Моделювання обводів у просторі можливого розташування монотонних кривих : дисертація

Abstract

UA: Дисертація присвячена моделюванню обводів точкових рядів довільної конфігурації в просторі можливого розташування дискретно представлених кривих (ДПК) із заданими характеристиками. ДПК моделюються згущенням точкового ряду по частинам, які можливо інтерполювати кривою постійного ходу, вздовж якої значення радіусів стичних кіл і сфер монотонно зростають або зменшуються. Запропонований спосіб аналізу точкового ряду заснований на використанні дискретних характеристик, які є аналогами стичних площин, кіл і сфер, значень скруту в точках ДПК. Дискретні характеристики та їх значення визначають діапазони можливих положень і значень геометричних характеристик монотонної ДПК, а також область її можливого розташування. В результаті призначення точок згущення в межах області можливого розташування монотонних частин ДПК область послідовно локалізується, в межі вироджуючись в безперервну криву. Абсолютна похибка дискретної інтерполяції оцінюється як максимально можлива відстань між монотонними кривими лініями, що інтерполюють точковий ряд. Після того, як похибка стає менше заданої величини, згущений точковий ряд інтерполюється обводом. На основі розроблених способів інтерполяції запропонована методика проектування поверхонь з підвищеними динамічними характеристиками. Поверхні моделюються на основі каркаса, лінійні елементи якого задають ДПК. Характеристики ДПК забезпечують задані функціональні якості поверхні. Розроблені алгоритми реалізовані в комп'ютерній програмі, яка впроваджена у виробничий процес. /// EN: 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. Determination of the boundaries of the area of possible location of spatial and plane DPCs made it possible to control the accuracy with which a point set represents a curved line with specified geometric properties. The absolute discrete interpolation error is estimated as the maximum possible distance between monotone curved lines interpolating a point series. After the absolute error in each of the sections becomes less than a predetermined value, the DPC is considered formed, and the resulting points set can be interpolated by a contour. Methods of interpolation of a point set by a broken line, a corrugated line of circles and a B-spline within the area of possible location of the DPC are proposed. This arrangement ensures that a monotonic line curve is replaced by a bypass with an error that does not exceed the discrete interpolation error. Interpolation by sections of curved lines, the formation of which is provided by the libraries of CAD systems, made it possible to create computer models based on the algorithms for thickening DPC. On the basis of the developed interpolation methods, a technique for designing surfaces with increased dynamic characteristics is proposed. Surfaces are modeled on the basis of a wireframe, the linear elements of which define the DPC interpolating point series, selected from the array of points belonging to the surface. The algorithms developed within the framework of the dissertation research are implemented in a computer program, which is introduced into the production process as a tool for the formation of linear elements of surface frames. The results of the implementation of the developed method indicate the prospects of its further development and application in solving various problems of geometric modeling.