Havrylenko Y. Contour modeling in the possible arrangement space of monotonic curves

The dissertation is devoted to the development of theoretical and methodological foundations for modeling the contours of point sets of arbitrary configuration in the space of the possible location of curved lines with a given combination of geometric characteristics, which is a further development of variable discrete geometric modeling of geometric objects according to given data. For this purpose, a method has been developed for modeling flat and spatial discretely presented curves (DPC), which provides control of the dynamics of changes along the curve of the values of its characteristics, the possibility of correcting the generated solution, and prevention of uncontrolled occurrence of singular points. The DPC is formed by condensing the initial point set in parts that define a monotonic curve - curve with constant course, along which the values of the radiuses of the tangent circles and spheres monotonically increase or decrease. An obligatory stage of modeling is the analysis of the initial point set in order to determine its parts, which can be interpolated by a monotonic curve line, as well as areas that necessarily contain special points. The proposed method for analyzing a point series is based on the use of discrete characteristics, which are analogs of the geometric characteristics of a curved line - touching planes, circles and spheres, values of torsion at the points of the DPC. Discrete characteristics are determined by the position of successive points that define the DPC. Discrete characteristics and their values determine the ranges of possible positions and values of the geometric characteristics of a monotonic DPC, and also the area of its possible location. It has been proven that the areas of monotonic DPC are located within spherical trihedrons, each of which is limited by areas of three consecutive adjacent spheres - analogs of the contiguous spheres of duodenum. All discrete characteristics, type and area of possible location of monotonic parts of the DPC are controlled through the characteristics of the broken line, the links of which connect the centers of successive adjacent spheres. It is proved that the areas of monotonic DPC are located within the spherical trihedrons, each of which is limited by areas of three consecutive adjacent spheres - analogs of the contacting spheres of duodenum. All discrete characteristics, type and area of possible location of monotonic parts of the DPC are controlled through the characteristics of the broken line, the links of which connect the centers of successive adjacent spheres. All discrete characteristics, type and area of possible location of monotonic parts of the DPC are controlled through the characteristics of the broken line, the links of which connect the centers of successive adjacent spheres. The developed methods of thickening point sets ensure the designation of thickening points within the area of possible location of monotonic parts of the DPC. As a result of thickening, the area of the location of the duodenum is sequentially localized, degenerating in the limit into a continuous curve.