative shaped solutions in the context of former defined criteria, which provides successful overview and implementation of the analyzed surfaces into geometrical forms of future architectonic objects. The possibilties of practical use are taking into account through the review of characteristic segments and cutouts of surfaces, their multiplication and fitting in complex space structures – assemblies. Special attention in the work is dedicated to computer generating and constructive treatment of ruled surfaces of higher degree and transcendent ruled surfaces. NURBS modeling is used because of the quality and accuracy of treated surfaces in computer programme Rhinoceros. This conducted research represents the connection of the existing knowledge from the geometrical surfaces spheres, new original methods and their constructive treatments and uses of inovative shaped solutions in architecture