Lysenko K. Theoretical bases of methods of formation of composite lines and surfaces

In the given dissertation work the composite geometrical modeling in the part of increase in quantity of basis points of an initial geometrical figure which are exposed to global compositional interpolations has received the further development. Composite geometric modeling is a new scientific field, in general, and in applied geometry, in particular, which uses non-empty finite sets of points that discretely represent the original geometric objects. CGM is designed for the formation of geometrically analytically formalized continuous models of geometric objects of arbitrary shape under predetermined conditions by performing compositional interpolation with point polynomials. The analysis of literature sources is carried out, according to the results of which it is determined that the method of dissertation research should be the method of compositional geometric modeling, the relevance is established, the problem is defined, the purpose of dissertation research is formulated and the range of problems necessary for solution is outlined. The theory of the three-point metric operator has been further developed, which is a scientific novelty and is important for further development and research of new methods of compositional geometric modeling. The scientific novelty of the obtained results is the development of methods for the formation of characteristic functions in parametric form for segments of flat and spatial composite curves, the method of transition from characteristic functions of composite curves to BN coordinates (Balyuby-Najdysha coordinates) of these curves. A method of forming segments of composite surfaces with the same and different parametric bases for the edges of their frames has been developed with the use of composite matrices. A method for the transition from characteristic functions to BN coordinates for segments of composite surfaces with the same and different parametric bases in the edges of their frames has been developed. A sequence has been developed, which is implemented on examples of local correction of the shape of a flat composite curve by changing the position of individual basis points of the original discrete curve (DPC). A graphoanalytical method for finding, with a predetermined accuracy, inflection points for segments of flat composite curves has been developed. The theoretical and practical significance of the results is to develop methods of global compositional interpolation for segments of flat and spatial discrete represented curves, the number of basis points of which is greater than three and for segments of discrete represented surfaces, the number of basis points of which is greater than nine, each of which is simultaneously defined in the coordinate systems of the coordinate 3-space and n-dimensional space of parameters. The developed methods can be used in information systems for process control and will improve the quality of process control. For the first time the method of one-parameter compositional interpolation was created in general form, for which the technique of algebraic formation of characteristic functions and Balyuba-Najdysh coordinates with different number of basis points of initial discrete represented curve both flat and spatial in 3-space coordinate was developed.